GRADE. New York State Common Core. Mathematics Curriculum. GRADE 8 β’
MODULE 4. Module 4: Linear Equations. Date: 11/8/13. 1. Β© 2013 Common Core
Β ...
New York State Common Core
8
GRADE
Mathematics Curriculum GRADE 8 β’ MODULE 4
Table of Contents1
Linear Equations Module Overview .................................................................................................................................................. 3 Topic A: Writing and Solving Linear Equations (8.EE.C.7) ................................................................................... 10 Lesson 1: Writing Equations Using Symbols ........................................................................................... 12 Lesson 2: Linear and Non-Linear Expressions in π₯.................................................................................. 20
Lesson 3: Linear Equations in π₯ .............................................................................................................. 28
Lesson 4: Solving a Linear Equation........................................................................................................ 38 Lesson 5: Writing and Solving Linear Equations ..................................................................................... 50 Lesson 6: Solutions of a Linear Equation ................................................................................................ 61 Lesson 7: Classification of Solutions ....................................................................................................... 71 Lesson 8: Linear Equations in Disguise ................................................................................................... 79 Lesson 9: An Application of Linear Equations......................................................................................... 92 Topic B: Linear Equations in Two Variables and Their Graphs (8.EE.B.5).......................................................... 104 Lesson 10: A Critical Look at Proportional Relationships ..................................................................... 105 Lesson 11: Constant Rate ..................................................................................................................... 117 Lesson 12: Linear Equations in Two Variables ...................................................................................... 132 Lesson 13: The Graph of a Linear Equation in Two Variables .............................................................. 147 Lesson 14: The Graph of a Linear EquationβHorizontal and Vertical Lines ........................................ 163 Mid-Module Assessment and Rubric ................................................................................................................ 179 Topics A through B (assessment 1 day, return 1 day, remediation or further applications 3 days) Topic C: Slope and Equations of Lines (8.EE.B.5, 8.EE.B.6) ............................................................................... 189 Lesson 15: The Slope of a Non-Vertical Line ........................................................................................ 191 Lesson 16: The Computation of the Slope of a Non-Vertical Line ........................................................ 215 Lesson 17: The Line Joining Two Distinct Points of the Graph π¦ = ππ₯ + π has Slope π.................... 239 Lesson 18: There is Only One Line Passing Through a Given Point with a Given Slope ....................... 257
1
Each lesson is ONE day, and ONE day is considered a 45-minute period.
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Linear Equations 11/8/13
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NYS COMMON CORE MATHEMATICS CURRICULUM
Module Overview
8β’4
Lesson 19: The Graph of a Linear Equation in Two Variables is a Line ................................................. 280 Lesson 20: Every Line is a Graph of a Linear Equation ......................................................................... 301 Lesson 21: Some Facts about Graphs of a Linear Equation in Two Variables ...................................... 320 Lesson 22: Constant Rates Revisited .................................................................................................... 334 Lesson 23: The Defining Equation of a Line .......................................................................................... 348 Topic D: Systems of Linear Equations and Their Solutions (8.EE.B.5, 8.EE.C.8) ................................................ 359 Lesson 24: Introduction to Simultaneous Equations ............................................................................ 361 Lesson 25: Geometric Interpretation of the Solutions of a Linear System .......................................... 379 Lesson 26: Characterization of Parallel Lines ....................................................................................... 395 Lesson 27: Nature of Solutions of a System of Linear Equations ......................................................... 409 Lesson 28: Another Computational Method of Solving a Linear System ............................................. 425 Lesson 29: Word Problems ................................................................................................................... 441 Lesson 30: Conversion Between Celsius and Fahrenheit ..................................................................... 455 Optional Topic E: Pythagorean Theorem (8.EE.C.8, 8.G.B.7) ............................................................................ 464 Lesson 31: System of Equations Leading to Pythagorean Triples ........................................................ 465 End-of-Module Assessment and Rubric ............................................................................................................ 476 Topics C through D (assessment 1 day, return 1 day, remediation or further applications 3 days)
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NYS COMMON CORE MATHEMATICS CURRICULUM
Module Overview
8β’4
Grade 8 β’ Module 4
Linear Equations OVERVIEW In Module 4, students extend what they already know about unit rates and proportional relationships (6.RP.A.2, 7.RP.A.2) to linear equations and their graphs. Students understand the connections between proportional relationships, lines, and linear equations in this module (8.EE.B.5, 8.EE.B.6). Also in this module, students learn to apply the skills they acquired in Grades 6 and 7, with respect to symbolic notation and properties of equality (6.EE.A.2, 7.EE.A.1, 7.EE.B.4) to transcribe and solve equations in one variable and then in two variables. In Topic A, students begin by transcribing written statements using symbolic notation. Then, students write linear and non-linear expressions leading to linear equations, which are solved using properties of equality (8.EE.C.7b). Students learn that not every linear equation has a solution. In doing so, students learn how to transform given equations into simpler forms until an equivalent equation results in a unique solution, no solution, or infinitely many solutions (8.EE.C.7a). Throughout Topic A students must write and solve linear equations in real-world and mathematical situations. In Topic B, students work with constant speed, a concept learned in Grade 6 (6.RP.A.3), but this time with proportional relationships related to average speed and constant speed. These relationships are expressed as linear equations in two variables. Students find solutions to linear equations in two variables, organize them in a table, and plot the solutions on a coordinate plane (8.EE.C.8a). It is in Topic B that students begin to investigate the shape of a graph of a linear equation. Students predict that the graph of a linear equation is a line and select points on and off the line to verify their claim. Also in this topic is the standard form of a linear equation, ππ₯ + ππ¦ = π, and when π, π β 0, a non-vertical line is produced. Further, when π or π = 0, then a vertical or horizontal line is produced.
In Topic C, students know that the slope of a line describes the rate of change of a line. Students first encounter slope by interpreting the unit rate of a graph (8.EE.B.5). In general, students learn that slope can be determined using any two distinct points on a line by relying on their understanding of properties of similar triangles from Module 3 (8.EE.B.6). Students verify this fact by checking the slope using several pairs of points and comparing their answers. In this topic, students derive π¦ = ππ₯ and π¦ = ππ₯ + π for linear equations by examining similar triangles. Students generate graphs of linear equations in two variables first by completing a table of solutions, then using information about slope and π¦-intercept. Once students are sure that every linear equation graphs as a line and that every line is the graph of a linear equation, students graph equations using information about π₯- and π¦-intercepts. Next, students learn some basic facts about lines and equations, such as why two lines with the same slope and a common point are the same line, how to write equations of lines given slope and a point, and how to write an equation given two points. With the concepts of slope and lines firmly in place, students compare two different proportional relationships
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Module Overview
8β’4
represented by graphs, tables, equations, or descriptions. Finally, students learn that multiple forms of an equation can define the same line. Simultaneous equations and their solutions are the focus of Topic D. Students begin by comparing the constant speed of two individuals to determine which has greater speed (8.EE.C.8c). Students graph simultaneous linear equations to find the point of intersection and then verify that the point of intersection is in fact a solution to each equation in the system (8.EE.C.8a). To motivate the need to solve systems algebraically, students graph systems of linear equations whose solutions do not have integer coordinates. Students use an estimation of the solution from the graph to verify their algebraic solution is correct. Students learn to solve systems of linear equations by substitution and elimination (8.EE.C.8b). Students understand that a system can have a unique solution, no solution, or infinitely many solutions, as they did with linear equations in one variable. Finally, students apply their knowledge of systems to solve problems in real-world contexts, including converting temperatures from Celsius to Fahrenheit. Optional Topic E is an application of systems of linear equations (8.EE.C.8b). Specifically, a system that generates Pythagorean triples. First, students learn that a Pythagorean triple can be obtained by multiplying any known triple by a positive integer (8.G.B.7). Then, students are shown the Babylonian method for finding a triple that requires the understanding and use of a system of linear equations.
Focus Standards Understand the connections between proportional relationships, lines, and linear equations. 8.EE.B.5
Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.
8.EE.B.6
Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.
Analyze and solve linear equations and pairs of simultaneous linear equations. 8.EE.C.7
Solve linear equations in one variable. a.
Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers).
b.
Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.
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Linear Equations 11/7/13
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Module Overview
NYS COMMON CORE MATHEMATICS CURRICULUM
8.EE.C.8
8β’4
Analyze and solve pairs of simultaneous linear equations. a.
Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously.
b.
Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. For example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6.
c.
Solve real-world and mathematical problems leading to two linear equations in two variables. For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair.
Foundational Standards Understand ratio concepts and use ratio reasoning to solve problems. 6.RP.A.2
Understand the concept of a unit rate a/b associated with a ratio a:b with b β 0, and use rate language in the context of a ratio relationship. For example, βThis recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar.β βWe paid $75 for 15 hamburgers, which is a rate of $5 per hamburger.β 2
6.RP.A.3
Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations. a.
Make tables of equivalent ratios relating quantities with whole-number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.
b.
Solve unit rate problems including those involving unit pricing and constant speed. For example, if it took 7 hours to mow 4 lawns, then at that rate, how many lawns could be mowed in 35 hours? At what rate were lawns being mowed?
Apply and extend previous understandings of arithmetic to algebraic expressions. 6.EE.A.2
Write, read, and evaluate expressions in which letters stand for numbers. a. b.
2
Write expressions that record operations with numbers and letters standing for numbers. For example, express the calculation βSubtract π¦ from 5β as 5 β π¦.
Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity. For example, describe the expression 2(8 + 7) as a product of two factors; view (8 + 7) as both a single entity and a sum of two terms.
Expectations for unit rates in this grade are limited to non-complex fractions.
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NYS COMMON CORE MATHEMATICS CURRICULUM
c.
Module Overview
8β’4
Evaluate expression at specific values of their variables. Include expressions that arise from formulas used in real-world problems. Perform arithmetic operations, including those involving whole-number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations). For example, use the formulas π = π 3 and π΄ = 6π 2to find the volume and surface area of a cube with side 1 length π = . 2
Analyze proportional relationships and use them to solve real-world and mathematical problems. 7.RP.A.2 Recognize and represent proportional relationships between quantities. a.
Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin.
b.
Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.
c.
Represent proportional relationships by equations. For example, if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as π‘ = ππ.
d. Explain what a point (π₯, π¦) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, π) where π is the unit rate.
Use properties of operations to generate equivalent expressions. 7.EE.A.1
Apply properties of operations as strategies to add, subtract, factor and expand linear expressions with rational coefficients.
Solve real-life and mathematical problems using numerical and algebraic expressions and equations. 7.EE.B.4
Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities. a. Solve word problems leading to equations of the form ππ₯ + π = π and π(π₯ + π) = π, where π, π and π are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. For example, the perimeter of a rectangle is 54 cm. Its length is 6 cm. What is its width?
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Linear Equations 11/7/13
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Module Overview
NYS COMMON CORE MATHEMATICS CURRICULUM
8β’4
Focus Standards for Mathematical Practice MP.1
Make sense of problems and persevere in solving them. Students analyze given constraints to make conjectures about the form and meaning of a solution to a given situation in onevariable and two-variable linear equations, as well as in simultaneous linear equations. Students are systematically guided to understand the meaning of a linear equation in one variable, the natural occurrence of linear equations in two variables with respect to proportional relationships, and the natural emergence of a system of two linear equations when looking at related, continuous proportional relationships.
MP.2
Reason abstractly and quantitatively. Students decontextualize and contextualize throughout the module as they represent situations symbolically and make sense of solutions within a context. Students use facts learned about rational numbers in previous grade levels to solve linear equations and systems of linear equations.
MP.3
Construct viable arguments and critique the reasoning of others. Students use assumptions, definitions, and previously established facts throughout the module as they solve linear equations. Students make conjectures about the graph of a linear equation being a line, then proceed to prove this claim. While solving linear equations, they learn that they must first assume that a solution exists, then proceed to solve the equation using properties of equality based on the assumption. Once a solution is found, students justify that it is in fact a solution to the given equation, thereby verifying their initial assumption. This process is repeated for systems of linear equations.
MP.4
Model with mathematics. Throughout the module, students represent real-world situations symbolically. Students identify important quantities from a context and represent their relationship in the form of an equation, table, and graph. Students analyze the various representations and draw conclusions and/or make predictions. Once a solution or prediction has been made, students reflect on whether the solution makes sense in the context presented. One example of this is when students determine how many buses are needed for a field trip. Students must interpret their fractional solution and make sense of it as it applies to the real world.
MP.7
Look for and make use of structure. Students use the structure of an equation to make sense of the information in the equation. For example, students write equations that represent the constant rate of motion for a person walking. In doing so, they interpret an equation like 3 3 π¦ = π₯ as the total distance a person walks, π¦, in π₯ amount of time at a rate of . Students 5 5 look for patterns or structure in tables and show that a rate is constant.
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Module Overview
NYS COMMON CORE MATHEMATICS CURRICULUM
8β’4
Terminology New or Recently Introduced Terms ο§
Slope (Slope is a number that describes the βsteepnessβ or βslantβ of a line. It is the constant rate of 3 7
change. Example: The slope, π, of the graph of line π below is π = .)
slope of π =
ο§
ο§
3 7
System of Linear Equations (A system of linear equations, also referred to as simultaneous linear π₯ + π¦ = 15 equations, is the set of at least two linear equations. Example: οΏ½ is a system of linear 3π₯ β 7π¦ = β2 equations.)
Solution to a system of linear equations (If an equation has two variables, then a solution is a pair of numbers from the domain of the variables that, when each number from the pair is substituted into all instances of its corresponding variable, makes the equation a true number sentence. Example: π₯ + π¦ = 15 103 47 The solution to the system of linear equations οΏ½ is the ordered pair οΏ½ 10 , 10οΏ½ because 3π₯ β 7π¦ = β2 the ordered pair is a solution to each linear equation of the system and it is the point on the plane where the graphs of the two equations intersect.)
Familiar Terms and Symbols 3 ο§ ο§ ο§ ο§ 3
Linear Expression Equation Unit rate Term
These are terms and symbols students have seen previously.
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Module Overview
NYS COMMON CORE MATHEMATICS CURRICULUM
ο§ ο§ ο§ ο§
8β’4
Coefficient Like terms Variable Solution
Suggested Tools and Representations ο§ ο§ ο§ ο§
Scientific calculator Online graphing calculator (for example: https://www.desmos.com/calculator) Graph paper Straight-edge
Assessment Summary Assessment Type Administered
Format
Standards Addressed
Mid-Module Assessment Task
After Topic B
Constructed response with rubric
8.EE.C.7, 8.EE.B.5
End-of-Module Assessment Task
After Topic D
Constructed response with rubric
8.EE.B.5, 8.EE.B.6, 8.EE.C.7, 8.EE.C.8
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Linear Equations 11/7/13
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New York State Common Core
8
Mathematics Curriculum
GRADE
GRADE 8 β’ MODULE 4
Topic A:
Writing and Solving Linear Equations 8.EE.C.7 Focus Standard:
Instructional Days:
8.EE.C.7
Solve linear equations in one variable. a.
Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b result (where a and b are different numbers).
b.
Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.
9
Lesson 1: Writing Equations Using Symbols (P)
1
Lesson 2: Linear and Non-Linear Expressions in π₯ (P) Lesson 3: Linear Equations in π₯ (P)
Lesson 4: Solving a Linear Equation (P)
Lesson 5: Writing and Solving Linear Equations (P) Lesson 6: Solutions of a Linear Equation (P) Lesson 7: Classification of Solutions (S) Lesson 8: Linear Equations in Disguise (P) Lesson 9: An Application of Linear Equations (S)
In Lesson 1, students begin by transcribing written statements into symbolic language. Students learn that before they can write a symbolic statement, they must first define the symbols they intend to use. In Lesson 2, students learn the difference between linear expressions in π₯ and non-linear expressions in π₯, a distinction that is necessary to know whether or not an equation can be solved (at this point). Also in Lesson 2 is a quick 1
Lesson Structure Key: P-Problem Set Lesson, M-Modeling Cycle Lesson, E-Exploration Lesson, S-Socratic Lesson
Topic A: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
Writing and Solving Linear Equations 11/7/13
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NYS COMMON CORE MATHEMATICS CURRICULUM
Topic A
8β’4
review of terms related to linear equations, such as constant, term, and coefficient. In Lesson 3, students learn that a linear equation in π₯ is a statement of equality between two linear expressions in π₯. Students also learn that an equation that contains a variable really a question: Is there a value of π₯ that makes the linear equation true? In Lesson 4, students begin using properties of equality to rewrite linear expressions, specifically using the distributive property to βcombine like terms.β Further, students practice substituting numbers into equations to determine if a true number sentence is produced. In Lesson 5, students practice the skills of the first few lessons in a geometric context. Students transcribe written statements about angles and triangles into symbolic language and use properties of equality to begin solving equations (8.EE.C.7b). More work on solving equations occurs in Lesson 6, where the equations are more complicated and require more steps to solve (8.EE.C.7b). It is in Lesson 6 that students learn that not every linear equation has a solution (8.EE.C.7a). This leads to Lesson 7, where students learn that linear equations either have a unique solution, no solution, or infinitely many solutions (8.EE.C.7a). In Lesson 8, students rewrite equations that are not obviously linear equations then solve them (8.EE.C.7b). In Lesson 9, students take another look at the Facebook problem from Module 1 in terms of linear equations (8.EE.C.7a).
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Lesson 1
NYS COMMON CORE MATHEMATICS CURRICULUM
8β’4
Lesson 1: Writing Equations Using Symbols Student Outcomes ο§
Students write mathematical statements using symbols to represent numbers.
ο§
Students know that written statements can be written as more than one correct mathematical sentence.
Lesson Notes The content of this lesson will continue to develop the skills and concepts presented in Grades 6 and 7. Specifically, this lesson builds on both 6.EE.B.7 (Solve real world and mathematical problems by writing and solving equations of the form π₯ + π = π and ππ₯ = π), and 7.EE.B.4 (Solve word problems leading to equations of the form ππ₯ + π = π and π(π₯ + π) = π. )
Classwork Discussion (4 minutes) Show students the text of a mathematical statement compared to the equation. The number 1,157 is the sum of the squares of two consecutive odd integers divided by the difference between the two consecutive odd integers.
MP.2
1,157 =
π₯ 2 + (π₯ + 2)2 (π₯ + 2) β π₯
Ask students to write or share aloud (a) how these two are related, (b) which representation they prefer, and (c) why. Then, continue with the discussion that follows. ο§
Using letters to represent numbers in mathematical statements was introduced by RenΓ© Descartes in the 1600s. In that era, people used only words to describe mathematical statements. Use of letters, or symbols, to represent numbers not only brought clarity to mathematical statements, it also expanded the horizons of mathematics.
ο§
The reason we want to learn how to write a mathematical statement using symbols is to save time and labor. Imagine having to write the sentence: βThe number 1,157 is the sum of the squares of two consecutive odd integers divided by the difference between the two consecutive odd integers.β Then, imagine having to write the subsequent sentences necessary to solve it; compare that to: Let π₯ represent the first odd integer. Then,
ο§ ο§
1,157 =
π₯ 2 + (π₯ + 2)2 (π₯ + 2) β π₯
Notice that π₯ is just a number. That means the square of π₯ is also a number, along with the square of the next odd integer and the difference between the numbers. This is a symbolic statement about numbers.
Writing in symbols is simpler than writing in words, as long as everyone involved is clear about what the symbols mean. This lesson focuses on accurately transcribing written statements into mathematical symbols. When we write mathematical statements using letters, we say we are using symbolic language.
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Writing Equations Using Symbols 11/7/13
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Lesson 1
NYS COMMON CORE MATHEMATICS CURRICULUM
ο§
8β’4
All of the βmathematical statementsβ in this lesson are equations. Recall that an equation is a statement of equality between two expressions. Developing equations from written statements forms an important basis for problem solving and is one of the most vital parts of algebra. Throughout this module, there will be work with written statements and symbolic language. Students work first with simple expressions, then with equations that gradually increase in complexity, and finally with systems of equations (more than one equation at a time).
Example 1 (3 minutes) Throughout Example 1, have students write their thoughts on personal white boards or a similar tool and show you their transcription(s). Scaffolding: ο§ We want to express the following statement using symbolic language: A whole number has the property that when the square of half the number is subtracted from five times the number, we get back the number itself. ο§
Do the first step, and hold up your white board. οΊ
ο§
First, we define the variable. Let π₯ be the whole number.
Using π₯ to represent the whole number, write βthe square of half the number.β οΊ
π₯ 2
2
1 2
2
π₯2 4
1 4
οΏ½ οΏ½ or οΏ½ π₯οΏ½ or οΏ½ οΏ½ or οΏ½ π₯ 2 οΏ½
ο§ Alternative written statement: ο§ A whole number has the property that when half the number is added to 15, we get the number itself. 1 ο§ π₯ + 15 = π₯ 2
Ask students to write their expression in more than one way. Then, have students share their expressions of βthe square of half the number.β Elicit the above responses from students (or provide them). Ask students why they are all correct. ο§
Write the entire statement: A whole number has the property that when the square of half the number is subtracted from five times the number, we get back the number itself. οΊ
π₯ 2
2
5π₯ β οΏ½ οΏ½ = π₯
Challenge students to write this equation in another form. Engage in a conversation about why they are both correct. For example, when a number is subtracted from itself, the difference is zero. For that reason, the equation above can be π₯ 2
2
written as 5π₯ β οΏ½ οΏ½ β π₯ = 0.
Example 2 (4 minutes) Throughout Example 2, have students write their thoughts on personal white boards or a similar tool and show you their transcription(s). ο§
We want to express the following statement using symbolic language: Paulo has a certain amount of money. If he spends π ππ₯ dollars, then he has original amount left.
ο§
1 4
of the
What is the first thing that must be done before we express this situation using symbols? οΊ
We need to define our variables; that is, we must decide what symbol to use and state what it is going to represent.
Lesson 1: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
Scaffolding: You may need to remind students that we do not have to put the multiplication symbol between a number and a symbol. It is not wrong if we include it, but by convention (a common agreement), it is not necessary.
Writing Equations Using Symbols 11/7/13
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Lesson 1
NYS COMMON CORE MATHEMATICS CURRICULUM
ο§
Suppose we decide to use the symbol π₯. We will let π₯ be the amount of money Paulo had originally. How do we show Pauloβs spending 6 dollars using symbols? οΊ
ο§
To show that Paulo spent 6 dollars, we write: π₯ β 6. 1
How do we express: βhe has of the original amountβ? οΊ
ο§
8β’4
4
We can express it as:
1 4
π₯.
Put the parts together to express: βPaulo has a certain amount of money. If he spends 6 dollars, then he has of the original amount left.β Use π₯ to represent the amount of money Paulo had originally. οΊ
1 4
π₯ β 6 = π₯.
1 4
Challenge students to write this equation in another form. Engage in a conversation about why they are both correct. For example, students may decide to show that the six dollars plus what he has left is equal to the amount of money he 1 4
now has. In symbols: π₯ = π₯ + 6.
Example 3 (8 minutes) Throughout Example 3, have students write their thoughts on personal white boards or a similar tool and show you their transcription(s). ο§
We want to write the following statement using symbolic language: When a fraction of 57 is taken away from 2
57, what remains exceeds of 57 by 4. 3
ο§
The first step is to clearly state what we want our symbol to represent. If we choose the letter π₯, then we would say: βLet π₯ be the fraction of 57β because that is the number that is unknown to us in the written statement. It is acceptable to use any letter to represent the unknown number, but regardless of which letter we use to symbolize the unknown number, we must clearly state what it means. This is called defining our variables.
ο§
The hardest part of transcription is figuring out exactly how to write the statement so that it is accurately represented in symbols. Begin with the first part of the statement: βWhen a fraction of 57 is taken away from 57ββhow can we capture that information in symbols? οΊ
ο§
2
How do we write of 57? οΊ
ο§
Students should write: 57 β π₯. 3
2
If we are trying to find of 57, then we multiply 3
Would it be accurate to write: 57 β π₯ = οΊ
2 β
57? 3
2 3
β
57. 2
No, because we are told that βwhat remains exceeds of 57 by 4.β
Lesson 1: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
3
Scaffolding:
ο§ Alternative written statement: ο§ When a number is taken away from 57, what remains is four more than 5 times the number. ο§ 57 β π₯ = 5π₯ + 4 Scaffolding: If students have a hard time thinking about these transcriptions, give them something easier to think about. One number, say 10, exceeds another number, say 6, by 4. Is it accurate to represent this by: 10 β 4 = 6?
10 = 6 + 4?
10 β 6 = 4?
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ο§
2
Where does the 4 belong? βWhat remains exceeds of 57 by 4.β Think about what the word exceeds means in 3
the context of the problem. Specifically, which is bigger: 57 β π₯ or οΊ
ο§
2
2 3
3
We either have to subtract 4 from 57 β π₯, or add 4 to
correct? ο§
3
of 57? How do you know?
β
57 by 4. What would make the two numbers equal? 2 3
2 3
β
57 to make them equal.
Now, if π₯ is the fraction of 57, then we could write (57 β π₯) β 4 = οΊ
2
57 β π₯ is bigger because 57 β π₯ exceeds of 57 by 4. That means that 57 β π₯ is 4 more than of 57.
We know that 57 β π₯ is bigger than οΊ
ο§
8β’4
Lesson 1
NYS COMMON CORE MATHEMATICS CURRICULUM
2 2 β
57, or 57 β π₯ = β
57 + 4. Which is 3 3
Both transcriptions are correct because both express the written statement accurately. 2 3
Consider this transcription: (57 β π₯) β β
57 = 4. Is it an accurate transcription of the information in the
written statement? οΊ
Yes, because 57 β π₯ is bigger than
2 3
β
57 by 4. That means that the difference between the two
numbers is 4. If we subtract the smaller number from the bigger number, we have a difference of 4 and that is what this version of the transcription shows.
Example 4 (4 minutes) Throughout Example 4, have students write their thoughts on personal white boards or a similar tool and show you their transcription(s). ο§ ο§
We want to express the following statement using symbolic language: The sum of three consecutive integers is 372. Do the first step, and hold up your white board. οΊ
ο§
If we let π₯ represent the first integer, what do we need to do to get the next consecutive integer? οΊ
ο§
Explain that consecutive means one after the next. For example, 18, 19, and 20 are consecutive integers. Consider giving students a number and asking them what the next consecutive integer would be.
If π₯ is the first integer, we add 1 to π₯ to get the next integer.
In symbols, the next integer would be π₯ + 1. What do we need to do now to get the next consecutive integer? οΊ
ο§
Let π₯ be the first integer.
Scaffolding:
We need to add 1 to that integer, or π₯ + 1 + 1; this is the same as π₯ + 2.
Now express the statement: The sum of three consecutive integers is 372. οΊ
π₯ + π₯ + 1 + π₯ + 2 = 372.
Students may also choose to rewrite the above equation as 3π₯ + 3 = 372. Transforming equations, such as this, will be a focus of the next few lessons when students begin to solve linear equations. Ask students why both equations are correct.
Lesson 1: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
Writing Equations Using Symbols 11/7/13
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Lesson 1
NYS COMMON CORE MATHEMATICS CURRICULUM
8β’4
Example 5 (3 minutes) Throughout Example 5, have students write their thoughts on personal white boards or a similar tool and show you their transcription(s). ο§ ο§
We want to express the following statement using symbolic language: The sum of three consecutive odd integers is 93.
Do the first step, and hold up your white board. οΊ
ο§
If we let π₯ represent the first integer, what do we need to do to get the next consecutive odd integer? οΊ
ο§
We need to add 2 to that integer, or π₯ + 2 + 2; this is the same as π₯ + 4.
Now express the statement: The sum of three consecutive odd integers is 372. οΊ
ο§
If π₯ is the first integer, we add 2 to π₯ to get the next odd integer.
Consider giving kids an odd number, then asking them what the next consecutive odd number is. Then ask student what they need to add to go from the first number to the second number.
In symbols, the next odd integer would be π₯ + 2. What do we need to do now to get the next odd consecutive integer? οΊ
ο§
Let π₯ be the first integer.
Scaffolding:
π₯ + π₯ + 2 + π₯ + 4 = 93.
Represent the statement βThe sum of three consecutive odd integers is 93β in another way. Be prepared to explain why both are correct. οΊ
Answers will vary. Accept any correct answers and justifications. For example, students may write (π₯ + π₯ + π₯) + 6 = 93 and state that it is equivalent to the equation π₯ + π₯ + 2 + π₯ + 4 = 93. Because the Associative and Commutative properties of addition were applied, those properties do not change the value of an expression.
Exercises 1β5 (10 minutes) Students complete Exercises 1β5 independently or in pairs. Exercises 1β5 Write each of the following statements using symbolic language. 1.
The sum of four consecutive even integers is βππ.
Let π be the first integer. Then, π + π + π + π + π + π + π = βππ. 2.
A number is four times larger than the square of half the number. π π
π
Let π be the number. Then, π = π οΏ½ οΏ½ . 3.
Steven has some money. If he spends ππππ dollars, then he will have π π
π π
of the amount he started with.
Let π be the amount of money Steven started with. Then, π β π = π.
Lesson 1: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
Writing Equations Using Symbols 11/7/13
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Lesson 1
NYS COMMON CORE MATHEMATICS CURRICULUM
4.
8β’4
The sum of a number squared and three less than twice the number is πππ. Let π be the number. Then, ππ + ππ β π = πππ.
5.
π
Miriam read a book with an unknown number of pages. The first week she read five less than of the pages. The π
second week she read πππ pages and finished the book. Write an equation that represents the total number of pages in the book. Let π be the total number of pages in the book. Then,
π π
π β π + πππ = π.
Closing (4 minutes) Summarize, or ask students to summarize, the main points from the lesson: ο§
We know how to write mathematical statements using symbolic language.
ο§
Written mathematical statements can be represented as more than one correct symbolic statement.
ο§
We must always begin writing a symbolic statement by defining our symbols (variables).
ο§
Complicated statements should be broken into parts or attempted with simple numbers to make the representation in symbolic notation easier.
Lesson Summary Begin all word problems by defining your variables. State clearly what you want each symbol to represent. Written mathematical statements can be represented as more than one correct symbolic statement. Break complicated problems into smaller parts or try working them with simpler numbers.
Exit Ticket (5 minutes)
Lesson 1: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
Writing Equations Using Symbols 11/7/13
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Lesson 1
NYS COMMON CORE MATHEMATICS CURRICULUM
Name ___________________________________________________
8β’4
Date____________________
Lesson 1: Writing Equations Using Symbols Exit Ticket Write each of the following statements using symbolic language. 1.
When you square five times a number you get three more than the number.
2.
Monica had some cookies. She gave seven to her sister. Then, she divided the remainder in half, and she still had five cookies.
Lesson 1: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
Writing Equations Using Symbols 11/7/13
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Lesson 1
NYS COMMON CORE MATHEMATICS CURRICULUM
8β’4
Exit Ticket Sample Solutions Write each of the following statements using symbolic language. 1.
When you square five times a number you get three more than the number.
2.
Let π be the number. Then, (ππ)π = π + π.
Monica had some cookies. She gave seven to her sister. Then, she divided the remainder in half, and she still had five cookies.
Let π be the original amount of cookies. Then,
π π
(π β π) = π.
Problem Set Sample Solutions Students practice transcribing written statements into symbolic language. Write each of the following statements using symbolic language. 1.
2.
Bruce bought two books. One book costs four dollars more than three times the other. Together, the two books cost him $ππ.
Let π be the cost of the less expensive book. Then, π + π + ππ = ππ.
Janet is three years older than her sister Julie. Janetβs brother is eight years younger than their sister Julie. The sum of all of their ages is ππ.
3.
Let π be Julieβs age. Then, π + π + π β π + π = ππ.
4.
Let π be the first integer. Then, π + π + π + π + π = π, πππ.
5.
6.
7.
8.
9.
The sum of three consecutive integers is π, πππ.
One number is six more than another number. The sum of their squares is ππ. Let π be the smaller number. Then ππ + (π + π)π = ππ.
When you add ππ to
π π
of a number, you get the number itself. π
Let π be the number. Then, π + ππ = π. π
When a fraction of ππ is taken away from ππ, what remains exceeds one-third of seventeen by six.
Let π be the fraction of ππ. Then, ππ β π =
π β
ππ + π. π
The sum of two consecutive even integers divided by four is πππ. π.
Let π be the first even integer. Then,
π+π+π π
= πππ. π.
Subtract seven more than twice a number from the square of one-third of the number to get zero. π π
π
Let π be the number. Then, οΏ½ ποΏ½ β (ππ + π) = π.
The sum of three consecutive integers is ππ. Let π be the middle of the three integers. Transcribe the statement accordingly. π β π + π + π + π = ππ. Lesson 1: Date:
Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
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Lesson 2
NYS COMMON CORE MATHEMATICS CURRICULUM
8β’4
Lesson 2: Linear and Non-Linear Expressions in π
Student Outcomes ο§ ο§
Students know the properties of linear and non-linear expressions in π₯. Students transcribe and identify expressions as linear or non-linear.
Classwork Discussion (4 minutes) ο§ ο§ ο§
A symbolic statement in π₯ with an equal sign is called an equation in π₯. The equal sign divides the equation into two parts, the left side and the right side. The two sides are called expressions.
For sake of simplicity, we will only discuss expressions in π₯, but know that we can write expressions in any symbol.
The following chart contains both linear and non-linear expressions in π₯. Sort them into two groups and be prepared to explain what is different about the two groups. 7 5π₯ + 3 9 β π₯2 β8π₯ + β 3 9 π₯ 3 0.31π₯ + 7 β 4.2π₯ 4π₯ 2 β 9 οΏ½ οΏ½ +1 2 11(π₯ + 2)
β(6 β π₯) + 15 β 9π₯
7 + π₯ β4 + 3π₯
5π₯ + 3
7 β8π₯ + β 3 9
9 β π₯2
Linear expressions are noted in red in the table below.
MP.3
4π₯ 2 β 9
ο§
11(π₯ + 2)
ο§
β(6 β π₯) + 15 β 9π₯
π₯ 3 οΏ½ οΏ½ +1 2
7 + π₯ β4 + 3π₯
Identify which equations you placed in each group. Explain your reasoning for grouping the equations. οΊ
ο§
0.31π₯ + 7 β 4.2π₯
Equations that contained an exponent of π₯ other than 1 were put into one group. The other equations were put into another group. That seemed to be the only difference between the types of equations given.
Linear expressions in π₯ are a special type of expression. Linear expressions are expressions that are sums of constants and products of a constant and π₯ raised to a power of 0, which simplifies to a value of 1. Non-linear expressions are also sums of constants and products of a constant and a power of π₯. However, non-linear expressions will have a power of π₯ that is not equal to 1 or 0.
The reason we want to be able to distinguish linear expressions from non-linear expressions is because we will soon be solving linear equations. Non-linear equations will be a set of equations you learn to solve in Algebra I, though we will begin to solve simple non-linear equations later this year (Module 7). We also want to be able to recognize linear equations in order to predict the shape of their graph, which is a concept we will learn more about later in this module.
Lesson 2: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
Linear and Non-Linear Expressions in π₯ 11/7/13
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Lesson 2
NYS COMMON CORE MATHEMATICS CURRICULUM
8β’4
Example 1 (3 minutes) ο§
A linear expression in π₯ is an expression where each term is either a constant, an π₯, or a product of a constant and π₯. For example, the expression (57 β π₯) is a linear expression. However, the expression 2π₯ 2 + 9π₯ + 5 is not a linear expression. Why is 2π₯ 2 + 9π₯ + 5 not a linear expression in π₯? οΊ
Students should say that 2π₯ 2 + 9π₯ + 5 is not a linear expression because the terms of linear expressions must either be a constant, an π₯, or the product of a constant and π₯. The term 2π₯ 2 does not fit the definition of a linear expression in π₯. Scaffolding:
Example 2 (4 minutes) ο§
Letβs examine the expression 4 + 3π₯ 5 more deeply. To begin, we want to identify the terms of the expression. How many terms are there, and what are they? οΊ
ο§
How many terms are comprised of just constants, and what are they? οΊ
ο§
There is one constant term, 4.
How many terms have coefficients, and what are they? οΊ
ο§
There are two terms, 4 and 3π₯ 5 .
There is one term with a coefficient, 3.
ο§ Terms are any product of an integer power of π₯ and a constant or just a constant. ο§ Constants are fixed numbers. ο§ When a term is the product of a constant(s) and a power of π₯, the constant is called a coefficient.
Is 4 + 3π₯ 5 a linear or non-linear expression in π₯? Why or why not? οΊ
The expression 4 + 3π₯ 5 is a non-linear expression in π₯ because it is the sum of a constant and the product of a constant and positive integer power of π₯ > 1.
Example 3 (4 minutes) ο§
How many terms does the expression 7π₯ + 9 + 6 + 3π₯ have? What are they? οΊ
ο§
As is, this expression has 4 terms: 7π₯, 9, 6, and 3π₯.
This expression can be transformed using some of our basic properties of numbers. For example, if we apply the Commutative Property of Addition, we can rearrange the terms from 7π₯ + 9 + 6 + 3π₯ to 7π₯ + 3π₯ + 9 + 6. First, we can apply the Associative Property of Addition: Next, we apply the Distributive Property:
(7 + 3)π₯ + (9 + 6)
Finally, ο§
10π₯ + 15
How many terms with coefficients does the expression 10π₯ + 15 have? What are they? οΊ
ο§
(7π₯ + 3π₯) + (9 + 6)
The expression has one term with a coefficient, 10π₯. For this term, the coefficient is 10.
Is 10π₯ + 15 a linear or non-linear expression in π₯? Why or why not? οΊ
The expression 10π₯ + 15 is a linear expression in π₯ because it is the sum of constants and products st that contain π₯ to the 1 power.
Lesson 2: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
Linear and Non-Linear Expressions in π₯ 11/7/13
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Lesson 2
NYS COMMON CORE MATHEMATICS CURRICULUM
8β’4
Example 4 (2 minutes) ο§
How many terms does the expression 5 + 9π₯ β
7 + 2π₯ 9 have? What are they? οΊ
ο§
How many terms with coefficients does the expression 5 + 9π₯ β
7 + 2π₯ 5 have? What are they? οΊ
ο§
The expression has three terms: 5, 9π₯ β
7, and 2π₯ 9 .
The expression has two terms with coefficients: 63π₯ and 2π₯ 5 . The coefficients are 63 and 2.
Is 5 + 9π₯ β
7 + 2π₯ 9 a linear or non-linear expression in π₯? Why or why not? οΊ
The expression 5 + 9π₯ β
7 + 2π₯ 9 is a non-linear expression in π₯ because it is the sum of constants and products that contain π₯ raised to a power that is greater than 1.
Example 5 (2 minutes) ο§
Is 94 + π₯ + 4π₯ β6 β 2 a linear or non-linear expression in π₯? Why or why not? οΊ
Students may first say that it is not a linear nor non-linear expression in π₯ because of the β2. Remind them that subtraction can be rewritten as a sum, i.e., + (β2); therefore, this expression does fit the definition of non-linear.
Example 6 (2 minutes) ο§
Is the expression π₯ 1 + 9π₯ β 4 a linear expression in π₯? οΊ
ο§
Yes, π₯ 1 + 9π₯ β 4 is a linear expression in π₯ because π₯ 1 is the same as π₯.
What powers of π₯ are acceptable in the definition of a linear expression in π₯? οΊ
Only the power of 1 is acceptable because π₯ 1 is, by definition, just π₯.
Exercises 1β12 (14 minutes) Students complete Exercises 1β12 independently. Exercises 1β12 Write each of the following statements in Exercises 1β12 as a mathematical expression. State whether or not the expression is linear or non-linear. If it is non-linear, then explain why. 1.
The sum of a number and four times the number. Let π be a number; then, π + ππ is a linear expression.
2.
The product of five and a number. Let π be a number, then ππ is a linear expression.
3.
Multiply six and the reciprocal of the quotient of a number and seven. Let π be a number, then π β
π π π is a non-linear expression. The number = π β
= π β
πβπ is the reason it is not a π π π
linear expression. The exponent of the π is the reason it is not linear. Lesson 2: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
Linear and Non-Linear Expressions in π₯ 11/7/13
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Lesson 2
NYS COMMON CORE MATHEMATICS CURRICULUM
4.
8β’4
Twice a number subtracted from four times a number, added to ππ. Let π be a number, then ππ + (ππ β ππ) is a linear expression.
5.
The square of the sum of six and a number. Let π be a number, then (π + π)π is a non-linear expression. When you multiply (π + π)π, you get ππ + πππ + ππ. The ππ is the reason it is not a linear expression.
6.
The cube of a positive number divided by the square of the same positive number. Let π be a number, then linear.
7.
ππ
ππ
is a non-linear expression. However, if you simplify the expression to just π, then it is
The sum of four consecutive numbers. Let π be a number, then π + π + π + π + π + π + π is a linear expression.
8.
Four subtracted from the reciprocal of a number. Let π be a number, then
π π
β π is a non-linear expression. The term
π π
is the same as πβπ , which is why this π
expression is not linear. It is possible that a student may let π be the reciprocal of a number, , which would make π
the expression linear. 9.
Half of the product of a number multiplied by itself, three times. Let π be a number; then,
π π
β
π β
π β
π is not a linear expression. The term
why this expression is not linear.
π π
β
π β
π β
π is the same as
π π π , which is π
π
10. The sum that shows how many pages Maria read if she read ππ pages of a book yesterday and of the remaining pages today.
Let π be the number of remaining pages of the book, then ππ + 11. An admission fee of $ππ plus an additional $π per game.
π π is a linear expression. π
π
Let π be the number of games, then ππ + ππ is a linear expression.
12. Five more than four times a number, then twice that sum. Let π be the number, then π(ππ + π) is a linear expression.
Closing (5 minutes) Summarize, or ask students to summarize, the main points from the lesson: ο§
We have definitions for linear and non-linear expressions.
ο§
We know how to use the definitions to identify expressions as linear or non-linear.
ο§
We can write expressions that are linear and non-linear.
Lesson 2: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
Linear and Non-Linear Expressions in π₯ 11/7/13
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Lesson 2
NYS COMMON CORE MATHEMATICS CURRICULUM
8β’4
Lesson Summary Linear expressions are sums of constants and products of constants and π raised to a power of π or π. For example,
π + ππ, ππ + π β ππ, and
π π
π + π β π are all linear expressions in π.
Non-linear expressions are also sums of constants and products of constants and a π raised to a power that is not π
or π. For example, πππ β π, βππβπ + π + π, and
π π
+ π are all non-linear expressions in π.
Exit Ticket (5 minutes)
Lesson 2: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
Linear and Non-Linear Expressions in π₯ 11/7/13
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Lesson 2
NYS COMMON CORE MATHEMATICS CURRICULUM
Name ___________________________________________________
8β’4
Date____________________
Lesson 2: Linear and Non-Linear Expressions in π Exit Ticket
Write each of the following statements as a mathematic expression. State whether the expression is a linear or nonlinear expression in π₯. 1.
Seven subtracted from five times a number, then the difference added to nine times a number.
2.
Three times a number subtracted from the product of fifteen and the reciprocal of a number.
3.
Half of the sum of two and a number multiplied by itself three times.
Lesson 2: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
Linear and Non-Linear Expressions in π₯ 11/7/13
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Lesson 2
NYS COMMON CORE MATHEMATICS CURRICULUM
8β’4
Exit Ticket Sample Solutions Write each of the following statements as a mathematic expression. State whether the expression is a linear or nonlinear expression in π. 1.
Seven subtracted from five times a number, then the difference added to nine times a number. Let π be a number. (ππ β π) + ππ.
The expression is a linear expression. 2.
Three times a number subtracted from the product of fifteen and the reciprocal of a number. Let π be a number. ππ β
π β ππ. π
The expression is a non-linear expression. 3.
Half of the sum of two and a number multiplied by itself three times. Let π be a number.
π π
(π + ππ ).
The expression is a non-linear expression.
Problem Set Sample Solutions Students practice writing expressions and identifying them as linear or non-linear. Write each of the following statements as a mathematic expression. State whether the expression is linear or non-linear. If it is non-linear, then explain why. 1.
A number decreased by three squared. Let π be a number, then π β ππ is a linear expression.
2.
The quotient of two and a number, subtracted from seventeen. Let π be a number, then ππ β
π π π π is a non-linear expression. The term is the same as π β
and = πβπ . That is π π π π
why it is not a linear expression. 3.
The sum of thirteen and twice a number. Let π be a number, then ππ + ππ is a linear expression.
4.
π. π more than the product of seven and a number.
Let π be a number, then π. π + ππ is a linear expression. 5.
The sum that represents tickets sold if ππ tickets were sold Monday, half of the remaining tickets were sold on Tuesday, and ππ tickets were sold on Wednesday. π π
Let π be the remaining number of tickets, then ππ + π + ππ is a linear expression.
Lesson 2: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
Linear and Non-Linear Expressions in π₯ 11/7/13
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Lesson 2
NYS COMMON CORE MATHEMATICS CURRICULUM
6.
The product of ππ and a number, subtracted from the reciprocal of the number cubed.
Let π be a number, then linear expression.
7.
8.
π π β πππ is a non-linear expression. The term π is the same as πβπ . That is why it is not a ππ π
The product of ππ and number, multiplied by itself four times.
Let π± be a number, then (πππ±)π is a non-linear expression. The expression can be written as πππ β
π± π . The exponent of π with a base of π± is the reason it is not linear. A number increased by five, divided by two.
Let π be a number, then 9.
8β’4
π+π π
is a linear expression.
Eight times the result of subtracting three from a number. Let π be a number, then π(π β π) is a linear expression.
10. The sum of twice a number and four times a number subtracted from the number squared. Let π± be a number, then π± π β (ππ± + ππ±) is a non-linear expression. The term π± π is the reason it is not linear.
11. One-third of the result of three times a number that is increased by ππ. Let π be a number, then
π π
(ππ + ππ) is a linear expression.
12. Five times the sum of one-half and a number. π π
Let π be a number, then π οΏ½ + π οΏ½ is a linear expression. 13. Three-fourths of a number multiplied by seven. π
Let π be a number, then π β
π is a linear expression. π
14. The sum of a number and negative three, multiplied by the number. Let π be a number, then οΏ½π + (βπ)οΏ½π is a non-linear expression because οΏ½π + (βπ)οΏ½π = ππ β ππ after using the distributive property. It is non-linear because the power of π in the term ππ is greater than π. 15. The square of the difference between a number and ππ.
Let π be a number, then (π β ππ)π is a non-linear expression because (π β ππ)π = ππ β πππ + πππ. The term ππ is a positive power of π > π; therefore, this is a not a linear expression.
Lesson 2: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
Linear and Non-Linear Expressions in π₯ 11/7/13
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Lesson 3
NYS COMMON CORE MATHEMATICS CURRICULUM
8β’4
Lesson 3: Linear Equations in π
Student Outcomes ο§
Students know that a linear equation is a statement of equality between two expressions.
ο§
Students know that a linear equation in π₯ is actually a question: Can you find all numbers π₯, if they exist, that satisfy a given equation? Students know that those numbers π₯ that satisfy a given equation are called solutions.
Classwork Concept Development (7 minutes) ο§
We want to define βlinear equation in π₯.β Here are some examples of linear equations in π₯. Using what you know about the words linear (from Lesson 2) and equation (from Lesson 1), develop a mathematical definition of βlinear equation in π₯.β
Scaffolding:
Show students the examples below, and provide them time to work individually or in small groups to develop an appropriate definition. Once students share their definitions, continue with the definition and discussion that follows. π₯ + 11 = 15
ο§ ο§
15 β 4π₯ = π₯ +
4 5
5+3 =8
3 β (π₯ + 2) = β12π₯
Consider developing a word bank or word wall to be used throughout the module.
1 β π₯ = 22 2
3 π₯ + 6(π₯ β 1) = 9(2 β π₯) 4
When two linear expressions are equal, they can be written as a linear equation in π₯. Consider the following equations. Which are true, and how do you know? 4+1 =5
6 + 5 = 16
21 β 6 = 15
6β2 =1 οΊ
ο§
The first and third equations are true because the value on the left side is equal to the number on the right side.
Is 4 + 15π₯ = 49 true? How do you know?
Have a discussion that leads to students developing a list of values for π₯ that make it false, along with one value of π₯ that makes it true. Then, conclude the discussion by making the two points below. ο§
A linear equation in π₯ is a statement about equality, but it is also an invitation to find all of the numbers π₯, if they exist, that make the equation true. Sometimes the question is asked in this way: What number(s) π₯ satisfy the equation? The question is often stated more as a directive: Solve. When phrased as a directive, it is still considered a question. Is there a number(s) π₯ that make the statement true? If so, what is the number(s) π₯? Lesson 3: Date:
Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
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Lesson 3
NYS COMMON CORE MATHEMATICS CURRICULUM
ο§
8β’4
Equations that contain a variable do not have a definitive truth value; in other words, there are values of the variable that make the equation true and values that make it false. When we say that we have βsolved an equation,β what we are really saying is that we have found a number (or numbers) π₯ that makes the linear equation true. That number π₯ is called the solution to the linear equation.
Example 1 (4 minutes) ο§
Here is a linear equation in π₯: 4 + 15π₯ = 49. The question is, is there a number π₯ that makes the linear expression 4 + 15π₯ equal to the linear expression 49? Suppose you are told this number π₯ has a value of 2, i.e., π₯ = 2. We replace any instance of π₯ in the linear equation with the value of 2, as shown: 4 + 15 β
2 = 49
Next, we evaluate each side of the equation. The left side is 4 + 15 β
2 = 4 + 30 = 34
ο§
The right side of the equation is 49. Clearly, 34 β 49. Therefore, the number 2 is not a solution to this equation.
Is the number 3 a solution to the equation, i.e., is this equation true when π₯ = 3? οΊ
Yes, because the left side of the equation equals the right side of the equation: The left side is
Scaffolding: Remind students that when a number and a symbol are next to one another, such as 15π₯, we do not need to use a symbol to represent the multiplication (it is a convention). For clarity, when two numbers are being multiplied, we do use a multiplication symbol. It is necessary to tell the difference between the number, 152 and the product, 15 β
2.
4 + 15 β
3 = 4 + 45 = 49
ο§
The right side is 49. Since 49 = 49, then we can say that π₯ = 3 is a solution to the equation 4 + 15π₯ = 49.
3 is a solution to the equation because it is a value of π₯ that makes the equation true.
Example 2 (4 minutes) ο§ ο§
Here is a linear equation in π₯: 8π₯ β 19 = β4 β 7π₯.
Is 5 a solution to the equation? That is, is π₯ = 5 a solution to the equation? οΊ
No, because the left side of the equation does not equal the right side of the equation: The left side is
8 β
5 β 19 = 40 β 19 = 21
The right side is
Since 21 β β39, then π₯ β 5.
Lesson 3: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
β4 β 7 β
5 = β4 β 35 = β39
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Lesson 3
NYS COMMON CORE MATHEMATICS CURRICULUM
ο§
8β’4
Is 1 a solution to the equation? That is, is this equation true when π₯ = 1? οΊ
Yes, the left side and right side of the equation are equal to the same number. The left side is
8 β
1 β 19 = 8 β 19 = β11
The right side is
Since β11 = β11, then π₯ = 1.
β4 β 7 β
1 = β4 β 7 = β11
Example 3 (4 minutes) ο§ ο§
Here is a linear equation in π₯: 3(π₯ + 9) = 4π₯ β 7 + 7π₯.
We can make our work simpler if we use some properties to transform the expression on the right side of the equation into an expression with fewer terms.
Provide students time to transform the equation into fewer terms, then proceed with the points below. MP.1
ο§
For example, notice that on the right side there are two terms that contain π₯. First, we will use the Commutative Property to rearrange the terms to better see what we are doing: 4π₯ + 7π₯ β 7
Next, we will use the distributive property to collect the terms that contain π₯: 4π₯ + 7π₯ β 7 = (4 + 7)π₯ β 7 = 11π₯ β 7
ο§
Finally, the transformed (but still the same) equation can be written as: 3(π₯ + 9) = 11π₯ β 7. Is
5 5 a solution to the equation? That is, is this equation true when π₯ = ? 4 4
οΊ
No, because the left side of the equation does not equal the right side of the equation: The left side is
41 5 3 οΏ½ + 9οΏ½ = 3 οΏ½ οΏ½ 4 4 123 = 4
The right side is
11 β
Since
123 4
β
27 4
5 4
, then π₯ β .
Lesson 3: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
55 5 β7= β7 4 4 27 = 4
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Lesson 3
NYS COMMON CORE MATHEMATICS CURRICULUM
8β’4
Example 4 (4 minutes) ο§ ο§
Here is a linear equation in π₯: β2π₯ + 11 β 5π₯ = 5 β 6π₯.
We want to check to see if 6 is a solution to the equation, i.e., is π₯ = 6 a solution to the equation? Before we do that, what would make our work easier? οΊ
ο§
We could use the Commutative and Distributive Properties to transform the left side of the equation into an expression with fewer terms. β2π₯ + 11 β 5π₯ = β2π₯ β 5π₯ + 11 = (β2 β 5)π₯ + 11 = β7π₯ + 11
The transformed equation can be written as: β7π₯ + 11 = 5 β 6π₯. Is 6 a solution to the equation, i.e., is this equation true when π₯ = 6? οΊ
Yes, because the left side of the equation is equal to the right side of the equation:
The left side is
β7π₯ + 11 = β7 β
6 + 11 = β42 + 11 = β31
The right side is
5 β 6π₯ = 5 β 6 β
6 = 5 β 36 = β31
Since β31 = β31, then π₯ = 6.
Exercises 1β6 (12 minutes) Students complete Exercises 1β6 independently. Exercises 1β6 1.
Is the equation true when π = βπ; in other words, is βπ a solution to the equation: ππ + π = ππ + π + ππ? Explain.
If we replace π with the number βπ, then the left side of the equation is and the right side of the equation is
π β
(βπ) + π = βππ + π = βππ
π β
(βπ) + π + π β
(βπ) = βππ + π β π = βπ β π = βππ
Since βππ = βππ, then π = βπ is a solution to the equation ππ + π = ππ + π + ππ.
Note: Some students may have transformed the equation.
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Lesson 3
NYS COMMON CORE MATHEMATICS CURRICULUM
2.
π π
8β’4
π π
Does π = ππ satisfy the equation: ππ β π = π + π? Explain.
If we replace π with the number ππ, then the left side of the equation is
and the right side of the equation is
π π ππ β π = ππ β β
(ππ) π π = ππ β π = ππ π π π + π = β
(ππ) + π π π =π+π = ππ π π
π π
Since ππ = ππ, then π = ππ is a solution to the equation ππ β π = π + π. 3.
Chad solved the equation πππ + π + ππ = π(πππ β π) and is claiming that π = π makes the equation true. Is Chad correct? Explain. If we replace π with the number π, then the left side of the equation is
and the right side of the equation is
πππ + π + ππ = ππ β
π + π + π β
π = ππ + π + π = ππ
π(πππ β π) = π(ππ β
π β π) = π(ππ β π) = π(ππ) = ππ Since ππ β ππ, then π = π is not a solution to the equation πππ + π + ππ = π(πππ β π), and Chad is not correct. 4.
π π
Lisa solved the equation π + π = π + ππ and claimed that the solution is π = β . Is she correct? Explain. π π
If we replace π with the number β , then the left side of the equation is
and the right side of the equation is
π π+π=β +π π π =π π
π π + ππ = π + π β
οΏ½β οΏ½ π π =πβ π ππ π = β π π ππ = π π ππ π Since π = , then π = β is a solution to the equation π + π = π + ππ, and Lisa is correct. π
π
π
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Linear Equations in π₯ 11/8/13
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Lesson 3
NYS COMMON CORE MATHEMATICS CURRICULUM
5.
8β’4
Angel transformed the following equation from ππ + π β π = π(π + π) to ππ = π(π + π). He then stated that the solution to the equation is π = π. Is he correct? Explain.
No, Angel is not correct. He did not transform the equation correctly. The expression on the left side of the equation ππ + π β π = π(π + π) would transform to ππ + π β π = ππ β π + π = (π β π)π + π = ππ + π
If we replace π with the number π, then the left side of the equation is
and the right side of the equation is
ππ + π = π β
π + π = ππ + π = ππ
π(π + π) = π(π + π) = π(π) = ππ Since ππ β ππ, then π = π is not a solution to the equation ππ + π β π = π(π + π), and Angel is not correct. 6.
Claire was able to verify that π = π was a solution to her teacherβs linear equation, but the equation got erased from the board. What might the equation have been? Identify as many equations as you can with a solution of π = π.
Answers will vary. Ask students to share their equations and justifications as to how they knew π = π would make a true number sentence. 7.
Does an equation always have a solution? Could you come up with an equation that does not have a solution? Answers will vary Expect students to write equations that are false. Ask students to share their equations and justifications as to how they knew the equation they wrote did not have a solution. The concept of βno solutionβ is introduced in Lesson 6 and solidified in Lesson 7.
Closing (5 minutes) Summarize, or ask students to summarize, the main points from the lesson: ο§
We know that equations are statements about equality. That is, the expression on the left side of the equal sign is equal to the expression on the right side of the equal sign.
ο§
We know that a solution to a linear equation in π₯ will be a number, and that when all instances of π₯ are replaced with the number, the left side will equal the right side.
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Lesson 3
NYS COMMON CORE MATHEMATICS CURRICULUM
8β’4
Lesson Summary Equations are statements about equality. If the expression on the left side of the equal sign has the same value as the expression on the right side of the equal sign, then you have a true equation. A solution to a linear equation in π is a number, such that when all instances of π are replaced with the number, the left side will equal the right side. For example, π = π is a solution to ππ + π = π + π because the left side of the equation is
and the right side of the equation is
ππ + π = π(π) + π =π+π = ππ π+π =π+π = ππ
Since ππ = ππ, then π = π is a solution to the linear equation ππ + π = π + π.
Exit Ticket (5 minutes)
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Lesson 3
NYS COMMON CORE MATHEMATICS CURRICULUM
Name ___________________________________________________
8β’4
Date____________________
Lesson 3: Linear Equations in π Exit Ticket
1
1.
Is 8 a solution to π₯ + 9 = 13? Explain.
2.
Write three different equations that have π₯ = 5 as a solution.
3.
Is β3 a solution to the equation 3π₯ β 5 = 4 + 2π₯? Explain.
2
Lesson 3: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
Linear Equations in π₯ 11/8/13
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Lesson 3
NYS COMMON CORE MATHEMATICS CURRICULUM
8β’4
Exit Ticket Sample Solutions 1.
π
Is π a solution to π + π = ππ? Explain. π
If we replace π with the number π, then the left side is
ππ = ππ, then π = π is a solution. 2.
π π
(π) + π = π + π = ππ and the right side is ππ.
Since
Write three different equations that have π = π as a solution.
Answers will vary. Accept equations where π = π makes a true number sentence.
3.
Is βπ a solution to the equation ππ β π = π + ππ? Explain.
If we replace π with the number βπ, then the left side is π(βπ) β π = βπ β π = βππ. The right side is π + π(βπ) = π β π = βπ. Since βππ β βπ, then π = βπ is not a solution to the equation.
Problem Set Sample Solutions Students practice determining whether or not a given number is a solution to the linear equation. 1.
Given that ππ + π = ππ and ππ + π = ππ, does ππ + π = ππ + π? Explain.
No, because a linear equation is a statement about equality. We are given that ππ + π = ππ, but ππ + π = ππ. Since each linear expression is equal to a different number, ππ + π β ππ + π.
2.
Is βπ a solution to the equation: ππ + π = ππ + π + ππ? Explain.
If we replace π with the number βπ, then the left side of the equation is and the right side of the equation is
π β
(βπ) + π = βππ + π = βππ
π β
(βπ) + π + π β
(βπ) = βππ + π β ππ = βππ β ππ = βππ
Since βππ β βππ, then π β βπ is not a solution to the equation ππ + π = ππ + π + ππ.
Note: Some students may have transformed the equation.
3.
π π
Does π = π. π satisfy the equation: π β ππ = β ? Explain.
If we replace π with the number π. π, then the left side of the equation is and the right side of the equation is
π β π β
π. π = π β π. π = βπ. π
π. π = βπ. π π π Since βπ. π = βπ. π, then π = π. π is a solution to the equation π β ππ = β . β
Lesson 3: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
π
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Lesson 3
NYS COMMON CORE MATHEMATICS CURRICULUM
4.
8β’4
Use the linear equation π(π + π) = ππ + π to answer parts (a)β(d). a.
Does π = π satisfy the equation above? Explain.
If we replace π with the number π, then the left side of the equation is π(π + π) = π(π) = ππ
and the right side of the equation is
ππ + π = π β
π + π = ππ + π = ππ Since ππ = ππ, then π = π is a solution to the equation π(π + π) = ππ + π. b.
Is π = βπ a solution the equation above? Explain.
If we replace π with the number βπ, then the left side of the equation is π(βπ + π) = π(βπ) = βππ
and the right side of the equation is
ππ + π = π β
(βπ) + π = βππ + π = βππ Since βππ = βππ, then π = βπ is a solution to the equation π(π + π) = ππ + π. c.
Is π =
π a solution the equation above? Explain. π π π
If we replace π with the number , then the left side of the equation is
and the right side of the equation is
Since
d.
π π
π
= , then π = π
π π π π οΏ½ + ποΏ½ = π οΏ½ + οΏ½ π π π π = ποΏ½ οΏ½ π π = π π ππ + π = π β
οΏ½ οΏ½ + π π π = +π π π π = + π π π = π
π is a solution to the equation π(π + π) = ππ + π. π
What interesting fact about the equation π(π + π) = ππ + π is illuminated by the answers to parts (a), (b), and (c)? Why do you think this is true? Note to teacher: Ideally students will notice that the equation π(π + π) = ππ + π is an identity under the distributive law. The purpose of this problem is to prepare students for the idea that linear equations can have more than one solution, which is a topic of Lesson 7.
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Lesson 4
NYS COMMON CORE MATHEMATICS CURRICULUM
8β’4
Lesson 4: Solving a Linear Equation Student Outcomes ο§
Students extend the use of the properties of equality to solve linear equations having rational coefficients.
Classwork Concept Development (13 minutes) ο§ ο§
To solve an equation means to find all of the numbers π₯, if they exist, so that the given equation is true.
In some cases, some simple guess work can lead us to a solution. For example, consider the following equation: 4π₯ + 1 = 13
What number π₯ would make this equation true? That is, what value of π₯ would make the left side equal to the right side? (Give students a moment to guess a solution.) οΊ
When π₯ = 3, we get a true statement. The left side of the equal sign is equal to 13 and so is the right side of the equal sign.
In other cases, guessing the correct answer is not so easy. Consider the following equation: 3(4π₯ β 9) + 10 = 15π₯ + 2 + 7π₯
Can you guess a number for π₯ that would make this equation true? (Give students a minute to guess.)
ο§
Guessing is not always an efficient strategy for solving equations. In the last example, there are several terms in each of the linear expressions comprising the equation. This makes it more difficult to easily guess a solution. For this reason, we want to use what we know about the properties of equality to transform equations into equations with fewer terms.
ο§
The ultimate goal of solving any equation is to get it into the form of π₯ (or whatever symbol is being used in the equation) equal to a constant.
Complete the activity described to remind students of the properties of equality, then proceed with the discussion that follows. Give students the equation: 4 + 1 = 7 β 2 and ask them the following questions. 1.
Is this equation true?
2.
Perform each of the following operations, and state whether or not the equation is still true:
MP.8
3.
a.
Add three to both sides of the equal sign.
b.
Add three to the left side of the equal sign, and add two to the right side of the equal sign.
c.
Subtract six from both sides of the equal sign.
d.
Subtract three from one side of the equal sign and subtract three from the other side.
e.
Multiply both sides of the equal sign by ten.
f.
Multiply the left side of the equation by ten and the right side by four.
g.
Divide both sides of the equation by two.
h.
Divide the left side of the equation by two and the right side of the equation by five.
What do you notice? Describe any patterns you see.
Lesson 4: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
Solving a Linear Equation 11/8/13
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Lesson 4
NYS COMMON CORE MATHEMATICS CURRICULUM
ο§
8β’4
There are four properties of equality that will allow us to transform an equation into the form we want. If π΄, π΅, and πΆ are any rational numbers, then β’
β’
β’ β’
If π΄ = π΅, then π΄ + πΆ = π΅ + πΆ.
If π΄ = π΅, then π΄ β πΆ = π΅ β πΆ. If π΄ = π΅, then π΄ β
πΆ = π΅ β
πΆ.
If π΄ = π΅, then
π΄ πΆ
=
π΅ πΆ
, where πΆ is not equal to zero.
All four of the properties require us to start off with π΄ = π΅. That is, we have to assume that a given equation has an expression on the left side that is equal to the expression on the right side. Working under that assumption, each time we use one of the properties of equality, we are transforming the equation into another equation that is also true, i.e., left side equals right side.
Example 1 (3 minutes) ο§ ο§
Solve the linear equation 2π₯ β 3 = 4π₯ for the number π₯.
Examine the properties of equality. Choose βsomethingβ to add, subtract, multiply, or divide on both sides of the equation.
Validate the use of the properties of equality by having students share their thoughts. Then, discuss the βbestβ choice for the first step in solving the equation with the points below. Be sure to remind students throughout this and the other examples that our goal is to get π₯ equal to a constant; therefore, the βbestβ choice is one that gets us to that goal most efficiently. ο§
First, we must assume that there is a number π₯ that makes the equation true. Working under that assumption, when we use the property, If π΄ = π΅, then π΄ β πΆ = π΅ β πΆ, we get an equation that is also true: 2π₯ β 3 = 4π₯ 2π₯ β 2π₯ β 3 = 4π₯ β 2π₯
Now, using the distributive property, we get another set of equations that is also true: (2 β 2)π₯ β 3 = (4 β 2)π₯ 0π₯ β 3 = 2π₯ β3 = 2π₯
Using another property, If π΄ = π΅, then
π΄ πΆ
2
After simplifying the fraction , we have 2
is also true. ο§
3 2
=
π΅ πΆ
, we get another true equation: β3 2π₯ = 2 2 β3 =π₯ 2
The last step is to check to see if π₯ = β satisfies the equation 2π₯ β 3 = 4π₯. 3 2
The left side of the equation is equal to 2 β
οΏ½β οΏ½ β 3 = β3 β 3 = β6. 3 2
The right side of the equation is equal to 4 β
οΏ½β οΏ½ = 2 β
(β3) = β6.
Since the left side equals the right side, we know we have found the correct number π₯ that solves the equation 2π₯ β 3 = 4π₯. Lesson 4: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
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Lesson 4
NYS COMMON CORE MATHEMATICS CURRICULUM
8β’4
Example 2 (4 minutes) ο§
3
Solve the linear equation π₯ β 21 = 15. Keep in mind that our goal is to transform the equation so that it is in 5
the form of π₯ equal to a constant. If we assume that the equation is true for some number π₯, which property should we use to help us reach our goal and how should we use it?
Again, provide students time to decide which property is βbestβ to use first. οΊ
We should use: If π΄ = π΅, then π΄ + πΆ = π΅ + πΆ, where the number πΆ is 21.
Note to teacher: If students suggest that we subtract 15 from both sides, i.e., make πΆ be β15, then remind them that we want the form of π₯ equal to a constant. Subtracting 15 from both sides of the equal sign puts the π₯ and all of the constants on the same side of the equal sign. There is nothing mathematically incorrect about subtracting 15; it just doesnβt get us any closer to reaching our goal. ο§
If we use π΄ + πΆ = π΅ + πΆ, then we have the true statement: 3 π₯ β 21 + 21 = 15 + 21 5 and 3 π₯ = 36. 5 Which property should we use to reach our goal, and how should we use it? οΊ
ο§
ο§
We should use: If π΄ = π΅, then π΄ β
πΆ = π΅ β
πΆ, where C is
5 3
.
If we use π΄ β
πΆ = π΅ β
πΆ, then we have the true statement: 3 5 5 π₯ β
= 36 β
5 3 3 and by the commutative property and the cancellation law we have: π₯ = 12 β
5 = 60.
3 Does π₯ = 60 satisfy the equation π₯ β 21 = 15? 5
οΊ
Yes, because the left side of the equation is equal to 3 5
180 β 21 = 36 β 21 = 15. Since the right side is 5
also 15, then we know that π₯ = 60 is a solution to π₯ β 21 = 15.
Example 3 (5 minutes) ο§
The properties of equality are not the only properties we can use with equations. What other properties do we know that could make solving an equation more efficient? οΊ
We know the distributive property, which allows us to expand and simplify expressions.
οΊ
We know the commutative and associative properties, which allow us to rearrange and group terms within expressions.
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Lesson 4
NYS COMMON CORE MATHEMATICS CURRICULUM
8β’4
1
Now we will solve the linear equation π₯ + 13 + π₯ = 1 β 9π₯ + 22. Is there anything we can do to the linear 5
expression on the left side to transform it into an expression with fewer terms?
Yes, we can use the commutative and distributive properties: 1 1 π₯ + 13 + π₯ = π₯ + π₯ + 13 5 5 6 = π₯ + 13 5 Is there anything we can do to the linear expression on the right side to transform it into an expression with fewer terms? οΊ
ο§
οΊ
ο§
Yes, we can use the commutative property: 1 β 9π₯ + 22 = 1 + 22 β 9π₯ = 23 β 9π₯
Now we have the equation: solve the equation?
6 5
π₯ + 13 = 23 β 9π₯. What should we do now to
Students should come up with the following four responses as to what should be done first. A case can be made for each of them being the βbestβ move. In this case, each first move gets us one step closer to our goal of having the solution in the form of π₯ equal to a constant. Select one option and move forward with solving the equation (the notes that follow align to the first choice, subtracting 13 from both sides of the equal sign). οΊ
οΊ
οΊ οΊ ο§
ο§
We should subtract 23 from both sides of the equal sign.
We should add 9π₯ to both sides of the equal sign. We should subtract
6 5
π₯ from both sides of the equal sign.
Letβs choose to subtract 13 from both sides of the equal sign. Though all options were generally equal with respect to being the βbestβ first step, I chose this one because when I subtract 13 on both sides, the value of the constant on the left side is positive. I prefer to work with positive numbers. Then we have 6 π₯ + 13 β 13 = 23 β 13 β 9π₯ 5 6 π₯ = 10 β 9π₯ 5
What should we do next? Why? οΊ
ο§
We should subtract 13 from both sides of the equal sign.
We should add 9π₯ to both sides of the equal sign. We want our solution in the form of π₯ equal to a constant, and this move puts all terms with an π₯ on the same side of the equal sign.
Adding 9π₯ to both sides of the equal sign, we have: 6 π₯ + 9π₯ = 10 β 9π₯ + 9π₯ 5 51 π₯ = 10 5
Lesson 4: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
Note to Teacher: There are many ways to solve this equation. Any of the actions listed below are acceptable. In fact, a student could say: βadd 100 to both sides of the equal sign,β and that, too, would be an acceptable action. It may not lead us directly to our answer, but it is still an action that would make a mathematically correct statement. Make clear to students that it doesnβt matter which option they choose or in which order; what matters is that they use the properties of equality to make true statements that lead to a solution in the form of π₯ equal to a constant.
Note to Teacher: We still have options. If students say we should 6
subtract π₯ from both sides of 5
the equal sign, remind them of our goal of obtaining the π₯ equal to a constant.
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Lesson 4
NYS COMMON CORE MATHEMATICS CURRICULUM
ο§
What do we need to do now? οΊ
ο§
We should multiply
Then we have:
5
51
on both sides of the equal sign. 51 5 5 π₯β
= 10 β
5 51 51
By the commutative property and the fact that
ο§
8β’4
5
51
π₯=
Γ
51 5
50 . 51
= 1, we have
Since all transformed versions of the original equation are true, we can select any of them to check our answer. However, it is best to check the solution in the original equation because we may have made a mistake transforming the equation. 1 π₯ + 13 + π₯ = 1 β 9π₯ + 22 5 1 50 50 50 οΏ½ οΏ½ + 13 + = 1 β 9 οΏ½ οΏ½ + 22 5 51 51 51 6 50 50 οΏ½ οΏ½ + 13 = 23 β 9 οΏ½ οΏ½ 5 51 51 300 450 + 13 = 23 β 255 51 3615 723 = 255 51 723 723 = 51 51 Since both sides of our equation equal
723 51
, then we know our answer is correct.
Exercises 1β5 (10 minutes) Students work on Exercises 1β5 independently. Exercises 1β5 For each problem, show your work and check that your solution is correct. 1.
Solve the linear equation: π + π + π + π + π + π + π = βππ. State the property that justifies your first step and why you chose it.
The left side of the equation can be transformed from π + π + π + π + π + π + π to ππ + ππ using the commutative and distributive properties. Using these properties decreases the number of terms of the equation. Now we have the equation: ππ + ππ = βππ ππ + ππ β ππ = βππ β ππ ππ = βππ π π β
ππ = βππ β
π π π = βππ
The left side of the equation is equal to (βππ) + (βππ) + π + (βππ) + π + (βππ) + π which is βππ. Since the left side is equal to the right side, then π = βππ is the solution to the equation. Note: Students could use the division property in the last step to get the answer.
Lesson 4: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
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Lesson 4
NYS COMMON CORE MATHEMATICS CURRICULUM
2.
8β’4
Solve the linear equation: π(ππ + π) = ππ β π + π. State the property that justifies your first step and why you chose it.
Both sides of equation can be rewritten using the distributive property. I have to use it on the left side to expand the expression. I have to use it on the right side to collect like terms. The left side is: π(ππ + π) = ππ + π
The right side is:
ππ β π + π = ππ + π β π = ππ β π
The equation is:
ππ + π = ππ β π ππ + π β π = ππ β π β π ππ = ππ β π ππ β ππ = ππ β ππ β π ππ = βπ π π β
ππ = β
(βπ) π π π π=β π The left side of the equation is equal to π(βπ + π) = π(βπ) = βπ. The right side of the equation is equal to βπ β π = βπ. Since both sides are equal to βπ, then π = β
π is a solution to π(ππ + π) = ππ β π + π. π
Note: Students could use the division property in the last step to get the answer. 3.
π π
Solve the linear equation: π β π = π. State the property that justifies your first step and why you chose it.
I chose to use the subtraction property of equality to get all terms with an π on one side of the equal sign. π πβπ= π π π πβπβπ= πβπ π π βπ = β π π π π π β β
(βπ) = β β
β π π π π ππ =π π The left side of the equation is
4.
ππ π
β
ππ π
=
ππ π
.
The right side is
ππ π the same number, then π = is a solution to π β π = π. π π
π ππ π
β
π
=
π π
β
π π
=
ππ π
. Since both sides are equal to
Solve the linear equation: ππ β ππ = ππ + π. State the property that justifies your first step and why you chose it.
I chose to use the addition property of equality to get all terms with an π on one side of the equal sign. ππ β ππ = ππ + π ππ β ππ + ππ = ππ + ππ + π ππ = ππ + π ππ β π = ππ + π β π ππ = ππ π π β
ππ = β
ππ π π π=π
The left side of the equal sign is ππ β π(π) = ππ β π = ππ. The right side is equal to π(π) + π = ππ + π = ππ. Since both sides are equal, π = π is a solution to ππ β ππ = ππ + π. Note: Students could use the division property in the last step to get the answer.
Lesson 4: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
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Lesson 4
NYS COMMON CORE MATHEMATICS CURRICULUM
5.
Solve the linear equation: it.
π π
8β’4
π β π + πππ = π. State the property that justifies your first step and why you chose
I chose to combine the constants βπ and πππ. Then, I used the subtraction property of equality to get all terms with an π on one side of the equal sign.
The left side of the equation is
π π
π π β π + πππ = π π π π + πππ = π π π π π π β π + πππ = π β π π π π π πππ = π π π π π πππ β
= β
π π π π ππ Γ π = π πππ = π
β πππ β π + πππ = ππ β π + πππ = ππ + πππ = πππ, which is exactly equal
to the right side. Therefore, π = πππ is a solution to
π π
π β π + πππ = π.
Closing (5 minutes) Summarize, or ask students to summarize, the main points from the lesson: ο§
We know that properties of equality, when used to transform equations, make equations with fewer terms that are simpler to solve.
ο§
When solving an equation, we want the answer to be in the form of the symbol π₯ equal to a constant. Lesson Summary The properties of equality, shown below, are used to transform equations into simpler forms. If π¨, π©, πͺ are rational numbers, then ο§
ο§ ο§ ο§
If π¨ = π©, then π¨ + πͺ = π© + πͺ
If π¨ = π©, then π¨ β πͺ = π© β πͺ If π¨ = π©, then π¨ β
πͺ = π© β
πͺ
If π¨ = π©, then
π¨ πͺ
=
π© πͺ
, where πͺ is not equal to zero
Addition Property of Equality
Subtraction Property of Equality Multiplication Property of Equality Division Property of Equality
To solve an equation, transform the equation until you get to the form of π equal to a constant (π = π, for example).
Exit Ticket (5 minutes)
Lesson 4: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
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Lesson 4
NYS COMMON CORE MATHEMATICS CURRICULUM
Name ___________________________________________________
8β’4
Date____________________
Lesson 4: Solving a Linear Equation Exit Ticket 1.
Guess a number for π₯ that would make the equation true. Check your solution. 5π₯ β 2 = 8
2.
Use the properties of equality to solve the equation: 7π₯ β 4 + π₯ = 12. State which property justifies your first step and why you chose it. Check your solution.
3.
Use the properties of equality to solve the equation: 3π₯ + 2 β π₯ = 11π₯ + 9. State which property justifies your first step and why you chose it. Check your solution.
Lesson 4: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
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Lesson 4
NYS COMMON CORE MATHEMATICS CURRICULUM
8β’4
Exit Ticket Sample Solutions 1.
Guess a number for π that would make the equation true. Check your solution. ππ β π = π
When π = π, the left side of the equation is π, which is the same as the right side. Therefore, π = π is the solution to the equation. 2.
Use the properties of equality to solve the equation: ππ β π + π = ππ. State which property justifies your first step and why you chose it. Check your solution.
I used the commutative and distributive properties on the left side of the equal sign to simply the expression to fewer terms. ππ β π + π = ππ ππ β π = ππ ππ β π + π = ππ + π ππ = ππ π ππ π= π π π=π
The left side of the equation is π(π) β π + π = ππ β π + π = ππ. The right side is also ππ. Since the left side equals the right side, π = π is the solution to the equation. 3.
Use the properties of equality to solve the equation: ππ + π β π = πππ + π. Check your solution.
I used the commutative and distributive properties on the left side of the equal sign to simplify the expression to fewer terms.
The left side of the equation is π οΏ½ ππ π = . π π
ππ + π β π = πππ + π ππ + π = πππ + π ππ β ππ + π = πππ β ππ + π π = ππ + π π β π = ππ + π β π βπ = ππ βπ π = π π π π β =π π
βπ βπ ππ ππ π π π βππ οΏ½+πβ =β + + = . The right side is ππ οΏ½β οΏ½ + π = + π π π π π π π π π π
Since the left side equals the right side, π = β is the solution to the equation.
Lesson 4: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
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Lesson 4
NYS COMMON CORE MATHEMATICS CURRICULUM
8β’4
Problem Set Sample Solutions Students solve equations using properties of equality. For each problem, show your work and check that your solution is correct. 1.
Solve the linear equation: π + π + ππ = ππ. State the property that justifies your first step and why you chose it.
I used the commutative and distributive properties on the left side of the equal sign to simplify the expression to fewer terms. π + π + ππ = ππ ππ + π = ππ ππ + π β π = ππ β π ππ = ππ π ππ π= π π π = ππ
The left side is equal to ππ + π + π(ππ) = ππ + ππ = ππ, which is what the right side is. Therefore, π = ππ is a solution to the equation π + π + ππ = ππ. 2.
Solve the linear equation: π + π + π β π + π = ππ. State the property that justifies your first step and why you chose it.
I used the commutative and distributive properties on the left side of the equal sign to simplify the expression to fewer terms. π + π + π β π + π = ππ ππ β π = ππ ππ β π + π = ππ + π ππ = ππ π ππ π= π π π = ππ
The left side is equal to ππ + π + ππ β π + ππ = ππ β π + ππ = ππ + ππ = ππ, which is equal to the right side. Therefore, π = ππ is a solution to π + π + π β π + π = ππ. 3.
Solve the linear equation: chose it.
π π
π
π + ππ = β π + ππ. State the property that justifies your first step and why you π
I chose to use the subtraction property of equality to get all of the constants on one side of the equal sign. π π π + ππ = π + ππ π π π π π + ππ β ππ = π + ππ β ππ π π π π π = β π + ππ π π π π π π π β π = π β π + ππ π π π π π π = ππ π π π β
π = π β
ππ π π = πππ The left side of the equation is
π π
(πππ) + ππ = ππ + ππ = ππ.
The right side of the equation is
ππ + ππ = ππ. Since both sides equal ππ, π = πππ is a solution to the equation Lesson 4: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
π π
π + ππ =
π π
π π
(πππ) + ππ =
π + ππ.
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Lesson 4
NYS COMMON CORE MATHEMATICS CURRICULUM
4.
Solve the linear equation:
π π
8β’4
π + ππ = π. State the property that justifies your first step and why you chose it.
I chose to use the subtraction property of equality to get all terms with an π on one side of the equal sign.
The left side of the equation is π = ππ is a solution to 5.
π π
π π
π π + ππ = π π π π π π β π + ππ = π β π π π π π ππ = π π π π π β
ππ = β
π π π π ππ = π
(ππ) + ππ = π + ππ = ππ, which is what the right side is equal to.
Therefore,
π + ππ = π.
Solve the linear equation: ππ β π = it.
π β
ππ + π. State the property that justifies your first step and why you chose π
The right side of the equation can be simplified to ππ. Then the equation is ππ β π = ππ
and π = π. Both sides of the equation equal ππ; therefore, π = π is a solution to the equation ππ β π =
I was able to solve the equation mentally without using the properties of equality.
6.
Solve the linear equation:
π+π+π
The left side of the equation is
π
= πππ. π. State the property that justifies your first step and why you chose it.
πππ+πππ+π
Therefore, π = πππ is a solution to
Lesson 4: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
π β
ππ + π. π
π π+π+π π
π+π+π = πππ. π π π + π + π = π(πππ. π) ππ + π = πππ ππ + π β π = πππ β π ππ = πππ π πππ π= π π π = πππ
=
πππ π
= πππ. π, which is equal to the right side of the equation.
= πππ. π.
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Lesson 4
NYS COMMON CORE MATHEMATICS CURRICULUM
7.
8β’4
Alysha solved the linear equation: ππ β π β ππ = ππ + π β π. Her work is shown below. When she checked her answer, the left side of the equation did not equal the right side. Find and explain Alyshaβs error, and then solve the equation correctly. ππ β π β ππ = ππ + π β π. βππ β π = ππ + ππ βππ β π + π = ππ + π + ππ βππ = ππ + ππ βππ + ππ = ππ βππ = ππ βπ ππ π= βπ βπ π = βπ
Alysha made a mistake on the πππ line. She added ππ to the left side of the equal sign and subtracted ππ on the right side of the equal sign. To use the property correctly, she should have subtracted ππ on both sides of the equal sign, making the equation at that point: βππ β ππ = ππ + ππ β ππ βππ = ππ βπ ππ π= βπ βπ π = βπ
Lesson 4: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
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Lesson 5
NYS COMMON CORE MATHEMATICS CURRICULUM
8β’4
Lesson 5: Writing and Solving Linear Equations Student Outcomes ο§
Students apply knowledge of geometry to writing and solving linear equations.
Lesson Notes All of the problems in this lesson relate to what students have learned about geometry in recent modules and previous years. The purpose is two-fold, first, we want to reinforce what students have learned about geometry, and second, we want to demonstrate a need for writing and solving an equation in a context that is familiar. Throughout the lesson students solve mathematical problems that relate directly to what students have learned about angle relationships, congruence, and the triangle sum theorem. Encourage students to draw a diagram to represent the situation presented in the word problems.
Classwork Example 1 (5 minutes) ο§
Solve the following: Example 1 One angle is five less than three times the size of another angle. Together they have a sum of πππΛ. What are the sizes of each angle?
MP.1
Provide students with time to make sense of the problem and persevere in solving it. They could begin their work by guessing and checking, drawing a diagram, or other methods as appropriate. Then move to the algebraic method shown below. ο§
What do we need to do first to solve this problem? οΊ
ο§
If π₯ is the size of the first angle, how do you represent the size of the second angle? οΊ
ο§
The second angle is 3π₯ β 5.
What is the equation that represents this situation? οΊ
ο§
First we need to define our variable (symbol). Let π₯ be the size of the first angle.
The equation is: π₯ + 3π₯ β 5 = 143.
Scaffolding: Model for students how to use diagrams to help make sense of the problems throughout this lesson. Encourage students to use diagrams to help them understand the situation.
The equation that represents this situation is: π₯ + 3π₯ β 5 = 143. Solve for π₯, then determine the size of each angle.
Lesson 5: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
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Lesson 5
NYS COMMON CORE MATHEMATICS CURRICULUM
8β’4
As students share their solutions for this and subsequent problems, ask them a variety of questions to reinforce the learning of the last few lessons. For example, you can ask students whether or not this is a linear equation and how they know, to justify their steps and explain why they chose their particular first step, what the solution means, or how they know their answer is correct. Student work: π + ππ β π = πππ (π + π)π β π = πππ ππ β π = πππ ππ β π + π = πππ + π ππ = πππ π = ππ The size of the first angle is ππΛ. The second angle is π(ππ) β π = πππ β π = πππΛ.
ο§
Compare the method you tried at the beginning of the problem with the algebraic method. What advantage does writing and solving an equation have? οΊ
ο§
Could we have defined π₯ to be the size of the second angle? If so, what, if anything, would change? οΊ
ο§
The first angle would be
π₯+5 3
.
The equation that represents the situation is π₯ + οΊ οΊ
ο§
If we let π₯ be the size of the second angle, then the equation would change, but the answers for the sizes of the angles should remain the same.
If π₯ is the size of the second angle, how would we write the size of the first angle? οΊ
ο§
Writing and solving an equation is a more direct method than the one I tried before. It allows me to find the answer more quickly.
π₯+5 = 143. How should we solve this equation? 3
We could add the fractions together, then solve for π₯.
We could multiply every term by 3 to change the fraction to a whole number.
Using either method, solve the equation. Verify that the sizes of the angles are the same as before. οΊ
π₯+5 = 143 3 3π₯ π₯ + 5 + = 143 3 3 3π₯ + π₯ + 5 = 143 3 3π₯ + π₯ + 5 = 143(3) (3 + 1)π₯ + 5 = 429 4π₯ + 5 = 429 4π₯ + 5 β 5 = 429 β 5 4π₯ = 424 π₯ = 106 π₯+
Lesson 5: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
Scaffolding: You may need to remind students how to add fractions by rewriting term(s) as equivalent fractions then adding the numerators. Provide support as needed.
Writing and Solving Linear Equations 11/8/13
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Lesson 5
NYS COMMON CORE MATHEMATICS CURRICULUM
οΊ
The size of the second angle is 106Λ. The measure of the first angle is
OR
π₯+5 = 143 3 π₯+5 3 οΏ½π₯ + = 143οΏ½ 3 3π₯ + π₯ + 5 = 429 4π₯ + 5 = 429 4π₯ + 5 β 5 = 429 β 5 4π₯ = 424 π₯ = 106 π₯+
ο§
The size of the second angle is 106Λ. The measure of the first angle is
π₯+5 3
π₯+5 3
=
=
106+5 3
106+5 3
=
=
111 3
111 3
8β’4
= 37Β°.
= 37Λ.
Whether we let π₯ represent the first angle or the second angle does not change our answers. Whether we solve the equation using the first or second method does not change our answers. What matters is that we accurately write the information in the problem and correctly use the properties of equality. You may solve a problem differently than your classmates or teachers. Again, what matters most is that what you do is accurate and correct.
Example 2 (12 minutes) ο§
Solve the following: Example 2 Given a right triangle, find the size of the angles if one angle is ten more than four times the other angle and the third angle is the right angle.
Give students time to work. As they work, walk around and identify students who are writing and solving the problem in different ways. The instructional goal of this example is to make clear that there are different ways to solve a linear equation as opposed to one βright wayβ. Select students to share their work with the class. If students donβt come up with different ways of solving the equation, talk them through the following student work samples. Again, as students share their solutions, ask them a variety of questions to reinforce the learning of the last few lessons. For example, you can ask students whether or not this is a linear equation and how they know, or to justify their steps and explain why they chose their particular first step, what the solution means, or how they know their answer is correct.
Lesson 5: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
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Lesson 5
NYS COMMON CORE MATHEMATICS CURRICULUM
8β’4
Solution One Let π be the size of the first angle. Then the second angle is ππ + ππ. The sum of the measures for the angles for this right triangle is as follows: π + ππ + ππ + ππ = πππ. π + ππ + ππ + ππ = πππ (π + π)π + πππ = πππ ππ + πππ = πππ ππ + πππ β πππ = πππ β πππ ππ = ππ π = ππ
The measure of the first angle is ππΛ, the measure of the second angle is π(ππ) + ππ = ππ + ππ = ππΛ, and the third angle is ππΛ. Solution Two Let π be the size of the first angle. Then the second angle is ππ + ππ. Since we have a right triangle we already know that one angle is ππΛ which means that the sum of the other two angles is ππ: π + ππ + ππ = ππ. π + ππ + ππ = ππ (π + π)π + ππ = ππ ππ + ππ = ππ ππ + ππ β ππ = ππ β ππ ππ = ππ π = ππ
The measure of the first angle is ππΛ, the measure of the second angle is π(ππ) + ππ = ππ + ππ = ππΛ, and the third angle is ππΛ. Solution Three Let π be the size of the second angle. Then the first angle is
πβππ π
. Since we have a right triangle we already know that
one angle is ππΛ which means that the sum of the other two angles is ππ: π + π β ππ = ππ π π β ππ π οΏ½π + = πποΏ½ π ππ + π β ππ = πππ (π + π)π β ππ = πππ ππ β ππ = πππ ππ β ππ + ππ = πππ + ππ ππ = πππ π = ππ
π+
The measure of the second angle is ππΛ, the measure of the first angle is
Lesson 5: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
πβππ = ππ. π
ππβππ π
=
ππ π
= ππΛ, and the third angle is ππΛ.
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Lesson 5
NYS COMMON CORE MATHEMATICS CURRICULUM
8β’4
Solution Four Let π be the size of the second angle. Then the first angle is
ππ = πππ.
πβππ π
. The sum of the three angles is as follows: π +
π β ππ + ππ = πππ π π β ππ π+ + ππ β ππ = πππ β ππ π π β ππ π+ = ππ π ππ π β ππ + = ππ π π ππ + π β ππ = ππ π ππ + π β ππ = πππ ππ β ππ + ππ = πππ + ππ ππ = πππ π = ππ
πβππ + π
π+
The second angle is ππΛ, the first angle is
ππβππ π
=
ππ π
= ππΛ, and the third angle is ππΛ.
Make sure students see at least four different methods of solving the problem. Conclude this example with the statements below. ο§
Each method was slightly different either in terms of how the variable was defined or how the properties of equality were used to solve the equation. The way you find the answer may be different than your classmates or your teacher.
ο§
As long as you write accurately and do what is mathematically correct, you will find the correct answer.
Example 3 (4 minutes) ο§
A pair of alternate interior angles are described as follows. One angle is fourteen more than half a number. The other angle is six less than half a number. Are the angles congruent?
ο§
We will begin by assuming that the angles are congruent. If the angles are congruent, what does that mean about their measures? οΊ
ο§
Write an expression that describes each angle. οΊ
ο§
It means that they are equal in measure. One angle is
π₯
2
+ 14 and the other angle is
π₯
2
β 6.
If the angles are congruent, we can write the equation is
π₯
2
+ 14 =
π₯
2
β 6. We know that our properties of
equality allow us to transform equations while making sure that they remain true. π₯ π₯ + 14 = β 6 2 2 π₯ π₯ π₯ π₯ β + 14 = β β 6, 2 2 2 2 14 β β6. Therefore, our assumption was not correct and the angles are not congruent.
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Lesson 5
NYS COMMON CORE MATHEMATICS CURRICULUM
8β’4
Exercises 1β6 (16 minutes) Students complete Exercises 1β6 independently or in pairs. Exercises 1β6 For each of the following problems, write an equation and solve. 1.
A pair of congruent angles are described as follows: the measure of one angle is three more than twice a number and the other angleβs measure is ππ. π less than three times the number. Determine the size of the angles.
Let π be the number. Then the measure of one angle is π + ππ and the measure of the other angle is ππ β ππ. π. Because the angles are congruent, their measures are equal. Therefore, π + ππ = ππ β ππ. π π + ππ β ππ = ππ β ππ β ππ. π π = π β ππ. π π + ππ. π = π β ππ. π + ππ. π ππ. π = π
2.
Then each angle is π + π(ππ. π) = π + πππ = πππΛ.
The measure of one angle is described as twelve more than four times a number. Its supplement is twice as large. Find the measure of each angle.
Let π be the number. Then the measure of one angle is ππ + ππ. The other angle is π(ππ + ππ) = ππ + ππ. Since the angles are supplementary, their sum must be πππ: ππ + ππ + ππ + ππ = πππ πππ + ππ = πππ πππ + ππ β ππ = πππ β ππ πππ = πππ π = ππ
3.
One angle is π(ππ) + ππ = ππ + ππ = ππΛ and the other angle is πππ β ππ = πππΛ.
A triangle has angles described as follows: the first angle is four more than seven times a number, another angle is four less than the first and the third angle is twice as large as the first. What are the sizes of each of the angles?
Let π be the number. The measure of the first angle is ππ + π. The measure of the second angle is ππ + π β π = ππ. The measure of the third angle is π(ππ + π) = πππ + π. The sum of the angles of a triangle must be πππΛ. ππ + π + ππ + πππ + π = πππ πππ + ππ = πππ πππ + ππ β ππ = πππ β ππ πππ = πππ π=π
4.
The measure of the first angle is π(π) + π = ππ + π = ππΛ. The measure of the second angle is π(π) = ππΛ. The measure of the third angle is π(ππ) = ππΛ.
One angle measures nine more than six times a number. A sequence of rigid motions maps the angle onto another angle that is described as being thirty less than nine times the number. What is the measure of the angles?
Let π be the number. Then the measure of one angle is ππ + π. The measure of the other angle is ππ β ππ. Since rigid motions preserve the measures of angles, then the measure of these angles is equal. ππ + π = ππ β ππ ππ + π β π = ππ β ππ β π ππ = ππ β ππ ππ β ππ = ππ β ππ β ππ βππ = βππ π = ππ
The angles measure π(ππ) + π = ππ + π = ππΛ. Lesson 5: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
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Lesson 5
NYS COMMON CORE MATHEMATICS CURRICULUM
5.
8β’4
A right triangle is described as having an angle of size βsix less than negative two times a number,β another angle that is βthree less than negative one-fourth the numberβ, and a right angle. What are the measures of the angles? π
Let π be a number. Then one angle is βππ β π. The other angle is β β π. The sum of the two angles must be ππΛ. π π βππ β π + οΏ½β οΏ½ β π = ππ π ππ π οΏ½β οΏ½ + οΏ½β οΏ½ β π = ππ π π ππ οΏ½β οΏ½ β π + π = ππ + π π ππ β = ππ π βππ = πππ π = βππ One of the angles is βπ(βππ) β π = ππ β π = ππΛ. The other angle is ππ β ππ = πΛ. 6.
One angle is one less than six times the size of another. The two angles are complementary angles. Find the size of each angle. Let π be the measure of the first angle. Then the measure of the second angle is ππ β π. The sum of the measures will be ππ because the angles are complementary. π + ππ β π = ππ ππ β π = ππ ππ β π + π = ππ + π ππ = ππ π = ππ
One angle is ππΛ and the other angle is π(ππ) β π = ππ β π = ππΛ.
Closing (4 minutes) Summarize, or ask students to summarize, the main points from the lesson: ο§
We know that an algebraic method for solving equations is more efficient than guessing and checking.
ο§
We know how to write and solve equations that relate to angles, triangles, and geometry in general.
ο§
We know that drawing a diagram can sometimes make it easier to understand a problem and that there is more than one way to solve an equation.
Exit Ticket (4 minutes)
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Lesson 5
NYS COMMON CORE MATHEMATICS CURRICULUM
Name
8β’4
Date
Lesson 5: Writing and Solving Linear Equations Exit Ticket For each of the following problems, write an equation and solve. 1.
Given a right triangle, find the measures of all the angles if one angle is a right angle and the measure of the second angle is six less than seven times the measure of the third angle.
2.
In a triangle, the measure of the first angle is six times a number. The measure of the second angle is nine less than the first angle. The measure of the third angle is three times the number more than the measure of the first angle. Determine the measures of each of the angles.
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Lesson 5
NYS COMMON CORE MATHEMATICS CURRICULUM
8β’4
Exit Ticket Sample Solutions 1.
Given a right triangle, find the measures of all of the angles if one angle is a right angle and the measure of a second angle is six less than seven times the measure of the third angle. Let π represent the measure of the third angle. Then ππ β π can represent the measure of the second angle. The sum of the two angles in the right triangles will be ππΛ. ππ β π + π = ππ ππ β π = ππ ππ β π + π = ππ + π ππ = ππ π ππ π= π π π = ππ
2.
The measure of the third angle is ππΛ and the measure of the second angle is π(ππ) β π = ππ β π = ππΛ,. The measure of the third angle is ππΛ.
In a triangle, the measure of the first angle is six times a number. The measure of the second angle is nine less than the first angle. The measure of the third angle is three times the number more than the measure of the first angle. Determine the measures of each of the angles.
Let π be the number. Then the measure of the first angles is ππ, the measure of the second angle is ππ β π, and the measure of the third angle is ππ + ππ. The sum of the measures of the angles in a triangle is πππΛ. ππ + ππ β π + ππ + ππ = πππ πππ β π = πππ πππ β π + π = πππ + π πππ = πππ ππ πππ π= ππ ππ π=π
The measure of the first angle is π(π) = ππΛ. The measure of the second angle is ππ β π = ππΛ. The measure of the third angle is ππ + π(π) = ππ + ππ = ππΛ.
Note to teacher: There are several ways to solve problems like these. For example, a student may let π₯ be the size of the first angle, and write the size of the other angles accordingly. Either way, make sure that students are defining their symbols and correctly using the properties of equality to solve.
Problem Set Sample Solutions Students practice writing and solving linear equations. For each of the following problems, write an equation and solve. 1.
The measure of one angle is thirteen less than five times the measure of another angle. The sum of the measures of the two angles is πππΛ. Determine the measures of each of the angles.
Let π be the size of the one angle. Then the measure of the other angle is ππ β ππ. π + ππ β ππ = πππ ππ β ππ = πππ ππ β ππ + ππ = πππ + ππ ππ = πππ π = ππ. π
The measure of one angle is ππ. πΛ and the measure of the other angle is πππ β ππ. π = πππ. πΛ. Lesson 5: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
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Lesson 5
NYS COMMON CORE MATHEMATICS CURRICULUM
2.
8β’4
An angle measures seventeen more than three times a number. Its supplement is three more than seven times the number. What is the measure of each angle? Let π be the number. Then one angle is ππ + ππ. The other angle is ππ + π. Since the angles are supplementary, the sum of their measures will be πππ. ππ + ππ + ππ + π = πππ πππ + ππ = πππ πππ + ππ β ππ = πππ β ππ πππ = πππ π = ππ
One angle is π(ππ) + ππ = ππΛ. The other angle is πππ β ππ = πππΛ. 3.
The angles of a triangle are described as follows: β π¨ is the largest angle, its measure is twice the measure of β π©. The measure of β πͺ is π less than half the measure of β π©. Find the measures of the three angles.
Let π be the measure of β π©. Then the measure of β π¨ = ππ and β πͺ =
must be πππ.
π β π. The sum of the measures of the angles π
π π + ππ + β π = πππ π π ππ + β π + π = πππ + π π π ππ + = πππ π ππ π + = πππ π π ππ = πππ π ππ = πππ π = ππ
The measures of the angles are as follows: β π¨ = πππΛ, β π© = ππΛ, πππ
β πͺ = 4.
ππ β π = ππ β π = ππΛ. π
A pair of corresponding angles are described as follows: the measure of one angle is five less than seven times a number and the measure of the other angle is eight more than seven times the number. Are the angles congruent? Why or why not? Let π be the number. Then measure of one angle is ππ β π and the measure of the other angle is ππ + π. Assume they are congruent, which means their measures are equal.
Since β π β π, the angles are not congruent. 5.
ππ β π = ππ + π ππ β ππ β π = ππ β ππ + π βπ β π
The measure of one angle is eleven more than four times a number. Another angle is twice the first angleβs measure. The sum of the measures of the angles is πππΛ. What is the measure of each angle?
Let π be the number. The measure of one angle can be represented with ππ + ππ and the other angleβs measure can be represented as π(ππ + ππ) = ππ + ππ. ππ + ππ + ππ + ππ = πππ πππ + ππ = πππ πππ + ππ β ππ = πππ β ππ πππ = πππ π = ππ. π
The measure of one angle is π(ππ. π) + ππ = ππ + ππ = ππΛ and the measure of the other angle is πππΛ.
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Lesson 5
NYS COMMON CORE MATHEMATICS CURRICULUM
6.
8β’4
Three angles are described as follows: β π© is half the size of β π¨. The measure of β πͺ is equal to one less than π times the measure of β π©. The sum of β π¨ and β π© is πππ. Can the three angles form a triangle? Why or why not?
Let π represent the measure of β π¨. Then the measure of β π© = The sum of β π¨ + β π© = πππ.
π π and the measure of β πͺ = π οΏ½ οΏ½ β π = π β π . π π
π = πππ π ππ = πππ π ππ = πππ π = ππ
π+
The measure of β π¨ = ππΛ, β π© =
ππ = ππΛ, and β πͺ = ππΛ. The sum of the three angles is ππ + ππ + ππ = πππ. π
Since the sum of the measures of the interior angles of a triangle must sum to πππΛ, these angles do not form a triangle. Their sum is too large.
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Lesson 6
NYS COMMON CORE MATHEMATICS CURRICULUM
8β’4
Lesson 6: Solutions of a Linear Equation Student Outcomes ο§
Students transform equations into simpler forms using the distributive property.
ο§
Students learn that not every linear equation has a solution.
Lesson Notes The distributive property can be used to both expand and simplify expressions. Students have already used the distributive Property to βcollect like terms.β For example, 2π₯ + 6π₯ = (2 + 6)π₯ = 8π₯. Students have also used the distributive property to expand expressions. For example, 2(π₯ + 5) = 2π₯ + 10. In this lesson students continue to use the distributive property to solve more complicated equations. Also highlighted in this lesson is a common error that students make when using the distributive property, which is multiplying a factor to terms that are not part of the group. For example, in the expression 3(π₯ + 1) β 5, students should know that they do not distribute the factor 3 to the term β5 because it is not in the group (π₯ + 1).
Classwork Example 1 (4 minutes) ο§
What value of π₯ would make the linear equation 4π₯ + 3(4π₯ + 7) = 4(7π₯ + 3) β 3 true? What is the βbestβ first step and why?
Have a discussion with students about what the βbestβ first step is and why. Make clear that the distributive property will allow us to better see and work with the terms of the linear equation. Proceed with the following discussion. ο§
In order to find out what that solution might be, we must use the distributive property. The left side of the equation has the following expression: 4π₯ + 3(4π₯ + 7)
Where and how will the distributive property be used? οΊ ο§
ο§
We will need to use the distributive property to expand 3(4π₯ + 7) and then again to collect like terms.
Our work for now is just on the left side of the equation, the right side will remain unchanged for the moment. 4π₯ + 3(4π₯ + 7) = 4(7π₯ + 3) β 3 4π₯ + 12π₯ + 21 = 4(7π₯ + 3) β 3 (4 + 12)π₯ + 21 = 4(7π₯ + 3) β 3 16π₯ + 21 = 4(7π₯ + 3) β 3
Now we need to rewrite the right side. Again we will use the distributive property. The left side of the equation will remain unchanged. 16π₯ + 21 = 4(7π₯ + 3) β 3 16π₯ + 21 = 28π₯ + 12 β 3
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Lesson 6
NYS COMMON CORE MATHEMATICS CURRICULUM
ο§
ο§
Notice that we did not apply the distributive property to the term β3. Since it was not part of the group (7π₯ + 3), it is not multiplied by 4. 16π₯ + 21 = 28π₯ + 9
Now we have the transformed the given equation into the following form: 16π₯ + 21 = 28π₯ + 9. Solve the equation. οΊ
ο§
8β’4
Student work:
16π₯ + 21 = 28π₯ + 9 16π₯ β 16π₯ + 21 = 28π₯ β 16π₯ + 9 21 = 12π₯ + 9 21 β 9 = 12π₯ + 9 β 9 12 = 12π₯ π₯=1
Is π₯ = 1 really a solution to the equation 4π₯ + 3(4π₯ + 7) = 4(7π₯ + 3) β 3? How do you know? οΊ
Yes, π₯ = 1 is a solution because 4 + 3(11) = 37 and 4(10) β 3 = 37. Since both expressions are equal to 37, then π₯ = 1 is a solution to the equation.
Example 2 (4 minutes) ο§
ο§
ο§
What value of π₯ would make the following linear equation true: 20 β (3π₯ β 9) β 2 = β(β11π₯ + 1)? Since we have a group of terms that is preceded by a βββ sign, we will simplify this first. The βββ sign means we need to take the opposite of each of the terms within the group, i.e., Scaffolding: parentheses. The equation in this example We begin with the left side of the equation: can be modified to 20 β (3π₯ β 20 β (3π₯ β 9) β 2 = β(β11π₯ + 1) 9 + 1) = 10 to meet the needs of diverse learners. Also We need only to take the opposite of the terms within the grouping symbols. Is consider having students fold a the term β2 within the grouping symbol? piece of paper in half, solving οΊ No. on the left side and justifying For that reason, we need only find the opposite of 3π₯ β 9. What is the opposite their steps on the right side. of 3π₯ β 9? οΊ
ο§
ο§
The opposite of 3π₯ β 9 is β3π₯ + 9.
The left side of the equation is rewritten as
20 β (3π₯ β 9) β 2 = β(β11π₯ + 1) 20 β 3π₯ + 9 β 2 = β(β11π₯ + 1) 20 + 9 β 2 β 3π₯ = β(β11π₯ + 1) 27 β 3π₯ = β(β11π₯ + 1)
Now we rewrite the right side of the equation: β(β11π₯ + 1).
27 β 3π₯ = β(β11π₯ + 1) 27 β 3π₯ = 11π₯ β 1
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Lesson 6
NYS COMMON CORE MATHEMATICS CURRICULUM
ο§
ο§
8β’4
The transformed equation is 27 β 3π₯ = 11π₯ β 1.
27 β 3π₯ = 11π₯ β 1 27 β 3π₯ + 3π₯ = 11π₯ + 3π₯ β 1 27 = 14π₯ β 1 27 + 1 = 14π₯ β 1 + 1 28 = 14π₯ 2=π₯
Check: The left side is: 27 β 3(2) = 27 β 6 = 21. The right side is: 11(2) β 1 = 22 β 1 = 21. Since 21 = 21, π₯ = 2 is the solution.
Example 3 (4 minutes) ο§
What value of π₯ would make the following linear equation true: transforming both sides of the equation into a simpler form. οΊ
Student work:
1 2
(4π₯ + 6) β 2 = β(5π₯ + 9)?
1 (4π₯ + 6) β 2 = β(5π₯ + 9) 2 2π₯ + 3 β 2 = β5π₯ β 9 1
Make sure that students do not distribute the factor to the term β2, and that they have, in general, transformed the equation correctly. ο§
Scaffolding:
The equation in this example can be modified to 2π₯ + 1 = β(5π₯ + 9) to meet the needs of diverse learners.
2
Now that we have the simpler equation, 2π₯ + 3 β 2 = β5π₯ β 9, complete the solution. οΊ
ο§
Begin by
Student work:
2π₯ + 3 β 2 = β5π₯ β 9 2π₯ + 1 = β5π₯ β 9 2π₯ + 1 β 1 = β5π₯ β 9 β 1 2π₯ + 0 = β5π₯ β 10 2π₯ = β5π₯ β 10 2π₯ + 5π₯ = β5π₯ + 5π₯ β 10 (2 + 5)π₯ = (β5 + 5)π₯ β 10 7π₯ = 0π₯ β 10 7π₯ = β10 7 10 π₯=β 7 7 10 π₯=β 7
10 20 7 13 10 50 63 13 οΏ½ + 1 = β + = β . The right side is: β5 οΏ½β οΏ½ β 9 = β =β . 7 7 7 7 7 7 7 7 13 13 10 is the solution. Since β = β , π₯ = β 7 7 7
Check: The left side is: 2 οΏ½β
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Lesson 6
NYS COMMON CORE MATHEMATICS CURRICULUM
8β’4
Example 4 (11 minutes) ο§
Consider the following equation: 2(π₯ + 1) = 2π₯ β 3. What value of π₯ makes the equation true? οΊ
ο§
Student work:
How should we interpret 2 = β3?
2(π₯ + 1) = 2π₯ β 3 2π₯ + 2 = 2π₯ β 3 2π₯ β 2π₯ + 2 = 2π₯ β 2π₯ β 3 2 = β3
Lead a discussion with the conclusion that since 2 β β3, then the equation has no solution. Allow students time to try to find a value of π₯ that would make it true by guessing and checking. After they realize that there is no such number π₯, MP.3 make it clear to students that some equations have no solution. Ask the following question. ο§
Why do you think this happened? οΊ
ο§
We know that an equation is a statement of equality. The linear expressions were such that they could not be equal to each other, no matter what value was substituted for π₯.
What value of π₯ would make the following linear equation true: 9(4 β 2π₯) β 3 = 4 β 6(3π₯ β 5)? Transform the equation by simplifying both sides. οΊ
Student work:
9(4 β 2π₯) β 3 = 4 β 6(3π₯ β 5) 36 β 18π₯ β 3 = 4 β 18π₯ + 30
Scaffolding:
The equation in this example can be modified to 9(4 β 2π₯) β 3 = β18π₯ to meet the needs of diverse learners.
Be sure to check that students did not subtract 4 β 6 on the right side, then distribute β2. This is a common error. Remind students that we must multiply first, then add or subtract, just like we would to simply expressions using the correct order of operations. ο§
The transformed equation is 36 β 18π₯ β 3 = 4 β 18π₯ + 30. Now complete the solution. οΊ
Student work:
36 β 18π₯ β 3 = 4 β 18π₯ + 30 33 β 18π₯ = 34 β 18π₯ 33 β 18π₯ + 18π₯ = 34 β 18π₯ + 18π₯ 33 = 34
ο§
Like the last problem, there is no value of π₯ that can be substituted into the equation to make it true. Therefore this equation has no solution.
Write at least one equation that has no solution. It does not need to be complicated, but the result should be similar to the last two problems. The result from the first equation was 2 β β3 and the second was 33 β 34.
Have students share their equations and verify that they have no solution.
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Lesson 6
NYS COMMON CORE MATHEMATICS CURRICULUM
8β’4
Optional Example 5 (4 minutes) ο§
So far we have used the distributive property to simplify expressions when solving equations. In some cases, we can use the distributive property to make our work even simpler. Consider the following equation: 3π₯ + 15 = β6
Notice that each term has a common factor of 3. We will use the distributive property and what we know about the properties of equality to solve this equation quickly. 3π₯ + 15 = β6 3(π₯ + 5) = 3 β
(β2)
Notice that the expressions on both sides of equal sign have a factor of 3. We can use the multiplication property of equality, if π΄ = π΅, then π΄ β
πΆ = π΅ β
πΆ as follows:
ο§
3(π₯ + 5) = 3 β
(β2) π₯ + 5 = β2 π₯ + 5 β 5 = β2 β 5 π₯ = β7
This isnβt something that we can expect to do every time we solve an equation, but it is good to keep an eye out for it.
Exercises 1β6 (12 minutes) Students complete Exercises 1β6 independently. Exercises Find the value of π that makes the equation true. 1.
ππ β π(ππ β π) = β(βππ + ππ) + π
Sample student work:
ππ β π(ππ β π) = β(βππ + ππ) + π ππ β πππ + ππ = ππ β ππ + π ππ β πππ = ππ β π ππ β πππ + πππ = ππ + πππ β π ππ = πππ β π ππ + π = πππ β π + π ππ = πππ ππ ππ = π ππ ππ ππ =π ππ ππ =π π
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Lesson 6
NYS COMMON CORE MATHEMATICS CURRICULUM
2.
β(π β π) +
8β’4
π = π(π + π) π
Sample student work:
π = π(π + π) π π βπ + π + = ππ + ππ π ππ βπ + = ππ + ππ π ππ βπ + π + = ππ + π + ππ π ππ = ππ + ππ π ππ β ππ = ππ + ππ β ππ π ππ β = ππ π π βππ π β
= β
ππ π π π ππ β =π π
β (π β π) +
3.
π π
+ π(π β π) =
ππ π
β (π β ππ) + π
Sample student work:
4.
ππ π + π(π β π) = β (π β ππ) + π π π π π ππ π β + π(π β π) = β β (π β ππ) + π π π π π ππ ππ β π = β π + ππ + π π ππ ππ β π = + ππ π ππ ππ ππ β π + π = + + ππ π π ππ ππ = + ππ π ππ ππ β ππ = π ππ βππ = π π π ππ β
βππ = β
βπ βπ π ππ π=β π
π(ππ + π) β ππ = ππ β π(βππ + ππ)
This equation has no solution.
Lesson 6: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
π(ππ + π) β ππ = ππ β π(βππ + ππ) πππ + ππ β ππ = ππ + ππ β ππ πππ + ππ = πππ β ππ πππ β πππ + ππ = πππ β πππ β ππ ππ β βππ
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Lesson 6
NYS COMMON CORE MATHEMATICS CURRICULUM
5.
8β’4
π π
ππ β (ππ + π) β π = (ππ + ππ) β ππ + π
Sample student work:
π (ππ + ππ) β ππ + π π ππ β ππ β π β π = ππ + ππ β ππ + π ππ β ππ = βππ + ππ ππ β ππ + ππ = βππ + ππ + ππ ππ = βππ + ππ ππ + ππ = βππ + ππ + ππ ππ = ππ π=π
ππ β (ππ + π) β π =
6.
Write at least three equations that have no solution. Answers will vary. Verify that the equations written have no solution.
Closing (5 minutes) Summarize, or ask students to summarize, the following main points from the lesson: ο§
We know how to transform equations into simpler forms using the distributive property.
ο§
We now know that there are some equations that do not have solutions.
Lesson Summary The distributive property is used to expand expressions. For example, the expression π(ππ β ππ) is rewritten as ππ β ππ after the distributive property is applied.
The distributive property is used to simply expressions. For example, the expression ππ + πππ is rewritten as (π + ππ)π and πππ after the distributive property is applied.
The distributive property is applied only to terms within a group:
π(ππ + π) β π = πππ + ππ β π
Notice that the term βπ is not part of the group and therefore not multiplied by π. When an equation is transformed into an untrue sentence, like π β ππ, we say the equation has no solution.
Exit Ticket (5 minutes)
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Solutions of a Linear Equation 11/8/13
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Lesson 6
NYS COMMON CORE MATHEMATICS CURRICULUM
Name
8β’4
Date
Lesson 6: Solutions of a Linear Equation Exit Ticket Transform the equation if necessary, then solve to find the value of π₯ that makes the equation true. 1 3
1.
5π₯ β (π₯ + 3) = (9π₯ + 18) β 5
2.
5(3π₯ + 9) β 2π₯ = 15π₯ β 2(π₯ β 5)
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Solutions of a Linear Equation 11/8/13
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Lesson 6
NYS COMMON CORE MATHEMATICS CURRICULUM
8β’4
Exit Ticket Sample Solutions Transform the equation if necessary, then solve to find the value of π that makes the equation true. 1.
2.
π π
ππ β (π + π) = (ππ + ππ) β π
π (ππ + ππ) β π π ππ β π β π = ππ + π β π ππ β π = ππ + π ππ β ππ β π = ππ β ππ + π πβπ=π π=π
ππ β (π + π) =
π(ππ + π) β ππ = πππ β π(π β π)
π(ππ + π) β ππ = πππ β π(π β π) πππ + ππ β ππ = πππ β ππ + ππ πππ + ππ = πππ + ππ ππ β ππ
Since ππ β ππ, then the equation has no solution.
Problem Set Sample Solutions Students practice using the distributive property to transform equations and solve. Transform the equation if necessary, then solve it to find the value of π that makes the equation true. 1.
π π
π β (ππ β ππ) + ππ = πππ + π οΏ½βππ + οΏ½
Sample student work:
2.
π π
ππ + π οΏ½π + οΏ½ = π(ππ β π) β π
Sample student work:
Lesson 6: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
π π β (ππ β ππ) + ππ = πππ + π οΏ½βππ + οΏ½ π π β ππ + ππ + ππ = πππ β ππ + π βππ + ππ = ππ + π βππ + ππ + ππ = ππ + ππ + π ππ = πππ + π ππ β π = πππ + π β π ππ = πππ ππ ππ = π ππ ππ ππ =π π π ππ + π οΏ½π + οΏ½ = π(ππ β π) β π π ππ + ππ + π = πππ β ππ β π πππ + π = πππ β ππ πππ β πππ + π = πππ β πππ β ππ π = ππ β ππ π + ππ = ππ β ππ + ππ ππ = ππ ππ π = π π π ππ =π π
Solutions of a Linear Equation 11/8/13
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Lesson 6
NYS COMMON CORE MATHEMATICS CURRICULUM
3.
βππ β π(ππ + π) = β(βππ β ππ)
8β’4
Sample student work:
4.
βππ β π(ππ + π) = β(βππ β ππ) βππ β πππ β π = ππ + ππ βπππ β π = ππ + ππ βπππ + πππ β π = ππ + πππ + ππ βπ = πππ + ππ βπ β ππ = πππ + ππ β ππ βππ = πππ ππ ππ β = π ππ ππ π β =π ππ
ππ(π + ππ) = πππ
Sample student work:
5.
π π
ππ(π + ππ) = πππ π + ππ = ππ π=π
π π
πππ + β οΏ½π + οΏ½ = π(ππ β π) + π
Sample student work:
π π β οΏ½π + οΏ½ = π(ππ β π) + π π π π π πππ + β π β = πππ β ππ + π π π π πππ + = πππ β ππ π π πππ β πππ + = πππ β πππ β ππ π π β βππ π
πππ +
This equation has no solution. 6.
π(ππ β ππ) + π = ππ β (βππ β π) Sample student work:
7.
π(ππ + π) = ππ
π(ππ β ππ) + π = ππ β (βππ β π) ππ β ππ + π = ππ + ππ + π ππ β ππ = ππ + ππ ππ β ππ β ππ = ππ + ππ β ππ βππ = ππ + ππ βππ β ππ = ππ β ππ + ππ βππ = ππ βππ = π
Sample student work:
Lesson 6: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
π(ππ + π) = ππ ππ + π = π ππ + π β π = π β π ππ = βπ π = βπ
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Lesson 7
NYS COMMON CORE MATHEMATICS CURRICULUM
8β’4
Lesson 7: Classification of Solutions Student Outcomes ο§
Students know the conditions for which a linear equation will have a unique solution, no solution, or infinitely many solutions.
Lesson Notes Part of the discussion on the second page in this lesson is optional. The key parts of the discussion are those that point out the characteristics of the constants and coefficients of an equation that allow students to see the structure in equations that lead to a unique solution, no solution, or infinitely many solutions. You can choose to go through the discussion with your students or choose to use the activity at the top of the fourth page of this lesson. The activity requires students to examine groups of equations and make conclusions about the nature of their solutions based on what they observe, leading to the same result as discussion.
Classwork Exercises 1β3 (6 minutes) Students complete Exercises 1β3 independently in preparation for the discussion that follows. Exercises 1β3 Solve each of the following equations for π. 1.
ππ β π = ππ + π
Sample student work: ππ β π = ππ + π ππ β π + π = ππ + π + π ππ = ππ + π ππ β ππ = ππ β ππ + π ππ = π π=π 2.
ππ β π = ππ + π
Sample student work:
This equation has no solution.
Lesson 7: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
ππ β π = ππ + π ππ β ππ β π = ππ β ππ + π βπ β π
Classification of Solutions 11/8/13
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Lesson 7
NYS COMMON CORE MATHEMATICS CURRICULUM
3.
8β’4
ππ β π = βπ + ππ
Sample student work: ππ β π = βπ + ππ ππ β π + π = βπ + π + ππ ππ = ππ
OR
ππ β π = βπ + ππ ππ β ππ β π = βπ + ππ β ππ βπ = βπ
Discussion (15 minutes) Display the three equations so that students can easily see and compare them throughout the discussion. 7π₯ β 3 = 5π₯ + 5 ο§
7π₯ β 3 = 7π₯ + 5
7π₯ β 3 = β3 + 7π₯
Was there anything new or unexpected about Exercise 1? οΊ
No. We solved the equation for π₯ and π₯ = 4.
You may choose to continue with the discussion of Exercise 1 or complete the activity described at the end of the discussion on page 74. Be sure to revisit Exercises 2 and 3 and classify the solution to those equations. ο§
What can you say about the coefficients of terms with π₯ in the equation? οΊ
ο§
They are different.
Do you think any number other than 4 would make the equation true? οΊ
I donβt think so, but Iβd have to try different numbers in order to find out.
ο§
Instead of having to check every single number to see if it makes the equation true, we can look at a general form of an equation to show there can be only one solution.
ο§
Given a linear equation that has been simplified on both sides, ππ₯ + π = ππ₯ + π, where π, π, π, and π are constants and π β π, we can use use our normal properties of equality to solve for π₯: ππ₯ + π = ππ₯ + π ππ₯ + π β π = ππ₯ + π β π ππ₯ = ππ₯ + π β π ππ₯ β ππ₯ = ππ₯ β ππ₯ + π β π (π β π)π₯ = π β π πβπ πβπ π₯= πβπ πβπ πβπ π₯= πβπ The only value of π₯ that will make the equation true is the division of the difference of the constants by the difference of the coefficients of π₯. In other words, if the coefficients of π₯ are different on each side of the equal sign, then the equation will have one solution.
ο§
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Lesson 7
NYS COMMON CORE MATHEMATICS CURRICULUM
ο§
ο§ ο§
As an aside, if the coefficients of π₯ are different and the value of the constants are the same, the only solution is π₯ = 0. Consider the following equation: 2π₯ + 12 = π₯ + 12 2π₯ + 12 β 12 = π₯ + 12 β 12 2π₯ = π₯ 2π₯ β π₯ = π₯ β π₯ π₯=0
The only value of π₯ that will make 2π₯ + 12 = π₯ + 12 true, is π₯ = 0.
Was there anything new or unexpected about Exercise 2? οΊ
ο§
ο§
The coefficients were the same.
What can you say about the constants in the second equation? οΊ
ο§
No. We tried to solve the equation and got an untrue statement, β3 β 5; therefore, the equation had no solutions.
What can you say about the coefficients of the terms with π₯ in the second equation? οΊ
ο§
8β’4
The constants were different.
It is time to begin formalizing what we have observed as we have solved equations. To begin with, we have learned that some linear equations have no solutions. That is, when we solve an equation using our properties of equality and end up with a statement like π = π, where π, π are different numbers, we say that the equation has no solution.
To produce and/or recognize an equation with no solutions, consider the equation with both sides simplified. The coefficients of the variable π₯ will be the same on both sides of the equal sign and the value of the constants on each side will be different. This type of equation has no solution because there is no value of π₯ in an equation such as π₯ + π = π₯ + π that will make the equation true.
Give students a moment to compare the equation π₯ + π = π₯ + π with Exercise 2. ο§
Was there anything new or unexpected about Exercise 3? οΊ οΊ
Yes. When solving the equation we got β3 = β3, which did not give us a value for π₯.
Yes. When solving the equation we got to 7π₯ = 7π₯ and we werenβt sure what to do next or what it meant.
Provide students with time to discuss in small groups how they should interpret the meaning of such statements. Students have yet to work with equations that yield such results and they need the time to make sense of the statements. If necessary, ask the questions in the next two bullet points to guide their thinking. Give them time to try different values of π₯ in the both equations to verify that it is always true. Next, have students share their thoughts and interpretations and resume the discussion below. ο§
Is β3 = β3 a true statement? Is it always true? οΊ
ο§
If 7π₯ = 7π₯, will we always have an equality, no matter which number we choose for π₯? οΊ
ο§
Yes.
Yes. This is an identity because the same expression is on both sides of the equal sign.
What can you say about the coefficients of the terms with π₯ in the equation in Exercise 3? οΊ
The coefficients are the same.
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Lesson 7
NYS COMMON CORE MATHEMATICS CURRICULUM
ο§
What can you say about the constants in the equation? οΊ
ο§
ο§
8β’4
The constants are the same.
We now know that there are equations that produce statements that are always true. That is, β3 will always equal β3, 0 is always equal to 0, and 7π₯ will always equal 7π₯ no matter what number π₯ happens to be (because 7 times a number is just a number). When we solve an equation using our properties of equality and end up with a statement like π = π where π is a number, then we say that the equation has infinitely many solutions. To produce and/or recognize an equation with infinitely many solutions, the coefficients of the variable π₯ and the value of the constants will be the same in the simplified expressions on both sides of the equal sign. This type of equation has infinitely many solutions because any value of π₯ in an equation like π₯ + π = π₯ + π will make the equation true.
Give students a moment to compare the equation π₯ + π = π₯ + π with Exercise 3. Activity: Display the table below. Provide students time to work independently to solve the equations, then in small groups to generalize what they notice about each of the equations, that is, they should answer the question: What can we see in an equation that will tell us about the solution to the equation? 3π₯ + 4 = 8π₯ β 9
MP.8
β4π₯ β 5 = 6 β 11π₯ 1 9 + π₯ = 5π₯ β 1 2
6π₯ + 5 = 8 + 6π₯
10π₯ β 4 = β4 + 10π₯
5 5 π₯β1=1+ π₯ 4 4
7 + 9π₯ = 9π₯ + 7
12 β 15π₯ = β2 β 15π₯
β2π₯ + 5 = β2π₯ + 5
The goal of the activity is for students to make the following three observations. (1) In equations where both sides are simplified, when the coefficients of π₯ and the constants in the equation are unique, i.e., ππ₯ + π = ππ₯ + π, there will be a unique solution, i.e., π₯ =
πβπ , as in the first column. (2) Similarly, we want students to observe that when an equation is πβπ
of the form π₯ + π = π₯ + π, that there will be no solution, as in the second column. (3) Lastly, when an equation is of the form π₯ + π = π₯ + π, then there are infinitely many solutions, as in the third column.
Exercises 4β10 (14 minutes) Students complete Exercises 4β10 independently or in pairs. Exercises 4β10 Give a brief explanation as to what kind of solution(s) you expect the following linear equations to have. Transform the equation into a simpler form if necessary. 4.
πππ β ππ + ππ = π + π + ππ
If I use the distributive property on the left side I notice that the coefficients of the π are the same, specifically π, and when I simplify the constants on the right side, I notice that they are the same. Therefore, this equation has infinitely many solutions.
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Lesson 7
NYS COMMON CORE MATHEMATICS CURRICULUM
5.
8β’4
π(π β ππ) + π = βππ + π
If I use the distributive property on the left side I notice that the coefficients of the π are the different. Therefore, the equation has one solution. 6.
βππ + ππ β ππ = βπ(ππ + ππ)
If I use the distributive property on the each side of the equation I notice that the coefficients of π are the same, but the constants are different. Therefore, this equation has no solutions. 7.
8.
π π
(ππ + ππ) = ππ + ππ
If I use the distributive property on the left side I notice that the coefficients of the π are the same, specifically π, and the constants are also the same, ππ. Therefore, this equation has infinitely many solutions. Write two equations that have no solutions.
Answers will vary. Verify that students have written equations where the coefficients of π on each side of the equal sign are the same and the constants on each side are unique. 9.
Write two equations that have one unique solution each. Answers will vary. Accept equations where the coefficients of π on each side of the equal sign are unique.
10. Write two equations that have infinitely many solutions.
Answers will vary. Accept equations where the coefficients of π and constants on each side of the equal sign are the same.
Closing (5 minutes) Summarize, or ask students to summarize, the main points from the lesson: ο§
We know that equations will either have a unique solution, no solution, or infinitely many solutions.
ο§
We know that equations with no solution will, after being simplified on both sides, have coefficients of π₯ that are the same on both sides of the equal sign and constants that are different.
ο§
We know that equations with infinitely many solutions will, after being simplified on both sides, have coefficients of π₯ and constants that are the same on both sides of the equal sign. Lesson Summary There are three classifications of solutions to linear equations: one solution (unique solution), no solution, or infinitely many solutions. Equations with no solution will, after being simplified, have coefficients of π that are the same on both sides of the equal sign and constants that are different. For example, π + π = π + π, where π, π are constants that are not equal. A numeric example is ππ + π = ππ β π.
Equations with infinitely many solutions will, after being simplified, have coefficients of π and constants that are the same on both sides of the equal sign. For example, π + π = π + π, where π is a constant. A numeric example is ππ + π = π + ππ.
Exit Ticket (5 minutes)
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Lesson 7
NYS COMMON CORE MATHEMATICS CURRICULUM
Name
8β’4
Date
Lesson 7: Classification of Solutions Exit Ticket Give a brief explanation as to what kind of solution(s) you expect the following linear equations to have. Transform the equation into a simpler form if necessary. 1.
3(6π₯ + 8) = 24 + 18π₯
2.
12(π₯ + 8) = 11π₯ β 5
3.
5π₯ β 8 = 11 β 7π₯ + 12π₯
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Classification of Solutions 11/8/13
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Lesson 7
NYS COMMON CORE MATHEMATICS CURRICULUM
8β’4
Exit Ticket Sample Solutions Give a brief explanation as to what kind of solution(s) you expect the following linear equations to have. Transform the equation into a simpler form if necessary. 1.
π(ππ + π) = ππ + πππ
If I use the distributive property on the left side I notice that the coefficients of the π are the same and the constants are the same. Therefore, this equation has infinitely many solutions. 2.
ππ(π + π) = πππ β π
If I use the distributive property on the left side I notice that the coefficients of the π are different and the constants are different. Therefore, this equation has a unique solution. 3.
ππ β π = ππ β ππ + πππ
If I collect the like terms on the right side I notice that the coefficients of π are the same, but the constants are different. Therefore, this equation has no solution.
Problem Set Sample Solutions Students apply their knowledge of solutions to linear equations by writing equations with unique solutions, no solutions, and infinitely many solutions. 1.
Give a brief explanation as to what kind of solution(s) you expect the linear equation to have: πππ +
π = π(ππ + ππ). Transform the equation into a simpler form if necessary. π
If I use the distributive property on the right side of the equation I notice that the coefficients of π are the same, but the constants are different. Therefore, this equation has no solutions. 2.
Give a brief explanation as to what kind of solution(s) you expect the linear equation to have: π β ππ = πππ + π + ππ. Transform the equation into a simpler form if necessary.
If I collect the like terms on the right side of the equation then I notice that the coefficients of π are different and so are the constants. Therefore, this equation will have a unique solution. 3.
Give a brief explanation as to what kind of solution(s) you expect the linear equation to have: π(π + π) = ππ + ππ. Transform the equation into a simpler form if necessary.
This is an identity under the distributive property. Therefore, this equation will have infinitely many solutions. 4.
Give three examples of equations where the solution will be unique, that is, only one solution is possible. Accept equations where the coefficients of π on each side of the equal sign are unique.
5.
Solve one of the equations you wrote in problem 4 and explain why it is the only solution. Verify that students solved one of the equations. They should have an explanation that includes the statement that there is only one possible number that could make the equation true. They may have referenced one of the simpler forms of their transformed equation to make their case.
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Lesson 7
NYS COMMON CORE MATHEMATICS CURRICULUM
6.
8β’4
Give three examples of equations where there will be no solution. Accept equations where the coefficients of π on each side of the equal sign are the same and the constants on each side are unique.
7.
Attempt to solve one of the equations you wrote in problem 6 and explain why it has no solution. Verify that students have solved one of the equations. They should have an explanation that includes the statement about getting a false equation, e.g., π β ππ.
8.
Give three examples of equations where there will be infinitely many solutions. Accept equations where the coefficients of π and constants on each side of the equal sign are the same.
9.
Attempt to solve one of the equations you wrote in problem 8 and explain why it has infinitely many solutions. Verify that students have solved one of the equations. They should have an explanation that includes the statement about the linear expressions being exactly the same, an identity; therefore, any rational number π would make the equation true.
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Lesson 8
NYS COMMON CORE MATHEMATICS CURRICULUM
8β’4
Lesson 8: Linear Equations in Disguise Student Outcomes ο§
Students rewrite and solve equations that are not obviously linear equations using properties of equality.
Lesson Notes In this lesson, students learn that some equations that may not look like linear equations are, in fact, linear. This lesson on solving rational equations is included because of the types of equations students will see in later topics of this module related to slope. Students will recognize these equations as proportions. It is not necessary to refer to these types of equations as equations that contain rational expressions. They can be referred to simply as proportions since students are familiar with this terminology. Expressions of this type will be treated carefully in algebra as they involve a discussion about why the denominator of such expressions cannot be equal to zero. That discussion is not included in this lesson.
Classwork Concept Development (3 minutes) ο§
ο§
Some linear equations may not look like linear equations upon first glance. A simple example that you should recognize is π₯ 6 = 5 12 What do we call this kind of problem and how do we solve it? οΊ
This is a proportion. We can solve this by multiplying both sides of the equation by 5. We can also solve it by multiplying each numerator with the other fractionβs denominator.
Students may not think of multiplying each numerator with the other fractionβs denominator because multiplying both sides by 5 requires fewer steps and uses the multiplication property of equality that has been used to solve other equations. If necessary, state the theorem and give a brief explanation. Theorem. Given rational numbers π΄, π΅, πΆ, and π·, so that π΅ β 0 and π· β 0, the property states If
ο§
π΄
π΅
=
πΆ
π·
, then π΄π· = π΅πΆ.
To find the value of π₯, we can multiply each numerator by the other fractionβs denominator. π₯ 6 = 5 12 12π₯ = 6(5)
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Linear Equations in Disguise 11/8/13
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NYS COMMON CORE MATHEMATICS CURRICULUM
ο§
ο§
Lesson 8
8β’4
It should be more obvious now that we have a linear equation. We can now solve it as usual using the properties of equality. 12π₯ = 30 30 π₯= 12 5 π₯= 2 In this lesson, our work will be similar, but the numerator and/or the denominator of the fractions may contain more than one term. However, the way we solve these kinds of problems remain the same.
Example 1 (5 minutes) ο§
ο§
ο§
Given a linear equation in disguise, we will try to solve it. To do so, we must first assume that the following equation is true for some number π₯. 1 π₯β1 π₯+3 = 4 2 We want to make this equation look like the linear equations we are used to. For that reason, we will multiply both sides of the equation by 2 and 4, as we normally do with proportions: 1 2 οΏ½π₯ + οΏ½ = 4(π₯ β 1) 3 Is this a linear equation? How do you know? οΊ
Yes, this is a linear equation because the expressions on the left and right of the equal sign are linear expressions.
ο§
Notice that the expressions that contained more than one term were put in parentheses. We do that so we donβt make a mistake and forget to use the distributive property.
ο§
Now that we have a linear equation. We will use the distributive property and solve as usual. 1 2 οΏ½π₯ + οΏ½ = 4(π₯ β 1) 3 2 2π₯ + = 4π₯ β 4 3 2 2π₯ β 2π₯ + = 4π₯ β 2π₯ β 4 3 2 = 2π₯ β 4 3 2 + 4 = 2π₯ β 4 + 4 3 14 = 2π₯ 3 1 14 1 β
= β
2π₯ 2 3 2 7 =π₯ 3
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Lesson 8
NYS COMMON CORE MATHEMATICS CURRICULUM
ο§
7
How can we verify that is the solution to the equation? 3
7
We can replace π₯ with in the original equation:
οΊ
ο§
8β’4
3
7
1 π₯β1 π₯+3 = 4 2 7 7 1 β1 + 3 =3 3 4 2 8 4 3 = 3 2 4 4 8 4οΏ½ οΏ½ = 2οΏ½ οΏ½ 3 3 16 16 = 3 3
Since made the equation true, then we know it is the solution to the equation. 3
Example 2 (4 minutes) ο§
ο§
ο§
Can we solve the following equation? Explain. 1 β π₯ 2π₯ + 9 5 = 3 7 οΊ We need to multiply each numerator with the other fractionβs denominator. So,
What would be the next step? οΊ
ο§
ο§
Use the distributive property.
1 β π₯ 2π₯ + 9 5 = 3 7 1 7(2π₯ + 9) = 3 οΏ½ β π₯οΏ½ 5
Now we have 1 7(2π₯ + 9) = 3 οΏ½ β π₯οΏ½ 5 3 14π₯ + 63 = β 3π₯ 5 Is this a linear equation? How do you know? οΊ
Yes, this is a linear equation because the left and right side are linear expressions.
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Lesson 8
NYS COMMON CORE MATHEMATICS CURRICULUM
8β’4
Example 3 (5 minutes) ο§
Can we solve the following equation? If so, go ahead and solve it. If not, explain why not. Example 3 Can this equation be solved? π π+π = π π π
ππ +
Give students a few minutes to work. Provide support to students as needed. οΊ
Yes, we can solve the equation because we can multiply each numerator with the other fractionβs denominator and then use the distributive property to begin solving it: π π+π = π π ππ + π
π (π + π)π = οΏ½ππ + οΏ½ π π ππ + ππ = πππ + π ππ + ππ β ππ = πππ β ππ + π ππ = πππ + π ππ β π = πππ + π β π ππ = πππ ππ =π ππ
Example 4 (5 minutes) ο§
Can we solve the following equation? If so, go ahead and solve it. If not, explain why not. Example 4 Can this equation be solved? π π = ππ + π π
Give students a few minutes to work. Provide support to students as needed. οΊ
Yes, we can solve the equation because we can multiply each numerator with the other fractions denominator and then use the distributive property to begin solving it. π π = ππ + π π π(π) = (ππ + π)π ππ = ππ + π ππ β π = ππ + π β π ππ = ππ ππ =π π Lesson 8: Date:
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Lesson 8
NYS COMMON CORE MATHEMATICS CURRICULUM
8β’4
Example 5 (5 minutes) Example 5 In the diagram below, β³ π¨π©πͺ~ β³ π¨β² π©β² πͺβ² . Using what we know about similar triangles, we can determine the value of π.
ο§
Begin by writing the ratios that represent the corresponding sides. πβπ π+π = π. π ππ
It is possible to write several different proportions in this case. If time, discuss this fact with students. ο§
Now that we have the ratios, solve for π₯ and find the lengths of π΄π΅ and π΄πΆ. πβπ π+π = π. π ππ (π β π)ππ = π. π(π + π) πππ β ππ = π. ππ + ππ πππ β ππ + ππ = π. ππ + ππ + ππ πππ = π. ππ + ππ πππ β π. ππ = π. ππ β π. ππ + ππ π. ππ = ππ ππ π= π. π π = ππ. π
The length of |π¨π©| = ππ. π cm and the length of |π¨πͺ| = ππ. π cm.
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Lesson 8
NYS COMMON CORE MATHEMATICS CURRICULUM
8β’4
Exercises 1β4 (8 minutes) Students complete Exercises 1β4 independently. Exercises 1β4 Solve the following equations of rational expressions, if possible. 1.
ππ+π π
=
πβπ
=
π
π
Sample student work:
2.
π+ππ
ππβπ
π
Sample student work:
3.
π+π ππ
=
π
βππβ π π
Sample student work:
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ππ + π π β π = π π π(π β π) = (ππ + π)π π β ππ = πππ + π π β ππ + ππ = πππ + ππ + π π = πππ + π π β π = πππ + π β π π = πππ π ππ = π ππ ππ π =π π
π + ππ π = ππ β π π (π + ππ)π = (ππ β π)π ππ + πππ = πππ β π ππ β ππ + πππ = πππ β π β ππ πππ = πππ β ππ πππ β πππ = πππ β πππ β ππ βππ = βππ βπ βππ π= βπ βπ ππ π= π
π π + π βππ β π = π ππ π ππ οΏ½βππ β οΏ½ = (π + π)π π βπππ β π = ππ + ππ βπππ + πππ β π = ππ + πππ + ππ βπ = πππ + ππ βπ β ππ = πππ + ππ β ππ βππ = πππ βππ ππ = π ππ ππ ππ β =π π Linear Equations in Disguise 11/8/13
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Lesson 8
NYS COMMON CORE MATHEMATICS CURRICULUM
4.
π
πβππ
=
π
8β’4
π π
ππ+
Sample student work:
π π = π β ππ ππ + π π π π οΏ½ππ + οΏ½ = (π β ππ)π π πππ + π = ππ β πππ πππ + π β π = ππ β π β πππ πππ = ππ β πππ πππ + πππ = ππ β πππ + πππ πππ = ππ ππ ππ π= ππ ππ ππ π= ππ
Closing (5 minutes) Summarize, or ask students to summarize, the main points from the lesson: ο§
We know that proportions that have more than one term in the numerator and/or denominator can be solved the same way we normally solve a proportion.
ο§
When multiplying a fraction with more than one term in the numerator and/or denominator by a number, we should put the expressions with more than one term in parentheses, so we are less likely to forget to use the distributive property.
Lesson Summary Proportions are linear equations in disguise and are solved the same way we normally solve proportions. When multiplying a fraction with more than one term in the numerator and/or denominator by a number, put the expressions with more than one term in parentheses so you remember to use the distributive property when transforming the equation. For example: π+π π = ππ β π π
π(π + π) = π(ππ β π)
The equation π(π + π) = π(ππ β π) is now clearly a linear equation and can be solved using the properties of equality.
Exit Ticket (5 minutes)
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Lesson 8
NYS COMMON CORE MATHEMATICS CURRICULUM
Name
8β’4
Date
Lesson 8: Linear Equations in Disguise Exit Ticket Solve the following equations for π₯. 1.
5π₯β8
2.
π₯+11
3.
βπ₯β2
3
7
β4
=
=
=
11π₯β9 5
2π₯+1 β8
3π₯+6 2
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Lesson 8
NYS COMMON CORE MATHEMATICS CURRICULUM
8β’4
Exit Ticket Sample Solutions 1.
ππβπ π
=
πππβπ π
ππ β π πππ β π = π π
π(ππ β π) = π(πππ β π) πππ β ππ = πππ β ππ
πππ β πππ β ππ = πππ β πππ β ππ βππ = ππ β ππ
βππ + ππ = ππ β ππ + ππ βππ = ππ
β 2.
π+ππ π
=
ππ+π βπ
ππ =π π
π + ππ ππ + π = π βπ
π(ππ + π) = βπ(π + ππ) πππ + π = βππ β ππ πππ + π β π = βππ β ππ β π πππ = βππ β ππ πππ + ππ = βππ + ππ β ππ πππ = βππ ππ π=β ππ 3.
βπβπ βπ
=
ππ+π π
βπ β π ππ + π = βπ π βπ(ππ + π) = π(βπ β π)
βπππ β ππ = βππ β π βπππ β ππ + ππ = βππ β π + ππ βπππ = βππ + ππ βπππ + ππ = βππ + ππ + ππ βπππ = ππ π = βπ
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Lesson 8
NYS COMMON CORE MATHEMATICS CURRICULUM
8β’4
Problem Set Sample Solutions Students practice solving equations with rational expressions, if a solution possible. Solve the following equations of rational expressions, if possible. If it cannot be solved, explain why. 1.
π
ππβπ
=
βπ
π+π
Sample student work: π βπ = ππ β π π + π π(π + π) = βπ(ππ β π) ππ + π = βππ + π ππ + π β π = βππ + π β π ππ = βππ β π ππ + ππ = βππ + ππ β π πππ = βπ π π=β ππ
2.
πβπ π
=
ππβπ π
Sample student work: π β π ππ β π = π π π(ππ β π) = (π β π)π πππ β π = ππ β ππ πππ β π + π = ππ + π β ππ πππ = ππ β ππ πππ + ππ = ππ β ππ + ππ πππ = ππ ππ ππ π= ππ ππ ππ π= ππ 3.
ππ
π+π
=
π π
Sample student work: π ππ = π+π π π(ππ) = (π + π)π πππ = ππ + ππ πππ β ππ = ππ β ππ + ππ πππ = ππ ππ ππ π= ππ ππ π π= ππ
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Lesson 8
NYS COMMON CORE MATHEMATICS CURRICULUM
4.
π π+π π
π
=
8β’4
πβπ π
Sample student work: π π+π πβπ π = π π π π(π β π) = π οΏ½ π + ποΏ½ π ππ β π = π + ππ ππ β π + π = π + ππ + π ππ = π + ππ ππ β π = π β π + ππ ππ = ππ ππ π= π 5.
πβππ π
=
πβπ π
Sample student work: π β ππ π β π = π π π(π β π) = (π β ππ)π ππ β ππ = π β ππ ππ β ππ + ππ = π + ππ β ππ ππ = ππ β ππ ππ + ππ = ππ β ππ + ππ ππ = ππ π ππ π= π π ππ π= π 6.
ππ+π π
=
ππβπ π
Sample student work: ππ + π ππ β π = π π π(ππ β π) = π(ππ + π) ππ β π = πππ + ππ ππ β π + π = πππ + ππ + π ππ = πππ + ππ ππ β πππ = πππ β πππ + ππ βππ = ππ ππ π=β π ππ π=β π
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Lesson 8
NYS COMMON CORE MATHEMATICS CURRICULUM
7.
ππ+π π
8β’4
πβπ
=
π
Sample student work: ππ + π π β π = π π (ππ + π)π = π(π β π) πππ + π = ππ β ππ πππ + π β π = ππ β π β ππ πππ = ππ β ππ πππ + ππ = ππ β ππ + ππ πππ = ππ ππ ππ π= ππ ππ π π= π 8.
π πβπ π
ππ
=
βπβπ ππ
Sample student work: π π β π βπ β π π = ππ ππ π ππ(βπ β π) = οΏ½ π β ποΏ½ ππ π βππ β πππ = ππ β πππ βππ β πππ + πππ = ππ + πππ β πππ βππ = πππ β πππ βππ + πππ = πππ β πππ + πππ ππ = πππ ππ ππ = π ππ ππ ππ =π ππ 9.
πβπ πβπ
=
π π
Sample student work: πβπ π = πβπ π (π β π)π = (π β π)π π β ππ = π β ππ π β ππ + ππ = π β ππ + ππ π=πβπ πβπ=πβπβπ π = βπ βπ = π
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Lesson 8
NYS COMMON CORE MATHEMATICS CURRICULUM
8β’4
10. In the diagram below, β³ π¨π©πͺ~ β³ π¨β² π©β² πͺβ² . Determine the lengths of π¨πͺ and π©πͺ.
Sample student work: π + π ππ β π = π π. π π(π + π) = π. π(ππ β π) ππ + ππ = ππ. ππ β π ππ + ππ + π = ππ. ππ β π + π ππ + ππ = ππ. ππ ππ β ππ + ππ = ππ. ππ β ππ ππ = π. ππ ππ = π
The length of |π¨πͺ| = ππ ππ and the length of |π©πͺ| = ππ ππ.
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Lesson 9
NYS COMMON CORE MATHEMATICS CURRICULUM
8β’4
Lesson 9: An Application of Linear Equations Student Outcomes ο§
Students know how to rewrite an exponential expression that represents a series as a linear equation.
Lesson Notes The purpose of this lesson is to expose students to applications of linear equations. The discussion below revisits the Facebook problem from Module 1, but this time in the context of a linear equation. This is an opportunity to highlight MP 1: Make sense of problems and persevere in solving them, as the discussion requires students to work with equations in ways they have not before. If you feel that the discussion is too challenging for your students you can choose to use the series of more accessible applications of linear equations beginning on page 94 of this lesson.
Classwork Discussion (30 minutes) ο§
In Module 1, you saw the following problem: You sent a photo of you and your family on vacation to seven Facebook friends. If each of them sends it to five of their friends, and each of those friends sends it to five of their friends, and those friends send it to five more, how many people (not counting yourself) will see your photo? Assume that no friend received the photo twice. In Module 1 you were asked to express your answer in exponential notation. The solution is the given here: (1) The number of friends you sent a photo to = 7.
(2) The number of friends 7 people sent the photo to = 7 β
5
(3) The number of friends 7 β
5 people sent the photo to = (7 β
5) β
5
(4) The number of friends (7 β
5) β
5 people sent the photo to = οΏ½(7 β
5) β
5οΏ½ β
5 Therefore, the total number of people who received the photo is:
ο§
7 + 7 β
5 + 7 β
5 2 + 7 β
53
Letβs refer to βyou sending the photoβ as the first step. Then βyour friends sending the photo to their friendsβ is the second step, and so on. In the original problem, there were four steps. Assuming the trend continues, how would you find the sum after ten steps? οΊ
ο§
What if I asked you how many friends received the photo after 100 steps? οΊ
ο§
We would continue the pattern until we got to the 10π‘β step.
It would take a long time to continue the pattern to the 100π‘β step.
We want to be able to answer the question for any number of steps. For that reason, we will work towards expressing our answer as a linear equation.
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Lesson 9
NYS COMMON CORE MATHEMATICS CURRICULUM
ο§
For convenience, letβs introduce some symbols. Since we are talking about steps, we will refer to the sum after step one as π1 . Then the sum after step two is π2 , the sum after step three is π3 , and so on. Thus, πΊπ πΊπ πΊπ πΊπ
ο§
ο§
(1) (2) (3) (4)
=π = π+πβ
π = π + π β
π + π β
ππ = π + π β
π + π β
ππ + π β
ππ
What patterns do you notice within each of the equations (1)β(4)? οΊ
ο§
8β’4
They contain some of the same terms. For example, equation (2) is the same as (1), except equation (2) has the term 7 β
5. Similarly, equation (3) is the same as (2), except equation (3) has the term 7 β
52 .
What you noticed is true. However, we want to generalize in a way that doesnβt require us to know one step before getting to the next step. Letβs see what other hidden patterns there are. Letβs begin with equation (2): π2 = 7 + 7 β
5 π2 β 7 = 7 β
5 π2 β 7 + 7 β
52 = 7 β
5 + 7 β
52 π2 β 7 + 7 β
52 = 5(7 + 7 β
5)
Scaffolding: Talk students through the manipulation of the equation. For example, βWe begin by subtracting 7 from both sides. Next, we will add the number 7 times 52 to both sides of the equation. Then, using the distributive propertyβ¦β
Notice that the grouping on the right side of the equation is exactly π2 , so we have the following: π2 β 7 + 7 β
52 = 5π2
ο§
This linear equation is a linear equation in π2 . It is an equation we know how to solve (pretend π2 is an π₯ if that helps). Letβs do something similar with equation (3):
π3 = 7 + 7 β
5 + 7 β
52 π3 β 7 = 7 β
5 + 7 β
52 π3 β 7 + 7 β
53 = 7 β
5 + 7 β
52 + 7 β
53 π3 β 7 + 7 β
53 = 5(7 + 7 β
5 + 7 β
52 )
Again, the grouping on the right side of the equation is exactly the equation we began with, π3 , so we have the following:
ο§ ο§
This is a linear equation in π3 .
Letβs work together to do something similar with equation (4): π4 = 7 + 7 β
5 + 7 β
52 + 7 β
53
What did we do first in each of the equations (2) and (3)? οΊ
ο§
π3 β 7 + 7 β
53 = 5π3
Subtract 7 from both sides of the equation.
Now we have the following:
π4 β 7 = 7 β
5 + 7 β
52 + 7 β
53
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Lesson 9
NYS COMMON CORE MATHEMATICS CURRICULUM
ο§
What did we do next? οΊ
ο§ ο§
ο§
ο§
π4 β 7 + 7 β
54 = 7 β
5 + 7 β
52 + 7 β
53 + 7 β
54
What did we do after that?
We used the distributive property to rewrite the right side of the equation.
Now we have What do we do now? οΊ
ο§
We added 7 β
5 raised to a power to both sides of the equation. When it was the second step, the power of 5 was 2. When it was the third step, the power of 5 was 3. Now that itβs the fourth step, the power of 5 should be 4.
Now we have
οΊ ο§
π4 β 7 + 7 β
54 = 5(7 β
5 + 7 β
52 + 7 β
53 )
We substitute the grouping on the right side of the equation with π4 .
Finally, we have a linear equation in π4 :
π4 β 7 + 7 β
54 = 5π4
Letβs look at the linear equations altogether:
Note to teacher:
π2 β 7 + 7 β
52 = 5π2
π3 β 7 + 7 β
53 = 5π3 π4 β 7 + 7 β
54 = 5π4
What do you think the equation would be for π10 ? οΊ
ο§
8β’4
According to the pattern, it would be π10 β 7 + 7 β
510 = 5π10 .
Now letβs solve the equation. (Note again that we do not simplify (1 β 5) for the reason explained above.): π10 β 7 + 7 β
510 = 5π10 π10 β 5π10 β 7 + 7 β
510 = 5π10 β 5π10 S10 (1 β 5) β 7 + 7 β
510 = 0 π10 (1 β 5) β 7 + 7 + 7 β
510 β 7 β
510 = 7 β 7 β
510 π10 (1 β 5) = 7(1 β 510 ) 7(1 β 510 ) π10 = (1 β 5) π10 = 17,089,842
This equation is solved differently than the equations we have been solving because we want to show students the normal form of an equation for the summation of a geometric series. In general, if π β 1, then 1 + π + π2 +β
β
β
+ππβ1 + ππ 1 β ππ+1 = 1βπ
After ten steps, 17,089,842 will see the photo!
Exercises 1β2 (5 minutes) Students complete Exercises 1β2 independently. Exercises 1β2 1.
Write the equation for the fifteenth step. πΊππ β π + π β
πππ = ππΊππ
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Lesson 9
NYS COMMON CORE MATHEMATICS CURRICULUM
2.
8β’4
How many people would see the photo after fifteen steps? Use a calculator if needed. πΊππ β π + π β
πππ = ππΊππ
πΊππ β ππΊππ β π + +π β
πππ = ππΊππ β ππΊππ πΊππ (π β π) β π + π β
πππ = π
πΊππ (π β π) β π + π + π β
πππ β π β
πππ = π β π β
πππ
πΊππ (π β π) = π(π β πππ ) πΊππ =
π(π β πππ ) (π β π)
πΊππ = ππ, πππ, πππ, πππ
Exercises 3β11 as Alternative to Discussion (30 minutes) Students should be able to complete the following problems independently as they are an application of skills learned to this point, namely transcription and solving linear equations in one-variable. You can choose to have students work on the problems one at a time and share their work with the whole class, or assign the entire set and allow students to work at their own pace. Provide correct solutions at the end of the lesson. Exercises 3β11 3.
Marvin paid an entrance fee of $π plus an additional $π. ππ per game at a local arcade. Altogether he spent $ππ. ππ. Write and solve an equation to determine how many games Marvin played.
Let π represent the number of games he played.
π + π. πππ = ππ. ππ π. πππ = ππ. ππ π=
π = ππ
Marvin played ππ games. 4.
ππ. ππ π. ππ
The sum of four consecutive numbers is βππ. What are the numbers? Let π be a number
The numbers are βπ, βπ, βπ, and βπ.
Lesson 9: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
π + π + π + π + π + π + π = βππ ππ + π = βππ ππ = βππ π = βπ
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Lesson 9
NYS COMMON CORE MATHEMATICS CURRICULUM
5.
π
A book has π pages. How many pages are in the book if Maria read ππ pages of a book Monday, the book π
Tuesday, and the remaining ππ pages Wednesday? Let π be the number of pages in the book.
A number increased by π and divided by π is equal to ππ. Let π be the number.
π+π = ππ π π + π = πππ π = πππ
The number is πππ. 7.
π π = ππ + π + ππ π π π = πππ + π π
π π = πππ π π = πππ
The book has πππ pages. 6.
8β’4
The sum of thirteen and twice a number is seven less than six times a number. What is the number? Let π be the number.
ππ + ππ = ππ β π
ππ + ππ = ππ
ππ = ππ π=π
The number is π. 8.
The width of a rectangle is π less than twice the length. If the perimeter of the rectangle is ππ. π inches, what is the area of the rectangle? Let π represent the length of the rectangle.
ππ + π(ππ β π) = ππ. π ππ + ππ β ππ = ππ. π
ππ β ππ = ππ. π
ππ = ππ. π π=
ππ. π π
π = π. π
The length of the rectangle is π. π inches and the width is ππ. π inches, so the area is πππ. ππ in2.
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Lesson 9
NYS COMMON CORE MATHEMATICS CURRICULUM
9.
8β’4
Two hundred and fifty tickets are available for sale for a school dance. On Monday ππ tickets were sold. An equal number of tickets were sold each day for the next π days. How many tickets were sold on one of those π days?
Let π be the number of tickets sold on one of those π days.
ππ tickets were sold on each of the π days.
πππ = ππ + ππ πππ = ππ ππ = π
10. Shonna skateboarded for some number of minutes on Monday. On Tuesday, she skateboarded for twice as many minutes as she did on Monday, and on Wednesday, she skateboarded for half the sum of minutes from Monday and Tuesday. Altogether she skateboarded for a total of π hours. How many minutes did she skateboard each day? Let π be the number of minutes she skateboarded on Monday.
ππ + π = πππ π ππ ππ ππ + π + + = πππ π π π ππ = πππ π ππ = πππ π = ππ π + ππ +
Shonna skateboarded ππ minutes on Monday, ππ minutes on Tuesday, and ππ minutes on Wednesday. 11. In the diagram below, β³ π¨π©πͺ~ β³ π¨β² π©β² πͺβ² . Determine the length of π¨πͺ and π©πͺ.
ππ ππ = ππ. π β π ππ β π. π
ππ(ππ β π. π) = ππ(ππ. π β π) πππ β π. π = πππ. π β πππ
ππππ β π. π = πππ. π
ππππ = πππ. π π=
πππ. π πππ
π = π. π
The length of π¨πͺ is π. π mm and the length of π©πͺ is ππ. π mm.
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8β’4
Closing (5 minutes) Summarize, or ask students to summarize, the main points from the lesson: ο§
We can rewrite equations to develop a pattern and make predictions.
ο§
We know that for problems like these we can generalize equations so that we do not have to do each step to get our answer.
ο§
We learned how equations can be used to solve problems.
Exit Ticket (5 minutes)
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Lesson 9
NYS COMMON CORE MATHEMATICS CURRICULUM
Name
8β’4
Date
Lesson 9: An Application of Linear Equations Exit Ticket 1.
Rewrite the equation that would represent the fifth step of the Facebook problem: π5 = 7 + 7 β
5 + 7 β
52 + 7 β
53 + 7 β
54
2.
The sum of four consecutive integers is 74. Write an equation and solve to find the numbers?
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Lesson 9
NYS COMMON CORE MATHEMATICS CURRICULUM
8β’4
Exit Ticket Sample Solutions 1.
Rewrite the equation that would represent the fifth step of the Facebook problem: πΊπ = π + π β
π + π β
ππ + π β
ππ + π β
ππ
πΊπ = π + π β
π + π β
ππ + π β
ππ + π β
ππ
πΊπ β π = π β
π + π β
ππ + π β
ππ + π β
ππ
πΊπ β π + π β
ππ = π β
π + π β
ππ + π β
ππ + π β
ππ + π β
ππ πΊπ β π + π β
ππ = π(π + π β
π + π β
ππ + π β
ππ + π β
ππ ) πΊπ β π + π β
ππ = π(πΊπ )
πΊπ β ππΊπ β π + π β
ππ = π
πΊπ β ππΊπ = π β (π β
ππ )
(π β π)πΊπ = π β (π β
ππ ) (π β π)πΊπ = π(π β ππ ) πΊπ =
2.
π(π β ππ ) πβπ
The sum of four consecutive integers is ππ. Write an equation and solve to find the numbers? Let π be the first number
The numbers are ππ, ππ, ππ, and ππ.
π + π + π + π + π + π + π = ππ ππ + π = ππ ππ = ππ π = ππ
Problem Set Sample Solutions Assign the problems that relate to the elements of the lesson you chose to use with students. 1.
You forward an e-card you found online to three of your friends. They liked it so much that they forwarded it on to four of their friends, who then forwarded it on to four of their friends, and so on. The number of people who saw the e-card is shown below. Let πΊπ represent the number of people who saw the e-card after one step, let πΊπ represent the number of people who saw the e-card after two steps, and so on. πΊπ = π
πΊπ = π + π β
π
πΊπ = π + π β
π + π β
ππ
πΊπ = π + π β
π + π β
ππ + π β
ππ a.
Find the pattern in the equations. πΊπ = π + π β
π πΊπ β π = π β
π πΊπ β π + π β
ππ = π β
π + π β
ππ πΊπ β π + π β
ππ = π(π + π β
π) πΊπ β π + π β
ππ = ππΊπ
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8β’4
πΊπ = π + π β
π + π β
ππ πΊπ β π = π β
π + π β
ππ πΊπ β π + π β
ππ = π β
π + π β
ππ + π β
ππ πΊπ β π + π β
ππ = π(π + π β
π + π β
ππ ) πΊπ β π + π β
ππ = ππΊπ
πΊπ = π + π β
π + π β
ππ + π β
ππ πΊπ β π = π β
π + π β
ππ + π β
ππ πΊπ β π + π β
ππ = π β
π + π β
ππ + π β
ππ + π β
ππ πΊπ β π + π β
ππ = π(π + π β
π + π β
ππ + π β
ππ ) πΊπ β π + π β
ππ = ππΊπ b.
Assuming the trend continues, how many people will have seen the e-card after ππ steps? πΊππ β π + π β
πππ = ππΊππ
πΊππ β ππΊππ β π + π β
πππ = π
πΊππ (π β π) = π β π β
πππ
πΊππ (π β π) = π(π β πππ ) πΊππ =
π(π β πππ ) (π β π)
πΊππ = πππππππ
After ten steps, π, πππ, πππ people will have seen the e-card. c.
How many people will have seen the e-card after π steps? πΊπ =
π(π β ππ ) (π β π)
For each of the following questions, write an equation and solve to find each answer. 2.
Lisa has a certain amount of money. She spent ππ dollars and has did she have originally?
Let π be the amount of money Lisa had originally.
Lisa had $πππ originally.
Lesson 9: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
π π
of the original amount left. How much money
π π π π βππ = β π π
π β ππ =
πππ = π
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3.
8β’4
The length of a rectangle is π more than π times the width. If the perimeter of the rectangle is ππ. π cm, what is the area of the rectangle?
Let π represent the width of the rectangle.
π(π + ππ) + ππ = ππ. π π + ππ + ππ = ππ. π π + ππ = ππ. π ππ = ππ. π ππ. π π= π π = π. π
The width of the rectangle is π. π cm and the length is π. π cm, so the area is ππ. ππ cm2. 4.
Eight times the result of subtracting π from a number is equal to the number increased by ππ. Let π be the number.
π(π β π) = π + ππ ππ β ππ = π + ππ ππ β ππ = ππ ππ = ππ π=π
The number is π. 5.
Three consecutive odd integers have a sum of π. What are the numbers? Let π be the first number.
The numbers are βπ, π, and π. 6.
π+π+π+π+π =π ππ + π = π ππ = βπ π = βπ
Each month Liz pays $ππ to her phone company just to use the phone. Each text she sends costs her an additional $π. ππ. In March her phone bill was $ππ. ππ. In April her phone bill was $ππ. ππ. How many texts did she send each month? Let π be the number of texts she sent in March.
She sent πππ texts in March.
ππ + π. πππ = ππ. ππ π. πππ = ππ. π ππ. π π= π. ππ π = πππ
Let π be the number of texts she sent in April.
She sent πππ texts in April.
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ππ + π. πππ = ππ. ππ π. πππ = ππ. ππ ππ. ππ π= π. ππ π = πππ
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7.
8β’4
Claudia is reading a book that has πππ pages. She has already read some of the book last week. She plans to read π
ππ pages today. When she does, she will be of the way through the book. How many pages did she read last π
week?
Let π be the number of pages she read last week.
π (πππ) π π + ππ = πππ π = πππ π + ππ =
Claudia read πππ pages last week. 8.
In the diagram below, β³ π¨π©πͺ~ β³ π¨β² π©β² πͺβ² . Determine the measure of β π¨.
ππ β π = π + ππ ππ β π = ππ ππ = ππ π=π
The measure of β π¨ is ππΛ.
9.
In the diagram below, β³ π¨π©πͺ~ β³ π¨β² π©β² πͺβ² . Determine the measure of β π¨.
The measure of β π¨ is πππΛ.
Lesson 9: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
πππ β π = ππ + ππ ππ β π = ππ ππ = ππ π = ππ
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New York State Common Core
8
Mathematics Curriculum
GRADE
GRADE 8 β’ MODULE 4
Topic B:
Linear Equations in Two Variables and Their Graphs 8.EE.B.5 Focus Standard:
8.EE.B.5
Instructional Days:
5
Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways.
Lesson 10: A Critical Look at Proportional Relationships (S)
1
Lesson 11: Constant Rate (P) Lesson 12: Linear Equations in Two Variables (E) Lesson 13: The Graph of a Linear Equation in Two Variables (S) Lesson 14: The Graph of a Linear EquationβHorizontal and Vertical Lines (S)
Topic B begins with students working with proportional relationships related to average speed and constant speed. In Lesson 10, students use information that is organized in the form of a table to write linear equations. In Lesson 11, students learn how to apply the concept of constant rate to a variety of contexts requiring two variables (8.EE.B.5). Lesson 12 introduces students to the standard form of an equation in two variables. At this point, students use a table to help them find and organize solutions to a linear equation in two variables. Students then use the information from the table to begin graphing solutions on a coordinate plane. Lesson 13 is when students begin to question whether or not the graph of a linear equation is a line as opposed to something that is curved. Lesson 14 presents students with equations in standard form, ππ₯ + ππ¦ = π, where π = 0 or π = 0, which produces lines that are either vertical or horizontal.
1
Lesson Structure Key: P-Problem Set Lesson, M-Modeling Cycle Lesson, E-Exploration Lesson, S-Socratic Lesson
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Lesson 10: A Critical Look at Proportional Relationships Student Outcomes ο§
Students work with proportional relationships in terms of average speed and constant speed in order to write a linear equation in two variables.
ο§
Students use linear equations in two variables to answer questions about distance and time.
Classwork Discussion (25 minutes) Example 1 ο§
Consider the word problem below. We can do several things to answer this problem, but letβs begin to organize our work using a table for time and distance: Example 1 Paul walks π miles in ππ minutes. How many miles can Paul walk in πππ. π minutes? Time (in minutes) ππ ππ ππ πππ πππ
Scaffolding: It may be necessary to remind students of the relationship between distance traveled, rate, and time spent traveling at that rate.
Distance (in miles) π π π π ππ
As students answer the questions below, fill in the table. ο§
How many miles would Paul be able to walk in 50 minutes? Explain. οΊ
ο§
How many miles would Paul be able to walk in 75 minutes? Explain. οΊ
ο§
He could walk 8 miles.
How many miles would he walk in 125 minutes? οΊ
ο§
Paul could walk 6 miles in 75 minutes because 75 minutes is three times the number of minutes we were given, so we can calculate three times the distance, which is 6.
How many miles would Paul be able to walk in 100 minutes? οΊ
ο§
Paul could walk 4 miles in 50 minutes because 50 minutes is twice the time we were given, so we can calculate twice the distance, which is 4.
He could walk 10 miles.
How could we determine the number of miles Paul could walk in 137.5 minutes? Lesson 10: Date:
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Provide students time to think about the answer to this question. They will likely say that they can write a proportion to figure it out. Allow them to share and demonstrate their solutions. Then proceed with the discussion below, if necessary. ο§
Since the relationship between the distance Paul walks and the time it takes him to walk that distance is proportional, we let π¦ represent the distance Paul walks in 137.5 minutes and write: ππ πππ. π = π π πππ = πππ. π(π) πππ = πππ π = ππ Therefore, Paul can walk ππ miles in πππ. π minutes.
ο§
How many miles π¦ can Paul walk in π₯ minutes?
Provide students time to think about the answer to this question. Allow them to share their ideas and then proceed with the discussion below, if necessary. ο§
We know for a fact that Paul can walk 2 miles in 25 minutes, so we can write the ratio
25 2
as we did with the π₯
proportion. We can write another ratio for the number of miles π¦ Paul walks in π₯ minutes. It is . For the π¦
same reason we could write the proportion before, we can write one now with these two ratios: 25 π₯ = 2 π¦ Does this remind you of something weβve done recently? Explain. οΊ
ο§
This is a linear equation in disguise. All we need to do is multiply each numerator by the other fractions denominator, then we will have a linear equation. 25π¦ = 2π₯
Recall our original question: How many miles π¦ can Paul walk in π₯ minutes? We need to solve this equation for π¦. 25π¦ = 2π₯ 2 π¦= π₯ 25 π¦ = 0.08π₯
Paul can walk π¦ miles in 0.08π₯ minutes. This equation will allow us to answer all kinds of questions about Paul with respect to any given number of minutes or miles.
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ο§
Letβs go back to the table and look for π¦ = 0.08π₯ or its equivalent π¦ =
οΊ
The fraction
2
25
8β’4
2 π₯ there. What do you notice? 25
Time (in minutes)
Distance (in miles)
25 50 75 100 125
2 4 6 8 10
came from the first row in the table. Itβs the distance traveled divided by the time it
took to travel that distance. It is also in between each row of the table. For example, the difference between 4 miles and 2 miles is 2, and the difference between the associated times 50 and 25 is 25. The pattern repeats throughout the table. Show on the table the +2 between each distance interval and the +25 between each time interval. Remind students that they have done work like this before, specifically finding a unit rate for a proportional relationship. Make clear that the unit rate found in the table was exactly the same as the unit rate found using the proportion and that the unit rate is the rate at which Paul walks. ο§
Letβs look at another problem where only a table is provided: Time (in hours)
Distance (in miles)
3
123
6 9 12
246 369 492 π¦
We want to know how many miles π¦ can be traveled in any number of hours π₯. Using our previous work, what should we do? οΊ
ο§
How many miles π¦ can be traveled in any number of hours π₯? οΊ
ο§
We can write and solve a proportion that contains both π₯ and π¦ or use the table to help us determine the unit rate.
Student work:
What does the equation π¦ = 41π₯ mean? οΊ
123 π¦ = 3 π₯ 123π₯ = 3π¦ 123 π₯=π¦ 3 41π₯ = π¦
It means that the distance traveled π¦ is equal to the rate of 41 multiplied by the number of hours π₯ traveled at that rate.
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Example 2 The point of this problem is to make clear to students that we must assume constant rate in order to write linear equations in two variables and to use those equations to answer questions about distance, time, and rate. ο§
Consider the following word problem: Alexxa walked from the Grand Central Station on 42ππ street to the Penn Station on 7π‘β avenue. The total distance traveled was 1.1 miles. It took Alexxa 25 minutes to make the walk. How many miles did she walk in the first 10 minutes?
Give students a minute to think and/or work on the problem. Expect them to write a proportion and solve the problem. The next part of the discussion will get them to think about what is meant by βconstantβ speed or rather lack of it. οΊ ο§
Are you sure about your answer? How often do you walk at a constant speed? Notice the problem did not even mention that she was walking at the same rate throughout the entire 1.1 miles. What if you have more information about her walk: Alexxa walked from Grand Central Station (GCS) along 42ππ street to an ATM machine 0.3 miles away in 8 minutes. It took her 2 minutes to get some money out of the machine. Do you think your answer is still correct? οΊ
ο§
ο§
She walked 0.44 miles. (Assuming students used a proportion to solve.)
Probably not since now that we know she had to stop at the ATM.
Letβs continue with Alexxaβs walk: She reached the 7π‘β avenue junction 13 minutes after she left GCS, a distance of 0.6 miles. There she met her friend Karen with whom she talked for 2 minutes. After leaving her friend she finally got to Penn Station 25 minutes after her walk began.
Isnβt this a more realistic situation than believing that she walked the exact same speed throughout the entire trip? What other events typically occur during walks in the city? οΊ
Stop lights at crosswalks, traffic, maybe a trip/fall, running an errand, etc.
ο§
This is precisely the reason we need to take a critical look at what we call βproportional relationshipsβ and constant speed, in general.
ο§
The following table shows an accurate picture of Alexxaβs walk:
ο§
Time(in minutes)
Distance Traveled (in miles)
π π ππ ππ ππ ππ
0 0.3 0.3 0.6 0.6 1.1
With this information, we can answer the question. Alexxa walked exactly 0.3 miles in 10 minutes.
Now that we have an idea of what could go wrong when we assume a person walks at a constant rate or that a proportion can give us the correct answer all of the time, letβs define what is called average speed. Suppose a person walks a distance of π (miles) in a given time interval π‘ (minutes). Then the average speed in the given time interval is
π π‘
in miles per minute.
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ο§
With this definition we can calculate Alexxaβs average speed: The distance that Alexxa traveled divided by the time interval she walked is
ο§
ο§
8β’4
Lesson 10
NYS COMMON CORE MATHEMATICS CURRICULUM
1.1 25
miles per minute.
If we assume that someone can actually walk at the same average speed over any time interval, then we say that the person is walking at a constant speed. Suppose the average speed of a person is the same constant πΆ for any given time interval. Then we say that the person is walking at a constant speed πͺ. If the original problem included information specifying constant speed, then we could write the following: Alexxaβs average speed for 25 minutes is
1.1 25
.
Let π¦ represent the distance Alexxa walked in 10 minutes. Then her average speed for 10 minutes is Since Alexxa is walking at a constant speed of πΆ miles per minute, then we know that 1.1 =πΆ 25
π¦ =πΆ 10
and
Since both fractions are equal to πΆ, then we can write
π¦
10
.
1.1 π¦ = . 25 10
With the assumption of constant speed, we now have a βproportional relationship,β which would make the answer you came up with in the beginning correct. ο§
We can go one step further and write a statement in general. If Alexxa walks π¦ miles in π₯ minutes, then π¦ =πΆ π₯
and
1.1 π¦ = 25 π₯
To find how many miles π¦, Alexxa walks in π₯ miles, we solve the equation for π¦: 1.1 π¦ = 25 π₯ 25π¦ = 1.1π₯
1.1 25 π¦= π₯ 25 25 1.1 π¦= π₯ 25
where the last equation is an example of a linear equation in two variables π₯ and π¦. With this general equation, we can find the distance π¦ Alexxa walks in any given time π₯. Since we have more information about Alexxaβs walk, that is where and when she stopped, we know that the equation cannot accurately predict the distance she walks after a certain number of minutes. To do so requires us to assume that she walks at a constant rate. This is an assumption we generally take for granted when solving problems about rate.
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Exercises 1β2 (5 minutes) Students complete Exercises 1β2 independently or in pairs. Exercises 1β2 1.
Wesley walks at a constant speed from his house to school π. π miles away. It took him ππ minutes to get to school. a.
b.
c.
d.
2.
What fraction represents his constant speed, πͺ?
π. π =πͺ ππ
You want to know how many miles he has walked after ππ minutes. Let π represent the distance he traveled after ππ minutes of walking at the given constant speed. Write a fraction that represents the constant speed, πͺ in terms of π.
π =πͺ ππ
Write the fractions from parts (a) and (b) as a proportion and solve to find out many miles Wesley walked after ππ minutes.
Wesley walks π. π miles in ππ minutes.
π. π π = ππ ππ πππ = ππ. π ππ ππ. π π= ππ ππ π = π. π
Let π be the distance in miles that Wesley traveled after π minutes. Write a linear equation in two variables that represents how many miles Wesley walked after π minutes. π. π π = ππ π πππ = π. ππ ππ π. π π= π ππ ππ π. π π= π ππ
Stefanie drove at a constant speed from her apartment to her friendβs house ππ miles away. It took her ππ minutes to reach her destination. a.
b.
What fraction represents her constant speed, πͺ?
ππ =πͺ ππ
What fraction represents constant speed, πͺ, if it takes her π number of minutes to get half-way to her friendβs house? ππ =πͺ π
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c.
8β’4
Write a proportion using the fractions from parts (a) and (b) to determine how many minutes it takes her to get to the half-way point. ππ ππ = ππ π πππ = πππ ππ πππ π= ππ ππ π = ππ. π
d.
Stefanie gets half-way to her friendβs house, ππ miles away, after ππ. π minutes.
Write a two variable equation to represent how many miles Stefanie can drive over any time interval. Let π represent the distance traveled over any time interval π. Then, ππ π = ππ π πππ = πππ ππ ππ π= π ππ ππ π π=π π
Discussion (4 minutes) ο§
ο§ ο§
Consider the problem: Dave lives 15 miles from town π΄. He is driving at a constant speed of 50 miles per hour from his home away from (in the opposite direction of) the city. How far away is Dave from the town after π₯ hours of driving?
Since we know he is driving at a constant speed of 50 miles per hour, then we need to determine the distance he travels over a time interval.
If we say that Dave is π¦ miles from town π΄ after driving π₯ hours, how can we express the actual number of miles that Dave traveled? οΊ
ο§
Dave is 15 miles from town π΄ to begin with, so the total number of miles Dave traveled is π¦ β 15.
If Daveβs average speed in π₯ hours is
the equation
ο§
π¦β15 π₯
, which is equal to a constant, i.e., his constant speed, then we have π¦ β 15 = 50 π₯
We want to know how many miles Dave is from town π΄, π¦, after driving for π₯ hours. Solve this equation for π¦. οΊ
Student work:
π¦ β 15 = 50 π₯ π¦ β 15 = 50π₯ π¦ β 15 + 15 = 50π₯ + 15 π¦ = 50π₯ + 15
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ο§
8β’4
With this equation, π¦ = 50π₯ + 15, we can find the distance Dave is from town π΄ for any given time π₯. How far away is Dave from town π΄ after one hour? οΊ
Student work: If π₯ = 1, then
π¦ = 50(1) + 15 π¦ = 65
Dave is 65 miles from town π΄ after one hour.
Exercise 3 (4 minutes) Students complete Exercise 3 independently or in pairs. Exercise 3 3.
The equation that represents how many miles, π, Dave travels after π hours is π = πππ + ππ. Use the equation to complete the table below. π (hours)
Linear equation in π: π = πππ + ππ
π (miles)
π
π = ππ(π) + ππ
πππ
π
π = ππ(π) + ππ
π
ππ
π = ππ(π) + ππ
π. π
πππ
π = ππ(π. π) + ππ
π. π
πππ
π = ππ(π. π) + ππ
πππ
Closing (3 minutes) Summarize, or ask students to summarize, the main points from the lesson: ο§
Average speed is found by taking the total distance traveled in a given time interval, divided by the time interval.
ο§
If we assume the same average speed over any time interval, then we have constant speed, which can then be used to express a linear equation in two variables relating distance and time.
ο§
We know how to use linear equations to answer questions about distance and time.
ο§
We cannot assume that a problem can be solved using a proportion unless we know that the situation involves constant speed (or rate). Lesson Summary Average speed is found by taking the total distance traveled in a given time interval, divided by the time interval. If π is the total distance traveled in a given time interval π, then
π π
is the average speed.
If we assume the same average speed over any time interval, then we have constant speed, which can then be used to express a linear equation in two variables relating distance and time. If
π π
= πͺ, where πͺ is a constant, then you have constant speed.
Exit Ticket (4 minutes)
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Lesson 10
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Name
8β’4
Date
Lesson 10: A Critical Look at Proportional Relationships Exit Ticket 1.
Alex skateboards at a constant speed from his house to school 3.8 miles away. It takes him 18 minutes. a.
What fraction represents his constant speed, πΆ?
b.
After school, Alex skateboards at the same constant speed to his friendβs house. It takes him 10 minutes. Write the fraction that represents constant speed, πΆ, if he travels a distance of π¦.
c.
Write the fractions from part (a) and (b) as a proportion, and solve to find out how many miles Alexβs friendβs house is from school. Round your answer to the tenths place.
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8β’4
Exit Ticket Sample Solutions 1.
Alex skateboards at a constant speed from his house to school π. π miles away. It takes him ππ minutes. a.
b.
c.
What fraction represents his constant speed, πͺ?
π. π =πͺ ππ
After school, Alex skateboards at the same constant speed to his friendβs house. It takes him ππ minutes. Write the fraction that represents constant speed, πͺ, if he travels a distance of π.
π =πͺ ππ
Write the fractions from part (a) and (b) as a proportion and solve to find out how many miles Alexβs friendβs house is from school. Round your answer to the tenths place. π. π π = ππ ππ π. π(ππ) = πππ ππ = πππ ππ =π ππ π. π β π
Alexβs friend lives about π. π miles from school.
Problem Set Sample Solutions Students practice writing and solving proportions to solve constant speed problems. Students write two variable equations to represent situations generally. 1.
Eman walks from the store to her friendβs house, π miles away. It takes her ππ minutes. a.
b.
c.
What fraction represents her constant speed, πͺ?
π =πͺ ππ
Write the fraction that represents her constant speed, πͺ, if she walks π miles in ππ minutes.
π =πͺ ππ
Write a proportion using the fractions from parts (a) and (b) to determine how many miles she walks after ππ minutes. Round your answer to the hundredths place. π π = ππ ππ πππ = ππ ππ ππ π= ππ ππ π = π. πππππ β¦
Eman walks about π. ππ miles after ππ minutes.
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d.
8β’4
Write a two-variable equation to represent how many miles Eman can walk over any time interval. Let π represent the distance Eman walks in π minutes.
2.
π π = ππ π πππ = ππ ππ π π= π ππ ππ π π= π ππ
Erika drives from school to soccer practice π. π miles away. It takes her π minutes. a.
b.
c.
What fraction represents her constant speed, πͺ? π. π =πͺ π
What fraction represents her constant speed, πͺ, if it takes her π minutes to drive exactly π mile? π =πͺ π
Write a proportion using the fractions from parts (a) and (b) to determine how much time it takes her to drive exactly π mile. Round your answer to the tenths place. π. π π = π π π. ππ = π π. π π π= π. π π. π π = π. πππππ β¦
It takes Erika about π. π minutes to drive exactly π mile. d.
Write a two-variable equation to represent how many miles Erika can drive over any time interval. Let π be the number of miles Erika travels in π minutes.
3.
π. π π = π π ππ = π. ππ π π. π π= π π π π. π π= π π
Darla drives at a constant speed of ππ miles per hour. a.
If she drives for π miles and it takes her π hours, write the two-variable equation to represent the number of miles Darla can drive in π hours. π = ππ π
π = πππ
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b.
8β’4
Darla plans to drive to the market ππ miles from her house, then to the post office π miles from the market, and then return home, which is ππ miles from the post office. Assuming she drives at a constant speed the entire time, how long will it take her to get back home after running her errands? Round your answer to the hundredths place. Altogether, Darla plans to drive ππ + π + ππ = ππ miles.
ππ = πππ ππ ππ = π ππ ππ π. πππππ β¦ = π
It will take Darla about π. ππ hours to get home after running her errands. 4.
Aaron walks from his sisterβs house to his cousinβs house, a distance of π miles, in ππ minutes. How far does he walk in ππ minutes?
I cannot say for sure how far Aaron walks in ππ minutes because I donβt know if he is walking at a constant speed. Maybe he stopped at his friendβs house for ππ minutes or something. 5.
Carlos walks π miles every night for exercise. It takes him exactly ππ minutes to finish his walk. a.
Assuming he walks at a constant rate, write an equation that represents how many miles, π, Carlos can walk in π minutes. Since
π
ππ
= πͺ and
π π
= πͺ, then
π π = ππ π
πππ = ππ
ππ π π= π ππ ππ π π= π ππ b.
Use your equation from part (a) to complete the table below. Use a calculator and round all values to the hundredths place. π (minutes) ππ ππ ππ ππ ππ
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Linear equation in π: π π= π ππ π (ππ) π= ππ π (ππ) π= ππ π (ππ) π= ππ π (ππ) π= ππ π (ππ) π= ππ
π (miles) π. ππ π. ππ π. ππ π. ππ π. ππ
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Lesson 11
NYS COMMON CORE MATHEMATICS CURRICULUM
8β’4
Lesson 11: Constant Rate Student Outcomes ο§
Students know the definition of constant rate in varied contexts as expressed using two variables where one is representing a time interval.
ο§
Students graph points on a coordinate plane related to constant rate problems.
Classwork Example 1 (6 minutes) Give students the first question below and allow them time to work. Ask them to share their solutions with the class and then proceed with the discussion, table, and graph to finish Example 1. Example 1 Pauline mows a lawn at a constant rate. Suppose she mows a feet, can she mow in minutes? minutes?
ο§
What is Paulineβs average rate in οΊ
ο§
What is Paulineβs average rate in οΊ
ο§
Paulineβs average rate in
Let
ο§
If we let minutes is
ο§
minutes is
square feet per minute.
minutes?
represent area mowed. Paulineβs average rate in
square feet of lawn in
minutes. What area, in square
minutes?
Since Pauline mows at a constant rate, , then
Pauline mows
square foot lawn in
minutes is
and
square feet per mintute.
Therefore,
minutes.
represent the number of square feet Pauline can mow in minutes, then Paulineβs average rate in square feet per minute.
Write the two variable equation that represents the area of lawn,
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Pauline can mow in minutes.
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NYS COMMON CORE MATHEMATICS CURRICULUM
8β’4
Sample student work:
ο§ MP.7
What is the meaning of οΊ
ο§
The number
in the equation represents the rate at which Pauline can mow a lawn.
We can organize the information in a table: Linear equation:
(time in minutes)
ο§
(area in square feet)
On a coordinate plane, we will let the -axis represent time , in minutes, and the -axis represent the area of mowed lawn in square feet. Then we have the following graph: 55
Lawn mowed in square feet, π
50 45 40 35 30 25 20 15 10 5
0
ο§
1
2
3
4
5
6
Time in minutes, π
7
Because Pauline mows at a constant rate, we would expect the square feet of mowed lawn to continue to rise as the time in minutes increases.
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NYS COMMON CORE MATHEMATICS CURRICULUM
8β’4
Concept Development (6 minutes) ο§
In the last lesson, we learned about average speed and constant speed. Constant speed problems are just a special case of a larger variety of problems known as constant rate problems. Some of these problems such as water pouring out of a faucet into a tub, painting a house, and mowing a lawn were topics in Grade 7.
ο§
First, we define the average rate: Suppose
gallons of water flow from a faucet in a given time interval (minutes). Then the average rate of
water flow in the given time interval is ο§
in gallons per minute.
Then, we define the constant rate: Suppose the average rate of water flow is the same constant the water is flowing at a constant rate, .
ο§
Similarly, suppose
square feet of lawn are mowed in a given time interval (minutes). Then the average rate
of lawn mowing in the given time interval is lawn mowing is the same constant constant rate, . ο§
square feet per minute. If we assume that the average rate of
for any given time interval, then we say that the lawn is mowed at a
Describe the average rate of painting a house. οΊ
Suppose
square feet of house are painted in a given time interval (minutes). Then the average rate
of house painting in the given time interval is ο§
If we assume that the average rate of house painting is the same constant interval, then we say that the wall is painted at a constant rate, .
over any given time
What is the difference between average rate and constant rate? οΊ
ο§
square feet per minute.
Describe the constant rate of painting a house. οΊ
ο§
for any given time interval. Then we say that
Average rate is the rate in which something can be done over a specific time interval. Constant rate assumes that the average rate is the same over any time interval.
As you can see, the way we define average rate and constant rate for a given situation is very similar. In each case, a transcription of the given information leads to an expression in two variables.
Example 2 (8 minutes) Example 2 Water flows at a constant rate out of a faucet. Suppose the volume of water that comes out in three minutes is gallons. How many gallons of water comes out of the faucet in minutes?
ο§
Write the linear equation that represents the volume of water, , that comes out in minutes.
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Lesson 11
NYS COMMON CORE MATHEMATICS CURRICULUM
Let
represent the constant rate of water flow. and
ο§ MP.7
then
.
What is the meaning of the number οΊ
ο§
8β’4
The number
in the equation
represents the rate at which water flows from a faucet.
Using the linear equation
, complete the table:
Linear equation: (time in minutes)
ο§
(in gallons)
On a coordinate plane, we will let the -axis represent time in minutes and the -axis represent the volume of water. Graph the data from the table.
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8β’4
17 16 15 14
Volume of water in gallons, π
13 12 11 10 9 8 7 6 5 4 3 2 1
0
1
2
3
4
5
6
7
8
9
10
Time in minutes, π‘
ο§
Using the graph, about how many gallons of water do you think would flow after οΊ
After
minutes, between
and
gallons of water will flow. Since the water is flowing at a constant
rate, we can expect the volume of water to rise between flows after minute and ο§
minutes. The number of gallons that
minutes then would have to be between the number of gallons that flows out after gallons of water to flow out of the faucet? Explain.
It would take between and minutes for gallons of water to flow out of the faucet. It takes minutes for gallons to flow; therefore, it must take more than minutes for gallons to come out. It must take less than
ο§
and
minutes.
Using the graph, about how long would it take for οΊ
minutes? Explain.
minutes because
gallons flow out every minute.
Graphing proportional relationships like these last two constant rate problems provide us more information than simply solving an equation and calculating one value. The graph provides information that is not so obvious in an equation.
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Lesson 11
NYS COMMON CORE MATHEMATICS CURRICULUM
8β’4
Exercises 1β3 (15 minutes) Students complete Exercises 1β3 independently. Exercises 1β3 1.
Jesus types at a constant rate. He can type a full page of text in
minutes. We want to know how many pages, ,
Jesus can type after minutes. a.
Write the linear equation in two variables that represents the number of pages Jesus types in any given time interval. Let
represent the constant rate that Jesus types. Then, and
b.
, therefore
Complete the table below. Use a calculator and round answers to the tenths place. Linear equation: (time in minutes)
c.
(pages typed)
Graph the data on a coordinate plane. 10 9
Pages typed, π
8 7 6 5 4 3 2 1 0
1
2
3
4
5
6
7
8
9
10
11 12 13 14 15
16 17
18 19
20 21
Time in minutes, π‘
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Lesson 11
NYS COMMON CORE MATHEMATICS CURRICULUM
d.
About how long would it take Jesus to type a -page paper? Explain. It would take him between and minutes. After minutes he will have typed pages. In minutes he can type pages. Since pages is between and , then it will take him between minutes.
2.
Emily paints at a constant rate. She can paint minutes? a.
square feet in five minutes. What area, , can she paint in
be the constant rate that Emily paints. Then, and
therefore
Complete the table below. Use a calculator and round answers to the tenths place. Linear equation:
(area painted in square feet)
(time in minutes)
c.
and
Write the linear equation in two variables that represents the number of square feet Emily can paint in any given time interval. Let
b.
8β’4
Graph the data on a coordinate plane. 26 24
Area painted in square feet, π΄
22 20 18 16 14 12 10 8 6 4 2
0
Lesson 11: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
1
2
3
4
5
6
7
Time in minutes, π‘
8
9
10
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Lesson 11
NYS COMMON CORE MATHEMATICS CURRICULUM
d.
About how many square feet can Emily paint in Emily can paint between feet and at
3.
and
minutes? Explain.
square feet in
minutes she will have painted
minutes. After
minutes she paints
minutes. How many miles,
,
Write the linear equation in two variables that represents the number of miles Joseph can walk in any given time interval, . Let
be the constant rate that Joseph walks. Then, and
b.
square
square feet.
Joseph walks at a constant speed. He walked to the store, one-half mile away, in can he walk in minutes? a.
8β’4
, therefore
.
Complete the table below. Use a calculator and round answers to the tenths place. Linear equation: (time in minutes)
c.
(distance in miles)
Graph the data on a coordinate plane. 16
Distance in miles, π
14 12 10 8 6 4 2
0
10
20
30
40
50
60
70
80
90
100
110 120 130 140
Time in minutes, π‘
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Lesson 11
NYS COMMON CORE MATHEMATICS CURRICULUM
d.
Josephβs friend lives house? Explain.
8β’4
miles away from him. About how long would it take Joseph to walk to his friendβs
It will take Joseph a little less than an hour to walk to his friendβs house. Since it takes minutes for him to walk miles, and minutes to walk miles, and is closer to than , it will take Joseph less than an hour to walk the miles.
Closing (5 minutes) Summarize, or ask students to summarize, the main points from the lesson: ο§
Constant rate problems appear in a variety of contexts like painting a house, typing, walking, water flow, etc.
ο§
We can express the constant rate as a two variable equation representing proportional change.
ο§
We can graph the constant rate situation by completing a table to compute data points.
Lesson Summary When constant rate is stated for a given problem, then you can express the situation as a two variable equation. The equation can be used to complete a table of values that can then be graphed on a coordinate plane.
Exit Ticket (5 minutes)
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NYS COMMON CORE MATHEMATICS CURRICULUM
Name
8β’4
Date
Lesson 11: Constant Rate Exit Ticket 1.
Vicky reads at a constant rate. She can read read after minutes.
pages in
minutes. We want to know how many pages, , Vicky can
a.
Write a linear equation in two variables that represents the number of pages Vicky reads in any given time interval.
b.
Complete the table below. Use a calculator and round answers to the tenths place.
(time in minutes)
c.
Linear equation:
About how long would it take Vicky to read
Lesson 11: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
(pages read)
pages? Explain.
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Lesson 11
NYS COMMON CORE MATHEMATICS CURRICULUM
8β’4
Exit Ticket Sample Solutions 1.
Vicky reads at a constant rate. She can read read after minutes. a.
minutes. We want to know how many pages, , Vicky can
Write a linear equation in two variables that represents the number of pages Vicky reads in any given time interval. Let
represent the constant rate that Vicky reads. Then, and
b.
pages in
, therefore
Complete the table below. Use a calculator and round answers to the tenths place. Linear equation: (time in minutes)
c.
(pages read)
About how long would it take Vicky to read
pages? Explain.
It would take her a little over minutes. After minutes she can read about pages, and after hour she can read about pages. Since pages is between and , then it will take her between and minutes to read pages.
Problem Set Sample Solutions Students practice writing two variable equations that represent a constant rate. 1.
A train travels at a constant rate of a.
miles per hour.
What is the distance, , in miles, that the train travels in hours? Let
be the constant rate the train travels. Then,
Lesson 11: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
and
; therefore,
.
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Lesson 11
NYS COMMON CORE MATHEMATICS CURRICULUM
b.
How many miles will it have traveled in
The train will travel
2.
miles in
What is the amount of water, Let
hours.
b.
, that is leaked from the faucet after minutes?
; therefore,
How much water is leaked after an hour?
The faucet will leak
gallons in one hour.
A car can be assembled on an assembly line in a.
hours. Assume that the cars are assembled at a constant rate.
How many cars, , can be assembled in hours? Let
be the constant rate the cars are assembled. Then, and
b.
gallons per minute.
be the constant rate the water leaks from the faucet. Then, and
3.
hours?
Water is leaking from a faucet at a constant rate of a.
8β’4
therefore
How many cars can be assembled in a week? A week is
Lesson 11: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
hours. So
Twenty-eight cars can be assembled in a week.
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Lesson 11
NYS COMMON CORE MATHEMATICS CURRICULUM
4.
A copy machine makes copies at a constant rate. The machine can make a.
minutes.
Write an equation to represent the number of copies, , that can be made over any time interval, . Let
be the constant rate that copies can be made. Then and
b.
copies in
8β’4
, therefore
Complete the table below.
Linear equation:
(time in minutes)
(number of copies) 0
c.
Graph the data on a coordinate plane. 36 34 32 30 28 26
Number of copies, π
24 22 20 18 16 14 12 10 8 6 4 2
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1
Time in minutes, π‘
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NYS COMMON CORE MATHEMATICS CURRICULUM
d.
The copy machine runs for occurred? Explain. Since seconds is is between and
5.
seconds, then jams. About how many copies were made before the jam
of a minute, then the number of copies made will be between .
Connor runs at a constant rate. It takes him a.
and
because
minutes to run four miles.
Write the linear equation in two variables that represents the number of miles Connor can run in any given time interval, . Let
be the constant rate that Connor runs and and
b.
8β’4
, therefore
represent the number of miles. Then
.
Complete the table below. Use a calculator and round answers to the tenths place.
Linear equation: (time in minutes)
Lesson 11: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
(distance in miles)
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NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 11
c.
Graph the data on a coordinate plane.
d.
Connor ran for minutes before tripping and spraining his ankle. About how many miles did he run before he had to stop? Explain.
8β’4
Since Connor ran for minutes, that means that he ran more than miles, but less than miles. Since is between and , then we can use those reference points to make an estimate of how many miles he ran in minutes, probably about miles.
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Lesson 12
NYS COMMON CORE MATHEMATICS CURRICULUM
8β’4
Lesson 12: Linear Equations in Two Variables Student Outcomes ο§
Students use a table to find solutions to a given linear equation and plot the solutions on a coordinate plane.
Lesson Notes In this lesson students will find solutions to a linear equation in two variables using a table, then plot the solutions as points on the coordinate plane. Students will need graph paper in order to complete the Exercises and the Problem Set.
Classwork Opening Exercise (5 minutes) Students complete the Opening Exercise independently in preparation for the discussion about standard form and solutions that follows. Opening Exercise Emily tells you that she scored ππ points in a basketball game with only two- and three-point baskets (no free throws). How many of each type of basket did she score? Use the table below to organize your work. Number of Two-Pointers
Number of Three-Pointers
ππ
π
ππ
π
ππ
π
π
π
π
π
π
ππ
Let π be the number of two-pointers and π be the number of three-pointers that Emily scored. Write an equation to represent the situation. ππ + ππ = ππ
Discussion (10 minutes) ο§
An equation in the form of ππ₯ + ππ¦ = π is called a linear equation in two variables, where π, π, and π are constants, and at least one of π and π are not zero. In this lesson, neither π or π will be equal to zero. In the opening exercise, what equation did you write to represent Emilyβs score at the basketball game?
ο§
The equation 2π₯ + 3π¦ = 32 is an example of a linear equation in two variables.
οΊ
2π₯ + 3π¦ = 32
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Lesson 12
NYS COMMON CORE MATHEMATICS CURRICULUM
ο§
An equation of this form, ππ₯ + ππ¦ = π, is also referred to as an equation in standard form. Is the equation you wrote in the opening exercise in standard form? οΊ
ο§
ο§
ο§
Students should identify the pairs of numbers in the table of the opening exercise. No, not just any pair of numbers worked. For example, I couldnβt say that Emily scored 15 two-pointers and 1 three-pointer because that would mean she scored 33 points in the game, and I know she only scored 32 points.
A solution to the linear equation in two variables is an ordered pair of numbers (π₯, π¦) so that π₯ and π¦ makes the equation a true statement. The pairs of numbers that you wrote in the table for Emily are solutions to the equation 2π₯ + 3π¦ = 32 because they are pairs of numbers that make the equation true. The question becomes, how do we find an unlimited number of solutions to a given linear equation? οΊ
ο§
No, because β50(1) + 2 = β50 + 2 = β48 β 15.
What pairs of numbers did you find that worked for Emilyβs basketball score? Did just any pair of numbers work? Explain. οΊ
ο§
The symbols π₯ and π¦ are numbers. Since they are not constants, it means they are unknown numbers, typically called variables, in the equation ππ₯ + ππ¦ = π.
For example, β50π₯ + π¦ = 15 is a linear equation in π₯ and π¦. As you can easily see, not just any pair of numbers π₯ and π¦ will make the equation true. Consider π₯ = 1 and π¦ = 2. Does it make the equation true? οΊ
ο§
Yes, it is in the same form as ππ₯ + ππ¦ = π.
In the equation ππ₯ + ππ¦ = π, the symbols π, π, and π are constants. What then are π₯ and π¦? οΊ
ο§
8β’4
Guess numbers until you find a pair that makes the equation true.
A strategy that will help us find solutions to a linear equation in two variables is as follows: We fix a number for π₯. That means we pick any number we want and call it π₯. Since we know how to solve a linear equation in one variable, then we solve for π¦. The number we picked for π₯ and the number we get when we solve for π¦ is the ordered pair (π₯, π¦), which is a solution to the two variable linear equation.
For example, let π₯ = 5. Then in the equation β50π₯ + π¦ = 15 we have β50(5) + π¦ β250 + π¦ β250 + 250 + π¦ π¦
= 15 = 15 = 15 + 250 = 265
Therefore, (5, 265) is a solution to the equation β50π₯ + π¦ = 15.
Similarly, we can fix a number for π¦ and solve for π₯. Let π¦ = 10, then
Therefore, (β
β50π₯ + 10 = 15 β50π₯ + 10 β 10 = 15 β 10 β50π₯ = 5 β50 5 π₯= β50 β50 1 π₯=β 10
1 , 10) is a solution to the equation β50π₯ + π¦ = 15. 10
Ask students to provide you with a number for π₯ or π¦ and demonstrate how to find a solution. You can do this more than once in order to prove to students that you can find a solution no matter which number you choose to fix for π₯ or π¦. Once they are convinced, allow them to work on the exploratory exercises.
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Lesson 12
NYS COMMON CORE MATHEMATICS CURRICULUM
8β’4
Exploratory Challenge/Exercises 1β5 (20 minutes) Students can work independently or in pairs to complete the exercises. Every few minutes, have students share their tables and graphs with the class. Make suggestions to students as they work as to which values for π₯ or π¦ they could choose. For example, in Exercises 1 and 2, small numbers would ease the mental math work. Exercise 3 may be made easier if they choose a number for π¦ and solve for π₯. Exercise 4 can be made easier if students choose values for π₯ that are multiples of 5. As you make suggestions, ask students why your suggestions would make the work easier. Exploratory Challenge/Exercises 1β5 1.
Find five solutions for the linear equation π + π = π, and plot the solutions as points on a coordinate plane. Sample student work:
π
Linear equation: π+π= π
π
π
π+π=π
π
π π π π
Lesson 12: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
π+π=π π+π=π π+π=π π+π=π
π π
βπ βπ
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Lesson 12
NYS COMMON CORE MATHEMATICS CURRICULUM
2.
8β’4
Find five solutions for the linear equation ππ β π = ππ, and plot the solutions as points on a coordinate plane. Sample student work: π π
π
π
π
π
Lesson 12: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
Linear equation: ππ β π = ππ
π(π) β π = ππ π β π = ππ π β π β π = ππ β π βπ = π π = βπ π(π) β π = ππ π β π = ππ π β π β π = ππ β π βπ = π π = βπ π(π) β π = ππ π β π = ππ π β π β π = ππ β π βπ = π π = βπ π(π) β π = ππ π β π = ππ π β π β π = ππ β π βπ = π π = βπ π(π) β π = ππ ππ β π = ππ π=π
π βπ
βπ
βπ
βπ
π
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Lesson 12
NYS COMMON CORE MATHEMATICS CURRICULUM
3.
8β’4
Find five solutions for the linear equation π + ππ = ππ, and plot the solutions as points on a coordinate plane. Sample student work: π
π
π + π(π) = ππ π + ππ = ππ π = ππ
π
π + π(π) = ππ π + π = ππ π = ππ
ππ ππ
π
π + π(π) = ππ π + ππ = ππ π=π
π
π
π + π(π) = ππ π + ππ = ππ π=π
π
π
π + π(π) = ππ π + ππ = ππ π = βπ
βπ
4.
Linear equation: π + ππ = ππ
π
π
Consider the linear equation π + π = ππ. a.
π
Will you choose to fix values for π or π? Explain.
If I fix values for π, it will make the computations easier. Solving for π can be done in one step. b.
Are there specific numbers that would make your computational work easier? Explain. Values for π that are multiples of π will make the computations easier. When I multiply will get an integer.
Lesson 12: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
π π
by a multiple of π, I
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Lesson 12
NYS COMMON CORE MATHEMATICS CURRICULUM
c.
8β’4
π
Find five solutions to the linear equation π + π = ππ, and plot the solutions as points on a coordinate
plane.
Sample student work: π π ππ ππ ππ ππ
Lesson 12: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
π
Linear equation: π π + π = ππ π π (π) + π = ππ π π + π = ππ π=π
π (ππ) + π = ππ π π + π = ππ π=π π (ππ) + π = ππ π π + π = ππ π=π π (ππ) + π = ππ π π + π = ππ π=π π (ππ) + π = ππ π ππ + π = ππ π=π
π π π π π π
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Lesson 12
NYS COMMON CORE MATHEMATICS CURRICULUM
5.
8β’4
At the store you see that you can buy a bag of candy for $π and a drink for $π. Assume you have a total of $ππ to spend. You are feeling generous and want to buy some snacks for you and your friends. a.
Write an equation in standard form to represent the number of bags of candy, π, and the number of drinks, π, you can buy with $ππ. ππ + π = ππ
b.
Find five solutions to the linear equation, and plot the solutions as points on a coordinate plane. Sample student work: Linear equation: π π π π ππ ππ
Lesson 12: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
ππ + π = ππ
π
π(π) + π = ππ π + π = ππ π = ππ
ππ
π(π) + π = ππ ππ + π = ππ π = ππ
ππ
π(π) + π = ππ ππ + π = ππ π = ππ
π(ππ) + π = ππ ππ + π = ππ π = ππ π(ππ) + π = ππ ππ + π = ππ π=π
ππ
ππ π
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Lesson 12
NYS COMMON CORE MATHEMATICS CURRICULUM
8β’4
Closing (5 minutes) Summarize, or ask students to summarize, the main points from the lesson: ο§ ο§ ο§
A two variable equation in the form of ππ₯ + ππ¦ = π is known as a linear equation in standard form.
A solution to a linear equation in two variables is an ordered pair (π₯, π¦) that makes the given equation true.
We can find solutions by fixing a number for π₯ or π¦, then solving for the other variable. Our work can be made easier by thinking about the computations we will need to make before fixing a number for π₯ or π¦. For
example, if π₯ has a coefficient of
1 3
we should select values for π₯ that are multiples of 3.
Lesson Summary A two-variable linear equation in the form ππ + ππ = π is said to be in standard form.
A solution to a linear equation in two variables is the ordered pair (π, π) that makes the given equation true. Solutions can be found by fixing a number for π and solving for π or fixing a number for π and solving for π.
Exit Ticket (5 minutes)
Lesson 12: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
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Lesson 12
NYS COMMON CORE MATHEMATICS CURRICULUM
Name
8β’4
Date
Lesson 12: Linear Equations in Two Variables Exit Ticket 1.
Is the point (1, 3) a solution to the linear equation 5π₯ β 9π¦ = 32? Explain.
2.
Find three solutions for the linear equation 4π₯ β 3π¦ = 1, and plot the solutions as points on a coordinate plane. π₯
Linear equation: 4π₯ β 3π¦ = 1
Lesson 12: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
π¦
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Lesson 12
NYS COMMON CORE MATHEMATICS CURRICULUM
8β’4
Exit Ticket Sample Solutions 1.
Is the point (π, π) a solution to the linear equation ππ β ππ = ππ? Explain.
No, (π, π) is not a solution to ππ β ππ = ππ because π(π) β π(π) = π β ππ = βππ and βππ β ππ.
2.
Find three solutions for the linear equation ππ β ππ = π, and plot the solutions as points on a coordinate plane. Sample student work.
π π π π
Linear equation: ππ β ππ = π
π
π(π) β ππ = π π β ππ = π βππ = βπ π=π
π
ππ β π(π) = π ππ β ππ = π ππ = ππ π=π
π
π(π) β ππ = π ππ β ππ = π βππ = βππ π=π
π
Problem Set Sample Solutions Students practice finding and graphing solutions for linear equations that are in standard form.
1.
π π
Consider the linear equation π β π = βπ. a.
Will you choose to fix values for π or π? Explain.
If I fix values for π, it will make the computations easier. Solving for π can be done in one step. b.
Are there specific numbers that would make your computational work easier? Explain. Values for π that are multiples of π will make the computations easier. When I multiply will get a whole number.
Lesson 12: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
π π
by a multiple of π, I
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Lesson 12
NYS COMMON CORE MATHEMATICS CURRICULUM
c.
8β’4
π π
Find five solutions to the linear equation π β π = βπ, and plot the solutions as points on a coordinate plane.
Sample student work: π π
π
π
ππ
ππ
Lesson 12: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
Linear equation: π π β π = βπ π
π π β (π) = βπ π π β π = βπ π β π + π = βπ + π π=π π π β (π) = βπ π π β π = βπ π β π + π = βπ + π π=π π π β (π) = βπ π π β π = βπ π β π + π = βπ + π π=π π π β (π) = βπ π π β ππ = βπ π β ππ + ππ = βπ + ππ π = ππ π π β (ππ) = βπ π π β ππ = βπ π β ππ + ππ = βπ + ππ π = ππ
π π
π
π
π
ππ
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Lesson 12
NYS COMMON CORE MATHEMATICS CURRICULUM
2.
8β’4
π
Find five solutions for the linear equation π + π = ππ, and plot the solutions as points on a coordinate plane.
Sample student work: π π π π ππ ππ
Lesson 12: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
π
Linear equation: π π + π = ππ π π (π) + π = ππ π π + π = ππ π = ππ π (π) + π = ππ π π + π = ππ π = ππ π (π) + π = ππ π π + π = ππ π=π
π (ππ) + π = ππ π π + π = ππ π=π π (ππ) + π = ππ π π + π = ππ π=π
π ππ ππ π π π
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Lesson 12
NYS COMMON CORE MATHEMATICS CURRICULUM
3.
8β’4
π π
Find five solutions for the linear equation βπ + π = βπ, and plot the solutions as points on a coordinate plane.
Sample student work:
π
π
ππ
ππ
ππ
ππ
Lesson 12: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
Linear equation: π βπ + π = βπ π π βπ + (π) = βπ π βπ + π = βπ βπ + π + π = βπ + π π = βπ β π π + π = βπ + π + π π=π
π
π
π βπ + (π) = βπ π βπ + π = βπ βπ + π + π = βπ + π π = βπ + π π + π = βπ + π + π ππ = π
π
π βπ + (ππ) = βπ π βπ + π = βπ βπ + π + π = βπ + π π = βπ + π π + π = βπ + π + π ππ = π
ππ
π βπ + (ππ) = βπ π βπ + ππ = βπ βπ + π + ππ = βπ + π ππ = βπ + π ππ + π = βπ + π + π ππ = π π βπ + (ππ) = βπ π βπ + ππ = βπ βπ + π + ππ = βπ + π ππ = βπ + π ππ + π = βπ + π + π ππ = π
ππ
ππ
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Lesson 12
NYS COMMON CORE MATHEMATICS CURRICULUM
4.
8β’4
Find five solutions for the linear equation ππ + π = π, and plot the solutions as points on a coordinate plane. Sample student work:
π π π π π π
Lesson 12: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
Linear equation: ππ + π = π π(π) + π = π π+π= π π=π π(π) + π = π π+π= π π=π
π π π
π(π) + π = π π+π =π π = βπ
βπ
π(π) + π = π ππ + π = π π = βπ
βπ
π(π) + π = π π+π =π π = βπ
βπ
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Lesson 12
NYS COMMON CORE MATHEMATICS CURRICULUM
5.
8β’4
Find five solutions for the linear equation ππ β ππ = ππ, and plot the solutions as points on a coordinate plane. Sample student work: π
ππ π
ππ π
ππ
ππ π
ππ π
Lesson 12: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
Linear equation: ππ β ππ = ππ
ππ β π(π) = ππ ππ β π = ππ ππ β π + π = ππ + π ππ = ππ π ππ π= π π ππ π= π ππ β π(π) = ππ ππ β ππ = ππ ππ β ππ + ππ = ππ + ππ ππ = ππ π ππ π= π π ππ π= π ππ β π(π) = ππ ππ β ππ = ππ ππ β ππ + ππ = ππ + ππ ππ = ππ π = ππ ππ β π(π) = ππ ππ β ππ = ππ ππ β ππ + ππ = ππ + ππ ππ = ππ π ππ π= π π ππ π= π ππ β π(π) = ππ ππ β ππ = ππ ππ β ππ + ππ = ππ + ππ ππ = ππ π ππ π= π π ππ π= π
π
π
π
π
π
π
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Lesson 13
NYS COMMON CORE MATHEMATICS CURRICULUM
8β’4
Lesson 13: The Graph of a Linear Equation in Two Variables Student Outcomes ο§
Students predict the shape of a graph of a linear equation by finding and plotting solutions on a coordinate plane.
ο§
Students informally explain why the graph of a linear equation is not curved in terms of solutions to the given linear equation.
Classwork Discussion (20 minutes) ο§
ο§
In the last lesson we saw that the solutions of a linear equation in two variables can be plotted on a coordinate plane as points. The collection of all points (π₯, π¦) in the coordinate plane so that each (π₯, π¦) is a solution of ππ₯ + ππ¦ = π is called the graph of ππ₯ + ππ¦ = π. Do you think it is possible to plot all of the solutions of a linear equation on a coordinate plane? οΊ
No, itβs not possible. There are an infinite number of values we can use to fix one of the variables.
ο§
For that reason, we cannot draw the graph of a linear equation. What we can do is plot a few points of an equation and make predictions about what the graph should look like.
ο§
Letβs find five solutions to the linear equation π₯ + π¦ = 6 and plot the points on a coordinate plane. Name a solution.
As students provide solutions (samples provided below) organize them in an π₯-π¦ table, as shown. It is most likely that students will give whole number solutions only. Accept them for now. π₯
π¦
1
5
0 2 3 4
Lesson 13: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
6 4 3 2
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Lesson 13
NYS COMMON CORE MATHEMATICS CURRICULUM
8β’4
ο§
Now letβs plot these points of the graph of π₯ + π¦ = 6 on a coordinate plane.
ο§
Can you predict the shape of the graph of this linear equation based on just the five points we have so far?
ο§
Yes, at this point it looks like the graph of the equation is a line, but for all we know, there can be some curved parts between some of these points.
οΊ
It looks like the points lie on a line.
Draw curves between each of the points on the graph to show that the line could look similar to the graph below.
ο§
The only thing we can do at this point is find more solutions that would show what the graph looks like between the existing points. That means we will have to look at some points with coordinates that are fractions. Name a solution that will plot as a point between the points we already have on the graph.
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Lesson 13
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Add to the π₯-π¦ table. Sample solutions provided below.
π₯
π¦
1
5
0 2 3 4 1 2
1
1 2
3
1 2
2
4
1 2 1 2
8β’4
6 4 3 2
5
1 2
3
1 2
4
2 1
1 2 1 2 1 2
ο§
Now letβs add these points of the graph of π₯ + π¦ = 6 on our coordinate plane.
ο§
Are you convinced that the graph of the linear equation π₯ + π¦ = 6 is a line? You shouldnβt be! What if it looked like this.
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149
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8β’4
Again, draw curves between each of the points on the graph to show what the graph could look like. Make sure to have curves in quadrants II and IV to illustrate the need to find points that fit (or donβt fit) the pattern of the graph when π₯ and π¦ values are negative. ο§
Time to find more solutions. This time, we will need to come up with solutions where the π₯-value is negative or the π¦-value is negative.
Add to the π₯-π¦ table. Sample solutions are provided below. π₯
π¦
1
5
0 2 3 4 1 2
1
1 2
3
1 2
2
4
1 2 1 2
β1 β2 β3 7 8
Lesson 13: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
6 4 3 2
5
1 2
3
1 2
4
2 1
1 2 1 2 1 2
7 8 9
β1 β2
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150
Lesson 13
NYS COMMON CORE MATHEMATICS CURRICULUM
ο§
Now we have 15 solutions to the equation π₯ + π¦ = 6. Are you convinced that the graph of this equation is a line?
οΊ ο§
8β’4
Students should say that they are still unsure, even though it looks like the graph is forming a line, there is a possibility for a curve between some of the existing points.
Based on the look of the graph, does the point οΏ½
Why or why not? οΊ
It looks like οΏ½
17 1 , οΏ½ belong to the graph of the linear equation π₯ + π¦ = 6? 3 3
17 1 , οΏ½ should be on the graph of the linear equation because it looks like the point would 3 3
follow the pattern of the rest of the points, i.e., be on the line.
ο§
Just by looking at the graph, we cannot be sure. We can predict that οΏ½
17 1 , οΏ½ is on the graph of π₯ + π¦ = 6 3 3
because it looks like it should be. What can we do to find out for sure? οΊ
ο§
ο§
17 3
+
1
= 6, and it does, then
3 17 1 we can say for sure that οΏ½ , οΏ½ is a point on the graph of π₯ + π¦ = 6. 3 3 3 Just by looking at the graph, would you say that οΏ½β2 , 4οΏ½ is a point on the graph of π₯ + π¦ = 6? 7 3 οΊ Based on the graph, it doesnβt look like the point οΏ½β2 , 4οΏ½ would be on the graph of π₯ + π¦ = 6. 7
How do you know that the graph doesnβt curve down at that point? How can you be sure? οΊ
ο§
Since each point on the graph represents a solution to the equation, if
3 7
3 7
I would have to see if οΏ½β2 , 4οΏ½ is a solution to π₯ + π¦ = 6, which it is not because β2 + 4 = 1 3 7
6. Therefore, the point οΏ½β2 , 4οΏ½ is not a solution to the equation.
4 , not 7
At this point, we can predict that the graph of this linear equation is a line. Does that mean that the graph of every linear equation is a line? Might there be some linear equations that have curves in them? For now, our only method of proving or disproving our prediction is plotting solutions on a coordinate plane. The more we learn about linear equations, the better we will be able to answer the questions just asked.
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151
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8β’4
Exercises 1β6 (15 minutes) Students will need graph paper to complete Exercises 1β2. Students work independently on Exercises 1β2 for the first 10 minutes. Then, they share their solutions with their partners, and plot more points on their graphs. As students work, verify through discussion that they are choosing a variety of rational numbers to get a good idea of what the graph of the linear equation will look like. Exercise 6 is an optional exercise as it challenges students to come up with an equation that does not graph as a line. Exercises 1β6 1.
Find at least ππ solutions to the linear equation ππ + π = βπ and plot the points on a coordinate plane.
Sample student work. π π
π
π π
π π
π
π π
π βπ βπ βπ βπ
Lesson 13: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
Linear Equation: ππ + π = βπ
π
π(π) + π = βπ π + π = βπ π = βππ
π π οΏ½π οΏ½ + π = βπ π π π + π = βπ π
π = βππ
π(π) + π = βπ π + π = βπ π = βππ
βππ
βππ
π π
βππ
π(π) + π = βπ π + π = βπ π = βππ
π π οΏ½π οΏ½ + π = βπ π π ππ + π = βπ π
π = βππ
π(π) + π = βπ ππ + π = βπ π = βππ
π π
βππ
βππ
π π
π π
βππ
π(βπ) + π = βπ βπ + π = βπ π = βπ
βπ
π(βπ) + π = βπ βπ + π = βπ π = βπ
βπ
π(βπ) + π = βπ βπ + π = βπ π=π
π
π(βπ) + π = βπ βππ + π = βπ π=π
π
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152
Lesson 13
NYS COMMON CORE MATHEMATICS CURRICULUM
8β’4
What shape is the graph of the linear equation taking? The graph appears to be the shape of a line. 2.
Find at least ππ solutions to the linear equation π β ππ = ππ and plot the points on a coordinate plane. Sample student work.
Linear Equation: π β ππ = ππ
π
ππ
π π
ππ
ππ
π π
ππ
Lesson 13: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
π
π π β π οΏ½ οΏ½ = ππ π π π β π = ππ π π π = ππ π
π π
π β π(π) = ππ π β π = ππ π = ππ
π
π π β π οΏ½π οΏ½ = ππ π π π β π = ππ π π π = ππ π
π
π β π(π) = ππ π β ππ = ππ π = ππ
π π
π
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ππ
π π
π π β π οΏ½π οΏ½ = ππ π π π β ππ = ππ π π π = ππ π
π
π β π(βπ) = ππ π + π = ππ π=π
βπ
π β π(βπ) = ππ π + ππ = ππ π = βπ
βπ
π β π(π) = ππ π β ππ = ππ π = ππ
ππ π
βπ
βπ
π β π(βπ) = ππ π + ππ = ππ π = βπ
βπ
π π
π
π β π(βπ) = ππ π + ππ = ππ π=π
π
8β’4
βπ
What shape is the graph of the linear equation taking? The graph appears to be the shape of a line. 3.
Compare the solutions you found in Exercise 1 with a partner. Add their solutions to your graph. Is the prediction you made about the shape of the graph still true? Explain. Yes. With the additional points, the graph still appears to be the shape of a line.
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4.
8β’4
Compare the solutions you found in Exercise 2 with a partner. Add their solutions to your graph. Is the prediction you made about the shape of the graph still true? Explain. Yes. With the additional points, the graph still appears to be the shape of a line.
5.
Joey predicts that the graph of βπ + ππ = π will look like the graph shown below. Do you agree? Explain why or why not.
Sample student work: No, I do not agree with Joey. The graph that Joey drew contains the point (π, π). If (π, π) is on the graph of the linear equation then it will be a solution to the equation; however, it is not. Therefore, the point cannot be on the graph of the equation which means Joeyβs prediction is incorrect. 6.
MP.3
We have looked at some equations that appear to be lines. Can you write an equation that has solutions that do not form a line? Try to come up with one and prove your assertion on the coordinate plane. Answers will vary. Any non-linear equation that students write will graph as something other than a line. For example, the graph of π = ππ or the graph of π = ππ will not be a line.
Closing (5 minutes) Summarize, or ask students to summarize, the main points from the lesson: ο§
All of the graphs of linear equations we have done so far appear to take the shape of a line.
ο§
We can show whether or not a point is on the graph of an equation by checking to see if it is a solution to the equation.
Lesson Summary One way to determine if a given point is on the graph of a linear equation is by checking to see if it is a solution to the equation. At this point, all graphs of linear equations appear to be lines.
Exit Ticket (5 minutes)
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Name
8β’4
Date
Lesson 13: The Graph of a Linear Equation in Two Variables Exit Ticket 1.
Ethan found solutions to the linear equation 3π₯ β π¦ = 8 and graphed them. What shape is the graph of the linear equation taking?
2.
Could the following be the graph of βπ₯ + 2π¦ = 5?
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Exit Ticket Sample Solutions 1.
Ethan found solutions to the linear equation ππ β π = π and graphed them. What shape is the graph of the linear equation taking?
It appears to take the shape of a line. 2.
Could the following be the graph of βπ + ππ = π?
Sample student answer: Student may have chosen any point to make the claim that this is not the graph of the equation βπ + ππ = π.
Although the graph appears to be a line, the graph contains the point (π, π). The point (π, π) is not a solution to the linear equation; therefore, this is not the graph of βπ + ππ = π. Note to teacher: Accept any point as not being a solution to the linear equation.
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Problem Set Sample Solutions In problem 1, students graph linear equations by plotting points that represent solutions. For that reason, they will need graph paper. Students informally explain why the graph of a linear equation is not curved by showing that a point on the curve is not a solution to the linear equation.
1.
π
Find at least ππ solutions to the linear equation π + π = π and plot the points on a coordinate plane.
Sample student work:
Linear Equation: π π+π =π π
π
π (π) + π = π π π +π= π π
π
π=π
π π
π π οΏ½π οΏ½ + π = π π π π +π=π π π=π
π (π) + π = π π π π +π= π π
π
π=π
βπ
π π
βπ
Lesson 13: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
π
π π
π (π) + π = π π π+π=π π=π
π
βπ
π
π (π) + π = π π π+π=π π=π
π
π
π
π π
π=π
π π
π π οΏ½βπ οΏ½ + π = π π π π β +π=π π π=π
π (βπ) + π = π π βπ + π = π π=π
π π
π
π π
π
π π
π (βπ) + π = π π π β +π=π π
π
π π
π
π π
π
π π
π
π π
π
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π (βπ) + π = π π π β +π=π π
βπ
βπ
π=π
π π
π π οΏ½βπ οΏ½ + π = π π π π β +π=π π
π π
π=π
π π
π
π π
π
π π
8β’4
What shape is the graph of the linear equation taking? The graph appears to be the shape of a line. 2.
Can the following be a graph of the equation π β π = π? Explain. The graph shown contains the point (π, βπ). If (π, βπ) is on the graph of the linear equation then it will be a solution to the equation. It is not; therefore, the point cannot be on the graph of the equation, which means the graph shown cannot be the graph of the equation π β π = π.
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3.
8β’4
Can the following be a graph of the equation π + ππ = π? Explain.
The graph shown contains the point (βπ, π). If (βπ, π) is on the graph of the linear equation then it will be a solution to the equation. It is not; therefore, the point cannot be on the graph of the equation which means the graph shown cannot be the graph of the equation π + ππ = π. 4.
Can the following be a graph of the equation π β π = π? Explain.
Yes, because each point on the graph represents a solution to the linear equation π β π = π.
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Can the following be a graph of the equation π + π = π? Explain.
Yes, because each point on the graph represents a solution to the linear equation π + π = π. 6.
Can the following be a graph of the equation ππ β π = π? Explain.
Yes, because each point on the graph represents a solution to the linear equation ππ β π = π.
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8β’4
Can the following be a graph of the equation π β π = π? Explain.
Sample student response: The graph shown contains the point (βπ, βπ). If (βπ, βπ) is on the graph of the linear equation then it will be a solution to the equation. It is not; therefore, the point cannot be on the graph of the equation which means the graph shown cannot be the graph of the equation π β π = π.
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Lesson 14: Graph of a Linear EquationβHorizontal and Vertical Lines Student Outcomes ο§
Students graph linear equations in standard form, ππ₯ + ππ¦ = π (π ππ π = 0), that produce a horizontal or a vertical line.
Lesson Notes The goal of this lesson is for students to know that the graph of an equation in the form of π₯ = π or π¦ = π, where π is a constant, is the graph of a vertical or horizontal line, respectively. In order to show this, we begin with linear equations in two variables, ππ₯ + ππ¦ = π, where one of the coefficients of π₯ or π¦ is equal to zero. The reason behind this is that students know an ordered pair (π₯, π¦) is a point on a coordinate plane, as well as a solution to the equation ππ₯ + ππ¦ = π. Frequently when students see an equation in the form of π₯ = π or π¦ = π, they do not think of it as a point nor a line, just a number. To avoid this, the approach to graphs of horizontal and vertical lines is embedded with what students already know about points and solutions to linear equations in two variables. In this lesson, students begin by graphing and exploring the connection between ππ₯ + ππ¦ = π and π₯ = π in the first three exercises, and then a discussion follows to solidify the concept that the graph of the linear equation π₯ = π is a vertical line. Similar exercises precede the discussion for the graph of π¦ = π as a horizontal line.
Classwork Exercises 1β3 (5 minutes) Students complete Exercises 1β3 independently or in pairs in preparation for the discussion that follows. Students will need graph paper to complete the exercises. Exercises 1β3 1.
Find at least π solutions to graph the linear equation ππ + ππ = π.
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2.
Find at least π solutions to graph the linear equation ππ + ππ = π.
3.
What was different about the equations in Exercises 1 and 2? What effect did this change have on the graph?
8β’4
In the first equation, the coefficient of π was π. In the second equation, the coefficient of π was π. The graph changed from being slanted to a vertical line.
Discussion (14 minutes) ο§
ο§
From Lesson 13, we can say that the graph of a linear equation in two variables looks like a line. We want to be able to prove that the graph is a line, not just predict. For that reason, we will begin with two special cases of linear equations in two variables. Given a linear equation in two variables ππ₯ + ππ¦ = π, where π, π, and π are constants, we will look first at Case 1, where π = 1 and π = 0. (In the past we have said that π, π β 0). To do some investigating, letβs say that π = 5. Then we have the following equation: 1β
π₯+0β
π¦ =5
For the linear equation 1 β
π₯ + 0 β
π¦ = 5, we want to find solutions, (π₯, π¦) to plot as points on a coordinate plane. How have we found solutions in prior lessons? οΊ
ο§
We find solutions by picking a number for π₯ or π¦, and then solve for the other variable. The numbers (π₯, π¦) are a solution to the equation.
What happens if we pick 7 for π₯? Explain. οΊ
If we pick 7 for π₯, then we get
1β
7+0β
π¦ = 5 7+0β
π¦ = 5 0 β
π¦ = β2 0 β β2 If we replace π₯ with 7, then we get an untrue statement. Therefore, (7, π¦) is not a solution to this linear equation.
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ο§
What happens if we pick 7 for π¦? Explain. οΊ
ο§
8β’4
If we pick 7 for π¦, then we get
1β
π₯+0β
7 =5 π₯+0 =5 π₯=5 If we replace π¦ with 7, then we see that π₯ = 5. Therefore, (5, 7) is a solution to this linear equation.
What happens if we pick β3 for π¦? Explain. οΊ
If we pick β3 for π¦, then we get
1 β
π₯ + 0 β
(β3) = 5 π₯+0=5 π₯=5 If we replace π¦ with β3, then we see that π₯ = 5. Therefore, (5, β3) is a solution to this linear equation.
ο§
1
What happens if we pick for π¦? Explain. οΊ
If we pick
1 2
2
for π¦, then we get
1 =5 2 π₯+0 =5 π₯=5
1β
π₯+0β
1
1 2
If we replace π¦ with , then we see that π₯ = 5. Therefore, οΏ½5, οΏ½ is a solution to this linear equation. ο§
2
What do you notice about the π₯-value each time we pick a number for π¦? οΊ
ο§
Look at the equation again, can we show that π₯ must always be equal to 5? οΊ
ο§
Yes, if we transform the equation, we see that π₯ = 5:
1β
π₯+0β
π¦ =5 π₯+0 =5 π₯=5
What does that mean for our π¦-values? Which number will produce a solution where π₯ = 5? οΊ
ο§
Each time we pick a number for π¦, we keep getting π₯ = 5.
Our π¦-values can be any number. No matter which number we pick for π¦, the value for π₯ will always be equal to 5.
Letβs graph the solutions we found and a few more.
Add points to the graph as a result of student responses to βIf π¦ = a number, then what value is π₯? " The graph should look something like what is shown below.
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Focus in on one unit, between 2 and 3 for example, and explain that all of the fractional values for π¦ between 2 and 3 will produce a solution where π₯ = 5.
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ο§
If this process is repeated between all integers, then the result is the vertical line π₯ = 5, as shown in blue.
ο§
The graph of the equation ππ₯ + ππ¦ = π, where π = 1, π = 0, and π = 5, is the vertical line passing through point (5, 0).
ο§
The above situation is not unique. That is, in the equation ππ₯ + ππ¦ = π, we chose the value for π to be 5. The same reasoning can be used for any value π. If we chose π to be 6, what do you think the graph would look like? οΊ
ο§
ο§
1 2
The graph would probably be a vertical line passing through the point οΏ½β , 0οΏ½.
Notice that the equation 1 β
π₯ + 0 β
π¦ = π is equivalent to π₯ = π. Therefore, we can make the following conclusion in the form of a theorem: Theorem: The graph of π₯ = π is the vertical line passing through (π, 0), where π is a constant.
To prove this theorem, we first want to show that the graph of π₯ = π will intersect the π₯-axis. To show that the graph of π₯ = π intersects the π₯-axis, we actually have to show that the graph of π₯ = π is not parallel to the π₯-axis because if it was parallel, it would not intersect the π₯-axis. Therefore, if we can show that π₯ = π is not parallel to the π₯-axis, then it must intersect it at some point. How do we know that the graph of π₯ = π is not parallel to the π₯-axis? οΊ
ο§
1 2
If we chose π to be β , what do you think the graph would look like? οΊ
ο§
The graph would probably be a vertical line passing through the point (6, 0).
We know itβs not parallel to the π₯-axis because it intersects the π₯-axis at (π, 0).
Then the graph of π₯ = π must be parallel to the π¦-axis. This is because of how we define/set up the coordinate plane. The plane is comprised of horizontal lines parallel to the π₯-axis and vertical lines parallel to the π¦-axis. Since π₯ = π is not parallel to the π₯-axis, then it must be parallel to the π¦-axis. Lesson 14: Date:
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ο§
Now we need to show that π₯ = π is the graph of the only line that is parallel to the π¦-axis going through the point (π, 0). How do we show that there is only one line parallel to the π¦-axis passing through a specific point? οΊ
ο§
8β’4
That goes back to what we learned about basic rigid motions. If we rotate a line around a center 180Λ (in this particular case we would rotate around the point (2.5, 0)) , then we will get an image of the line parallel to the line itself. There exists only one line parallel to a given line that goes through a specific point.
For that reason, we know that the graph of π₯ = π is the vertical line that passes through (π, 0).
Exercises 4β6 (3 minutes) Students complete Exercises 4β6 independently. Students will need graph paper to complete the exercises. 4.
Graph the linear equation π = βπ.
5.
Graph the linear equation π = π.
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8β’4
What will the graph of π = π look like?
The graph of π = π will look like a vertical line that goes through the point (π, π). It will be the same as the π-axis.
Exercises 7β9 (5 minutes) Students complete Exercises 7β9 independently or in pairs in preparation for the discussion that follows. Students will need graph paper to complete the exercises. 7.
Find at least π solutions to graph the linear equation ππ + ππ = π.
8.
Find at least π solutions to graph the linear equation ππ + ππ = π.
9.
What was different about the equations in Exercises 7 and 8? What effect did this change have on the graph? In the first equation, the coefficient of π was π. In the second equation, the coefficient of π was π. The graph changed from being a slanted line to a horizontal line.
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Discussion (8 minutes) ο§ ο§ ο§
Now for Case 2. We need to look at the graph of the linear equation in two variables ππ₯ + ππ¦ = π, where π = 0 and π = 1, and π is a constant.
Letβs pick a number for π, as we did for Case 1. Let π = 2. Then we have the equation 0 β
π₯ + 1 β
π¦ = 2.
Now letβs find a few solutions for this linear equation. What happens if we pick 7 for π¦? Explain. If we pick 7 for π¦, then we get
οΊ
0β
π₯+1β
7 =2 0+7 =2 7β 2 If we replace π¦ with 7, then we get an untrue statement. Therefore, (π₯, 7) is not a solution to this linear equation.
ο§
If π₯ = 5, what value does π¦ have? Explain. The value of π¦ is 2 because
οΊ
ο§
Therefore, (5, 2) is a solution to the linear equation.
If π₯ = β5, what value does π¦ have? Explain. The value of π¦ is 2 because
οΊ
ο§
0β
5+1β
π¦ = 2 0+π¦ = 2 π¦=2
1 2
0 β
(β5) + 1 β
π¦ = 2 0+π¦=2 π¦=2
Therefore, (β5, 2) is a solution to the linear equation.
If π₯ = , what value does π¦ have? Explain. οΊ
The value of π¦ is 2 because
1 2
0β
1 +1β
π¦ = 2 2 0+π¦ = 2 π¦=2
Therefore, οΏ½ , 2οΏ½ is a solution to the linear equation. ο§
Do you see a similar pattern emerging for this linear equation? Explain. οΊ
ο§
Yes, no matter which value we pick for π₯ the π¦-value is always 2. Therefore, the graph of 0 β
π₯ + 1 β
π¦ = 2 is equivalent to π¦ = 2.
What do you think the graph of π¦ = 2 will look like? οΊ
Students may predict that the graph of π¦ = 2 is the horizontal line that goes through the point (0, 2).
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As before, place the solutions you found as points of a graph. Add points to the graph as a result of student responses to βIf π₯ =a number, then what value is π¦?β The graph should look something like what is shown below.
Again, focus in on one unit, between 5 and 6 for example, and explain that all of the fractional values for π₯ between 5 and 6 will produce a solution where π¦ = 2.
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ο§
If this process is repeated between all integers, then the result is the horizontal line π¦ = 2, as shown in blue.
ο§
The graph of the equation ππ₯ + ππ¦ = π, where π = 0, π = 1, and π = 2 is the horizontal line passing through point (0, 2).
ο§
The above situation is not unique. As before in the equation ππ₯ + ππ¦ = π, we chose the value for π to be 2. The same reasoning can be used for any value π. If we chose π to be 6, what do you think the graph would look like? οΊ
ο§
ο§ ο§
1 2
If we chose π to be β , what do you think the graph would look like? οΊ
ο§
The graph would probably be a horizontal line passing through the point (0, 6).
1 2
The graph would probably be a horizontal line passing through the point οΏ½0, β οΏ½.
We can generalize this to a theorem:
Theorem: The graph of π¦ = π is the horizontal line passing through the point (0, π).
We can also say that there is only one line with the equation π¦ = π whose graph is parallel to the π₯-axis that goes through the point (0, π). The proofs of these statements are similar to the proofs for vertical lines.
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Exercises 10β12 (3 minutes) Students complete Exercises 10β12 independently. Students will need graph paper to complete the exercises. Exercises 10β12 10. Graph the linear equation π = βπ.
11. Graph the linear equation π = π.
12. What will the graph of π = π look like?
The graph of π = π will look like a horizontal line that goes through the point (π, π). It will be the same as the πaxis.
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Closing (3 minutes) Summarize, or ask students to summarize, the main points from the lesson: ο§ ο§
The graph of the linear equation in two variables ππ₯ + ππ¦ = π, where π = 1 and π = 0, is the graph of the equation π₯ = π. The graph of π₯ = π is the vertical line that passes through the point (π, 0).
The graph of the linear equation in two variables ππ₯ + ππ¦ = π, where π = 0 and π = 1, is the graph of the equation π¦ = π. The graph of π¦ = π is the horizontal line that passes through the point (0, π).
Lesson Summary A linear equation in standard form, ππ + ππ = π where π = π and π = π is the graph of the equation π = π. The graph of π = π is the vertical line passing through the point (π, π).
A linear equation in standard form, ππ + ππ = π where π = π and π = π is the graph of the equation π = π. The graph of π = π is the horizontal line passing through the point (π, π).
Exit Ticket (4 minutes)
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Name
8β’4
Date
Lesson 14: Graph of a Linear EquationβHorizontal and Vertical Lines Exit Ticket 1.
Graph the linear equation ππ₯ + ππ¦ = π where π = 0, π = 1, and π = 1.5.
2.
Graph the linear equation ππ₯ + ππ¦ = π, where π = 1, π = 0, and π = β .
3.
What linear equation represents the graph of the line that coincides with the π₯-axis?
4.
What linear equation represents the graph of the line that coincides with the π¦-axis?
5 2
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Exit Ticket Sample Solutions 1.
Graph the linear equation ππ + ππ = π, where π = π, π = π, and π = π. π.
2.
Graph the linear equation ππ + ππ = π, where π = π, π = π, and π = β .
3.
What linear equation represents the graph of the line that coincides with the π-axis?
π π
π=π 4.
What linear equation represents the graph of the line that coincides with the π-axis? π=π
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176
Lesson 14
NYS COMMON CORE MATHEMATICS CURRICULUM
8β’4
Problem Set Sample Solutions Students will need graph paper to complete the Problem Set. 1.
Graph the two variable linear equation ππ + ππ = π when π = π, π = π, and π = βπ.
2.
Graph the two variable linear equation ππ + ππ = π when π = π, π = π, and π = π.
3.
Graph the linear equation π = π.
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177
Lesson 14
NYS COMMON CORE MATHEMATICS CURRICULUM
4.
Graph the linear equation π = π.
5.
Explain why the graph of a linear equation in the form of π = π is the horizontal line, parallel to the π-axis passing through the point (π, π).
8β’4
The graph of π = π passes through the point (π, π), that means the graph of π = π cannot be parallel to π-axis because the graph intersects it. For that reason, the graph of π = π must be the horizontal line parallel to the πβ axis. 6.
Explain why there is only one line with the equation π = π that passes through the point (π, π).
There can only be one line parallel to another that goes through a given point. Since the graph of π = π is parallel to the π-axis and it goes through the point (π, π), then it must be the only line that does. Therefore, there is only one line that is the graph of the equation π = π that passes through (π, π).
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Mid-Module Assessment Task
NYS COMMON CORE MATHEMATICS CURRICULUM
Name
8β’4
Date
1. Write and solve each of the following linear equations. a.
1
Ofelia has a certain amount of money. If she spends $12, then she has of the original amount left. How much money did Ofelia have originally?
5
b.
Three consecutive integers have a sum of 234. What are the three integers?
c.
Gil is reading a book that has 276 pages. He has already read some of it last week. He plans to read 2
20 pages tomorrow. By then, he will be of the way through the book. How many pages did Gil 3
read last week?
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2.
a.
Mid-Module Assessment Task
8β’4
Without solving, identify which of the following equations has a unique solution, no solution, or infinitely many solutions. 3π₯ + 5 = β2
6(π₯ β 11) = 15 β 4π₯
12π₯ + 9 = 8π₯ + 1 + 4π₯
2(π₯ β 3) = 10π₯ β 6 β 8π₯ 5π₯ + 6 = 5π₯ β 4
b.
Solve the following equation for a number π₯. Verify that your solution is correct. β15 = 8π₯ + 1
c.
Solve the following equation for a number π₯. Verify that your solution is correct. 7(2π₯ + 5) = 4π₯ β 9 β π₯
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Mid-Module Assessment Task
NYS COMMON CORE MATHEMATICS CURRICULUM
3.
a.
Parker paid $4.50 for three pounds of gummy candy. Assuming each pound of gummy candy costs the same amount, complete the table of values representing the cost of gummy candy in pounds.
Gummy Candy in pounds (π₯)
1
2
3
Cost (π¦)
b.
8β’4
4
5
6
7
8
9
$4.50
Graph the data on the coordinate plane. 14 13 12 11 10 9 8 7 6 5 4 3 2 1
0
1
Module 4: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
2
3
4
5
6
7
8
9
10
11
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NYS COMMON CORE MATHEMATICS CURRICULUM
c.
Mid-Module Assessment Task
8β’4
1
On the same day, Parkerβs friend Peggy was charged $5 for 1 2 lb. of gummy candy. Explain in terms of the graph why this must be a mistake.
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NYS COMMON CORE MATHEMATICS CURRICULUM
Mid-Module Assessment Task
8β’4
A Progression Toward Mastery Assessment Task Item
1
a 8.EE.C.7b
b 8.EE.C.7b
c 8.EE.C.7b
STEP 1 Missing or incorrect answer and little evidence of reasoning or application of mathematics to solve the problem.
STEP 2 Missing or incorrect answer but evidence of some reasoning or application of mathematics to solve the problem.
STEP 3 A correct answer with some evidence of reasoning or application of mathematics to solve the problem, or an incorrect answer with substantial evidence of solid reasoning or application of mathematics to solve the problem.
STEP 4 A correct answer supported by substantial evidence of solid reasoning or application of mathematics to solve the problem.
Student made no attempt to solve the problem or left the problem blank. Student may or may not have identified the variable.
Student did not set up an equation (i.e., guessed the answer). Student may or may not have identified the variable.
Student may or may not have set up correct equation. Student may or may not have identified the variable. Student may have made calculation errors.
Student made no attempt to solve the problem or left the problem blank. May or may not have identified the variables.
Student did not set up an equation and guessed the answer. Student may or may not have identified the variable. Student may have made calculation errors. Student may have only answered part of the question, like the first number is 77, and not given all three numbers.
Student made no attempt to solve the problem or left the problem blank. Student may or may not have identified the variable.
Student did not set up an equation and guessed the answer. Student may or may not have identified the variable.
Student attempted to set up an equation. Student may have set up an incorrect equation. Student may or may not have identified the variables. Student may have made calculation errors. Student may have only answered part of the question, like the first number is 77, and not given all three numbers.
Student identified the variable as βLet π₯ be the amount of money Ofelia hadβ or something similar. AND Student set up a correct 1 equation π₯ β 12 = π₯ or 5 other equivalent version. AND Solved for the variable correctly, π₯ = 15.
Module 4: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
Student attempted to set up an equation. Student may have set up an incorrect equation. Student may or may not have identified the variable.
Student identified the variable as βLet π₯ be the first integer.β AND Student set up a correct equation 3π₯ + 3 = 234 or other equivalent version. AND Student solved the equation correctly. Identified all three numbers correctly. The first number is 77, the second is 78, and the third is 79.
Student identified the variable as βLet π₯ be the number of pages Gil read last weekβ or something similar. AND Student set up a correct equation π₯ + 20 = 184
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NYS COMMON CORE MATHEMATICS CURRICULUM
2
a 8.EE.C.7a
b 8.EE.C.7b
c 8.EE.C.7b
3
a 8.EE.B.5
Mid-Module Assessment Task
8β’4
Student may have made calculation errors leading to an incorrect answer.
or other equivalent version. AND Student solved for the number of pages Gil read last week as 164 pages.
Student made no attempt to determine the type of solution or left the problem blank. Student determined 0 of the solution types correctly. Student may have attempted to determine the solutions by solving.
Student determined 1β2 of the solution types correctly. Student may have attempted to determine the solutions by solving.
Student determined 3β5 of the solution types correctly. Student may have attempted to determine the solutions by solving.
Determined 5 of the solutions types correctly. Equations 1 and 2 have unique solutions, equation 3 has no solution, equation 4 has infinitely many solutions, and equation 5 has no solution. AND Student determined the solutions by observation only.
Student made no attempt to solve the problem or left the problem blank.
Student used properties of equality incorrectly, e.g., subtracted 1 from just one side of the equation, or divided by 8 on just one side of the equation, leading to an incorrect solution.
Student correctly used properties of rational numbers to solve the equation, but may have made a computational error leading to an incorrect solution. For example, student may have subtracted 1 from each side of the equation, but β15 β 1 led to an incorrect answer. Student may or may not have verified their answer.
Student correctly used properties of rational numbers to solve the equation (i.e., found π₯ = β2). There is evidence that the student verified their solution.
Student made no attempt to solve the problem or left the problem blank.
Student used the distributive property incorrectly on both sides of the equation, e.g., 7(2π₯ + 5) = 14π₯ + 5 or 4π₯ β π₯ = 4, leading to an incorrect solution.
Student used the distributive property correctly on one or both sides of the equation, but made a computational error leading to an incorrect solution. Student may or may not have verified their answer.
Student used the distributive property correctly on both sides of the equation leading to a correct solution (i.e., π₯ = β4). There is evidence that the student verified their solution.
Student made no attempt to complete the table or used completely random numbers in the blanks.
Student completed the table incorrectly, but only because of a simple computational error in finding the cost of one pound of candy, leading to all other parts being incorrect.
Student completed 6β7 parts of the table correctly. A computational error led to 1β2 parts being incorrect.
Student completed all 8 parts of the table correctly. (See table below for correct answers.)
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Mid-Module Assessment Task
NYS COMMON CORE MATHEMATICS CURRICULUM
Gummy candy in pounds (π₯) Cost (π¦)
b
1
$1.50
8.EE.B.5
Student made no attempt to put the data on the graph or points were put in randomly.
c
Student left item blank.
8.EE.B.5
Module 4: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
2
$3.00
3
$4.50
4
$. 6.00
Student plotted data points on the graph, but may have misplaced a few points. OR Student may have inversed the data (i.e., plotted points according to (π¦, π₯) instead of (π₯, π¦)).
Student may have performed a computation to prove the mistake. Little or no reference to the graph was made in the argument.
5
$7.50
6
$9.00
7
$10.50
8
$12.00
8β’4
9
$13.50
Student plotted 6β7 data points correctly according to the data in their table.
Student plotted all 8 data points correctly according to the data in their table.
Student made a weak argument as to why (1.5, 5) could not be correct. Student may have connected the dots on the graph to show (1.5, 5) could not be correct.
Student made a convincing argument as to why the point (1.5, 5) could not be correct. Student referenced the relationship being proportional and/or predicted that all points should fall into a line based on the existing pattern of points on the graph.
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Module 4: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
Mid-Module Assessment Task
8β’4
Linear Equations 11/8/13
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NYS COMMON CORE MATHEMATICS CURRICULUM
Module 4: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
Mid-Module Assessment Task
8β’4
Linear Equations 11/8/13
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NYS COMMON CORE MATHEMATICS CURRICULUM
Module 4: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
Mid-Module Assessment Task
8β’4
Linear Equations 11/8/13
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New York State Common Core
8
Mathematics Curriculum
GRADE
GRADE 8 β’ MODULE 4
Topic C:
Slope and Equations of Lines 8.EE.B.5, 8.EE.B.6 Focus Standard:
Instructional Days:
8.EE.B.5
Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways.
8.EE.B.6
Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation π¦ = ππ₯ for a line through the origin and the equation π¦ = ππ₯ +b of a line intercepting the vertical axis at b.
9
Lesson 15: The Slope of a Non-Vertical Line (P)
1
Lesson 16: The Computation of the Slope of a Non-Vertical Line (S) Lesson 17: The Line Joining Two Distinct Points of the Graph π¦ = ππ₯ + π has Slope π (S) Lesson 18: There is Only One Line Passing through a Given Point with a Given Slope (P) Lesson 19: The Graph of a Linear Equation in Two Variables is a Line (S) Lesson 20: Every Line is a Graph of a Linear Equation (P) Lesson 21: Some Facts about Graphs of a Linear Equation in Two Variables (P) Lesson 22: Constant Rates Revisited (P) Lesson 23: The Defining Equation of a Line (E)
Topic C begins with students examining the slope of non-vertical lines. Students relate what they know about unit rate in terms of the slope of the graph of a line (8.EE.B.5). In Lesson 16, students learn the formula for computing slope between any two points. Students reason that any two points on the same line can be used to determine slope because of what they know about similar triangles (8.EE.B.6). In Lesson 17, students transform the standard form of an equation into slope-intercept form. Further, students learn that the slope of a line joining any two distinct points is the graph of a linear equation with slope, π. In Lesson 18, students investigate the concept of uniqueness of a line and recognize that if two lines have the same slope and a common point, the two lines are the same. 1
Lesson Structure Key: P-Problem Set Lesson, M-Modeling Cycle Lesson, E-Exploration Lesson, S-Socratic Lesson
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Topic C
8β’4
Lessons 19 and 20 prove to students that the graph of a linear equation is a line and that a line is a graph of a linear equation. In Lesson 21, students learn that the π¦-intercept is the location on the coordinate plane where the graph of a linear equation crosses the π¦-axis. Also in this lesson, students learn to write the equation of a line given the slope and a point. In Lesson 22, constant rate problems are revisited. Students learn that any constant rate problem can be described by a linear equation in two variables where the slope of the graph is the constant rate, i.e., rate of change. Lesson 22 also presents students with two proportional relationships expressed in different ways. Given a graph and an equation, students must use what they know about slope to determine which of the two has a greater rate of change. Lesson 23 introduces students to the symbolic representation of two linear equations that would graph as the same line.
Topic C: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
Slope and Equations of Lines 11/8/13
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Lesson 15
NYS COMMON CORE MATHEMATICS CURRICULUM
8β’4
Lesson 15: The Slope of a Non-Vertical Line Student Outcomes ο§
Students know slope is a number that describes the steepness or slant of a line.
ο§
Students interpret the unit rate as the slope of a graph.
Lesson Notes In Lesson 13, we made some predictions about what the graph of a linear equation would look like. In all cases, we predicted that the graph of a linear equation in two variables would be a line. In Lesson 14, we learned that the graph of the linear equation is the vertical line passing through the point and the graph of the linear equation is the horizontal line passing through the point . We would like to prove that our predictions are true: That the graph of a linear equation in two variables is a line. Before we do, we need some tools: 1.
We must define a number for each non-vertical line that can be used to measure the steepness or slant of the line. Once defined, this number will be called the slope of the line and is often referred to as the rate of change.
Rate of change is terminology that will be used in later lessons. In this first exposure to slope, it is referred to as steepness or slant of a line. In this lesson, students make observations about the steepness of a line. Further, students give directions about how to get from one point on the line to another point on the line. This leads students to the conclusion that the units in the directions have the same ratio. Students then compare ratios between graphs and describe lines as steeper or flatter. 2.
We must show that any two points on a non-vertical line can be used to find the slope of the line.
3.
We must show that the line joining two points on a graph of a linear equation of the form the slope
4.
We must show that there is only one line passing through a given point with a given slope.
has
These tools will be developed over the next few lessons. It is recommended that students are made aware of the fourpart plan to achieve the goal of proving that the graph of a linear equation in two variables is a line as these parts are referenced in the next few lessons to help students make sense of the problem and persevere in solving it. In this lesson, we will look specifically at what is meant by the terms steepness and slant by defining the slope of a line. This lesson defines slope in the familiar context of unit rate; that is, slope is defined when the horizontal distance between two points is fixed at one. Defining slope this way solidifies the understanding that the unit rate is the slope of a graph. Further, students see that the number that describes slope is the distance between the -coordinates leading to the general slope formula.
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Lesson 15
NYS COMMON CORE MATHEMATICS CURRICULUM
8β’4
Opening (8 minutes) To develop conceptual understanding of slope, have students complete the opening activity where they make informal observations about the steepness of lines. Model for students how to answer the questions with the first pair of graphs. Then have students work independently or in pairs to describe how to move from one point to another on a line in terms of units up or down and units right or left. Students also compare the ratios of their descriptions and relate the ratios to the steepness or flatness of the graph of the line. Examine each pair of graphs and answer the questions that follow. Example Graph
a.
Graph
Which graph is steeper? It looks like Graph
b.
Write directions that explain how to move from one point on the graph to the other for each of Graph Graph . For Graph , move
c.
is steeper.
units up and
units right. For Graph , move
units up and
and
units right.
Write the directions from part (b) as ratios, and then compare the ratios. How does this relate to which graph was steeper in part (a)? Graph
was steeper and had the greater ratio.
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Lesson 15
NYS COMMON CORE MATHEMATICS CURRICULUM
8β’4
Pair 1: Graph A
a.
Graph B
Which graph is steeper? It looks like Graph
b.
is steeper.
Write directions that explain how to move from one point on the graph to the other for each of Graph Graph . For Graph , move
c.
units up and
units right. For Graph , move
unit up and
and
units right.
Write the directions from part (b) as ratios, and then compare the ratios. How does this relate to which graph was steeper in part (a)? Graph
was steeper and had the greater ratio.
Pair 2: Graph A
a.
Graph B
Which graph is steeper? It looks like Graph
b.
Write directions that explain how to move from one point on the graph to the other for each of Graph Graph . For Graph , move
c.
is steeper.
units up and
unit right. For Graph , move
units up and
and
units right.
Write the directions from part (b) as ratios, and then compare the ratios. How does this relate to which graph was steeper in part (a)? Graph
was steeper and had the greater ratio.
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Lesson 15
NYS COMMON CORE MATHEMATICS CURRICULUM
8β’4
Pair 3: Graph A
a.
Graph B
Which graph is steeper? It looks like Graph
b.
Write directions that explain how to move from one point on the graph to the other for each of Graph Graph . For Graph , move
c.
is steeper.
unit up and
units right. For Graph , move
units up and
and
unit right.
Write the directions from part (b) as ratios, and then compare the ratios. How does this relate to which graph was steeper in part (a)? Graph
was steeper and had the greater ratio.
Pair 4: Graph A
a.
Graph B
Which graph is steeper? They look about the same steepness.
b.
Write directions that explain how to move from one point on the graph to the other for each of Graph Graph . For Graph , move
c.
units up and
units right. For Graph , move
units up and
and
units right.
Write the directions from part (b) as ratios, and then compare the ratios. How does this relate to which graph was steeper in part (a)? The graphs have equal ratios, which may explain why they look like the same steepness.
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The Slope of a Non-Vertical Line 11/8/13
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NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 15
8β’4
Classwork Example 1 (2 minutes) ο§
Putting horizontal lines off to the side for a moment, there are two other types of non-vertical lines. Those that are left-to-right inclining, as in the graph of , and those that are left-to-right declining, as in the graph of shown below.
ο§
We want to use a number to describe the amount of steepness or slant that each line has. The definition should be written in such a way that a horizontal line has a slope of , that is, no steepness and no slant.
ο§
We begin by stating that lines that are left-to-right inclining are said to have a positive slope and lines that are left-to-right declining are said to have negative slope. We will discuss this more in a moment.
Example 2 (5 minutes) ο§
Now letβs look more closely at finding the number that will be the slope of a line. Suppose a non-vertical line is given in the coordinate plane. We let be the point on line that goes through the origin.
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Lesson 15
NYS COMMON CORE MATHEMATICS CURRICULUM
8β’4
ο§
Next, we locate a point , exactly one unit to the right of point .
ο§
Next, we draw a line, , through point will be named point
that is parallel to the -axis. The point of intersection of line and
ο§
Then the slope of line is the number,
, associated with the -coordinate of point
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NYS COMMON CORE MATHEMATICS CURRICULUM
ο§
Lesson 15
8β’4
The question remains, what is that number? Think of as a number line with point as zero, then however many units is from is the number we are looking for, the slope . Another way of thinking about this is through our basic rigid motion translation. We know that translation preserves lengths of segments. If we translate everything in the plane one unit to the left so that point maps onto the origin, then the segment will coincide with the -axis and the -coordinate of point is the slope of the line.
Consider tracing the graph of the line and points , , onto a transparency. Demonstrate for students the translation along vector βββββ so that point is at the origin. Make clear that the translation moves point to the -axis, which is why the -coordinate of point is the number that represents the slope of the line, This is how students would use the transparency, if needed, to complete the exercises in this lesson. ο§
Letβs look at an example with real numbers. We have the same situation as just described. We have translated everything in the plane one unit to the left so that point maps onto the origin and the segment coincides with the -axis. What is the slope of this line?
οΊ ο§
The slope of the line is , or
This explains why the slope of lines that are left-to-right inclining are positive. When we consider the number line, the point is at positive ; hence this line has a positive slope.
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Lesson 15
NYS COMMON CORE MATHEMATICS CURRICULUM
8β’4
Example 3 (4 minutes) ο§
Suppose a non-vertical line is given in the coordinate plane. As before, we mark a point one unit to the right of and mark point
ο§
Then we draw a line, , through point
ο§
We mark the intersection of lines and as point . Again, recall that we consider the line to be a vertical number line where point is at zero. Then the number associated with the -coordinate of point is the slope of the line.
Lesson 15: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
on the line and go
that is parallel to the -axis.
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NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 15
8β’4
Again, consider tracing the graph of the line and points , , onto a transparency. Demonstrate for students the translation along vector βββββ so that point is at the origin. Make clear that the translation moves point to the -axis, which is why the -coordinate of point is the number that represents the slope of the line, This is how students would use the transparency, if needed, to complete the exercises in this lesson. ο§
Letβs look at this example with real numbers. We have the same situation as just described. We have translated everything in the plane one unit to the left so that point maps onto the origin and the segment coincides with the -axis. What is the slope of this line?
οΊ ο§
The slope of the line is
, or
This explains why the slope of lines that are left-to-right declining are negative. When we consider the number line, the point is at negative ; hence this line has a negative slope.
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Lesson 15
NYS COMMON CORE MATHEMATICS CURRICULUM
8β’4
Example 4 (4 minutes) ο§
Now we have a line that does not go through the origin of the graph.
ο§
Our process for finding slope changes only slightly. We will mark the point Other than that, the work remains the same.
ο§
We will go one unit to the right of and mark point , then draw a line through parallel to the -axis. We mark the intersection of lines and point . Again, recall that we consider the line to be a vertical number line where point is at zero. Then, the number associated with the -coordinate of point is the slope of the line.
Lesson 15: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
at any location on the line .
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Lesson 15
NYS COMMON CORE MATHEMATICS CURRICULUM
ο§
Just as before, we translate so that point axis.
maps onto the origin and the segment
8β’4
coincides with the -
Again, consider tracing the graph of the line and points , , onto a transparency. Demonstrate for students the translation so that point is at the origin (along a vector from to the origin). Make clear that the translation moves point to the -axis, which is why the -coordinate of point is the number that represents the slope of the line, This is how students would use the transparency, if needed, to complete the exercises in this lesson. ο§
What is the slope of this line?
οΊ ο§
The slope of the line is
; that is,
In general, we describe slope as an integer or a fraction. Given that we know the -coordinate of can we express that number as a fraction? οΊ
The slope of the line is ; that is,
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is
, how
.
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NYS COMMON CORE MATHEMATICS CURRICULUM
8β’4
Exercises 1β6 (4 minutes) Students complete Exercises 1β6 independently. The exercises are referenced in the discussion that follows. Exercises 1β6 Use your transparency to find the slope of each line if needed. 1.
What is the slope of this non-vertical line?
The slope of this line is ,
2.
What is the slope of this non-vertical line?
The slope of this line is ,
3.
Which of the lines in Exercises 1 and 2 is steeper? Compare the slopes of each of the lines. Is there a relationship between steepness and slope? The graph in Exercise 1 seems steeper. The slopes are line.
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and . It seems like the greater the slope, the steeper the
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NYS COMMON CORE MATHEMATICS CURRICULUM
4.
,
What is the slope of this non-vertical line?
The slope of this line is
6.
8β’4
What is the slope of this non-vertical line?
The slope of this line is
5.
Lesson 15
,
What is the slope of this non-vertical line?
The slope of this line is ,
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8β’4
Discussion (6 minutes) ο§
When we began, we said that we wanted to define slope in a way so that a horizontal line would have a slope of because a horizontal line has no steepness or slant. What did you notice in Exercise 6? οΊ
ο§
In Exercise 3, you were asked to compare the steepness of the graphs of two lines and then compare their slopes. What did you notice? οΊ
ο§
The slope of the horizontal line was zero, like we wanted it to be.
The steeper the line, the greater the number that describes the slope. For example, Exercise 1 had the steeper line and the greater slope.
Does this same relationship exist for lines with negative slopes? Look specifically at Exercises 4 and 5.
Provide students a minute or two to look back at Exercises 4 and 5. They should draw the conclusion that the absolute value of the slopes of lines that are left-to-right declining will determine which is steeper. Use the points below to bring this fact to light. οΊ
ο§
A similar relationship exists. The line in Exercise 5 was steeper with a slope of had a slope of . Based on our previous reasoning weβd say that because a slope of would have more steepness, but this is not the case.
. The line in Exercise 4 , that the line with
We want to generalize the idea of steepness. When the slopes are positive we expect the line with greater steepness to have the greater slope. When the slopes are negative, it is actually the smaller number that has more steepness. Is there a way that we can compare the slopes so that our reasoning is consistent? That is, we want to say that a line is steeper than another when the number that describes the slope is larger than the other. How can we describe that mathematically, specifically when the slopes are negative? οΊ
We can compare just the absolute value of the slopes. That way, we can say that the steeper line will be the slope with the greater absolute value.
Example 5 (4 minutes) ο§
Letβs take another look at one of the proportional relationships that we graphed in Lesson 11. Here is the problem and the work that we did: Pauline mows a lawn at a constant rate. Suppose she mowed a Linear equation: (time in minutes)
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square foot lawn in
minutes.
(area in square feet)
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8β’4
Now if we assume that the points we plot on the coordinate plane make a line and the origin of the graph is point then we have
ο§
What is the slope of this line? Explain. οΊ
ο§
That makes the point
the location of point
What is the unit rate of mowing the lawn? οΊ
ο§
One unit to the right of point is the point Therefore the slope of this line is , Paulineβs unit rate of mowing a lawn is
square feet per
minute.
When we graph proportional relationships, the unit rate is interpreted as the slope of the graph of the line, which is why slope is referred to as the rate of change.
Closing (3 minutes) Summarize, or ask students to summarize, the main points from the lesson: ο§
We know that slope is a number that describes the steepness of a line.
ο§
We know that lines that are left-to-right inclining have a positive slope and lines that are left-to-right declining have a negative slope.
ο§
We can find the slope of a line by looking at the graphβs unit rate.
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NYS COMMON CORE MATHEMATICS CURRICULUM
8β’4
Lesson Summary Slope is a number that describes the steepness of a line. Slope is represented by the symbol Lines that are left-to-right inclining have a positive slope, as shown below.
Lines that are left-to-right declining have a negative slope, as shown below.
Determine the slope of a line when the horizontal distance between points is fixed at origin of the graph, and then identifying the -coordinate of point The slope of the line shown below is , or
, because point
is at
by translating point
to the
on the -axis.
Exit Ticket (3 minutes)
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NYS COMMON CORE MATHEMATICS CURRICULUM
Name
8β’4
Date
Lesson 15: The Slope of a Non-Vertical Line Exit Ticket 1.
What is the slope of this non-vertical line? Use your transparency if needed.
2.
What is the slope of this non-vertical line? Use your transparency if needed.
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NYS COMMON CORE MATHEMATICS CURRICULUM
8β’4
Exit Ticket Sample Solutions 1.
What is the slope of this non-vertical line? Use your transparency if needed.
The slope of the line is ,
2.
What is the slope of this non-vertical line? Use your transparency if needed.
The slope of the line is
which is equal to
,
.
Problem Set Sample Solutions Students practice identifying graphs of lines as having positive or negative slope. Students interpret the unit rate of a graph as the slope of the graph. The following solutions indicate an understanding of the objectives of this lesson: 1.
Does the graph of the line shown below have a positive or negative slope? Explain.
The graph of this line has a negative slope. First of all it is left-to-right declining, which is an indication of negative slope. Also, if we were to mark a point and a point one unit to the right of , then draw a line parallel to the axis through , then the intersection of the two lines would be below , making the number that represents slope negative.
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NYS COMMON CORE MATHEMATICS CURRICULUM
2.
8β’4
Does the graph of the line shown below have a positive or negative slope? Explain.
The graph of this line has a positive slope. First of all it is left-to-right inclining, which is an indication of positive slope. Also, if we were to mark a point and a point one unit to the right of , then draw a line parallel to the axis through , then the intersection of the two lines would be above , making the number that represents slope positive.
3.
What is the slope of this non-vertical line? Use your transparency if needed.
The slope of this line is ,
4.
What is the slope of this non-vertical line? Use your transparency if needed.
The slope of this line is β ,
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.
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NYS COMMON CORE MATHEMATICS CURRICULUM
5.
What is the slope of this non-vertical line? Use your transparency if needed.
The slope of this line is ,
6.
.
What is the slope of this non-vertical line? Use your transparency if needed.
The slope of this line is 7.
8β’4
,
What is the slope of this non-vertical line? Use your transparency if needed.
The slope of this line is
,
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NYS COMMON CORE MATHEMATICS CURRICULUM
8.
What is the slope of this non-vertical line? Use your transparency if needed.
The slope of this line is
9.
8β’4
,
What is the slope of this non-vertical line? Use your transparency if needed.
The slope of this line is ,
10. What is the slope of this non-vertical line? Use your transparency if needed.
The slope of this line is
,
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NYS COMMON CORE MATHEMATICS CURRICULUM
8β’4
11. What is the slope of this non-vertical line? Use your transparency if needed.
The slope of this line is , 12. What is the slope of this non-vertical line? Use your transparency if needed.
The slope of this line is , 13. What is the slope of this non-vertical line? Use your transparency if needed.
The slope of this line is ,
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.
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Lesson 15
NYS COMMON CORE MATHEMATICS CURRICULUM
8β’4
14. What is the slope of this non-vertical line? Use your transparency if needed.
The slope of this line is ,
In Lesson 11, you did the work below involving constant rate problems. Use the table and the graphs provided to answer the questions that follow. 15. Suppose the volume of water that comes out in three minutes is
Linear equation:
(time in minutes)
a.
(in gallons)
How many gallons of water flow out of the faucet per minute, that is, what is the unit rate of water flow? The unit rate of water flow is
b.
gallons.
gallons per minute.
Assume that the graph of the situation is a line, as shown in the graph. What is the slope of the line? The slope of the line is
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,
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16. Emily paints at a constant rate. She can paint
Linear equation:
(time in minutes)
a.
square feet in five minutes.
(area painted in square feet)
How many square feet can Emily paint in one minute; that is, what is her unit rate of painting? The unit rate at which Emily paints is
b.
square feet per minute.
Assume that the graph of the situation is a line, as shown in the graph. What is the slope of the line? The slope of the line is
,
17. A copy machine makes copies at a constant rate. The machine can make
(time in minutes)
a.
Linear equation:
copies in
minutes.
(number of copies)
How many copies can the machine make each minute; that is, what is the unit rate of the copy machine? The unit rate of the copy machine is
b.
8β’4
copies per minute.
Assume that the graph of the situation is a line, as shown in the graph. What is the slope of the line? The slope of the line is
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,
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Lesson 16
NYS COMMON CORE MATHEMATICS CURRICULUM
8β’4
Lesson 16: The Computation of the Slope of a Non-Vertical
++
Line Student Outcomes ο§ ο§
Students use similar triangles to explain why the slope π is the same between any two distinct points on a non-vertical line in the coordinate plane.
Students use the slope formula to compute the slope of a non-vertical line.
Lesson Notes Throughout the lesson, we use the phrase βrate of changeβ in addition to slope. The goal is for students to know that these are references to the same thing with respect to the graph of a line. At this point in studentsβ learning, the phrases rate of change and slope are interchangeable, but it could be said that rate of change refers to constant rate problems where time is involved, i.e., rate of change over time. In Module 5, when students work with non-linear functions, they will learn that the rate of change is not always constant, as it is with linear equations and linear functions. The points π = (π1 , π2 ) and π
= (π1 , π2 ) are used throughout the lesson in order to make clear that we are looking at two distinct points. Using points π and π
should decrease the chance of confusion compared to using the traditional (π₯1 , π¦1 ) and (π₯2 , π¦2 ). When considering what this this looks like in the formula, it should be clear that distinguishing the points by using different letters will clarify for students that we in fact have two distinct points. It is immediately
recognizable that π =
π2βπ2 is written incorrectly compared to the traditional way of seeing the slope formula. π1βπ1
Further, there should be less mixing up of the coordinates in the formula when it is presented with π and π
. There are several ways of representing the slope formula: π=
π2βπ2 , π1βπ1
π=
π2βπ2 , π1βπ1
π=
πππ π , ππ’π
π=
ππππππππππ ππ π¦βπ£πππ’ππ ππππππππππ ππ π₯βπ£πππ’ππ
and each representation has merit. Please make clear to students throughout this and subsequent lessons that no matter how slope is represented, it should result in the same value for a given line.
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Classwork Example 1 (1 minute) This example requires students to find the slope a line where the horizontal distance between two points with integer coordinates is fixed at 1. Example 1
Using what you learned in the last lesson, determine the slope of the line with the following graph:
The slope of the line is π.
Example 2 (1 minute) This example requires students to find the slope a line where the horizontal distance between two points with integer coordinates is fixed at 1. Example 2
Using what you learned in the last lesson, determine the slope of the line with the following graph:
The slope of this line is π. Lesson 16: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
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Example 3 (3 minutes) This example requires students to find the slope a line where the horizontal distance can be fixed at one, but determining the slope will be difficult because it is not an integer. The point of this example is to make it clear to students that we will need to develop a strategy that will allow us to determine the slope of a line no matter what the horizontal distance is between the two points that are selected. Example 3 What is different about this line compared to the last two examples?
This time, if we choose two points on the line that have a horizontal distance at π, we cannot precisely determine the slope of the line because the vertical change is not an integer. It is some fractional amount.
ο§
Make a conjecture about how you could find the slope of this line.
Have students write their conjectures and share their ideas about how to find the slope of the line in this example; then continue with the discussion that follows.
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Discussion (10 minutes) ο§
In the last lesson, we found a number that described the slope or rate of change of a line. In each case, we were looking at a special case of slope because the horizontal distance between the two points used to determine the slope, π and π, was always 1. Since the horizontal distance was 1, the difference between the π¦- coordinates of points π and π
is equal to the slope or rate of change. For example, in the following graph we thought of point π as zero on a vertical number line and noted how many units point π
was from point π.
ο§
Also in the last lesson we found that the unit rate of a problem was equal to the slope. Using that knowledge
ο§
we can say that the slope or rate of change of a line π =
οΏ½ππ
οΏ½
.
οΏ½πποΏ½
Now the task is to determine the rate of change of a non-vertical line when the distance between points π and π is a number other than 1. We can use what we know already to guide our thinking.
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ο§
8β’4
Letβs take a closer look at Example 2. There are several points on the line with integer coordinates that we could use to help us determine the slope of the line. Recall that we want to select points with integer coordinates because our calculation of slope will be simpler. In each case, from one point to the next we have a horizontal distance of 1 unit noted by the red segment and the difference in the π¦-values between the two points, which is a distance of 2, noted by the blue segments. When we compare the change in the π¦-values to 2
the change in the π₯-values, we have a ratio which is equal to the slope of the line. 1
2
Each of the βslope trianglesβ shown have the ratio . Using the same line, letβs look at a different pair of βslope 1
trianglesβ.
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ο§
What is the ratio of the larger slope triangle? οΊ
ο§
4
.
2
The ratios are equivalent
2 1
=
4 2
= 2.
We have worked with triangles before where the ratios of corresponding sides were equal. We called them similar triangles. Are the slope triangles in this diagram similar? How do you know? οΊ
ο§
The ratio of the larger slope triangle is
What do you notice about the ratio of the smaller slope triangle and the ratio of the larger slope triangle? οΊ
ο§
8β’4
Yes, the triangles are similar by the AA criterion. Each triangle has a right angle (at the intersection of the blue and red segments), and both triangles have a common angle (the angle formed by the red segment and the line itself).
When we have similar triangles we know that the ratios of corresponding side lengths must be equal. That is the reason that both of the slope triangles result in the same number for slope. Notice that we still got the correct number for the slope of the line even though the points chosen did not have a horizontal distance of 1. We now can find the slope of a line given any two points on the line. The horizontal distance between the points does not have to be 1.
Acknowledge any students that may have written or shared this strategy for finding slope from their work with Example 3. ο§
Now letβs look again at Example 3. We did not have a strategy for finding slope before, but we do now. What do you think the slope of this line is? Explain.
οΊ
2
The slope of this line is . 3
Ask students to share their work and explanations with the class. Specifically have them show the slope triangle they used to determine the slope of the line. Select several students to share their work, ideally students will pick different points and different slope triangles. Whether they do or not, have a discussion similar to the previous one that demonstrates that all slope triangles that could be drawn will be similar and that the ratios of corresponding sides are equal.
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Exercise (4 minutes) Students complete the Exercise independently. Exercise Letβs investigate concretely to see if the claim that we can find slope between any two points is true.
a.
Select any two points on the line to label as π· and πΉ.
Sample points selected on graph. b.
Identify the coordinates of points π· and πΉ.
Sample points labeled on graph. c.
Find the slope of the line using as many different points as you can. Identify your points and show your work below. Points selected by students will vary, but the slope should always equal π. Students could choose to use points (π, π), (βπ, π), (βπ, π), (βπ, βπ), (βπ, βπ), and (βπ, βπ).
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Discussion (10 minutes) ο§ ο§
ο§
ο§
We want to show that the slope of line π can be found using any two points π and π
on the line.
Let point π = (π1 , π2 ), where π1 is the π₯-coordinate of point π and π2 is the π¦-coordinate of point π. Also let point π = (π1 , π2 ) and π
= (π1 , π2 ).
Then we claim that the slope of line π is:
π=
|ππ
| . |ππ|
From the last lesson, we found the length of segment ππ
by looking at the π¦-coordinate. Without having to translate, we can find the length of segment ππ
by finding the difference between the π¦-coordinates of points π
and π (vertical distance). Similarly, we can find the length of the segment ππ by finding the difference between the π₯-coordinates of π and π (horizontal distance). |ππ
| (π2 β π2 ) π= = |ππ| (π1 β π1 )
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ο§
We can simplify this further by looking more closely at the coordinates of point π. Consider the following concrete example:
ο§
What do you notice about the π¦-coordinates of points π and π? οΊ
ο§
That means that π2 = π2 . What do you notice about the π₯-coordinates of points π
and π? οΊ
ο§
ο§
The π¦-coordinates of points π and π are the same, 2.
The π₯-coordinates of points π
and π are the same, 6.
That means that π1 = π1 . Then by substitution: |ππ
| (π2 β π2 ) (π2 β π2 ) π= = = |ππ| (π1 β π1 ) (π1 β π1 )
Then we claim that the slope can be calculated regardless of the choice of points. Also, we have discovered something called βthe slope formulaβ. With the formula for slope, or rate of change, π =
of a line can be found using any two points π and π
!
(π2βπ2) , the slope (π1βπ1)
Ask students to translate the slope formula into words and provide them with the traditional ways of describing slope. For example, students may say the slope of a line is the βheight of the slope triangle over the base of the slope triangle,β or βthe difference in the π¦-coordinates over the difference in the π₯-coordinatesβ. Tell students that slope can be referred to as βrise over runβ as well.
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Discussion (3 minutes) Show that the formula to calculate slope is true for horizontal lines. ο§
Suppose we are given a horizontal line. Based on our work in the last lesson, what do we expect the slope of this line to be?
οΊ
ο§
As before, the coordinates of points π and π
are represented as π = (π1 , π2 ) and π
= (π1 , π2 ). Since this is a horizontal line, what do we know about the π¦-coordinates of each point? οΊ
ο§
The slope should be zero because if we go one unit to the right of π, then identify the vertical difference between that point and point π
, there is no difference. Therefore, the slope is zero.
Horizontal lines are graphs of linear equations in the form of π¦ = π, where the π¦-value does not change. Therefore, π2 = π2 .
By the slope formula:
π=
(π2 β π2 ) 0 = =0 (π1 β π1 ) π1 β π1
The slope of the horizontal line is zero, as expected, regardless of the value of the horizontal change.
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Discussion (7 minutes) ο§
Now for the general case. We want to show that we can choose any two points π and π
to find the slope not just a point like π
β², where we have fixed the horizontal distance at 1. Consider the diagram below:
Now we have a situation where point π is an unknown distance from point π. We know that if β³ ππβ²π
β² is similar to β³ πππ
, then the ratio of the corresponding sides will be equal and the ratios are equal to the slope of the line πΏ. Are β³ ππβ²π
β² and β³ πππ
similar? Explain. οΊ
Yes, the triangles are similar, i.e., β³ ππβ² π
β² ~ β³ πππ
. Both triangles have a common angle, β π
ππ, and both triangles have a right angle, β π
β²πβ²π and β π
ππ. By the AA criterion, β³ ππβ² π
β² ~ β³ πππ
.
Now what we want to do is find a way to express this information in a formula. Because we have similar triangles we know the following: |π
β² πβ² | |ππβ² | |ππ
β² | = = =π |π
π| |ππ| |ππ
|
Based on our previous knowledge, we know that we know that |π
β² πβ² | = π and |ππβ² | = 1. By substitution we have π 1 = |π
π| |ππ| which is equivalent to the following:
π |π
π| = 1 |ππ| π=
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|π
π| |ππ|
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We also know from our work earlier that |π
π| = π2 β π2 and |ππ| = π1 β π1 . By substitution we have π2 β π2 π= π1 β π1 The slope of a line can be computed using any two points!
Closing (3 minutes) Summarize, or ask students to summarize, the main points from the lesson: ο§
We know that the slope of a line can be calculated using any two points on the line because of what we know about similar triangles.
ο§
Slope is referred to as the difference in π¦-values compared to the difference in π₯-values, or as the height compared to the base of the slope triangle, or as rise over run.
ο§
We know that the formula to calculate slope is π = line.
π2βπ2 , where (π1 , π2 ) and (π1 , π2 ) are two points on the π1βπ1
Lesson Summary The slope of a line can be calculated using any two points on the same line because the slope triangles formed are similar and corresponding sides will be equal in ratio.
The formula to calculate slope between points π· = (ππ , ππ ) and πΉ = (ππ , ππ ) is π=
π π β ππ π π β ππ
The numerator in the formula is referred to as the difference in π-values and the denominator as the difference in π-values.
Exit Ticket (3 minutes)
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Name
8β’4
Date
Lesson 16: The Computation of the Slope of a Non-Vertical Line Exit Ticket 1.
Find the rate of change of the line by completing parts (a) and (b).
a.
Select any two points on the line to label as π and π
. Name their coordinates.
b.
Compute the rate of change of the line.
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Exit Ticket Sample Solutions 1.
Find the rate of change of the line by completing parts (a) and (b).
a.
Select any two points on the line to label as π· and πΉ. Name their coordinates.
Sample student work. Other points on the graph may have been chosen. π· = (βπ, π) and πΉ = (π, π) b.
Compute the rate of change of the line. π=
(ππ β ππ ) (ππ β ππ )
πβπ βπ β π βπ = βπ π = π =
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Problem Set Sample Solutions Students practice finding slope between any two points on a line. Students also see that π =
result as π = 1.
π2βπ2 . π1βπ1
π2βπ2 yields the same π1βπ1
Calculate the slope of the line using two different pairs of points.
Sample student work: π=
π π β ππ π π β ππ
ππ β (βππ) βππ β ππ ππ = βππ π =β π =
= βπ π=
π π β ππ π π β ππ
ππ β (βππ) ππ β ππ ππ = βππ =
= βπ
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2.
8β’4
Calculate the slope of the line using two different pairs of points.
Sample student work: π=
π π β ππ π π β ππ
π=
π π β ππ π π β ππ
πβπ βπ β π π = βπ π =β π =
πβπ πβπ βπ = π π =β π =
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3.
8β’4
Calculate the slope of the line using two different pairs of points.
Sample student work: π π β ππ π= π π β ππ πβπ πβπ βπ = βπ π = π =
=π π=
π π β ππ π π β ππ
βπ β π πβπ βπ = βπ =
=π
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4.
8β’4
Calculate the slope of the line using two different pairs of points.
Sample student work: π=
π π β ππ π π β ππ
πβπ πβπ π = βπ π =β π =
= βπ π=
π π β ππ π π β ππ
πβπ πβπ π = βπ =
= βπ
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5.
8β’4
Calculate the slope of the line using two different pairs of points.
Sample student work: π=
π π β ππ π π β ππ
πβπ πβπ π = βπ π =β π =
π π β ππ π π β ππ πβπ = ππ β π βπ = π π =β π
π=
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6.
8β’4
Calculate the slope of the line using two different pairs of points.
a.
Select any two points on the line to compute the slope. Sample student work: π π β ππ π π β ππ βπ β (βπ) = βπ β (βπ) βπ = βπ π = π
π=
b.
Select two different points on the line to calculate the slope. Sample student work: Let the two new points be (βπ, π) and (βπ, π). π π β ππ π π β ππ πβπ = βπ β (βπ) βπ = βπ π = π
π=
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8β’4
What do you notice about your answers in parts (a) and (b)? Explain. The slopes are equal in parts (a) and (b). This is true because of what we know about similar triangles, that is, the slope triangle that is drawn between the two points selected in part (a) is similar to the slope triangle that is drawn between the two points in part (b) by the AA criterion. Then because the corresponding sides of similar triangles are equal in ratio, the slopes are equal.
7.
Calculate the slope of the line in the graph below.
Sample student work: π=
π π β ππ π π β ππ
βπ β π βπ β π βπ = βπ π = π =
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8β’4
π π
Your teacher tells you that a line goes through the points οΏ½βπ, οΏ½ and (βπ, π). Calculate the slope of this line.
Sample student work: π π β ππ π π β ππ π βπ = π βπ β (βπ) π β π = βπ π = π π π = Γ·π π π π = Γ π π π = π
π=
Do you think the slope will be the same if the order of the points is reversed? Verify by calculating the slope and explain your result. The slope should be the same because we are joining the same two points. π π β ππ π π β ππ π πβ π = βπ β (βπ) π = π π π = π
π=
Since the slope of a line can be computed using any two points on the same line, it makes sense that it doesnβt matter which point we name as π· and which point we name as πΉ.
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9.
8β’4
Use the graph below to complete parts (a)β(c).
a.
Select any two points on the line to calculate the slope. Sample student work: π π β ππ π π β ππ π β (βπ) = βπβ π π = βπ π =β π
π=
b.
Compute the slope again, this time reversing the order of the coordinates. Sample student work: (ππ β ππ ) (ππ β ππ ) βπ β π = π β (βπ) βπ = π π =β π
π=
c.
What do you notice about the slopes you computed in parts (a) and (b)? The slopes are equal.
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d.
οΏ½ππ βππ οΏ½
Why do you think π = οΏ½
ππ βππ οΏ½
If I multiply the first fraction by
οΏ½π
βπ
8β’4
οΏ½
= οΏ½ππβπποΏ½?
βπ
π
π
, then I get the second fraction:
βπ
(ππ β ππ ) (ππ β ππ ) βπ ΓοΏ½ . οΏ½= (ππ β ππ ) (ππ β ππ ) βπ
I can do the same thing to the second fraction to obtain the first:
(ππ β ππ ) (ππ β ππ ) βπ ΓοΏ½ . οΏ½= (ππ β ππ ) (ππ β ππ ) βπ
Also, since I know that I can find the slope between any two points, it shouldnβt matter which point I pick first. 10. Each of the lines in the lesson was non-vertical. Consider the slope of a non-vertical line, π = π. Select two points on the line to calculate slope. Based on your answer, why do you think the topic of slope focuses only on nonvertical lines?
Students can use any points on the line π = π to determine that the slope is undefined, that is, the computation of slope using the formula leads to a fraction with zero as its denominator. Since the slope of a vertical line is undefined, there is no need to focus them. π
11. Challenge: A certain line has a slope of . Name two points that may be on the line. π
Answers will vary. Accept any answers that have a difference in π-values equal to π and a difference of π-values equal to π. Points (π, π) and (π, π) for example.
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Lesson 17: The Line Joining Two Distinct Points of the Graph π = ππ + π has Slope π
Student Outcomes ο§ ο§
Students show that the slope of a line joining any two distinct points of the graph of π¦ = ππ₯ + π has slope, π. π π
π π
Students transform the standard form of an equation into π¦ = β π₯ + .
Lesson Notes In the previous lesson, we determined that slope can be calculated using any two points on the same line. In this lesson, we show that equations of the form π¦ = ππ₯ and π¦ = ππ₯ + π generate lines with slope π. Students need graph paper to complete some of the Exercises and Problem Set items.
Classwork Exercises 1β3 (8 minutes) Students work independently to complete Exercises 1β3. Exercises 1.
Find at least three solutions to the equation π = ππ, and graph the solutions as points on the coordinate plane. Connect the points to make a line. Find the slope of the line.
The slope of the line is π, π = π. Lesson 17: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
The Line Joining Two Distinct Points of the Graph π¦ = ππ₯ + π has Slope π 11/8/13
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8β’4
Find at least three solutions to the equation π = ππ β π, and graph the solutions as points on the coordinate plane. Connect the points to make a line. Find the slope of the line.
The slope of the line is 3, π = π. 3.
Find at least three solutions to the equation π = ππ + π, and graph the solutions as points on the coordinate plane. Connect the points to make a line. Find the slope of the line.
The slope of the line is 3, π = π.
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Discussion (12 minutes) Recall the goal from Lesson 15: We want to prove that the graph of a linear equation is a line. To do so, we needed some tools, specifically two related to slope. Now that facts are known about slope, we will focus on showing that the line that joins two distinct points is a linear equation with slope π. ο§
We know from our previous work with slope that when the horizontal distance between two points is fixed at one, then the slope of the line is the difference in the π¦-coordinates. We also know that when the horizontal distance is not fixed at one, we can find the slope of the line using any two points because the ratio of corresponding sides of similar triangles will be equal. We can put these two facts together to prove that the graph of the line π¦ = ππ₯ has slope π. Consider the diagram below:
ο§
Examine the diagram and think of how we could prove that
MP.2 π¦
π
π₯
= . 1
Provide students time to work independently, then time for them to discuss in pairs a possible proof of necessary, use the four bullet points below to guide studentsβ thinking. ο§
Yes. Each of the triangles has a common angle at the origin, and each triangle has a right angle. By the AA criterion these triangles are similar. The slope of the line is π. By our definition of slope and the information in the diagram, when the horizontal distance between two points is fixed at one, the slope is π.
Write the ratio of the corresponding sides. Then solve for π¦. οΊ
ο§
1
What is the slope of the line? Explain. οΊ
ο§
π
π₯
= . If
Do we have similar triangles? Explain. οΊ
ο§
π¦
π¦
π
π₯
= , π¦ = ππ₯ 1
Therefore, the slope of the graph of π¦ = ππ₯ is π.
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Point students to their work in Exercise 1 where the graph of π¦ = 2π₯ was a line with a slope of 2. ο§
We know that the graph of π¦ = ππ₯ has slope π, where π is a number. The π¦ in the equation π¦ = ππ₯ is equal to the difference in π¦-coordinates as shown in the diagram below:
ο§
Consider the diagram below. How does this compare to the graph of π¦ = ππ₯ that we just worked on?
οΊ
The graph is the same except for the fact that the line has been translated π units up the π¦-axis.
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ο§
8β’4
We want to write the ratio of corresponding sides of the slope triangles. We know the lengths of two sides of the smaller slope triangle; they are 1 and π. We also know one of the lengths of the larger slope triangle, π₯. What we need to do now is express the length of the larger slope triangle that corresponds to side π. When our line passed through the origin, the length was simply π¦, but that is not the case here. How can we express the length we need, noted in blue in the diagram below?
Provide students time to think and share in small groups. If necessary, show students the diagram below and ask them what the length of the dashed orange segment is. They should recognize that it is the length of π. Then ask them how they can use that information to determine the length of the blue segment.
οΊ
The length of the blue segment is π¦ β π.
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ο§
8β’4
Now that we know the length of the blue segment, we can write ratios that represent the corresponding sides of the triangles: π 1 = π¦βπ π₯ Now we can solve for π¦:
ππ₯ = π¦ β π
ππ₯ + π = π¦ β π + π
ππ₯ + π = π¦
Therefore, the slope of an equation of the form π¦ = ππ₯ + π is π. Point students to their work in Exercises 2 and 3 where the graph of π¦ = 3π₯ β 1 and π¦ = 3π₯ + 1 was a line with slope 3. ο§
We can show this algebraically using two distinct points on the graph of the line π¦ = 3π₯ β 1. Points (1, 2) and (2, 5) were on the graph of π¦ = 3π₯ β 1. Using the slope formula, we see that 5β2 π= 2β1 3 = 1 =3
The next four bullet points generalize the slope between any two points of the graph of π¦ = 3π₯ β 1 is 3. This is an optional part of the discussion of slope between any two points. The last bullet point of this section should be shared with students as it orients them to our progress towards our ultimate goal: showing that the graph of a linear equation is a line. ο§
In general, let π = (π1 , π2 ) and π
= (π1 , π2 ) be any two distinct points of the graph of π¦ = 3π₯ β 1. Recall what is known about points on a graph; it means that they are solutions to the equation. Since π is on the graph of π¦ = 3π₯ β 1, then π2 = 3π1 β 1
Similarly, since π
is on the graph of π¦ = 3π₯ β 1, then
π2 = 3π1 β 1
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The Line Joining Two Distinct Points of the Graph π¦ = ππ₯ + π has Slope π 11/8/13
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ο§
By the slope formula, π = equality to simplify: π= = = = =
π2βπ2 , we can substitute the above values of π2 and π2 and use our properties of π1βπ1
π2 β π2 π1 β π1
(3π1 β 1) β (3π1 β 1) π1 β π1 3π1 β 1 β 3π1 + 1 π1 β π1 3π1 β 3π1 π1 β π1
3(π1 β π1 ) π1 β π1
=3 ο§ ο§
8β’4
By substitution
By taking the opposite of the terms in the second grouping By simplifying (β1 + 1) By the Distributive Property "collecting like terms"
By division, the number π1 β π1 divided by itself
Thus we have shown that the line passing through points π and π
has a slope of 3.
To truly generalize our work, we need only replace 3 in the calculations with π and the β1 with π.
Make clear to students that we are still working toward our goal of proving that the graph of a linear equation is a line using the summary of work below. ο§
Our goal, ultimately, is to prove that the graph of a linear equation is a line. In Lesson 15, we said we have to develop the following tools: 1.
We must define a number for each non-vertical line that can be used to measure the βsteepnessβ or βslantβ of the line. Once defined, this number will be called the slope of the line and is often referred to as the βrate of changeβ.
2.
We must show that any two points on a non-vertical line can be used to find the slope of the line.
3.
We must show that the line joining two points on a graph of a linear equation of the form π¦ = ππ₯ + π has the slope π.
4.
We must show that there is only one line passing through a given point with a given slope.
At this point in our work, we just finished #3. Number 4, is the topic of the next lesson.
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Discussion (5 minutes) ο§
When an equation is in the form π¦ = ππ₯ + π, the slope π is easily identifiable compared to an equation in the form ππ₯ + ππ¦ = π. Note that π in each of the equations is unique. That is, the number represented by π in the equation π¦ = ππ₯ + π is not necessarily the same as the number π in ππ₯ + ππ¦ = π. For example, we will solve the equation 8π₯ + 2π¦ = 6 for π¦. Our goal is to have π¦ equal to an expression: 8π₯ + 2π¦ = 6
First we use our properties of equality to remove 8π₯ from the left side of the equation 8π₯ β 8π₯ + 2π¦ = 6 β 8π₯ 2π¦ = 6 β 8π₯
Now we divide both sides by 2
2π¦ 6 β 8π₯ = 2 2 6 8π₯ π¦= β 2 2 π¦ = 3 β 4π₯ π¦ = β4π₯ + 3
The slope of the graph of this equation is β4.
By convention (an agreed upon way of doing things) we place the term ππ₯ before the term π. This is a version of the standard form of a linear equation that is referred to as the βslope-interceptβ form. It is called βslopeinterceptβ form because it makes clear the number that describes slope, i.e., π, and the π¦-intercept, which is something that will be discussed later. Also, notice the value of π is different in both forms of the equation.
Exercises 4β11 (11 minutes) Students work independently or in pairs to identify the slope from an equation and to transform the standard form of an equation into slope-intercept form. 4.
The graph of the equation π = ππ β π has what slope? The slope is π.
5.
π π
The slope is β . 6.
π π
The graph of the equation π = β π β π has what slope?
You have $ππ in savings at the bank. Each week, you add $π to your savings. Let π represent the total amount of money you have saved at the end of π weeks. Write an equation to represent this situation and identify the slope of the equation. What does that number represent? π = ππ + ππ
The slope is π. It represents how much money is saved each week.
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7.
8β’4
A friend is training for a marathon. She can run π miles in ππ minutes. Assume she runs at a constant rate. Write an equation to represent the total distance, π, your friend can run in π minutes. Identify the slope of the equation. What does that number represent?
π π = π ππ π π= π ππ π π= π π
π
The slope is . It represents the rate at which my friend can run, one mile in seven minutes. π
8.
Four boxes of pencils cost $π. Write an equation that represents the total cost, π, for π boxes of pencils. What is the slope of the equation? What does that number represent? π=
π π π
π
The slope is . It represents the cost of one box of pencils, $π. ππ. 9.
π
Solve the following equation for π: ππ β ππ = ππ, then identify the slope of the line. ππ β ππ = ππ
ππ β ππ β ππ = ππ β ππ
βππ = ππ β ππ
βπ ππ β ππ π= βπ βπ ππ ππ π= β βπ βπ π = βπ + ππ π = ππ β π
The slope of the line is π. 10. Solve the following equation for π: ππ + ππ = π, then identify the slope of the line. ππ + ππ = π
ππ β ππ + ππ = π β ππ
ππ = π β ππ
π
The slope of the line is β .
π π β ππ π= π π π π π= β π π π π π π=β π+ π π
π
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The Line Joining Two Distinct Points of the Graph π¦ = ππ₯ + π has Slope π 11/8/13
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11. Solve the following equation for π: ππ + ππ = π, then identify the slope of the line. ππ + ππ = π
ππ β ππ + ππ = π β ππ
ππ = π β ππ
π π β ππ π= π π π ππ π= β π π π π π= β π π π π π π= β π+ π π π
The slope of the line is β . π
Closing (4 minutes) Summarize, or ask students to summarize, the main points from the lesson: ο§ ο§ ο§
We know that the line that joins any two distinct points of the graph π¦ = ππ₯ + π has slope π. We know how to identify the slope for any equation in the form π¦ = ππ₯ + π.
We know how to transform the standard form of a linear equation into another form to more easily identify the slope.
Lesson Summary The line joining two distinct points of the graph of the linear equation π = ππ + π has slope π.
The π of π = ππ + π is the number that describes the slope. For example, in the equation π = βππ + π, the slope of the graph of the line is βπ.
Exit Ticket (5 minutes)
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The Line Joining Two Distinct Points of the Graph π¦ = ππ₯ + π has Slope π 11/8/13
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Name
8β’4
Date
Lesson 17: The Line Joining Two Distinct Points of the Graph π = ππ + π has Slope π
Exit Ticket 1.
Solve the following equation for π¦: 35π₯ β 7π¦ = 49.
2.
What is the slope of the equation in problem 1?
3.
Show, using similar triangles, why the graph of an equation of the form π¦ = ππ₯ is a line with slope π.
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The Line Joining Two Distinct Points of the Graph π¦ = ππ₯ + π has Slope π 11/8/13
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8β’4
Exit Ticket Sample Solutions 1.
Solve the following equation for π: πππ β ππ = ππ. πππ β ππ = ππ
πππ β πππ β ππ = ππ β πππ
βππ = ππ β πππ
βπ ππ ππ π= β π βπ βπ βπ π = βπ β (βππ) π = ππ β π
2.
What is the slope of the equation in problem 1? The slope of π = ππ β π is π.
3.
Show, using similar triangles, why the graph of an equation of the form π = ππ is a line with slope π. Solutions will vary. Sample solution is shown below.
The line shown has slope π. When we compare the corresponding side lengths of the similar triangles we have the ratios
π π
=
π π
= π. In general, the ratios would be
Lesson 17: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
π
π
=
π
π
, equivalently π = ππ, which is a line with slope π.
The Line Joining Two Distinct Points of the Graph π¦ = ππ₯ + π has Slope π 11/8/13
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Problem Set Sample Solutions Students practice transforming equations from standard form into slope-intercept form and showing that the line joining two distinct points of the graph π¦ = ππ₯ + π has slope π. Students graph the equation and informally note the π¦intercept. 1.
Solve the following equation for π: βππ + ππ = ππ. βππ + ππ = ππ
βππ + ππ + ππ = ππ + ππ
ππ = ππ + ππ
π ππ π π= + π π π π π π=π+ π π π π = π+π π
a.
Based on your transformed equation, what is the slope of the linear equation βππ + ππ = ππ? The slope is
b.
π π
.
Complete the table to find solutions to the linear equation.
π βπ π π π
Lesson 17: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
Transformed linear equation: π π= π+π π π π = (βπ) + π π = βπ + π =π π π = (π) + π π =π+π =π π π = (π) + π π =π+π =π π π = (π) + π π =π+π =π
π π π π π
The Line Joining Two Distinct Points of the Graph π¦ = ππ₯ + π has Slope π 11/8/13
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c.
Graph the points on the coordinate plane.
d.
Find the slope between any two points.
8β’4
Using points (π, π) and (π, π): πβπ πβπ π = π
π=
e.
f.
The slope you found in part (d) should be equal to the slope you noted in part (a). If so, connect the points to make the line that is the graph of an equation of the form π = ππ + π that has slope π.
Note the location (ordered pair) that describes where the line intersects the π-axis. (π, π) is the location where the line intersects the π-axis.
2.
Solve the following equation for π: ππ + ππ = ππ. ππ + ππ = ππ
ππ β ππ + ππ = ππ β ππ
ππ = ππ β ππ
π ππ π π= β π π π π π = π β ππ
π = βππ + π
a.
Based on your transformed equation, what is the slope of the linear equation ππ + ππ = ππ? The slope is βπ.
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The Line Joining Two Distinct Points of the Graph π¦ = ππ₯ + π has Slope π 11/8/13
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b.
8β’4
Complete the table to find solutions to the linear equation. Transformed linear equation: π = βππ + π π = βπ(βπ) + π = π+π = ππ π = βπ(π) + π =π+π =π π = βπ(π) + π = βπ + π =π π = βπ(π) + π = βπ + π =π
π βπ π π π c.
Graph the points on the coordinate plane.
d.
Find the slope between any two points.
π ππ π π π
Using points (π, π) and (π, π): πβπ πβπ π = βπ = βπ
π=
e.
f.
The slope you found in part (d) should be equal to the slope you noted in part (a). If so, connect the points to make the line that is the graph of an equation of the form π = ππ + π that has slope π. Note the location (ordered pair) that describes where the line intersects the π-axis. (π, π) is the location where the line intersects the π-axis.
Lesson 17: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
The Line Joining Two Distinct Points of the Graph π¦ = ππ₯ + π has Slope π 11/8/13
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3.
8β’4
Solve the following equation for π: ππ + ππ = βπ.
ππ + ππ = βπ ππ β ππ + ππ = βπ β ππ ππ = βπ β ππ π βπ π π= β π π π π π π = βπ β π π π π =β πβπ π a.
Based on your transformed equation, what is the slope of the linear equation ππ + ππ = βπ? The slope is β
b.
π π
.
Complete the table to find solutions to the linear equation.
π βπ βπ π π c.
Transformed linear equation: π π= β πβπ π π π = β (βπ) β π π =πβπ =π π π = β (βπ) β π π =πβπ =π π π = β (π) β π π =πβπ = βπ π π = β (π) β π π = βπ β π = βπ
π π π βπ βπ
Graph the points on the coordinate plane.
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d.
8β’4
Find the slope between any two points. Using points (βπ, π) and (π, βπ): π β (βπ) βπ β π π = βπ π =β π
π=
e.
f.
The slope you found in part (d) should be equal to the slope you noted in part (a). If so, connect the points to make the line that is the graph of an equation of the form π = ππ + π that has slope π. Note the location (ordered pair) that describes where the line intersects the π-axis. (π, βπ) is the location where the line intersects the π-axis.
4.
Solve the following equation for π: ππ β π = π.
ππ β π = π ππ β ππ β π = π β ππ βπ = π β ππ π = βπ + ππ π = ππ β π a.
Based on your transformed equation, what is the slope of the linear equation ππ β π = π? The slope is π.
b.
Complete the table to find solutions to the linear equation.
π βπ π π π
Lesson 17: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
Transformed linear equation: π = ππ β π π = π(βπ) β π = βπ β π = βπ π = π(π) β π =πβπ = βπ π = π(π) β π =πβπ =π π = π(π) β π = ππ β π =π
π βπ βπ π π
The Line Joining Two Distinct Points of the Graph π¦ = ππ₯ + π has Slope π 11/8/13
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c.
Graph the points on the coordinate plane.
d.
Find the slope between any two points.
8β’4
Using points (π, βπ) and (π, π): βπ β π πβπ βπ = βπ =π
π=
e.
f.
The slope you found in part (d) should be equal to the slope you noted in part (a). If so, connect the points to make the line that is the graph of an equation of the form π = ππ + π that has slope π.
Note the location (ordered pair) that describes where the line intersects the π-axis. (π, βπ) is the location where the line intersects the π-axis.
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The Line Joining Two Distinct Points of the Graph π¦ = ππ₯ + π has Slope π 11/8/13
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Lesson 18: There is Only One Line Passing Through a Given Point with a Given Slope Student Outcomes ο§ ο§
Students graph equations in the form of π¦ = ππ₯ + π using information about slope and π¦-intercept.
Students know that if they have two straight lines with the same slope and a common point that the lines are the same.
Lesson Notes The Opening Exercise requires students to examine part (f) from the problem set of Lesson 17. Each part (f) requires students to identify the point where the graph of the line intersects the π¦-axis. Knowing that this point represents the π¦intercept and that it is the point (0, π) in the equation π¦ = ππ₯ + π is integral for the content in the lesson. To keep consistent, throughout the lesson students form the slope triangle with points π, π, and π
. Since slope has been defined in terms of lengths of sides of similar triangles, there is notation and verbiage about numbers in terms of distances. As you work through the examples, make clear that the lengths of the slope triangle are positive distances, but that does not necessarily mean that the slope must be positive (see Example 2). Remind students that graphs of lines with positive slopes are left-to-right inclining and graphs of lines with negative slopes are left-to-right declining.
Coordinate planes are provided for students in the exercises of this lesson, but they will need graph paper to complete the Problem Set.
Classwork Opening Exercise (5 minutes) Examine each of the graphs and their equations below. Identify the coordinates of the point where the line intersects the π-axis. Describe the relationship between the point and the equation π = ππ + π. a.
π
π= π+π π
b.
π = βππ + π
MP.8
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c.
π
d.
π=β πβπ π
8β’4
π = ππ β π
MP.8
Points noted in graphs above where the line intersects the π-axis: π π
a.
π = π + π,
b.
π = βππ + π,
π π = β π β π, π
c. d.
π = ππ β π,
(π, π)
(π, π)
(π, βπ)
(π, βπ)
In each equation, the number π was the π-coordinate of the point where the line intersected the π-axis.
Discussion (5 minutes) ο§
ο§ MP.7
In the last lesson, we transformed the standard form of a linear equation ππ₯ + ππ¦ = π, into what was referred to as the slope-intercept form π¦ = ππ₯ + π. We know that the slope is represented by π, but we did not discuss the meaning of π. In the Opening Exercise you were asked to note the location (ordered pair) that describes where the line intersected the π¦-axis. What do you notice about the value of π in relation to the point where the graph of the equation intersected the π¦-axis? οΊ
ο§
ο§ MP.3 ο§ ο§
The value of π was the same number as the π¦-coordinate of each location.
When a linear equation is in the form π¦ = ππ₯ + π, it is known as the slope-intercept form because this form provides information about the slope, π, and π¦-intercept, (0, π) of the graph. The π¦-intercept is defined as the location on the graph where a line intersects the π¦-axis.
In this lesson, we develop the last tool that we need in order to prove that the graph of a linear equation in two variables ππ₯ + ππ¦ = π, where π, π, and π are constants is a straight line. We will show that if two straight lines have the same slope and pass through the same point, then they are the same line.
Since an equation of the form π¦ = ππ₯ + π provides information about both the π¦-intercept and slope, we will use this equation to graph lines. Recall that we began discussing slope graphically, π = Lesson 18: Date:
Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
οΏ½ππ
οΏ½
:
οΏ½πποΏ½
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Example 1 (6 minutes) Graph equation in the form of π¦ = ππ₯ + π. Example 1
π π
Graph the equation π = π + π. Name the slope and π-intercept. The slope, π =
ο§
π and the π-intercept is (π, π). π
To graph the equation, we must begin with the known point. In this case the π¦-intercept. We cannot begin with the slope because the slope describes the rate of change between two points. That means we need a point to begin with. On a graph, we plot the point (0, 1).
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Next, we use the slope to find the second point. The slope tells us exactly how many units to go to the right to find point π, and then how many vertical units we need to go from π to find point π
. How many units will we go to the right in order to find point π? How do you know? οΊ
ο§
How many vertical units from point π must we go to find point π
? How do you know? οΊ
ο§
We need to go 3 units to the right of point π to find π. We go 3 units because |ππ| = 3.
We need to go 2 units from point π to find π
. We go 2 units because |ππ
| = 2.
Will we go up from point π or down from point π to find π
? How do you know? οΊ
We need to go up because the slope is positive. That means that the line will be left-to-right inclining.
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ο§
8β’4
Since we know that the line joining two distinct points of the form π¦ = ππ₯ + π has slope π, and we specifically constructed points π and π
with the slope in mind, we can join the points with a line.
Example 2 (4 minutes) Graph equation in the form of π¦ = ππ₯ + π. Example 2 π π
Graph the equation π = β π β π. Name the slope and π-intercept. π π
The slope, π = β and the π-intercept is (π, βπ).
ο§
How do we begin? οΊ
We must begin by putting a known point on the graph, (0, β2).
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ο§
How many units will we go to the right in order to find point π? How do you know? οΊ
ο§
We need to go 4 units to the right of point π to find π. We go 4 units because |ππ| = 4.
How many units from point π must we go to find point π
? How do you know? οΊ
ο§
8β’4
We need to go 3 units from point π to find π
. We go 3 units because |ππ
| = 3.
Will we go up from point π or down from point π to find π
? How do you know? οΊ
We need to go down from point π to point π
because the slope is negative. That means that the line will be left-to-right declining.
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ο§
8β’4
Now we draw the line through the points π and π
.
Example 3 (4 minutes) Graph equation in the form of π¦ = ππ₯ + π. ο§
Graph the equation π¦ = 4π₯ β 7. Name the slope and π¦-intercept. οΊ
ο§
The slope, π = 4 and the π¦-intercept is (0, β7).
How do we begin? οΊ
We must begin by putting a known point on the graph, (0, β7).
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ο§
Notice this time that the slope is the integer 4. In the last two examples, our slopes have been in the form of a fraction so that we can use the information in the numerator and denominator to determine the lengths of |ππ| and |ππ
|. If π = οΊ
ο§
οΏ½πποΏ½
= 4, then what fraction can we use to represent slope to help us graph? 4
The number 4 is equivalent to the fraction . 1
We need to go 1 unit to the right of point π to find π. We go 1 unit because |ππ| = 1.
How many vertical units from point π must we go to find point π
? How do you know? οΊ
ο§
οΏ½ππ
οΏ½
How many units will we go to the right in order to find point π? How do you know? οΊ
ο§
8β’4
We need to go 4 units from point π to find π
. We go 4 units because |ππ
| = 4.
Will we go up from point π or down from point π to find π
? How do you know? οΊ
We need to go up from point π to point π
because the slope is positive. That means that the line will be left-to-right inclining.
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ο§
8β’4
Now we join the points π and π
to make the line.
Exercises 1β4 (9 minutes) Students complete Exercises 1β4 individually or in pairs. Exercises 1β4 1.
π π
Graph the equation π = π β π. a.
Name the slope and the π-intercept. The slope is π =
b.
π and the π-intercept is (π, βπ). π
Graph the known point, then use the slope to find a second point before drawing the line.
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2.
8β’4
Graph the equation π = βππ + π. a.
Name the slope and the π-intercept.
The slope is π = βπ and the π-intercept is (π, π).
b.
3.
Graph the known point, then use the slope to find a second point before drawing the line.
The equation π = ππ + π can be simplified to π = π. Graph the equation π = π. a.
Name the slope and the π-intercept.
The slope is π = π and the π-intercept is (π, π). b.
Graph the known point, then use the slope to find a second point before drawing the line.
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4.
8β’4
Graph the point (π, π).
π π
a.
Find another point on the graph using the slope, π = .
b.
Connect the points to make the line.
c.
Draw a different line that goes through the point (π, π) with slope π = . What do you notice?
π π
Only one line can be drawn through the given point with the given slope.
Discussion (8 minutes) The following proof is optional. An exercise, below the discussion, can be used as an alternative. ο§
Now we must show that if two straight lines have the same slope and pass through the same point, then they are the same line, which is what you observed in Exercise 4.
ο§
We let π and πβ² be two lines with the same slope π passing through the same point π. Since π is a number, π could be positive, negative, or equal to zero. For this proof, we will let π > 0. Since π > 0, both lines π and πβ² are left-to-right inclining.
ο§
Since we are trying to show that π and πβ² are the same line, letβs assume that they are not and prove that our assumption is false.
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ο§
Using our first understanding of slope, we will pick a point π one unit to the right of point π. Then draw a line parallel to the π¦-axis going through point π, as shown.
ο§
Now we label the point of intersection of the line we just drew and line π and πβ² as π
and π
β², respectively.
ο§
By definition of slope, the length |ππ
| is the slope of line π and the length of |ππ
β² | is the slope of line πβ². What do we know about the slopes of lines π and π β² ? οΊ
ο§
οΊ
ο§
The slopes are the same.
Then what do we know about the lengths |ππ
| and |ππ
β² |? They must be equal.
For that reason points π
and π
β² must coincide. This means lines π and πβ² are the same line, not as we originally drew them. Therefore, there is just one line passing through a given point with a given slope.
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8β’4
Exercises 5β6 (8 minutes) Students complete Exercises 5β6 individually or in pairs. Exercises 5β6 5.
A bank put $ππ into a savings account when you opened the account. Eight weeks later you have a total of $ππ. Assume you saved the same amount every week. a.
b.
If π is the total amount of money in the savings account and π represents the number of weeks, write an equation in the form π = ππ + π that describes the situation.
ππ = π(π) + ππ ππ = ππ ππ =π π π =π π π π = π + ππ π
Identify the slope and the π-intercept. What do these numbers represent? π
The slope is and the π-intercept is (π, ππ). The π-intercept represents the amount of money the bank gave π
me, in the amount of $ππ. The slope represents the amount of money I save each week,
π π
= $π. ππ.
c.
Graph the equation on a coordinate plane.
d.
Could any other line represent this situation? For example, could a line through point (π, ππ) with slope represent the amount of money you save each week? Explain.
No, a line through point (π, ππ) with slope
π π
π π
cannot represent this situation. That line would show that at
the end of the π weeks I would have $ππ. ππ, but I was told that I would have $ππ by the end of the π weeks. Lesson 18: Date:
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6.
8β’4
A group of friends are on a road trip. So far they have driven πππ miles. They continue their trip and drive at a constant rate of ππ miles per hour. a.
Let π represent the total distance traveled in π hours. Write an equation to represent the total number of miles driven in π hours. π = πππ + πππ
b.
Identify the slope and the π-intercept. What do these numbers represent?
The slope is ππ and represents the rate of driving. The π-intercept is πππ and represents the number of miles they had already driven before driving at the given constant rate.
c.
Graph the equation on a coordinate plane.
d.
Could any other line represent this situation? For example, could a line through point (π, πππ) with slope 75 represent the total distance the friends drive? Explain. No, a line through point (π, πππ) with a slope of ππ could not represent this situation. That line would show that after an hour the friends traveled a total distance of πππ miles. According to the information given, the friends would only have traveled πππ miles after one hour.
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8β’4
Closing (4 minutes) Summarize, or ask students to summarize, the main points from the lesson: ο§ ο§ ο§
We know that in the equation π¦ = ππ₯ + π, (0, π) is the location where the graph of the line intersects the π¦axis. We know how to graph a line using a point, namely the π¦-intercept, and the slope.
We know that there is only one line with a given slope passing through a given point.
Lesson Summary The equation π = ππ + π is in slope-intercept form. The number π represents the slope of the graph and the point (π, π) is the location where the graph of the line intersects the π-axis.
To graph a line from the slope-intercept form of a linear equation, begin with the known point, (π, π), then use the slope to find a second point. Connect the points to graph the equation. There is only one line passing through a given point with a given slope.
Exit Ticket (4 minutes)
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Name
8β’4
Date
Lesson 18: There is Only One Line Passing Through a Given Point with a Given Slope Exit Ticket Mrs. Hodson said that the graphs of the equations below are incorrect. Find the studentβs errors and correctly graph the equations. 1.
2.
1 2
Student graph of π¦ = π₯ + 4.
3 5
Student graph of π¦ = β π₯ β 1.
Lesson 18: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
Error: Correct graph of equation:
Error: Correct graph of equation:
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Exit Ticket Sample Solutions Mrs. Hodson said that the graphs of the equations below are incorrect. Find the studentβs errors and correctly graph the equations. 1.
2.
π π
Student graph of the equation π = π + π.
π π
Student graph of the equation π = β π β π.
Lesson 18: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
Error: The student should have gone up π unit when finding |πΈπΉ| since the slope is positive. Correct graph of equation:
Error: The student did not find the y intercept correctly. It should be the point (π, βπ).
Correct graph of equation:
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Problem Set Sample Solutions Students practice graphing equations using π¦-intercept and slope. Students need graph paper to complete the problem set. Optional problem 11 has students show that there is only one line passing through a point with a given negative slope.
1.
π π
Graph the equation π = π β π. a.
Name the slope and the π-intercept. The slope is π =
b.
2.
π and the π-intercept is (π, βπ). π
Graph the known point, and then use the slope to find a second point before drawing the line.
Graph the equation π = π + π. a.
Name the slope and the π-intercept.
The slope is π = π and the π-intercept is (π, π). b.
Graph the known point, and then use the slope to find a second point before drawing the line.
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3.
8β’4
π π
Graph the equation π = β π + π. a.
Name the slope and the π-intercept. π π
The slope is π = β and the π-intercept is (π, π).
b.
4.
Graph the known point, and then use the slope to find a second point before drawing the line.
π π
Graph the equation π = π. a.
Name the slope and the π-intercept. The slope is π =
b.
π and the π-intercept is (π, π). π
Graph the known point, and then use the slope to find a second point before drawing the line.
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5.
8β’4
Graph the equation π = ππ β π. a.
Name the slope and the π-intercept.
The slope is π = π and the π-intercept is (π, βπ). b.
6.
Graph the known point, and then use the slope to find a second point before drawing the line.
Graph the equation π = βππ + π. a.
Name the slope and the π-intercept.
The slope is π = βπ and the π-intercept is (π, π).
b.
Graph the known point, and then use the slope to find a second point before drawing the line.
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7.
π π
Graph the equation π = π + π. a.
Name the slope and the π-intercept. The slope is π =
b.
8.
8β’4
π and the π-intercept is (π, π). π
Graph the known point, and then use the slope to find a second point before drawing the line.
Graph the equation ππ + ππ = π. (Hint: transform the equation so that it is of the form π = ππ + π.) a.
Name the slope and the π-intercept.
π π
ππ + ππ = π ππ β ππ + ππ = π β ππ ππ = π β ππ π π π π= β π π π π π π=πβ π π π π=β π+π π
The slope is π = β and the π-intercept is (π, π). b.
Graph the known point, and then use the slope to find a second point before drawing the line.
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9.
8β’4
Graph the equation βππ + ππ = ππ. a.
Name the slope and the π-intercept.
βππ + ππ = ππ ππ + ππ + ππ = ππ + ππ ππ = ππ + ππ π ππ π π= + π π π π π π=π+ π π π π = π+π π The slope is π = b.
π and the π-intercept is (π, π). π
Graph the known point, and then use the slope to find a second point before drawing the line.
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10. Let π and πβ² be two lines with the same slope π passing through the same point π·. Show that there is only one line with a slope π, where π < π, passing through the given point π·. Draw a diagram if needed.
First assume that there are two different lines π and πβ² with the same negative slope passing through π·. From point π·, I mark a point πΈ one unit to the right. Then I draw a line parallel to the π-axis through point πΈ. The intersection of this line and line π and πβ² are noted with points πΉ and πΉβ² . By definition of slope, the lengths |πΈπΉ| and |πΈπΉβ² | represent the slopes of lines π and πβ² , respectively. We are given that the lines have the same slope, which means that lengths |πΈπΉ| and |πΈπΉβ² | are equal. Since that is true, then points πΉ and πΉβ² coincide and so do lines π and πβ² . Therefore, our assumption that they are different lines is false; π and πβ² must be the same line. Therefore, there is only one line with slope π passing through the given point π·.
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Lesson 19: The Graph of a Linear Equation in Two Variables is a Line Student Outcomes ο§ ο§
Students prove that any point on the graph of π¦ = ππ₯ + π is on a line π and that any point on a line π is a point on the graph of π¦ = ππ₯ + π. Students graph linear equations on the coordinate plane.
Lesson Notes Included in this and the next several lessons are an analytical approach to understanding linear equations and their graphs. Though the analytical understandings are not required for Grade 8 students, it is of tremendous value for students to understand the proofs of common assertions about lines. For that reason, these lessons are structured to offer extensions (i.e., the proofs) in cases where the students are ready, and also to provide alternate activities that deepen conceptual understanding. The theorems are presented to students and, based on the readiness of each class, the teacher can choose to discuss the proof of the theorem or have students explore the theorem experimentally by completing the exercises of the alternate activity. This lesson is the result of many lessons leading up to this one. In previous lessons, it was stated that there were four things that needed to be proved about lines before we could state, definitively, that the graph of a linear equation in two variables is in fact a line. 1. We must define a number for each non-vertical line that can be used to measure the βsteepnessβ or βslantβ of the line. Once defined, this number will be called the slope of the line and is often referred to as the βrate of changeβ. 2. We must show that any two points on a non-vertical line can be used to find the slope of the line. 3. We must show that the line joining two points on the graph of a linear equation of the form π¦ = ππ₯ + π has slope π.
4. We must show that there is only one line passing through a given point with a given slope.
This lesson is the conclusion of that work, and it is an important result for students to observe. There are two options for the first part of the lesson. You may choose to use the formal proof in the first discussion beginning on page 281 and Exercise 1 beginning on page 283. This lesson path skips Exercises 2β5 and resumes the lesson with Example 1 on page 291. The other option is to skip the Discussion and Exercise 1 on the first five pages, having students examine several equations in the form of ππ₯ + ππ¦ = π, create their graphs, and informally conclude that the graphs of equations of this type always result in lines. The alternate activity begins on page 285 of this lesson with Exercises 2β8. Following these exercises is Example 1 on page 291 and Exercises 9β11 where students learn how to graph a linear equation using intercepts. Students need graph paper to complete the Exercises and the Problem Set.
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Classwork Discussion (15 minutes) Now that all of the necessary tools have been developed, we are ready to show that the graph of a linear equation in two variables in the form π¦ = ππ₯ + π, where π is always a constant, is a non-vertical line. ο§
To prove that the graph of a linear equation is a line, we need to show the following:
(1) any point on the graph of π¦ = ππ₯ + π is a point on a line π, and
ο§
(2) any point on the line π is a point on the graph of π¦ = ππ₯ + π.
Why do we need to show both (1) and (2)? Consider the following: Let π and π be two distinct points on the graph of π¦ = ππ₯ + π. Let π be the line passing through points π and π. Also, let the graph of π¦ = ππ₯ + π be the collection of red segments denoted by πΊ, as shown.
MP.2
ο§
ο§
If πΊ is the collection of red segments, as shown, then (1) is true, but (2) is not. Therefore, if we can show that both (1) and (2) are true, then we know that the graph of π¦ = ππ₯ + π is exactly the same as the graph of line π.
Letβs look at a specific case before proving (1) and (2) in general. Show that the graph of π¦ = 2π₯ + 3 is a nonvertical line.
We need two points that we know are on the graph of π¦ = 2π₯ + 3. Are points (0, 3) and (4, 11) solutions to the equation π¦ = 2π₯ + 3? Explain. οΊ
ο§
Yes, because 3 = 2(0) + 3 and 11 = 2(4) + 3 are true statements.
Let line π pass through the points π = (0, 3) and π = (1, 5). By our work in Lesson 17, we know that an equation in the form of π¦ = ππ₯ + π has slope π. Then, the slope of π¦ = 2π₯ + 3 is 2 and therefore the slope of line π is 2. We claim that the graph of π¦ = 2π₯ + 3 is the line π.
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ο§
Now we will show that any point on the graph of π¦ = 2π₯ + 3 is on line π. Let π
be any point on the graph of π¦ = 2π₯ + 3 (other than points π and π because if π
= π or if π
= π there would be nothing to prove; we already showed that they are solutions to the equation π¦ = 2π₯ + 3). Let πβ² be the line passing through points π and π
, as shown.
ο§
We need to show that line πβ² has a slope of 2. We have done this before in Lesson 16. Since π and π
are on the graph of π¦ = 2π₯ + 3, then the coordinates of π = (π1 , 2π1 + 3) and π
= (π1 , 2π1 + 3). We also know the following: (2π1 + 3) β (2π1 + 3) slope of π β² = π1 β π1
MP.2
=
=
2π1 β 2π1 π1 β π1
2(π1 β π1 ) π1 β π1
=2
ο§
Now that we know the slope of πβ² is 2, what can we say about lines π and πβ²? Hint: What do we know about lines that have the same slope and pass through the same point? οΊ
Lines π and πβ² must be the same line. There can be only one line with a slope of 2 that passes through a given point, in this case π. Lesson 19: Date:
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8β’4
ο§
Therefore, point π
must be on line π and the proof that any point on the graph of π¦ = 2π₯ + 3 is on line π is finished.
ο§
Now we must show that any point on π is on the graph of π¦ = 2π₯ + 3. Let π be any point on π (except for π and π for the same reason as before). Using π = (π 1 , π 2 ) and π = (0, 3) in the slope formula, we get the following: π 2 β 3 slope of π = π 1 β 0 We know the slope of π is 2.
2=
π 2 β 3 π 1 β 0
2(π 1 β 0) = π 2 β 3 2π 1 = π 2 β 3
2π 1 + 3 = π 2 β 3 + 3 2π 1 + 3 = π 2
The equation above shows that point π = (π 1 , π 2 ) is a solution to the equation π¦ = 2π₯ + 3 and must be on the graph of π¦ = 2π₯ + 3.
ο§
We have just shown that any point π
on the graph of π¦ = 2π₯ + 3 is on line π and any point π on line π is on the graph of π¦ = 2π₯ + 3. Therefore, the proof is complete.
Exercise 1 (10 minutes) Students work in pairs or small groups to prove the theorem in general. This is an optional exercise and related directly to the optional discussion above. If you chose the alternative path for this lesson, then this exercise should not be assigned. Exercise 1
MP.2
Theorem: The graph of a linear equation π = ππ + π is a non-vertical line with slope π and passing through (π, π), where π is a constant. 1.
Prove the theorem by completing parts (a)β(c). Given two distinct points, π· and πΈ, on the graph of π = ππ + π and let π be the line passing through π· and πΈ. You must show:
(1) (2)
any point on the graph of π = ππ + π is on line π, and
any point on the line π is on the graph of π = ππ + π.
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a.
8β’4
Proof of (1): Let πΉ be any point on the graph of π = ππ + π. Show that πΉ is on π. Begin by assuming it is not. Assume the graph looks like the diagram below where πΉ is on πβ².
What is the slope of line π?
Since the points π· and πΈ are on the graph of π = ππ + π, then we know that the slope of the line passing through those points must have slope π. Therefore, line π has slope π. What is the slope of line πβ² ?
We know that point πΉ is on the graph of π = ππ + π. Then the coordinates of point πΉ = (ππ , πππ + π) because πΉ is a solution to π = ππ + π and ππ = πππ + π. Similarly, the coordinates of π· = (ππ , πππ + π). Then, (πππ + π) β (πππ + π) π π β ππ πππ + π β πππ β π = π π β ππ πππ β πππ = π π β ππ π(ππ β ππ ) = π π β ππ =π
πππππ ππ πβ² =
What can you conclude about lines π and πβ²? Explain.
Lines π and πβ²are the same line. Both lines go through point π· and have slope π. There can be only one line with a given slope going through a given point; therefore, line π = πβ².
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b.
8β’4
Proof of (2): Let πΊ be any point on line π, as shown.
Show that πΊ is a solution to π = ππ + π. Hint: Use the point (π, π). Point πΊ is on line π. Let πΊ = (ππ , ππ ). Then, We know the slope of π is π; therefore,
πππππ ππ π = π=
ππ β π ππ β π
ππ β π ππ β π
π(ππ β π) = ππ β π πππ = ππ β π
πππ + π = ππ β π + π πππ + π = ππ which shows πΊ is a solution to π = ππ + π. c.
ππ = πππ + π
Now that you have shown that any point on the graph of π = ππ + π is on line π (part a), and any point on line π is on the graph of π = ππ + π (part b), what can you conclude about the graphs of linear equations? The graph of a linear equation is a line.
Exercises 2β8 (25 minutes) These exercises should be completed in place of the Discussion and Exercise 1 above. Students need graph paper to complete the following exercises. This is the alternate activity that was described in the Lesson Notes. For these exercises, students graph a linear equation and informally verify that any point on the graph of the equation is on the line and that any point on the line is a point on the graph of the equation. Exercises 6β8 ask students to draw these conclusions based on their work in Exercises 2β5. Consider having a class discussion based on Exercises 6β8, where students share their conclusions. End the discussion by making it clear that the graph of a linear equation is a line. Then proceed with Example 1 and Exercises 9β11.
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2.
8β’4
Use π = π and π = βπ to find two solutions to the equation π + ππ = π. Plot the solutions as points on the coordinate plane and connect the points to make a line. The solutions are (π, π) and (βπ, π).
a.
Identify two other points on the line with integer coordinates. Verify that they are solutions to the equation π + ππ = π. Choice of points and verifications will vary. Several possibilities are noted in the graph above.
b.
When π = π, what is the value of π? Does this solution appear to be a point on the line? π + ππ = π
ππ = π
π π
Yes, οΏ½π, οΏ½ does appear to be a point on the line. c.
π=
π π
When π = βπ, what is the value of π? Does this solution appear to be a point on the line? βπ + ππ = π
ππ = π
π π
Yes, οΏ½βπ, οΏ½ does appear to be a point on the line. d.
π=
π π
Is the point (π, π) on the line?
No, (π, π) is not a point on the line.
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e.
8β’4
Is the point (π, π) a solution to the linear equation π + ππ = π? No, (π, π) is not a solution to π + ππ = π because
π + π(π) = π π+π= π πβ π
3.
Use π = π and π = π to find two solutions to the equation ππ β π = π. Plot the solutions as points on the coordinate plane and connect the points to make a line. The solutions are (π, π) and (π, βπ).
a.
Identify two other points on the line with integer coordinates. Verify that they are solutions to the equation ππ β π = π. Choice of points and verifications will vary. Several possibilities are noted in the graph above.
b.
When π = π. π, what is the value of π? Does this solution appear to be a point on the line? π(π. π) β π = π ππ. π β π = π βπ = βπ. π π = π. π
Yes, (π. π, π. π) does appear to be a point on the line.
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c.
π π
8β’4
When π = , what is the value of π? Does this solution appear to be a point on the line? π ποΏ½ οΏ½ β π = π π π βπ=π π βπ = π. π π = βπ. π
π π
Yes, οΏ½ , βπ. ποΏ½ does appear to be a point on the line. d.
Is the point (π, π) on the line?
No, (π, π) is not a point on the line.
e.
Is the point (π, π) a solution to the linear equation ππ β π = π?
No, (π, π) is not a solution to ππ β π = π because
π(π) β π = π πβπ= π πβ π
4.
Use π = π and π = βπ to find two solutions to the equation ππ + ππ = ππ. Plot the solutions as points on the coordinate plane and connect the points to make a line. The solutions are (βπ, π) and (π, π).
a.
Identify two other points on the line with integer coordinates. Verify that they are solutions to the equation ππ + ππ = ππ. Choice of points and verifications will vary. Several possibilities are noted in the graph above.
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b.
8β’4
When π = π, what is the value of π? Does this solution appear to be a point on the line? π(π) + ππ = ππ π + ππ = ππ ππ = π π π= π
π π
Yes, οΏ½π, οΏ½ does appear to be a point on the line. c.
When π = βπ, what is the value of π? Does this solution appear to be a point on the line?
Yes, οΏ½βπ, d.
π(βπ) + ππ = ππ βπ + ππ = ππ ππ = ππ ππ π= π
ππ οΏ½ does appear to be a point on the line. π
Is the point (π, βπ) on the line?
No, (π, βπ) is not a point on the line.
e.
Is the point (π, βπ) a solution to the linear equation ππ + ππ = ππ? No, (π, βπ) is not a solution to ππ + ππ = ππ because
π(π) + π(βπ) = ππ ππ β π = ππ π β ππ
5.
Use π = π and π = βπ to find two solutions to the equation π β ππ = π. Plot the solutions as points on the coordinate plane and connect the points to make a line. The solutions are (π, βπ) and (βπ, βπ).
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a.
8β’4
Identify two other points on the line with integer coordinates. Verify that they are solutions to the equation π β ππ = π. Choice of points and verifications will vary. Several possibilities are noted in the graph above.
b.
When π = π, what is the value of π? Does this solution appear to be a point on the line? π β ππ = π βππ = π
π=β
π π
Yes, οΏ½π, β οΏ½ does appear to be a point on the line. c.
When π = βπ, what is the value of π? Does this solution appear to be a point on the line?
Yes, οΏ½βπ, β d.
π π
βπ β ππ = π βππ = ππ ππ π=β π
ππ οΏ½ does appear to be a point on the line. π
Is the point (βπ, βπ) on the line?
No, (βπ, βπ) is not a point on the line.
e.
Is the point (βπ, βπ) a solution to the linear equation π β ππ = π? No, (βπ, βπ) is not a solution to π β ππ = π because
βπ β π(βπ) = π βπ + π = π πβ π
6.
Based on your work in Exercises 2β5, what conclusions can you draw about the points on a line and solutions to a linear equation? It appears that all points on the line represent a solution to the equation. That is, any point identified on the line is a solution to the linear equation.
MP.8
7.
Based on your work in Exercises 2β5, will a point that is not a solution to a linear equation be a point on the graph of a linear equation? Explain. No, each time we were given a point off the line in part (d), we verified that it was not a solution to the equation in part (e). For that reason, I would expect that all points not on the line would not be a solution to the equation.
8.
Based on your work in Exercises 2β5, what conclusions can you draw about the graph of a linear equation? The graph of a linear equation is a line.
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Example 1 (4 minutes) ο§
Now that we know that the graph of a linear equation in two variables is a line and that there is only one line that can pass through two points, then we can easily graph equations using intercepts.
ο§
We already know that the π¦-intercept is the location on the graph of a line where the line intersects the π¦-axis. The point of intersection will have coordinates (0, π¦). Similarly, the π₯-intercept is the location on the graph of a line where the line intersects the π₯-axis. The point of intersection will have coordinates (π₯, 0).
ο§ ο§
To graph using intercepts, simply replace the symbols π₯ and π¦ with zero, one at a time, and solve.
Graph the equation: 2π₯ + 3π¦ = 9.
Replace π₯ with zero and solve for π¦ to determine the π¦-intercept: 2(0) + 3π¦ = 9 3π¦ = 9
The π¦-intercept is at (0, 3).
π¦=3
Replace π¦ with zero and solve for π₯ to determine the π₯-intercept: 2π₯ + 3(0) = 9 2π₯ = 9
9
The π₯-intercept is at οΏ½ , 0οΏ½. 2
ο§
π₯=
9 2
Now that we know the intercepts, we can place those two points on the graph and connect them to graph the linear equation 2π₯ + 3π¦ = 9:
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ο§
8β’4
Graphing using intercepts is an efficient way of graphing linear equations that are in standard form. Graphing using the slope and π¦-intercept is the most efficient way of graphing linear equations that are in slopeintercept form. Creating a table and finding solutions is another way we have learned how to graph linear equations. All three methods work, but some methods will save time depending on the form of the equation.
Exercise 9β11 (5 minutes) 9.
Graph the equation βππ + ππ = ππ using intercepts. βππ + ππ = ππ βπ(π) + ππ = ππ ππ = ππ π=π
The π-intercept is (π, π). βππ + ππ = ππ βππ + π(π) = ππ βππ = ππ π = βπ
The π-intercept is (βπ, π). 10. Graph the equation π β ππ = ππ using intercepts.
π π
The π-intercept is οΏ½π, β οΏ½.
The π-intercept is (ππ, π).
Lesson 19: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
π β ππ = ππ π β ππ = ππ βππ = ππ ππ π=β π π π=β π π β ππ = ππ π β π(π) = ππ π = ππ
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11. Graph the equation ππ + ππ = ππ using intercepts.
ππ + ππ = ππ π(π) + ππ = ππ ππ = ππ π=π
The π-intercept is (π, π).
The π-intercept is οΏ½
ππ , ποΏ½. π
ππ + ππ = ππ ππ + π(π) = ππ ππ = ππ ππ π= π
Closing (5 minutes) Summarize, or ask students to summarize, the main points from the lesson: ο§
We know that to prove the graph of a linear equation is a line we have to show that any point on the graph of the equation is on the line, and any point on the line is on the graph of the equation.
ο§
We can use what we know about slope and the fact that there is only one line with a given slope that goes through a given point, to prove that the graph of a linear equation is a line.
ο§
We have another method for graphing linear equations: using intercepts.
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Lesson Summary The graph of a linear equation is a line. A linear equation can be graphed using two-points: the π-intercept and the π-intercept. Example:
Graph the equation: ππ + ππ = π.
Replace π with zero and solve for π to determine the π-intercept:
The π-intercept is at (π, π).
π(π) + ππ = π ππ = π π=π
Replace π with zero and solve for π to determine the π-intercept:
π π
The π-intercept is at οΏ½ , ποΏ½.
ππ + π(π) = π ππ = π π π= π
Exit Ticket (5 minutes)
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Name
8β’4
Date
Lesson 19: The Graph of a Linear Equation in Two Variables is a Line Exit Ticket 5 4
1.
Graph the equation π¦ = π₯ β 10 using the π¦-intercept and slope.
2.
Graph the equation 5π₯ β 4π¦ = 40 using intercepts.
3.
What can you conclude about the equations π¦ = π₯ β 10 and 5π₯ β 4π¦ = 40?
5 4
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Exit Ticket Sample Solutions π π
1.
Graph the equation π = π β ππ using the π-intercept and slope.
2.
Graph the equation ππ β ππ = ππ using intercepts.
3.
What can you conclude about the equations π = π β ππ and ππ β ππ = ππ?
π π
Since the points (π, βππ), (π, βπ), and (π, π) are common to both graphs, then the lines must be the same. There is π π
only one line that can pass through two points. If you transform the equation π = π β ππ so that it is in standard
form, it is the equation ππ β ππ = ππ.
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Problem Set Sample Solutions Now that students know the graph of a linear equation is a line, students practice graphing linear equations in two variables using an appropriate method. Students need graph paper to complete the Problem Set. 1.
Graph the equation: π = βππ + ππ.
2.
Graph the equation: ππ + ππ = ππ. π(π) + ππ = ππ ππ = ππ π=π
The π-intercept is (π, π).
ππ + π(π) = ππ
ππ = ππ π=π
The π-intercept is (π, π).
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3.
Graph the equation: π = ππ + π.
4.
Graph the equation: π = β π + π.
8β’4
π π
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5.
8β’4
Graph the equation:
π π
π + π = π.
π (π) + π = π π π=π
The π-intercept is (π, π). π π+π=π π π π=π π ππ π= π
The π-intercept
is οΏ½
6.
ππ , ποΏ½. π
Graph the equation: ππ β ππ = ππ. π(π) β ππ = ππ βππ = ππ π = βπ
The π-intercept is (π, βπ). ππ β π(π) = ππ ππ = ππ π=π
The π-intercept is (π, π).
7.
Graph the equation: π = π. What is the slope of the graph of this line? The slope of this line is zero.
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8β’4
Graph the equation: π = βπ. What is the slope of the graph of this line? The slope of this line is undefined.
9.
Is the graph of ππ + ππ =
π a line? Explain. π
Yes, the graph of ππ + ππ = the equal sign.
π is a line because it is a linear equation comprised of linear expressions on both side of π
10. Is the graph of πππ β ππ = π a line? Explain.
Maybe. The equation πππ β ππ = π is not a linear equation because the expression on the left side of the equal sign is not a linear expression. If this were a linear equation, then I would be sure that it graphs as a line, but because it is not, Iβm not sure what the graph of this equation would look like.
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Lesson 20: Every Line is a Graph of a Linear Equation Student Outcomes ο§ ο§
Students know that any non-vertical line is the graph of a linear equation in the form of π¦ = ππ₯ + π, where π is a constant. Students write the equation that represents the graph of a line.
Lesson Notes The proof that every line is the graph of a linear equation in the first discussion below is optional. If you choose to use the Discussion below, you will need to skip the Opening Exercise that begins on page 302 and resume the lesson with Example 1 on page 303. Complete all other examples and exercises that follow. As an alternative to the discussion below, you may choose to complete the opening activity by showing a graph of a line on the coordinate plane and having students attempt to name the equation of the line. Two graphs are provided for your use beginning on page 302. Have students write their equations and strategies for determining the equation of the line; then lead the discussion described on page 302. Once the Opening Exercise is complete, work through Example 1 on page 303 and the remaining examples and exercises in the lesson. Revisit the equations and strategies students developed by having them critique their reasoning in comparison to the work in the example; then continuing with the remainder of the lesson.
Classwork Discussion (10 minutes) ο§
ο§ ο§ ο§ ο§ MP.3
Now that we are confident that the graph of every linear equation is a line, can we say that every line is the graph of a linear equation? We can say yes with respect to vertical and horizontal lines, recall π₯ = π and π¦ = π. But what about other non-vertical lines?
We must prove that any non-vertical (and non-horizontal) line is a graph of a linear equation in the form of π¦ = ππ₯ + π, where π β 0 and π are constants.
Let π be any non-vertical (and non-horizontal) line. Suppose the slope of the line is π and that the line intersects the π¦-axis at point π = (0, π).
First, we show that any point on the line π is a point on the graph of the linear equation π¦ = ππ₯ + π.
Let π = (π₯, π¦) be any point on line π. We need to show that π is a solution to π¦ = ππ₯ + π. Think about how we did this in the last lesson. What should we do? οΊ
Use the points π and π in the slope formula:
π¦βπ π₯β0 ππ₯ = π¦ β π ππ₯ + π = π¦ β π + π ππ₯ + π = π¦ π=
That shows that point π is a point on the graph of π¦ = ππ₯ + π. Point π is also on the graph of π¦ = ππ₯ + π because π = π β
0 + π. Therefore, any point on the line π is a point on the graph of the linear equation π¦ = ππ₯ + π. Lesson 20: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
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ο§ ο§
MP.3
Now, we want to show that any point on the graph of π¦ = ππ₯ + π is on π.
Let π
be any point on the graph of the linear equation π¦ = ππ₯ + π. We know that the graph of π¦ = ππ₯ + π is a line with slope π. Letβs call this line πβ². We know that π is on πβ² because π = π β
0 + π. Therefore, πβ² is a line with slope π that passes through point π. But π is a line with slope π that passes through point π. What does that mean about lines π and π β² ? οΊ
ο§ ο§
8β’4
The lines π and π β² are the same line because there is only one line with a given slope that can go through a given point.
Therefore, π
is a point on π.
Now we can be certain that every line is a graph of a linear equation.
Opening (10 minutes) Show students Graph 1 below and challenge them to write the equation for the line. Provide students time to work independently, then in pairs. Lead a discussion where students share their strategies for developing the equation of the line. Ask students how they knew their equations were correct, i.e., did they verify that the points with integer coordinates were solutions to the equation they wrote? Ask students what kind of equation they wrote: linear or nonlinear. Ask students if they were given another line, could they write an equation for it using their strategy. Show them Graph 2 and, again, ask them to write the equation of the line. Verify that they wrote the correct equation, and conclude the discussion by stating that every line is the graph of a linear equation. Opening Exercise Graph 1
π π
The equation for the line in Graph 1 is π = π β π.
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Graph 2
π π
The equation for the line in Graph 2 is π = β π + π.
Example 1 (5 minutes) ο§
Given the graph of a line, we want to be able to write the equation that represents it.
ο§
Which form of a linear equation do you think will be most valuable for this task, the standard form ππ₯ + ππ¦ = π or slope-intercept form π¦ = ππ₯ + π? οΊ
ο§
The slope-intercept form because we can easily identify the slope and π¦-intercept from both the equation and the graph.
Write the equation that represents the graph of the line shown below:
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ο§
First, identify the y-intercept. οΊ
ο§
ο§
The point π is represents our π¦-intercept. Letβs locate a point π
on the line with integer coordinates. We can use the point (5, 2) or (β5, 6).
We can use either point. For this example, letβs use (5, 2).
Now we can locate point π. It must be to the right of point π and be on a line parallel to the π¦-axis that goes through point π
. What is the location of π? οΊ
ο§
The line intersects the π¦-axis at (0, 4).
Now we must use what we know about slope to determine the slope of the line. Recall the following: |ππ
| π= |ππ| οΊ
ο§
8β’4
Point π must be (5, 4).
What fraction represents the slope of this line?
οΊ
2 5
The slope π = β . Lesson 20: Date:
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ο§
8β’4
2 5
The slope of the line is π = β and the π¦-intercept is (0, 4). What must the equation of the line be?
οΊ
2 5
The line is the graph of π¦ = β π₯ + 4.
Example 2 (5 minutes) ο§
What is the π¦-intercept of the line?
οΊ
The π¦-intercept is (0, β2).
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ο§
Select another point, π
, on the line with integer coordinates.
οΊ ο§
8β’4
Let π
be the point located at (1, 2).
Now place point π and find the slope of the line.
Q
οΊ
The slope of the line is π = 4.
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ο§
8β’4
Write the equation for the line.
οΊ
The line is the graph of π¦ = 4π₯ β 2.
Example 3 (5 minutes) ο§
What is the π¦-intercept of the line? Notice the units on the coordinate plane have increased.
οΊ
The π¦-intercept is (0, β40).
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ο§
Select another point, π
, on the line with integer coordinates.
οΊ ο§
8β’4
Let π
= (50, 0).
Now place point π and find the slope of the line.
οΊ
The slope of the line is π =
Lesson 20: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
40 4 = . 50 5
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ο§
Write the equation for the line.
οΊ ο§
8β’4
4 5
The line is the graph of π¦ = π₯ β 40.
The last thing we will do to this linear equation is rewrite it in standard form ππ₯ + ππ¦ = π where π, π, and π are integers and π is not negative. That means we must multiply the entire equation by a number that 4
will turn into an integer. What number should we multiply by? οΊ ο§
5 4 5
(5) = 4.
We multiply the entire equation by 5:
4 π₯ β 40οΏ½ 5 5 5π¦ = 4π₯ β 200 β4π₯ + 5π¦ = 4π₯ β 4π₯ β 200 β4π₯ + 5π¦ = β200 β1(β4π₯ + 5π¦ = β200) 4π₯ β 5π¦ = 200 οΏ½π¦ =
The standard form of the linear equation is 4π₯ β 5π¦ = 200.
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Exercises 1β6 (10 minutes) Students complete Exercises 1β6 independently. Exercises 1.
Write the equation that represents the line shown. π = ππ + π Use the properties of equality to change the equation from slope intercept form, π = ππ + π, to standard form, ππ + ππ = π, where π, π, and π are integers and π is not negative. π = ππ + π
βππ + π = ππ β ππ + π βππ + π = π
βπ(βππ + π = π)
ππ β π = βπ
2.
Write the equation that represents the line shown. π π =β πβπ π
Use the properties of equality to change the equation from slope intercept form, π = ππ + π, to standard form, ππ + ππ = π, where π, π, and π are integers and π is not negative. π π =β πβπ π π οΏ½π = β π β ποΏ½ π π ππ = βππ β π ππ + ππ = βππ + ππ β π ππ + ππ = βπ
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3.
8β’4
Write the equation that represents the line shown. π π= β πβπ π Use the properties of equality to change the equation from slope intercept form, π = ππ + π, to standard form, ππ + ππ = π, where π, π, and π are integers and π is not negative. π π=β πβπ π π οΏ½π = β π β ποΏ½ π π
ππ = βπ β ππ
π + ππ = βπ + π β ππ π + ππ = βππ π + ππ = βππ 4.
Write the equation that represents the line shown. π=π Use the properties of equality to change the equation from slope intercept form, π = ππ + π, to standard form, ππ + ππ = π, where π, π, and π are integers and π is not negative. π=π
βπ + π = π β π βπ + π = π
βπ(βπ + π = π) πβπ= π
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5.
8β’4
Write the equation that represents the line shown.
π=
π π+π π
Use the properties of equality to change the equation from slope intercept form, π = ππ + π, to standard form, ππ + ππ = π, where π, π, and π are integers and π is not negative. π π+π π π οΏ½π = π + ποΏ½ π π π=
ππ = π + ππ
βπ + ππ = π β π + ππ βπ + ππ = ππ
βπ(βπ + ππ = ππ)
π β ππ = βππ
6.
Write the equation that represents the line shown. π π= β πβπ π Use the properties of equality to change the equation from slope intercept form, π = ππ + π, to standard form, ππ + ππ = π, where π, π, and π are integers and π is not negative. π π=β πβπ π π οΏ½π = β π β ποΏ½ π π
ππ = βππ β ππ
ππ + ππ = βππ + ππ β ππ ππ + ππ = βππ
Closing (5 minutes) Summarize, or ask students to summarize, the main points from the lesson: ο§
We know that every line is a graph of a linear equation.
ο§
We know how to use the π¦-intercept and the slope of a line to write the equation that represents the graph of a line.
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Lesson Summary Write the equation of a line by determining the π-intercept, (π, π) and the slope, π, and replacing the numbers π and π into the equation π = ππ + π. Example:
The π-intercept of this graph is (π, βπ). The slope of this graph is π =
π = π. π
The equation that represents the graph of this line is π = ππ β π.
Exit Ticket (5 minutes)
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Name
8β’4
Date
Lesson 20: Every Line is a Graph of a Linear Equation Exit Ticket 1.
Write an equation in slope intercept form that represents the line shown.
2.
Use the properties of equality to change the equation you wrote for problem 1 from slope intercept form, π¦ = ππ₯ + π, to standard form, ππ₯ + ππ¦ = π, where π, π, and π are integers and π is not negative.
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3.
Write an equation in slope intercept form that represents the line shown.
4.
Use the properties of equality to change the equation you wrote for problem 3 from slope intercept form, π¦ = ππ₯ + π, to standard form, ππ₯ + ππ¦ = π, where π, π, and π are integers and π is not negative.
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8β’4
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Exit Ticket Sample Solutions 1.
2.
3.
Write an equation in slope intercept form that represents the line shown.
π π= β π+π π
Use the properties of equality to change the equation you wrote for problem 1 from slope intercept form, π = ππ + π, to standard form, ππ + ππ = π, where π, π, and π are integers and π is not negative. π π=β π+π π π οΏ½π = β π + ποΏ½ π π ππ = βπ + π π + ππ = βπ + π + π π + ππ = π
Write an equation in slope intercept form that represents the line shown.
π= 4.
π π+π π
Use the properties of equality to change the equation you wrote for problem 3 from slope intercept form, π = ππ + π, to standard form, ππ + ππ = π, where π, π, and π are integers and π is not negative. π π+π π π οΏ½π = π + ποΏ½ π π ππ = ππ + π βππ + ππ = ππ β ππ + π βππ + ππ = π βπ(βππ + ππ = π) ππ β ππ = βπ π=
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Problem Set Sample Solutions Students practice writing equations for lines. 1.
Write the equation that represents the line shown. π π= β πβπ π
Use the properties of equality to change the equation from slope intercept form, π = ππ + π, to standard form, ππ + ππ = π, where π, π, and π are integers and π is not negative. π π= β πβπ π π οΏ½π = β π β ποΏ½ π π
ππ = βππ β ππ
ππ + ππ = βππ + ππ β ππ ππ + ππ = βππ 2.
Write the equation that represents the line shown. π = ππ + π Use the properties of equality to change the equation from slope intercept form, π = ππ + π, to standard form, ππ + ππ = π, where π, π, and π are integers and π is not negative. π = ππ + π
βππ + π = ππ β ππ + π βππ + π = π
βπ(βππ + π = π)
ππ β π = βπ
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3.
8β’4
Write the equation that represents the line shown.
π=
π πβπ π
Use the properties of equality to change the equation from slope intercept form, π = ππ + π, to standard form, ππ + ππ = π, where π, π, and π are integers and π is not negative. π πβπ π π οΏ½π = π β ποΏ½ π π ππ = π β π βπ + ππ = π β π β π βπ + ππ = βπ βπ(βπ + ππ = βπ) π β ππ = π π=
4.
Write the equation that represents the line shown. π = βππ β π Use the properties of equality to change the equation from slope intercept form, π = ππ + π, to standard form, ππ + ππ = π, where π, π, and π are integers and π is not negative. π = βππ β π ππ + π = βππ + ππ β π ππ + π = βπ
5.
Write the equation that represents the line shown. π = ππ β ππ Use the properties of equality to change the equation from slope intercept form, π = ππ + π, to standard form, ππ + ππ = π, where π, π, and π are integers and π is not negative. π = ππ β ππ βππ + π = ππ β ππ β ππ βππ + π = βππ βπ(βππ + π = βππ) ππ β π = ππ
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6.
8β’4
Write the equation that represents the line shown. π = βππ + ππ Use the properties of equality to change the equation from slope intercept form, π = ππ + π, to standard form, ππ + ππ = π, where π, π, and π are integers and π is not negative. π = βππ + ππ ππ + π = βππ + ππ + ππ ππ + π = ππ
Lesson 20: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
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Lesson 21: Some Facts about Graphs of Linear Equations in Two Variables Student Outcomes ο§
Students write the equation of a line given two points or the slope and a point on the line.
ο§
Students know the traditional forms of the slope formula and slope-intercept equation.
Classwork Example 1 (10 minutes) Students determine the equation of a line from a graph by using information about slope and a point. ο§ MP.1
Let a line π be given in the coordinate plane. Our goal is to find the equation that represents the graph of line π. Can we use information about the slope and intercept to write the equation of the line, like we did in the last lesson?
Provide students time to attempt to write the equation of the line. Ask students to share their equations and explanations. Consider having the class vote on whose explanation/equation they think is correct. Scaffolding:
Example 1 Let a line π be given in the coordinate plane. What linear equation is the graph of line?
οΊ
If necessary, include another point, as done in Lesson 15, to help students determine the slope of the line.
We can pick two points to determine the slope, but the precise location of the π¦-intercept cannot be determined from the graph.
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ο§
Calculate the slope of the line. οΊ
ο§
8β’4
Using points, (β2, 2) and (5, 4), the slope of the line is 2β4 π= β2 β 5 β2 = β7 2 = 7
Now we need to determine the π¦-intercept of the line. We know that it is a point with coordinates (0, π) and 2 7
we know that the line that goes through points (β2, 2) and (5, 4) and has slope π = . Using this
information, we can determine the coordinates of the π¦-intercept and the value of π that we need in order to write the equation of the line. ο§
Recall what it means for a point to be on a line; the point is a solution to the equation. In the equation π¦ = ππ₯ + π, (π₯, π¦), is a solution and π is the slope. Can we find the value of π? Explain. οΊ
ο§
Do you think it matters which point we choose to substitute into the equation? That is, will we get a different equation if we use the point (β2, 2) compared to (5, 4)? οΊ
ο§
Yes, we can substitute one of the points and the slope into the equation and solve for π.
No, because there can be only one line with a given slope that goes through a point. 2 7
2 7
Verify this claim by using π = and (β2, 2) to find the equation of the line, then using π = and (5, 4) to see if the result is the same equation. οΊ
Sample student work:
2 (β2) + π 7 4 2=β +π 7 4 4 4 2+ =β + +π 7 7 7 18 =π 7 2=
2 (5) + π 7 10 4= +π 7 10 10 10 4β = β +π 7 7 7 18 =π 7 4=
The π¦-intercept is at οΏ½0,
Lesson 21: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
18 2 18 οΏ½ and the equation of the line is π¦ = π₯ + . 7 7 7
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ο§
8β’4
The equation of the line is:
Write it in standard form: οΊ
Sample student work:
π¦=
2 18 π₯+ 7 7
οΏ½π¦ =
2 18 π₯ + οΏ½7 7 7
7π¦ = 2π₯ + 18
β2π₯ + 7π¦ = 2π₯ β 2π₯ + 18
β2π₯ + 7π¦ = 18
β1(β2π₯ + 7π¦ = 18)
2π₯ β 7π¦ = β18
Example 2 (5 minutes) Students determine the equation of a line from a graph by using information about slope and a point. ο§
Let a line π be given in the coordinate plane. What information do we need to write the equation of the line? οΊ
We need to know the slope, so we must identify two points we can use to calculate the slope. Then we can use the slope and a point to determine the equation of the line.
Example 2 Let a line π be given in the coordinate plane. What linear equation is the graph of line?
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οΊ
ο§
Using points, (β1, 4) and (4, 1), the slope of the line is 4β1 π= β1 β 4 3 = β5 3 =β 5
Now to determine the π¦-intercept of the line. οΊ
Sample student work:
The π¦-intercept is at οΏ½0, ο§
17 οΏ½. 5
3 4 = οΏ½β οΏ½ (β1) + π 5 3 4 = +π 5 3 3 3 4β = β +π 5 5 5 17 =π 5 3 5
Now that we know the slope, π = β and the π¦-intercept, οΏ½0, intercept form. οΊ
ο§
8β’4
3 5
π¦=β π₯+
17 5
17 οΏ½, write the equation of the line π in slope5
Transform the equation so that it is written in standard form. οΊ
Sample student work: 3 17 π¦=β π₯+ 5 5 3 17 οΏ½π¦ = β π₯ + οΏ½ 5 5 5
5π¦ = β3π₯ + 17
3π₯ + 5π¦ = β3π₯ + 3π₯ + 17
3π₯ + 5π¦ = 17
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Example 3 (5 minutes) Students determine the equation of a line from a graph by using information about slope and a point. ο§
Let a line π be given in the coordinate plane. Assume the π¦-axis intervals are units of one (like the visible π₯axis). What information do we need to write the equation of the line? οΊ
We need to know the slope, so we must identify two points we can use to calculate the slope. Then we can use the slope and a point to determine the equation of the line.
Example 3 Let a line π be given in the coordinate plane. What linear equation is the graph of line?
οΊ
ο§
Using points, (12, 2) and (13, 7), the slope of the line is 2β7 π= 12 β 13 β5 = β1 =5
Now to determine the π¦-intercept of the line and write the equation of the line in slope-intercept form. οΊ
Sample student work:
2 = 5(12) + π 2 = 60 + π
π = β58
The π¦-intercept is at (0, β58) and the equation of the line is π¦ = 5π₯ β 58.
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ο§
8β’4
Now that we know the slope, π = 5 and the π¦-intercept, (0, β58), write the equation of the line π in standard form. οΊ
Sample student work:
π¦ = 5π₯ β 58
β5π₯ + π¦ = 5π₯ β 5π₯ β 58
β5π₯ + π¦ = β58
β1(β5π₯ + π¦ = β58) 5π₯ β π¦ = 58
Example 4 (3 minutes) Students determine the equation of a line from a graph by using information about slope and a point. ο§
Let a line π be given in the coordinate plane. Write the equation for this line.
οΊ
ο§
1 3
The π¦-intercept is at (0, 0) and the equation of the line is π¦ = π₯.
The π¦-intercept is the origin of the graph. What value does π have when this occurs? οΊ
ο§
Using points, (3, 1) and (β3, β1), the slope of the line is β1 β 1 π= β3 β 3 β2 = β6 1 = 3
When the line goes through the origin the value of π is zero.
All linear equations that go through the origin have the form π¦ = ππ₯ + 0 or simply π¦ = ππ₯. We have done a lot of work with equations in this form. Which do you remember? οΊ
All problems that describe proportional relationships have equations of this form.
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Concept Development (5 minutes) ο§
The following are some facts about graphs of linear equations in two variables.
ο§
Let (π₯1 , π¦1 ) and (π₯2 , π¦2 ) be the coordinates of two distinct points on the graph of a line π. We find the slope of the line by: π¦2 β π¦1 π= π₯2 β π₯1
ο§
Where this version of the slope formula, using coordinates of π₯ and π¦ instead of π and π, is a commonly accepted version.
As soon as you multiply the slope by the denominator of the fraction above you get the following equation: π(π₯2 β π₯1 ) = π¦2 β π¦1
This form of an equation is referred to as the point-slope form of a linear equation. As you can see, it doesnβt convey any more information than the slope formula. It is just another way to look at it. ο§ ο§
ο§
Given a known (π₯, π¦), then the equation is written as
π(π₯ β π₯1 ) = (π¦ β π¦1 )
The following is the slope-intercept form of a line:
π¦ = ππ₯ + π
In this equation, π is slope and (0, π) is the π¦-intercept.
What information must you have in order to write the equation of a line? οΊ
ο§
We need two points or one point and slope.
The names and symbols used are not nearly as important as your understanding of the concepts. Basically, if you can remember a few simple facts about lines, namely the slope formula and the fact that slope is the same between any two points on a line, you can derive the equation of any line.
Exercises 1β5 (7 minutes) Students complete Exercises 1β5 independently. Exercises 1β5 1.
Write the equation for the line π shown in the graph.
Using the points (βπ, βπ) and (π, βπ), the slope of the line is βπ β (βπ) βπ β π βπ = βπ π = π π βπ = (π) + π π π βπ = + π π π π π βπ β = β + π π π π π β =π π π=
The equation of the line is π = Lesson 21: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
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2.
8β’4
Write the equation for the line π shown in the graph.
Using the points (βπ, π) and (π, π), the slope of the line is πβπ βπ β π βπ = βπ π = π π π = (π) + π π π π= +π π π π π πβ = β +π π π π ππ =π π π=
π π
The equation of the line is π = π + 3.
ππ . π
Determine the equation of the line that goes through points (βπ, π) and (π, π). The slope of the line is
πβπ βπ β π π = βπ π =β π
π=
The π-intercept of the line is
π π
The equation of the line is π = β π + 4.
ππ . π
π π = β (π) + π π π π=β +π π ππ =π π
Write the equation for the line π shown in the graph.
Using the points (βπ, π) and (βπ, βπ), the slope of the line is π β (βπ) βπ β (βπ) π = βπ = βπ βπ = βπ(βπ) + π βπ = ππ + π βππ = π π=
The equation of the line is π = βππ β ππ.
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5.
8β’4
A line goes through the point (π, π) and has slope π = π. Write the equation that represents the line. π = π(π) + π π = ππ + π
The equation of the line is π = ππ β ππ.
βππ = π
Closing (5 minutes) Summarize, or ask students to summarize, the main points from the lesson: ο§ ο§ ο§
We know how to write an equation for a line from a graph, even if the line does not intersect the π¦-axis at integer coordinates. We know how to write the equation for a line given two points or one point and the slope of the line.
We know other versions of the formulas and equations that we have been using related to linear equations.
Lesson Summary Let (ππ , ππ ) and (ππ , ππ ) be the coordinates of two distinct points on the graph of a line π. We find the slope of the line by: ππ β ππ ππ β ππ Where this version of the slope formula, using coordinates of π and π instead of π and π, is a commonly accepted version. π=
As soon as you multiply the slope by the denominator of the fraction above you get the following equation: π(ππ β ππ ) = ππ β ππ
This form of an equation is referred to as the point-slope form of a linear equation. Given a known (π, π), then the equation is written as The following is slope-intercept form of a line:
π(π β ππ ) = (π β ππ ) π = ππ + π
In this equation, π is slope and (π, π) is the π-intercept.
To write the equation of a line you must have two points, one point and slope, or a graph of the line.
Exit Ticket (5 minutes)
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Name
8β’4
Date
Lesson 21: Some Facts about Graphs of Linear Equations in Two Variables Exit Ticket 1.
Write the equation for the line π shown in the graph below.
2.
A line goes through the point (5, β7) and has slope π = β3. Write the equation that represents the line.
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Exit Ticket Sample Solutions Note that some students may write equations in standard form. 1.
Write the equation for the line π shown in the graph below.
Using the points (βπ, π) and (π, π), the slope of the line is
πβπ π β (βπ) π π= π π π = (π) + π π π π= +π π π π π πβ = β +π π π π π =π π π=
π π
π π
The equation of the line is π = π + . 2.
A line goes through the point (π, βπ) and has slope π = βπ. Write the equation that represents the line. βπ = βπ(π) + π βπ = βππ + π The equation of the line is π = βππ + π.
Lesson 21: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
π=π
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Problem Set Sample Solutions Students practice writing equations from graphs of lines. Students write the equation of a line given only the slope and a point. 1.
Write the equation for the line π shown in the graph.
Using the points (βπ, π) and (π, βπ), the slope of the line is π β (βπ) βπ β π π = βπ π =β π π π = οΏ½β οΏ½ (βπ) + π π ππ π= +π π ππ ππ ππ πβ = β +π π π π π β =π π π=
π π
π π
The equation of the line is π = β π β .
2.
Write the equation for the line π shown in the graph.
Using the points (βπ, π) and (βπ, π), the slope of the line is πβπ βπ β (βπ) βπ = βπ =π π = π(βπ) + π π = βππ + π ππ = π π=
The equation of the line is π = ππ + ππ.
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3.
8β’4
Write the equation for the line π shown in the graph.
Using the points (βπ, π) and (π, π), the slope of the line is πβπ βπ β π βπ = βπ π = π π π = (π) + π π π π= +π π π π π πβ = β +π π π π π =π π π=
π π
π π
The equation of the line is π = π + .
4.
Triangle β³ π¨π©πͺ is made up line segments formed from the intersection of lines π³π¨π© , π³π©πͺ , and π³π¨πͺ . Write the equations that represents the lines that make up the triangle. π¨ = (βπ, βπ), π© = (π, π), πͺ = (π, βπ)
The equation of π³π¨π© :
The slope of π³π¨π© is βπ β π π= βπ β π βπ = βπ π = π π π = (π) + π π π π= +π π π π π πβ = β +π π π π π β =π π π π The equation of π³π¨π© is π = π β .
The slope of π³π©πͺ :
π
π
π β (βπ) πβπ π = βπ = βπ π = βπ(π) + π π = βπ + π π=π π=
The equation of π³π©πͺ is π = βππ + π.
The slope of π³π¨πͺ :
βπ β (βπ) βπ β π βπ = βπ π = π π βπ = (π) + π π π βπ = + π π π π π βπ β = β + π π π π ππ β =π π π=
π π
The equation of π³π¨πͺ is π = π β
Lesson 21: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
ππ . π
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5.
Write the equation for the line that goes through point (βππ, π) with slope π = π.
The equation of the line is π = ππ + ππ. 6.
ππ = βπ(ππ) + π ππ = βππ + π ππ = π
Write the equation for the line that goes through point (π, π) with slope π = βπ.
The equation of the line is π = βππ + ππ. 8.
π = π(βππ) + π π = βππ + π ππ = π
Write the equation for the line that goes through point (ππ, ππ) with slope π = βπ.
The equation of the line is π = βππ + ππ. 7.
8β’4
π = βπ(π) + π π = βπ + π ππ = π
Determine the equation of the line that goes through points (π, π) and (π, π).
The slope of the line is
πβπ πβπ βπ = βπ =π
π=
The π-intercept of the line is
The equation of the line is π = ππ β π.
Lesson 21: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
π = π(π) + π π= π+π βπ = π
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8β’4
Lesson 22: Constant Rates Revisited Student Outcomes ο§
Students know that any constant rate problem can be described by a linear equation in two variables where the slope of the graph is the constant rate.
ο§
Students compare two different proportional relationships represented by graphs, equations, and tables to determine which has a greater rate of change.
Classwork Example 1 (8 minutes) ο§
ο§
Recall our definition of constant rate: If the average speed of motion over any time interval is equal to the same constant, then we say the motion has constant speed. Erika set her stopwatch to zero and switched it on at the beginning of her walk. She walks at a constant speed of 3 miles per hour. Work in pairs to express this situation as an equation, table of values, and a graph.
Provide students time to analyze the situation and represent it in the requested forms. If necessary, provide support to students with the next few bullet points. Once students have finished working resume the work on the example (bullets below the graph). ο§
π¦ π₯
. Since Erika walks at a constant speed of 3 miles per hour, we can express her motion as
Solve the equation for π¦. οΊ
ο§
You may need to review the definition of average speed as well: Average speed is the distance (or area, pages typed, etc.) divided by the time interval spent moving that distance (or painting the given area, or typing a specific number of pages, etc.)
Suppose Erika walked a distance of π¦ miles in π₯ hours. Then her average speed in the time interval from 0 to π₯
hours is ο§
Scaffolding:
π¦ π₯
π¦ π₯
= 3.
=3
π¦ = 3π₯
We can use the equation to develop a table of values where π₯ is the number of hours Erika walks and π¦ is the distance she walks in miles. Complete the table for the given π₯-values. π₯
0
π
2
π
1 3
Lesson 22: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
π¦
π π
Constant Rates Revisited 11/8/13
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Lesson 22
NYS COMMON CORE MATHEMATICS CURRICULUM
MP.7
ο§
In terms of π₯ and π¦, π¦ = 3π₯ is a linear equation in two variables. Its graph is a line.
ο§
What is the meaning of the 3 in the equation π¦ = 3π₯? οΊ
ο§
The 3 is the slope of the line. It represents the rate at which Erika walks.
We can prove that her motion is constant, i.e., she walks at a constant speed, by showing that the slope is 3 over any time interval that she walks: Assume Erika walks π¦1 miles in the time interval from 0 to π₯1 , and she walks π¦2 miles in the time interval from 0 to π₯2 . Her average distance between time interval π₯1 and π₯2 =
π¦2βπ¦1 π₯2βπ₯1
8β’4
Scaffolding: Explain that we are looking at the distance traveled between times π₯1 and π₯2 ; therefore, we must subtract the distances traveled to represent the total distance traveled in π₯2 β π₯1 . Concrete numbers and a diagram like the following may be useful as well:
We know that π¦ = 3π₯; therefore, π¦2 = 3π₯2 and π¦1 = 3π₯1 . Then by substitution we have 3π₯2 β 3π₯1 = π₯2 β π₯1 =
3(π₯2 β π₯1 ) π₯2 β π₯1
=3
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NYS COMMON CORE MATHEMATICS CURRICULUM
ο§
ο§
8β’4
Therefore, Erikaβs average speed in the time interval between π₯1 and π₯2 is 3 mph. Since π₯1 and π₯2 can be any two times, then the average speed she walks over any time interval is 3 mph and we have confirmed that she walks at a constant speed of 3 mph. We can also show that her motion is constant by looking at the slope between different points on the graph:
Are triangles β³ π΄ππ΅, β³ π΄β² ππ΅β² , and β³ π΄β²β²ππ΅β²β² similar? οΊ
ο§
Lesson 22
Yes, because each triangle has a common angle, β π΄ππ΅ and a right angle. By the AA criterion the triangles are similar.
We used similar triangles before to show that the slope of a line would be the same between any two points. Since we have similar triangles, we can conclude that the rate at which Erika walks is constant because the ratio of the corresponding sides will be equal to a constant, i.e., the scale factor of dilation. For example, when we compare the ratio of corresponding sides for triangles β³ π΄β² ππ΅β² and β³ π΄β²β²ππ΅β²β² we see that 9 3 = 6 2
Make clear to students that the ratio of corresponding sides will not be equal to the slope of the line; rather, the ratios will be equal to one another and the scale factor of dilation. ο§
ο§
Or we could compare the ratio of corresponding side lengths for triangles β³ AOB and β³ Aβ² OB β² : 6 2 = 3 1 Again, because the ratio of corresponding sides are equal, we know that the rate at which Erika walks is constant.
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Lesson 22
NYS COMMON CORE MATHEMATICS CURRICULUM
8β’4
Example 2 (5 minutes) MP.4
ο§
A faucet leaks at a constant rate of 7 gallons per hour. Suppose π¦ gallons leak in π₯ hours from 6:00 a.m. Express the situation as a linear equation in two variables.
ο§
If we say the leak began at 6:00 a.m. and the total number of gallons leaked over the time interval from 6:00 a.m. to π₯ hours is π¦, then we know the average rate of the leak is
ο§
π¦ π₯
.
Since we know that the faucet leaks at a constant rate, then we can write the linear equation in two variables as
π¦ π₯
= 7, which can then be transformed to π¦ = 7π₯.
ο§
Again, take note of the fact that the number that represents slope in the equation is the same as the constant rate of water leaking.
ο§
Another faucet leaks at a constant rate and the table below shows the number of gallons π¦, that leak in π₯ hours. Hours (π₯)
Gallons (π¦)
4
26
2
13
7
45.5
10 ο§
How can we determine the rate at which this faucet leaks? οΊ
ο§
65
If these were points on a graph, we could find the slope between them.
Using what you know about slope, determine the rate the faucet leaks. οΊ
Sample student work: 26 β 13 4β2 13 = 2 = 6.5
π=
ο§
The number of gallons that leak from each faucet is dependent on the amount of time the faucet leaks. The information provided for each faucetβs rate of change, i.e., slope allows us to answer a question, such as the following: Which faucet has the worse leak? That is, which faucet leaks more water over a given time interval? οΊ
The first faucet has the worse leak because the rate is greater, 7 compared to 6.5.
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Lesson 22
NYS COMMON CORE MATHEMATICS CURRICULUM
8β’4
Example 3 (4 minutes) ο§
The graph below represents the constant rate at which Train A travels.
ο§
What is the constant rate of travel for Train A? οΊ
ο§
We know that the constant rate of travel is the same as the slope. On the graph, we can see that the train is traveling 60 miles every hour. That means that the slope of the line is 60.
Train B travels at a constant rate. The train travels at an average rate of 95 miles every one and a half hours. We want to know which train is traveling at a greater speed. Letβs begin by writing a linear equation that represents the constant rate of Train B. If π¦ represents the total distance traveled over any time period π₯, then 95 π¦ = 1.5 π₯ 95 π₯ 1.5 1 π¦ = 63 π₯ 3 Which train is traveling at a greater speed? Explain. π¦=
MP.4
ο§
Provide students time to talk to their partners about how to determine which train is traveling at a greater speed. οΊ
ο§
Train B is traveling at a greater speed. The graph provided the information about the constant rate of Train A, that is, Train A travels at a constant rate of 60 miles per hour. The given rate for Train B, the 1 3
1 3
slope of the graph for Train B, is equal to 63 . Since 63 > 60, Train B is traveling at a greater speed.
Why do you think the strategy of comparing each rate of change allows us to determine which train is travelling at a greater speed? οΊ
The distance that each train travels is proportional to the amount of time the train travels. Therefore, if we can describe the rate of change as a number, i.e. slope, and compare the numbers, we can determine which train travels at a greater speed.
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Lesson 22
NYS COMMON CORE MATHEMATICS CURRICULUM
8β’4
Example 4 (5 minutes) ο§
The graph below represents the constant rate at which Kristina can paint.
ο§
Her sister Tracee paints at an average rate of 45 square feet in 12 minutes. Assuming Tracee paints at a constant rate, determine which sister paints faster.
Provide students time to determine the linear equation that represents Traceeβs constant rate and then discuss with their partner who is the faster painter. οΊ
If we let π¦ represent the total area Tracee paints in π₯ minutes, then her constant rate is 45 π¦ = 12 π₯ 45 π₯=π¦ 12
We need to compare the slope of the line for Kristina with the slope of the equation for Tracee. The slope of the line is 4 and the slope in the equation that represents Traceeβs rate is Kristina paints at a faster rate.
ο§
45 12
. Since 4 >
45 , 12
How does the slope provide the information we need to answer the question about which sister paints faster? οΊ
The slope describes the rate of change for proportional relationships. If we know which rate of change is greater, then we can determine which sister paints faster.
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Lesson 22
NYS COMMON CORE MATHEMATICS CURRICULUM
8β’4
Example 5 (5 minutes) ο§
The graph below represents the constant rate of watts of energy produced from a single solar panel produced by Company A.
ο§
Company B offers a solar panel that produces energy at an average rate of 325 watts in 2.6 hours. Assuming solar panels produce energy at a constant rate, determine which company produces more efficient solar panels (solar panels that produce more energy per hour).
Provide students time to work with their partners to answer the question. οΊ
If we let π¦ represent the energy produced by a solar panel made by Company B in π₯ minutes, then the constant rate is 325 π¦ = 2.6 π₯ 325 π₯=π¦ 2.6 125π₯ = π¦
We need to compare the slope of the line for Company A with the slope in the equation that represents the rate for Company B. The slope of the line representing Company A is 120, and the slope of the line representing Company B is equal to 125. Since 125 > 120, Company B produces the more efficient solar panel.
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Lesson 22
NYS COMMON CORE MATHEMATICS CURRICULUM
8β’4
Exercises 1β5 (10 minutes) Students work in pairs or small groups to complete Exercises 1β5. Discuss Exercise 5 as it generalizes a strategy for comparing proportional relationships. Exercises 1.
Peter paints a wall at a constant rate of π square-feet per minute. Assume he paints an area π, in square feet after π minutes. a.
Express this situation as a linear equation in two variables. π π = π π
π = ππ b.
Graph the linear equation.
c.
Using the graph or the equation, determine the total area he paints after π minutes, π hours, and π hours.
π π
Note that the units are in minutes and hours.
In π minutes, he paints ππ square feet. π π
In π hours, he paints πππ square feet.
In π hours, he paints
π = π(πππ) = πππ
πππ square feet.
Lesson 22: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
π = π(ππ) = πππ
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Lesson 22
NYS COMMON CORE MATHEMATICS CURRICULUM
2.
8β’4
The graph below represents Nathanβs constant rate of walking.
a.
b.
Nicole just finished a π mile walkathon. It took her π. π hours. Assume she walks at a constant rate. Let π represent the distance Nicole walks in π hours. Describe Nicoleβs walking at a constant rate as a linear equation in two variables.
Who walks at a greater speed? Explain.
π π = π π. π ππ π= π π
Nathan walks at a greater speed. The slope of the graph for Nathan is π, and the slope or rate for Nicole is ππ
3.
a.
π
. When you compare the slopes, you see that π >
ππ . π
Susan can type π pages of text in ππ minutes. Assuming she types at a constant rate, write the linear equation that represents the situation. Let π represent the total number of pages Susan can type in π minutes. We can write
b.
π π
=
π
ππ
π π
and π = π.
The table of values below represents the number of pages that Anne can type, π, in π minutes. Assume she types at a constant rate. Minutes (π) π 5 8 10
Who types faster? Explain.
Pages Typed (π) π
ππ π ππ π ππ π
Anne types faster. Using the table we can determine the slope that represents Anneβs constant rate of typing π
π
is . The slope or rate for Nicole is . When you compare the slopes you see that π
π
Lesson 22: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
π π
π
> . π
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Lesson 22
NYS COMMON CORE MATHEMATICS CURRICULUM
4.
a.
Phil can build π birdhouses in π days. Assuming he builds birdhouses at a constant rate, write the linear equation that represents the situation. Let π represent the total number of birdhouses Phil can build in π days. We can write
b.
8β’4
π π
=
π π
π π
and π = π.
The graph represents Karlβs constant rate of building the same kind of birdhouses. Who builds birdhouses faster? Explain. Karl can build birdhouses faster. The slope of the graph for Karl is π
π π
, and the slope or rate of
change for Phil is . When you compare the slopes,
5.
π π
π
> . π
π
Explain your general strategy for comparing proportional relationships. When comparing proportional relationships we look specifically at the rate of change for each situation. The relationship with the greater rate of change will end up producing more, painting a greater area, or walking faster when compared to the same amount of time with the other proportional relationship.
Closing (4 minutes) Summarize, or ask students to summarize, the main points from the lesson: ο§
We know how to write a constant rate problem as a linear equation in two variables.
ο§
We know that in order to determine who has a greater speed (or is faster at completing a task), we need only to compare the slopes (rates of change) that apply to two situations. Whichever has the greater slope has the greater rate. We can determine rate (slope) from an equation, a graph, or a table.
Lesson Summary Problems involving constant rate can be expressed as linear equations in two variables. When given information about two proportional relationships, determine which is less or greater by comparing their slopes (rates of change).
Exit Ticket (4 minutes)
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Lesson 22
NYS COMMON CORE MATHEMATICS CURRICULUM
Name
8β’4
Date
Lesson 22: Constant Rates Revisited Exit Ticket 1.
Water flows out of Pipe A at a constant rate. Pipe A can fill 3 buckets of the same size in 14 minutes. a.
Write a linear equation that represents the situation.
The graph below represents the rate at which Pipe B can fill the same sized buckets.
b.
Which pipe fills buckets faster? Explain.
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Lesson 22
NYS COMMON CORE MATHEMATICS CURRICULUM
8β’4
Exit Ticket Sample Solutions 1.
Water flows out of Pipe A at a constant rate. Pipe A can fill π buckets of the same size in ππ minutes. a.
Write a linear equation that represents the situation.
Let π represent the total number of buckets that Pipe A can fill in x minutes. We can write π π = π. ππ
π π
=
π
ππ
and
The graph below represents the rate at which Pipe B can fill the same sized buckets.
b.
Which pipe fills buckets faster? Explain. π
Pipe A fills the same sized buckets faster than Pipe B. The slope of the graph for Pipe B is , the slope or rate for Pipe A is
π
ππ
. When you compare the slopes you see that
π
ππ
π
π
> . π
Problem Set Sample Solutions Students practice writing constant rate problems as linear equations in two variables. Students determine which of two proportional relationships is greater. 1.
a.
Train A can travel a distance of πππ miles in π hours. Assuming the train travels at a constant rate, write the linear equation that represents the situation. Let π represent the total number of miles Train A travels in π minutes. We can write
b.
π π
=
πππ π
and π =
πππ π. π
The graph represents the constant rate of travel for Train B. Which train is faster? Explain. Train B is faster than Train A. The slope or rate for Train A is is
πππ π
πππ π
, and the slope of the line for Train B
. When you compare the slopes,
Lesson 22: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
πππ π
>
πππ π
.
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Lesson 22
NYS COMMON CORE MATHEMATICS CURRICULUM
2.
a.
Natalie can paint ππ square feet in π minutes. Assuming she paints at a constant rate, write the linear equation that represents the situation. Let π represent the total square feet Natalie can paint in π minutes. We can write
b.
8β’4
π π
=
ππ π
and π =
ππ π. π
The table of values below represents Stevenβs constant rate of painting. Minutes (π)
Area Painted (π)
π
ππ ππ π
5 6
ππ ππ π
8
Who paints faster? Explain.
Natalie paints faster. Using the table of values I can find the slope that represents Stevenβs constant rate of painting:
3.
a.
ππ π
. The slope or rate for Natalie is
ππ π
. When you compare the slopes you see that
π
>
ππ π
.
Bianca can run π miles in ππ minutes. Assuming she runs at a constant rate, write the linear equation that represents the situation. Let π represent the total number of miles Bianca can run in π minutes. We can write
b.
ππ
π π
=
π
ππ
and π =
π π. ππ
The graph below represents Cynthiaβs constant rate of running.
Who runs faster? Explain. π
Cynthia runs faster. The slope of the graph for Cynthia is , and the slope or rate for Nicole is compare the slopes you see that
Lesson 22: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
π π
>
π
ππ
.
π
π
ππ
. When you
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Lesson 22
NYS COMMON CORE MATHEMATICS CURRICULUM
4.
a.
Geoff can mow an entire lawn of πππ square feet in ππ minutes. Assuming he mows at a constant rate, write the linear equation that represents the situation. Let π represent the total number of square feet Geoff can mow in π minutes. We can write
π = πππ. b.
8β’4
π π
=
πππ ππ
and
The graph represents Markβs constant rate of mowing a lawn. Who mows faster? Explain. Geoff mows faster. The slope of the graph for Mark is is
πππ ππ
ππ π
= π, and the slope or rate for Geoff
= ππ. When you compare the slopes
you see that ππ > π.
5.
a.
Juan can walk to school, a distance of π. ππ miles, in π minutes. Assuming he walks at a constant rate, write the linear equation that represents the situation. Let π represent the total distance in miles that Juan can walk in π minutes. We can write π π = π. ππ
b.
π π
=
π.ππ π
and
The graph below represents Lenaβs constant rate of walking.
Who walks faster? Explain. π
Lena walks faster. The slope of the graph for Lena is , and the slope of the equation for Juan is When you compare the slopes you see that
Lesson 22: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
π π
>
π
ππ
.
π
π.ππ π
or
π
ππ
.
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Lesson 23
NYS COMMON CORE MATHEMATICS CURRICULUM
8β’4
Lesson 23: The Defining Equation of a Line Student Outcomes ο§
Students know that two equations in the form of ππ₯ + ππ¦ = π and πβ²π₯ + πβ²π¦ = πβ² graph as the same line
when ο§
πβ² πβ² πβ² = = and at least one of π or π is nonzero. π π π
Students know that the graph of a linear equation ππ₯ + ππ¦ = π, where π, π, and π are constants and at least one of π or π is nonzero, is the line defined by the equation ππ₯ + ππ¦ = π.
Lesson Notes
Following the Exploratory Challenge is a discussion that presents a theorem about the defining equation of a line (page 351) and then a proof of the theorem. The proof of the theorem is optional. The discussion can end with the theorem and in place of the proof, students can complete Exercises 4β8 beginning on page 353. Whether you choose to discuss the proof or have students complete Exercises 4β8, it is important that students understand that two equations that are written differently can be the same and their graph will be the same line. This reasoning becomes important when you consider systems of linear equations. In order to make sense of βinfinitely many solutionsβ to a system of linear equations, students must know that equations that might appear different can have the same graph and represent the same line. Further, students should be able to recognize when two equations define the same line without having to graph each equation, which is the goal of this lesson. Students need graph paper to complete the Exploratory Challenge.
Classwork Exploratory Challenge/Exercises 1β3 (20 minutes) Students need graph paper to complete the Exercises in the Exploratory Challenge. Students complete Exercises 1β3 in pairs or small groups. Exploratory Challenge/Exercises 1β3 1.
Graph the equation ππ + ππ = ππ using intercepts. Then answer parts (a)β(f) that follow.
π(π) + ππ = ππ ππ = ππ π=π
The π-intercept is (π, π). ππ + π(π) = ππ ππ = ππ π=π
The π-intercept is (π, π).
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Lesson 23
NYS COMMON CORE MATHEMATICS CURRICULUM
a. b.
8β’4
Graph the equation π = βππ + π on the same coordinate plane.
What do you notice about the graphs of ππ + ππ = ππ and π = βππ + π? Why do you think this is so?
The graphs of the equations produce the same line. Both equations go through the same two points so they are the same line.
c.
Rewrite π = βππ + π in standard form. π = βππ + π
ππ + π = π d.
Identify the constants, π, π, π of the equation in standard form from part (c). π = π, π = π, and π = π.
e.
Identify the constants of the equation ππ + ππ = ππ. Note them as πβ² , πβ², and πβ². πβ² = π, πβ² = π, and πβ² = ππ
f.
What do you notice about πβ² π = = π, π π
πβ² πβ² π
,
π
, and
πβ² π
?
πβ² π πβ² ππ = = π, πππ
= =π π π π π
Each fraction is equal to the number π. 2.
π π
Graph the equation π = π + π using the π-intercept and the slope. Then answer parts (a)β(f) that follow. a.
Graph the equation ππ β ππ = βππ using intercepts on the same coordinate plane. π(π) β ππ = βππ βππ = βππ π=π
The π-intercept is (π, π). ππ β π(π) = βππ
ππ = βππ π = βπ
The π-intercept is (βπ, π). b.
π π
What do you notice about the graphs of π = π + π and ππ β ππ = βππ? Why do you think this is so?
The graphs of the equations produce the same line. Both equations go through the same two points, so they are the same line.
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Lesson 23
NYS COMMON CORE MATHEMATICS CURRICULUM
c.
π π
8β’4
Rewrite π = π + π in standard form. π π+π π π οΏ½π = π + ποΏ½ π π ππ = π + π βπ + ππ = π βπ(βπ + ππ = π) π β ππ = βπ
d.
π=
Identify the constants, π, π, π of the equation in standard form from part (c). π = π, π = βπ, and π = βπ.
e.
Identify the constants of the equation ππ β ππ = βππ. Note them as πβ² , πβ², and πβ². πβ² = π, πβ² = βπ, and πβ² = βππ
f.
What do you notice about β²
π π = = π, π π
β²
πβ² πβ² π
,
π βπ = = π, π βπ
π
, and
πβ² π
?
πππ
Each fraction is equal to the number π. 3.
πβ² βππ = =π π βπ
π π π Rewrite π = π β π in standard form. π
The equations π = π β π and ππ β ππ = ππ graph as the same line. a.
π πβπ π π οΏ½π = π β ποΏ½ π π ππ = ππ β ππ βππ + ππ = βππ βπ(βππ + ππ = βππ) ππ β ππ = ππ
b.
π=
Identify the constants, π, π, π, of the equation in standard form from part (a). π = π, π = βπ, and π = ππ.
c.
Identify the constants of the equation ππ β ππ = ππ. Note them as πβ² , πβ², and πβ². πβ² = π, πβ² = βπ, and πβ² = ππ
d.
What do you notice about πβ² π = = π, π π
πβ² πβ² π
,
πβ² βπ = = π, π βπ
π
, and
Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
π
πππ
Each fraction is equal to the number π.
Lesson 23: Date:
πβ²
? πβ² ππ = =π π ππ
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Lesson 23
NYS COMMON CORE MATHEMATICS CURRICULUM
e.
8β’4
You should have noticed that each fraction was equal to the same constant. Multiply that constant by the standard form of the equation from part (a). What do you notice? ππ β ππ = ππ
π(ππ β ππ = ππ) ππ β ππ = ππ
After multiplying the equation from part (a) by π, I noticed that it is the exact same equation that was given.
Discussion (15 minutes) Following the statement of the theorem is an optional proof of the theorem. Below the proof are Exercises 4β8 (beginning on page 353) that can be completed instead of the proof. ο§
What did you notice about the equations you graphed in each of the Exercises 1β3? οΊ
ο§ MP.8
In each case, the equations graphed as the same line.
What you observed in Exercises 1β3 can be summarized in the following theorem: Theorem. Suppose π, π, π, πβ² , π β² , and πβ² are constants, where at least one of π or π is nonzero, and one of πβ² or πβ² is nonzero . (1) If there is a non-zero number π , so that πβ² = π π, π β² = π π, and π β² = π π, then the graphs of the equations ππ₯ + ππ¦ = π and πβ² π₯ + π β² π¦ = πβ² are the same line.
(2) If the graphs of the equations ππ₯ + ππ¦ = π and πβ² π₯ + π β² π¦ = πβ² are the same line, then there exists a nonzero number π so that πβ² = π π, π β² = π π, and π β² = π π.
The optional part of the discussion begins here. ο§
We want to show that (1) is true. We need to show that the equations ππ₯ + ππ¦ = π and πβ² π₯ + π β² π¦ = πβ² are the same. What information are we given in (1) that will be useful in showing that the two equations are the same? οΊ
ο§
We are given that πβ² = π π, π β² = π π, and π β² = π π. We can use substitution in the equation πβ² π₯ + π β² π¦ = πβ² since we know what πβ² ,π β² , and πβ² are equal to.
Then by substitution, we have
πβ² π₯ + π β² π¦ = π β² π ππ₯ + π ππ¦ = π π
By the distributive property Divide both sides of the equation by π β²
β²
π (ππ₯ + ππ¦) = π π ππ₯ + ππ¦ = π
Which is what we wanted π π₯ + π π¦ = πβ² to be equal to. Therefore, we have proved (1).
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Lesson 23
NYS COMMON CORE MATHEMATICS CURRICULUM
ο§
To prove (2), we will assume that π, π β 0, that is, we are not dealing with horizontal or vertical lines. Proving (2) will require us to rewrite the given equations, ππ₯ + ππ¦ = π and πβ² π₯ + π β² π¦ = πβ², in slope-intercept form. Rewrite the equations. οΊ
οΊ
ππ₯ + ππ¦ = π
ππ¦ = βππ₯ + π π π π π¦=β π₯+ π π π π π π¦=β π₯+ π π
πβ² π₯ + π β² π¦ = πβ²
π π
π β² π¦ = βπβ² π₯ + π β² πβ² πβ² πβ² π¦ = β π₯ + πβ² πβ² πβ² πβ² πβ² π¦ = β β²π₯+ β² π π
π πβ² πβ² as (A) and π¦ = β π₯ + as (B). π πβ² πβ²
ο§
We will refer to π¦ = β π₯ +
ο§
What are the slopes of (A) and (B)? οΊ
ο§
π π
πβ² πβ²
The slope of (A) is β and the slope of (B) is β .
Since we know that the two lines are the same, we know the slopes must be the same. πβ² π =β πβ² π and when we multiply both sides of the equation by (β1), we have β
π πβ² = π πβ²
By using the multiplication property of equality we can rewrite πβ² πβ² = π π Notice that this proportion is equivalent to the original form. ο§
8β’4
π π
π π
What are the π¦-intercepts of π¦ = β π₯ + (A) and π¦ = β οΊ
The π¦-intercept of (A) is
Lesson 23: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
π
π π
=
Scaffolding: πβ² πβ²
Students may need to see the intermediate steps in the as
rewriting of
β²
π πβ² πβ² π = as = . π πβ² π π
πβ² πβ² π₯ + (B)? πβ² πβ² πβ²
, and the π¦-intercept of (B) is πβ² . π
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Lesson 23
NYS COMMON CORE MATHEMATICS CURRICULUM
ο§
Because we know that the two lines are the same, the π¦-intercepts will be the same. π πβ² = π πβ² We can rewrite the proportion.
Using the transitive property and the fact that
ο§
8β’4
πβ² π
=
β²
πβ² π β² = π π πβ² π
and β²
πβ² π
=
πβ² π
, we can make the following statement:
π πβ² π = = π π π Therefore, all three ratios are equal to the same number. Let this number be π . πβ² πβ² πβ² = π , = π , = π π π π We can rewrite this to state that πβ² = π π, π β² = π π, and π β² = ππ . Therefore, (2) is proved.
When two equations are the same, i.e., they graph as the same line, we say that any one of those equations is the defining equation of the line.
Exercises 4β8 (15 minutes) Students complete Exercises 4β8 independently or in pairs. Consider having students share the equations they write for each exercise while others in the class verify which equations graph as the same line. Exercises 4β8 4.
Write three equations that would graph as the same line as the equation ππ + ππ = π.
Answers will vary. Verify that students have multiplied π, π, and π by the same constant when they write the new equation.
5.
π π
Write three equations that would graph as the same line as the equation π β ππ = .
Answers will vary. Verify that students have multiplied π, π, and π by the same constant when they write the new equation. 6.
Write three equations that would graph as the same line as the equation βππ + ππ = βπ.
Answers will vary. Verify that students have multiplied π, π, and π by the same constant when they write the new equation.
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Lesson 23
NYS COMMON CORE MATHEMATICS CURRICULUM
7.
8β’4
Write at least two equations in the form ππ + ππ = π that would graph as the line shown below.
Answers will vary. Verify that students have the equation βπ + ππ = βπ in some form. 8.
Write at least two equations in the form ππ + ππ = π that would graph as the line shown below.
Answers will vary. Verify that students have the equation ππ + ππ = π in some form.
Closing (5 minutes) Summarize, or ask students to summarize, the main points from the lesson: ο§
We know that when two equations graph as the same line it is because they are the same equation in different forms.
ο§
We know that even if the equations that graph as the same line look different (i.e., different constants or different forms) that any one of those equations can be referred to as the defining equation of the line.
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NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 23
8β’4
Lesson Summary Two equations that graph as the same line are said to define the same line. Two equations that define the same line are the same equation, just in different forms. The equations may look different (different constants, different coefficients, or different forms). When two equations are written in standard form, ππ + ππ = π and πβ² π + πβ² π = πβ², they define the same line when
πβ² π
=
πβ² π
=
πβ² π
is true.
Exit Ticket (5 minutes)
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Lesson 23
NYS COMMON CORE MATHEMATICS CURRICULUM
Name
8β’4
Date
Lesson 23: The Defining Equation of a Line Exit Ticket 1.
Do the equations β16π₯ + 12π¦ = 33 and β4π₯ + 3π¦ = 8 graph as the same line? Why or why not?
2.
Given the equation 3π₯ β π¦ = 11, write another equation that will graph as the same line. Explain why.
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Lesson 23
NYS COMMON CORE MATHEMATICS CURRICULUM
8β’4
Exit Ticket Sample Solutions 1.
Do the equations βπππ + πππ = ππ and βππ + ππ = π graph as the same line? Why or why not?
No, in the first equation π = βππ, π = ππ, and π = ππ, and in the second equation πβ = βπ, πβ = π, and πβ = π. Then, βπ π πβ² π π πβ² = = , = = , π βππ π π ππ π πππ
π π πβ² = = . π ππ ππ
Since each fraction does not equal the same number, then they do not graph to the same line. 2.
Given the equation ππ β π = ππ, write another equation that will graph as the same line. Explain why.
Answers will vary. Verify that the student has written an equation that defines the same line by showing that the fractions
πβ² π
=
πβ² π
=
πβ² π
= π, where π is some constant.
Problem Set Sample Solutions Students practice identifying pairs of equations as the defining equation of a line or two distinct lines. 1.
Do the equations π + π = βπ and ππ + ππ = βπ define the same line? Explain.
Yes, these equations define the same line. When you compare the constants from each equation you get: πβ² π = = π, π π
πβ² π = = π, π π
πππ
When I multiply the first equation by π, I get the second equation:
πβ² βπ = =π π βπ
(π + π = βπ)π ππ + ππ = βπ
Therefore, these equations define the same line.
2.
π π
Do the equations π = β π + π and πππ + ππ = ππ define the same line? Explain.
Yes, these equations define the same line. When you rewrite the first equation in standard form you get: π π=β π+π π π οΏ½π = β π + ποΏ½ π π ππ = βππ + π ππ + ππ = π
When you compare the constants from each equation you get: πβ² ππ = = π, π π
πβ² π = = π, π π
πππ
When I multiply the first equation by π, I get the second equation:
πβ² ππ = =π π π
(ππ + ππ = π)π πππ + ππ = ππ
Therefore, these equations define the same line.
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Lesson 23
NYS COMMON CORE MATHEMATICS CURRICULUM
3.
8β’4
Write an equation that would define the same line as ππ β ππ = π.
Answers will vary. Verify that the student has written an equation that defines the same line by showing that the fractions
4.
πβ² π
πβ²
=
π
=
πβ² π
= π, where π is some constant.
Challenge: Show that if the two lines given by ππ + ππ = π and πβ² π + πβ² π = πβ² are the same when π = π (vertical lines), then there exists a non-zero number π, so that πβ² = ππ, πβ² = ππ, and πβ² = ππ. When π = π, then the equations are ππ = π and πβ² π = πβ². We can rewrite the equations as π = Because the equations graph as the same line, then we know that
and we can rewrite those fractions as
π πβ² = π πβ² π β² πβ² = π π
These fractions are equal to the same number. Let that number be π. Then β²
β²
π = ππ and π = ππ, as desired.
5.
πβ² π
= π and
πβ² π
π πβ² and π = β² . π π
= π, and therefore,
Challenge: Show that if the two lines given by ππ + ππ = π and πβ² π + πβ² π = πβ² are the same when π = π (horizontal lines), then there exists a non-zero number π, so that πβ² = ππ, πβ² = ππ, and πβ² = ππ. When π = π, then the equations are ππ = π and πβ²π = πβ². We can rewrite the equations as π = Because the equations graph as the same line, then we know that their slopes are the same.
We can rewrite the proportion.
π πβ² = π πβ² π β² πβ² = π π
These fractions are equal to the same number. Let that number be π. Then
and πβ² = ππ.
Lesson 23: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
πβ² π
= π and
π πβ² and π = β² . π π
πβ² = π. Therefore, πβ² = ππ π
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New York State Common Core
8
Mathematics Curriculum
GRADE
GRADE 8 β’ MODULE 4
Topic D:
Systems of Linear Equations and Their Solutions 8.EE.B.5, 8.EE.C.8 Focus Standards:
Instructional Days:
8.EE.B.5
Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways.
8.EE.C.8
Analyze and solve pairs of simultaneous linear equations. a.
Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously.
b.
Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. For example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6.
c.
Solve real-world and mathematical problems leading to two linear equations in two variables. For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair.
7
Lesson 24: Introduction to Simultaneous Equations (P)
1
Lesson 25: Geometric Interpretation of the Solutions of a Linear System (E) Lesson 26: Characterization of Parallel Lines (S) Lesson 27: Nature of Solutions of a System of Linear Equations (P) Lesson 28: Another Computational Method of Solving a Linear System (P) Lesson 29: Word Problems (P) Lesson 30: Conversion Between Celsius and Fahrenheit (M) 1
Lesson Structure Key: P-Problem Set Lesson, M-Modeling Cycle Lesson, E-Exploration Lesson, S-Socratic Lesson
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359
Topic D
NYS COMMON CORE MATHEMATICS CURRICULUM
8β’4
Topic D introduces students to systems of linear equations by comparing distance-time graphs to determine which of two objects has greater speed (8.EE.B.5, 8.EE.C.8c). Lessons 25β27 expose students to the possibilities for solutions of a system of linear equations. In Lesson 25, students graph two linear equations on a coordinate plane and identify the point of intersection of the two lines as the solution to the system (8.EE.C.8a). Next, students look at systems of equations that graph as parallel lines (8.EE.C.8b). In Lesson 26, students learn that a system can have no solutions because parallel lines do not have a point of intersection (8.EE.C.8b). Lesson 27 continues this thinking with respect to systems that have infinitely many solutions (8.EE.C.8b). In Lesson 28, students learn how to solve a system of equations using computational methods like elimination and substitution (8.EE.C.8b). In Lesson 29, students must use all of the skills of the module to transcribe written statements into a system of linear equations, find the solution(s) if it exists, and then verify that it is correct. Lesson 30 is an application of what students have learned about linear equations. Students develop a linear equation that represents the conversion between temperatures in Celsius to temperatures in Fahrenheit.
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360
Lesson 24
NYS COMMON CORE MATHEMATICS CURRICULUM
8β’4
Lesson 24: Introduction to Simultaneous Equations Student Outcomes ο§
Students know that a system of linear equations, also known as simultaneous equations, is when two or more equations are involved in the same problem and work must be completed on them simultaneously. Students also learn the notation for simultaneous equations.
ο§
Students compare the graphs that comprise a system of linear equations, in the context of constant rates, to answer questions about time and distance.
Lesson Notes In the Opening Exercise students complete Exercises 1β3 as an introduction to simultaneous linear equations in a familiar context. Example 1 demonstrates what happens to the graph of a line when there is change in the circumstances involving time with constant rate problems. It is in preparation for the examples that follow. It is not necessary that Example 1 be shown to students, but it is provided as a scaffold. If Example 1 is used, consider skipping Example 3.
Classwork Opening Exercises 1β3 (5 minutes) Students complete the opening activity in pairs. Once students are finished, continue with the discussion about systems of linear equations. Opening Exercises 1β3 1.
Derek scored ππ points in the basketball game he played and not once did he go to the free throw line. That means that Derek scored two point shots and three point shots. List as many combinations of two and three pointers as you can that would total ππ points. Number of Two-Pointers
Number of Three-Pointers
ππ
π
π
ππ
π
π
ππ π π
π π π
Write an equation to describe the data. Let π represent the number of π pointers and π represent the number of π pointers. ππ = ππ + ππ
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Lesson 24
NYS COMMON CORE MATHEMATICS CURRICULUM
2.
8β’4
Derek tells you that the number of two-point shots that he made is five more than the number of three-point shots. How many combinations can you come up with that fit this scenario? (Donβt worry about the total number of points.)
Number of Two-Pointers
Number of Three-Pointers
π
π
π
π
π
π
π
π
ππ
π
ππ
π
Write an equation to describe the data. Let π represent the number of two-pointers and π represent the number of three-pointers. π=π+π
3.
Which pair of numbers from your table in Exercise 2 would show Derekβs actual score of ππ points? The pair π and π would show Derekβs actual score of ππ points.
Discussion (5 minutes) ο§
There are situations where we need to work with two linear equations simultaneously. Hence, the phrase simultaneous linear equations. Sometimes a pair of linear equations is referred to as a system of linear equations.
ο§
The situation with Derek can be represented as a system of linear equations: Let π₯ represent the number of two-pointers and π¦ represent the number of three-pointers, then 2π₯ + 3π¦ = 30 οΏ½ π₯ =5+π¦
ο§
The notation for simultaneous linear equations letβs us know that we are looking for the ordered pair (π₯, π¦) that makes both equations true. That point is called the solution to the system.
ο§
Just like equations in one variable, some systems of equations have exactly one solution, no solution, or infinitely many solutions. This is a topic for later lessons.
ο§
Ultimately our goal is to determine the exact location on the coordinate plane where the graphs of the two linear equations intersect, giving us the ordered pair (π₯, π¦) that is the solution to the system of equations. This too is a topic for a later lesson.
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Lesson 24
NYS COMMON CORE MATHEMATICS CURRICULUM
8β’4
ο§
We can graph both equations on the same coordinate plane:
ο§
Because we are graphing two distinct lines on the same graph, we identify the lines in some manner. In this case, by the equations.
ο§
Note the point of intersection. Does it satisfy both equations in the system? οΊ
The point of intersection of the two lines is (9, 4).
2(9) + 3(4) = 30 18 + 12 = 30 30 = 30
9= 4+5 9=9
ο§
Yes, π₯ = 9 and π¦ = 4 satisfies both equations of the system.
Therefore, Derek made 9 two-point shots and 4 three-point shots.
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Lesson 24
NYS COMMON CORE MATHEMATICS CURRICULUM
8β’4
Example 1 (6 minutes) ο§
Pia types at a constant rate of 3 pages every 15 minutes. Suppose she types π¦ pages in π₯ minutes. Piaβs 1 5
constant rate can be expressed as the linear equation π¦ = π₯. ο§
We can complete a table to make sure that our graph is correct: Number of Minutes (π₯)
Pages Typed (π¦)
10
2
0 5
15 20 25
0 1 3 4 5
ο§
Here is the graph that represents Piaβs constant rate of typing.
ο§
Pia typically begins work at 8:00 a.m. every day. On our graph, her start time is reflected as the origin of the graph (0, 0), that is, zero minutes worked and zero pages typed. For some reason, she started working 5 minutes earlier today. How can we reflect her extra time working today on our graph? οΊ
The π₯-axis represents the time worked so we need to do something on the π₯-axis to reflect the additional 5 minutes of work.
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Lesson 24
NYS COMMON CORE MATHEMATICS CURRICULUM
ο§
1 5
If we translate the graph of π¦ = π₯ to the right 5 units to reflect the βadditionalβ 5 minutes of work, then we
have
ο§
Does a translation of 5 units to the right reflect her working an additional 5 minutes? οΊ
No, it makes it look like she got nothing done the first 5 minutes she was at work.
ο§
Letβs see what happens when we translate 5 units to the left.
ο§
Does a translation of 5 units to the left reflect her working an additional 5 minutes? οΊ
ο§
8β’4
Yes, it shows that she was able to type 1 page by the time she normally begins work.
What is the equation that represents the graph of the translated line? οΊ
1 5
The equation that represents the graph of the red line is π¦ = π₯ + 1.
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Lesson 24
NYS COMMON CORE MATHEMATICS CURRICULUM
8β’4
ο§
Note again, that even though the graph has been translated, the slope of the line is still equal to the constant rate of typing.
ο§
When you factor out
1 5
1
from π₯ + 1, we can better see the βadditionalβ 5 minutes of work time:
5 1 π¦ = (π₯ + 5). Pia typed for an additional 5 minutes so it makes sense that we are adding 5 to the number of 5
minutes, π₯ that she types. However, on the graph, we translated 5 units to the left of zero. How can we make sense of that? οΊ
ο§
Since her normal start time was 8:00, then 5 minutes before 8:00 is 5 minutes less than 8:00, which means we would need to go to the left of 8:00, in this case the origin of the graph, to mark her start time.
If Pia started work 20 minutes early, what equation would represent the number of pages she could type in π₯ minutes? οΊ
1 5
The equation that represents the graph of the red line is π¦ = (π₯ + 20).
Example 2 (10 minutes) ο§
ο§
Now we will look at an example of a situation that requires simultaneous linear equations. Sandy and Charlie walk at constant speeds. Sandy walks from their school to the train station in 15 minutes and Charlie walks the same distance in 10 minutes. Charlie starts 4 minutes after Sandy left the school. Can Charlie catch up to Sandy? The distance between the school and the station is 2 miles. What is Sandyβs average speed in 15 minutes? Explain. οΊ
MP.2
Sandyβs average speed in 15 minutes is
2
15
miles per minute because she walks 2 miles in 15 minutes.
ο§
Since we know Sandy walks at a constant speed, then her constant speed is
ο§
What is Charlieβs average speed in 10 minutes? Explain. οΊ
Charlieβs average speed in 10 minutes is because he walks 1 mile in 5 minutes.
2
10
2
15
miles per minute.
miles per minute, which is the same as
1 5
miles per minute
1
ο§
Since we know Charlie walks at a constant speed, then his constant speed is miles per minute.
ο§
Suppose the distance walked by Charlie in π₯ minutes is π¦ miles. Then we can write a linear equation to represent Charlieβs motion: π¦ 1 = π₯ 5 1 π¦= π₯ 5
5
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Lesson 24
NYS COMMON CORE MATHEMATICS CURRICULUM
ο§
Letβs put some information about Charlieβs walk in a table: Number of Minutes (π₯)
Miles Walked (π¦)
5
1
0
0
10
2
15
3
20
4
25 ο§
8β’4
5
At π₯ minutes, Sandy has walked 4 minutes longer than Charlie. Then the distance that Sandy walked in π₯ + 4 minutes is π¦ miles. Then the linear equation that represents Sandyβs motion is: π¦
π₯+4
=
2
15
2 (π₯ + 4) 15 2 8 π¦= π₯+ 15 15
π¦=
MP.2
Letβs put some information about Sandyβs walk in a table: Number of Minutes (π₯) 0 5
10 15 20 25
Lesson 24: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
Miles Walked (π¦) 8 15 18 3 1 =1 =1 15 15 5 28 13 =1 15 15 38 8 =2 15 15 48 3 1 =3 =3 15 15 5 58 13 =3 15 15
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Lesson 24
NYS COMMON CORE MATHEMATICS CURRICULUM
ο§
Now letβs graph each linear equation on a coordinate plane.
ο§
A couple of comments about our graph: β’
β’
ο§
Since the π¦-axis represents the distance traveled, then the point of intersection of the graphs of the two lines represents the moment they have both walked the same distance.
It looks like the lines intersect at a point between 1.5 and 2 miles; therefore, the answer is yes, Charlie can catch up to Sandy.
At approximately what point do the graphs of the lines intersect? οΊ
ο§
The π¦-intercept of the graph of Sandy shows exactly the distance she has walked at the moment that Charlie starts walking. Notice that the π₯-intercept of the graph of Sandy shows that she starts walking 4 minutes before Charlie.
Recall the original question that was asked, can Charlie catch up to Sandy? Keep in mind that the train station is 2 miles from the school. οΊ
ο§
8β’4
The lines intersect at approximately (10, 1.8). 1 5
A couple of comments about our equations, π¦ = π₯ and π¦ = β’
β’ β’
2 8 π₯+ : 15 15
Notice that π₯ (the representation of time) is the same in both equations.
Notice that we are interested in finding out when π¦ = π¦, that is, when the distance traveled by both Sandy and Charlie is the same (thatβs when Charlie catches up to Sandy). We write the pair of simultaneous linear equations as 8 2 π₯+ π¦= 15 15 οΏ½ 1 π¦= π₯ 5
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Lesson 24
NYS COMMON CORE MATHEMATICS CURRICULUM
8β’4
Example 3 (6 minutes) ο§
Randi and Craig ride their bikes at constant speeds. It takes Randi 25 minutes to bike 4 miles. Craig can bike 4 miles in 32 minutes. If Randi gives Craig a 20 minute head start, about how long will it take Randi to catch up to Craig?
ο§
We want to express the information about Randi and Craig in terms of a system of linear equations. Write the linear equations that represent their constant speeds. οΊ
οΊ
4
miles per minute. Since Randi bikes at a constant speed, if
4
miles per minute, which is equivalent to miles per minute.
Randiβs average speed in 25 minutes is
25
Craigβs average speed in 32 minutes is
32
we let π¦ be the distance Randi travels in π₯ minutes, then the linear equation that represents her motion is π¦ 4 = π₯ 25 4 π¦= π₯ 25 1 8
Since Craig bikes at a constant speed, if we let π¦ be the distance Craig travels in π₯ minutes, then the linear equation that represent his motion is π¦ 1 = π₯ 8 1 π¦= π₯ 8 ο§
We want to account for the head start that Craig is getting. Since Craig gets a head start of 20 minutes, then we need to add that time to his total number of minutes traveled at his constant rate: 1 π¦ = (π₯ + 20) 8 1 20 π¦= π₯+ 8 8 1 5 π¦= π₯+ 8 2
ο§
The system of linear equations that represents this situation is 5 1 π¦= π₯+ 2 8 οΏ½ 4 π¦= π₯ 25
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Lesson 24
NYS COMMON CORE MATHEMATICS CURRICULUM
ο§
ο§
Now we can graph the system of equations on a coordinate plane.
Notice again that the π¦-intercept of Craigβs graph shows the distance that Craig was able to travel at the moment Randi began biking. Also notice that the π₯-intercept of Craigβs graph shows us that he started biking 20 minutes before Randi.
Now to answer the question: About how long will it take Randi to catch up to Craig? We can give two answers, one in terms of time and the other in terms of distance. What are those answers? It will take Randi about 70 minutes or about 11 miles to catch up to Craig.
οΊ ο§
8β’4
At approximately what point do the graphs of the lines intersect? The lines intersect at approximately (70, 11).
οΊ
Exercises 4β5 (10 minutes) Students complete Exercises 4β5 individually or in pairs. Exercises 4.
Efrain and Fernie are on a road trip. Each of them drives at a constant speed. Efrain is a safe driver and travels ππ miles per hour for the entire trip. Fernie is not such a safe driver. He drives ππ miles per hour throughout the trip. Fernie and Efrain left from the same location, but Efrain left at 8:00 a.m. and Fernie left at 11:00 a.m. Assuming they take the same route, will Fernie ever catch up to Efrain? If so, approximately when? a.
Write the linear equation that represents Efrainβs constant speed. Make sure to include in your equation the extra time that Efrain was able to travel. Efrainβs average speed over one hour is
ππ π
miles per hour, which is the same as ππ miles per hour. If π
represents the distance he travels in π hours, then we have
π π
= ππ and the linear equation π = πππ. To
account for his additional π hours of driving time that Efrain gets, we write the equation π = ππ(π + π)
π = πππ + πππ
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NYS COMMON CORE MATHEMATICS CURRICULUM
b.
Write the linear equation that represents Fernieβs constant speed. Fernieβs average speed over one hour is
ππ π
miles per hour, which is the same as ππ miles per hour. If
π represents the distance he travels in π hours, then we have c.
8β’4
π
= ππ and the linear equation π = πππ.
π
Write the system of linear equations that represents this situation. οΏ½
π = πππ + πππ π = πππ
d.
Sketch the graph.
e.
Will Fernie ever catch up to Efrain? If so, approximately when? π π
Yes, Fernie will catch up to Efrain after about π hours of driving or after traveling about πππ miles. f.
At approximately what point do the graphs of the lines intersect? The lines intersect at approximately (π. π, πππ).
5.
Jessica and Karl run at constant speeds. Jessica can run π miles in ππ minutes. Karl can run π miles in π minutes. They decide to race each other. As soon as the race begins, Karl realizes that he did not tie his shoes properly and takes π minute to fix them. a.
Write the linear equation that represents Jessicaβs constant speed. Make sure to include in your equation the extra time that Jessica was able to run.
Jessicaβs average speed over ππ minutes is
π
ππ
π
miles per minute, which is equivalent to miles per minute. If π
represents the distance she runs in π minutes, then we have
π π
=
π π
π
account for her additional π minute of running that Jessica gets, we write the equation 1 y = (x + 1) 5 1 1 y= x+ 5 5
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π π
and the linear equation π = π. To
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NYS COMMON CORE MATHEMATICS CURRICULUM
b.
8β’4
Write the linear equation that represents Karlβs constant speed. π
π
Karlβs average speed over π minutes is miles per minute, which is the same as miles per minute. If π
π represents the distance he runs in π minutes, then we have c.
π π
=
π π
π
π π
and the linear equation π = π.
Write the system of linear equations that represents this situation. π π π+ π π οΏ½ π π= π π π=
d.
Sketch the graph.
e.
Use the graph to answer the questions below. i.
If Jessica and Karl raced for π miles. Who would win? Explain.
If the race were π miles, then Karl would win. It only takes Karl 8 minutes to run π miles, but it takes Jessica π minutes to run the distance of π miles. ii.
π
If the winner of the race was the person who got to a distance of mile first, who would the winner π
be? Explain. π
π
At miles, Jessica would be the winner. She would reach the distance of miles between π and π π
minutes, but Karl wouldnβt get there until about π minutes have passed.
iii.
π
At approximately what point would Jessica and Karl be tied? Explain. Jessica and Karl would be tied after about π minutes or a distance of π mile. That is where the graphs of the lines intersect.
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8β’4
Closing (4 minutes) Summarize, or ask students to summarize, the main points from the lesson: ο§
We know that some situations require two linear equations. In those cases, we have what is called a system of linear equations or simultaneous linear equations.
ο§
The solution to a system of linear equations, similar to a linear equation, is all of the points that make the equations true.
ο§
We can recognize a system of equations by the notation used, for example: οΏ½
1
π¦= π₯+
π¦=
8 4
25
π₯
5 2
.
Lesson Summary Simultaneous linear equations, or a system of linear equations, is when two or more linear equations are involved in the same problem. Simultaneous linear equations are graphed on the same coordinate plane. The solution to a system of linear equations is the set of all points that make the equations of the system true. If given two equations in the system, the solution(s) must make both equations true. 1
5
π¦= π₯+ 8 2. Systems of linear equations are identified by the notation used, for example: οΏ½ 4 π¦= π₯ 25
Exit Ticket (5 minutes)
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Lesson 24
NYS COMMON CORE MATHEMATICS CURRICULUM
Name
8β’4
Date
Lesson 24: Introduction to Simultaneous Equations Exit Ticket 1.
Darnell and Hector ride their bikes at constant speeds. Darnell leaves Hectorβs house to bike home. He can bike the 8 miles in 32 minutes. Five minutes after Darnell leaves, Hector realizes that Darnell left his phone. Hector rides to catch up. He can ride to Darnellβs house in 24 minutes. Assuming they bike the same path, will Hector catch up to Darnell before he gets home? a.
Write the linear equation that represents Darnellβs constant speed.
b.
Write the linear equation that represents Hectorβs constant speed. Make sure to take into account that Hector left after Darnell.
c.
Write the system of linear equations that represents this situation.
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d.
Sketch the graph.
e.
Will Hector catch up to Darnell before he gets home? If so, approximately when?
f.
At approximately what point do the graphs of the lines intersect?
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8β’4
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8β’4
Exit Ticket Sample Solutions 1.
Darnell and Hector ride their bikes at constant speeds. Darnell leaves Hectorβs house to bike home. He can bike the π miles in ππ minutes. Five minutes after Darnell leaves, Hector realizes that Darnell left his phone. Hector rides to catch up. He can ride to Darnellβs house in ππ minutes. Assuming they bike the same path, will Hector catch up to Darnell before he gets home? a.
Write the linear equation that represents Darnellβs constant speed. π
Darnellβs average speed over ππ minutes is miles per minute. If π represents the distance he bikes in π π π
π
minutes, then the linear equations is π = π.
b.
Write the linear equation that represents Hectorβs constant speed. Make sure to take into account that Hector left after Darnell. π
Hectorβs average speed over ππ minutes is miles per minute. If π represents the distance he bikes in π π π
π
minutes, then the linear equation is π = π. To account for the extra time Darnell has to bike, we write the
equation
c.
π (π β π) π π π π= πβ π π
π=
Write the system of linear equations that represents this situation. π π π οΏ½ π π π= πβ π π π=
d.
Sketch the graph.
e.
Will Hector catch up to Darnell before he gets home? If so, approximately when?
f.
Hector will catch up ππ minutes after Darnell left his house (or ππ minutes of biking by Hector) or approximately π miles. At approximately what point do the graphs of the lines intersect? The lines intersect at approximately (ππ, π). Lesson 24: Date:
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NYS COMMON CORE MATHEMATICS CURRICULUM
8β’4
Problem Set Sample Solutions 1.
Jeremy and Gerardo run at constant speeds. Jeremy can run π mile in π minutes and Gerardo can run π miles in ππ minutes. Jeremy started running ππ minutes after Gerardo. Assuming they run the same path, when will Jeremy catch up to Gerardo? a.
Write the linear equation that represents Jeremyβs constant speed. π
Jeremyβs average speed over π minutes is miles per minute. If π represents the distance he runs in π minutes, then we have
b.
π π
=
π π
π
π π
and the linear equation π = π.
Write the linear equation that represents Gerardoβs constant speed. Make sure to include in your equation the extra time that Gerardo was able to run. Gerardoβs average speed over ππ minutes is
π
ππ
miles per minute, which is the same as
π represents the distance he runs in π minutes, then we have
π π
=
π
ππ
miles per minute. If
and the linear equation π =
account for the extra time that Gerardo gets to run, we write the equation π (π + ππ) ππ π ππ π= π+ ππ ππ
π
ππ
π π. To ππ
π=
c.
Write the system of linear equations that represents this situation. π π π οΏ½ ππ π π+ π= ππ ππ π=
d.
Sketch the graph.
e.
Will Jeremy ever catch up to Gerardo? If so, approximately when? Yes, Jeremy will catch up to Gerardo after about ππ minutes or about π miles.
f.
At approximately what point do the graphs of the lines intersect? The lines intersect at approximately (ππ, π). Lesson 24: Date:
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NYS COMMON CORE MATHEMATICS CURRICULUM
2.
8β’4
Two cars drive from town π¨ to town π© at constant speeds. The blue car travels ππ miles per hour and the red car travels ππ miles per hour. The blue car leaves at 9:30 a.m. and the red car leaves at noon. The distance between the two towns is πππ miles. Who will get there first? Write and graph the system of linear equations that represents this situation.
The linear equation that represents the blue car is π = ππ(π + π. π), which is the same as π = πππ + ππ. π. The linear equation that represents the red car is π = πππ. The system of linear equations that represents this situation is οΏ½
π = πππ + ππ. π π = πππ
The red car will get to town π© first. a.
At approximately what point do the graphs of the lines intersect? The lines intersect at approximately (π. π, πππ).
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Lesson 25
NYS COMMON CORE MATHEMATICS CURRICULUM
8β’4
Lesson 25: Geometric Interpretation of the Solutions of a Linear System Student Outcomes ο§
Students graph two equations and find the point of intersection.
ο§
Students identify the point of intersection of the two lines as the solution to the system.
ο§
Students verify by computation that the point of intersection is a solution to each of the equations in the system.
Lesson Notes In the last lesson students were introduced to the concept of simultaneous equations. Students compared the graphs of two different equations and found the point of intersection. Students estimated the coordinates of the point of intersection of the lines in order to answer questions about the context related to time and distance. In this lesson, students first graph systems of linear equations and find the precise point of intersection. Then, students verify that the MP.6 ordered pair that is the intersection of the two lines is a solution to each equation in the system. Finally, students informally verify that there can be only one point of intersection of two distinct lines in the system by checking to see if another point that is a solution to one equation is a solution to both equations. Students need graph paper to complete the problem set.
Classwork Exercises 1β5 (25 minutes) Students work independently on Exercises 1β5. Exercises 1.
ππ + π = ππ Graph the linear system on a coordinate plane: οΏ½ π π = πβπ
For the equation ππ + π = ππ: The π-intercept is (π, π).
ππ + π = ππ ππ = ππ π=π
π(π) + π = ππ π = ππ π = ππ
The π-intercept is (ππ, π). π π
π
For the equation π = π β π: π
The slope is , and the π-intercept is (π, βπ). π
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a.
8β’4
Name the ordered pair where the graphs of the two linear equations intersect. (π, π)
b.
Verify that the ordered pair named in (a) is a solution to ππ + π = ππ. π(π) + π = ππ π + π = ππ ππ = ππ
The left and right sides of the equation are equal.
c.
π π
Verify that the ordered pair named in (a) is a solution to π = π β π. π (π) β π π π=πβπ π=π π=
The left and right sides of the equation are equal. d.
Could the point (π, π) be a solution to the system of linear equations? That is, would (π, π) make both equations true? Why or why not?
No. The graphs of the equations represent all of the possible solutions to the given equations. The point (π, π) is a solution to the equation ππ + π = ππ because it is on the graph of that equation. However, the π π
point (π, π) is not on the graph of the equation π = π β π. Therefore, (π, π) cannot be a solution to the system of equations.
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Lesson 25
NYS COMMON CORE MATHEMATICS CURRICULUM
2.
Graph the linear system on a coordinate plane: οΏ½
For the equation π + π = βπ:
8β’4
π + π = βπ π = ππ + π
π + π = βπ π = βπ The π-intercept is (π, βπ). π + π = βπ π = βπ
The π-intercept is (βπ, π).
For the equation π = ππ + π: π
The slope is , and the π-intercept is π (π, π).
a.
Name the ordered pair where the graphs of the two linear equations intersect. (βπ, βπ)
b.
Verify that the ordered pair named in (a) is a solution to π + π = βπ. βπ + (βπ) = βπ βπ = βπ
The left and right sides of the equation are equal. c.
Verify that the ordered pair named in (a) is a solution to π = ππ + π. βπ = π(βπ) + π βπ = βπ + π βπ = βπ
The left and right sides of the equation are equal. d.
Could the point (βπ, π) be a solution to the system of linear equations? That is, would (βπ, π) make both equations true? Why or why not?
No. The graphs of the equations represent all of the possible solutions to the given equations. The point (βπ, π) is a solution to the equation π + π = βπ because it is on the graph of that equation. However, the point (βπ, π) is not on the graph of the equation π = ππ + π. Therefore, (βπ, π) cannot be a solution to the system of equations.
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Lesson 25
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3.
Graph the linear system on a coordinate plane: οΏ½
For the equation ππ + π = βπ:
8β’4
ππ + π = βπ βππ + π = π
π(π) + π = βπ π = βπ
The π-intercept is (π, βπ). ππ + π = βπ ππ = βπ π = βπ
The π-intercept is (βπ, π). For the equation βππ + π = π: βπ(π) + π = π π=π
The π-intercept is (π, π).
βππ + π = π βππ = π π = βπ The π-intercept is (βπ, π). a.
Name the ordered pair where the graphs of the two linear equations intersect. (βπ, π)
b.
Verify that the ordered pair named in (a) is a solution to ππ + π = βπ. π(βπ) + π = βπ βπ = βπ
The left and right sides of the equation are equal. c.
Verify that the ordered pair named in (a) is a solution to βππ + π = π. βπ(βπ) + π = π π=π
The left and right sides of the equation are equal. d.
Could the point (π, π) be a solution to the system of linear equations? That is, would (π, π) make both equations true? Why or why not?
No. The graphs of the equations represent all of the possible solutions to the given equations. The point (π, π) is a solution to the equation βππ + π = π because it is on the graph of that equation. However, the point (π, π) is not on the graph of the equation ππ + π = βπ. Therefore, (π, π) cannot be a solution to the system of equations.
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Lesson 25
NYS COMMON CORE MATHEMATICS CURRICULUM
4.
Graph the linear system on a coordinate plane: οΏ½
For the equation ππ β ππ = ππ:
The π-intercept is (π, βπ). The π-intercept is (π, π). For the equation ππ + π = π: The π-intercept is (π, π). The π-intercept is (π, π).
a.
8β’4
ππ β ππ = ππ ππ + π = π π(π) β ππ = ππ βππ = ππ π = βπ ππ β π(π) = ππ ππ = ππ π=π
π(π) + π = π π=π ππ + π = π ππ = π π=π
Name the ordered pair where the graphs of the two linear equations intersect. (π, βπ)
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b.
8β’4
Verify that the ordered pair named in (a) is a solution to ππ β ππ = ππ. π(π) β π(βπ) = ππ π + ππ = ππ ππ = ππ
The left and right sides of the equation are equal. c.
Verify that the ordered pair named in (a) is a solution to ππ + π = π. π(π) + (βπ) = π πβπ= π π=π
The left and right sides of the equation are equal. d.
Could the point (π, βπ) be a solution to the system of linear equations? That is, would (π, βπ) make both equations true? Why or why not? No. The graphs of the equations represent all of the possible solutions to the given equations. The point (π, βπ) is not on the graph of either line; therefore, it is not a solution to the system of linear equations.
5.
Graph the linear system on a coordinate plane: οΏ½
For the equation π β π = π:
πβπ =π π = βππ β π πβπ= π π=π
The π-intercept is (π, π).
πβπ=π βπ = π π = βπ
The π-intercept is (βπ, π).
For the equation π = βππ β π: π π
The slope is β , and the π-intercept is (π, βπ).
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Lesson 25
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a.
8β’4
Name the ordered pair where the graphs of the two linear equations intersect. (βπ, π)
b.
Verify that the ordered pair named in (a) is a solution to π β π = π. π β (βπ) = π π=π
The left and right sides of the equation are equal. c.
Verify that the ordered pair named in (a) is a solution to π = βππ β π. π = βπ(βπ) β π π =πβπ π=π
The left and right sides of the equation are equal. d.
Could the point (βπ, π) be a solution to the system of linear equations? That is, would (βπ, π) make both equations true? Why or why not?
No. The graphs of the equations represent all of the possible solutions to the given equations. The point (βπ, π) is a solution to the equation π = βππ β π because it is on the graph of that equation. However, the point (βπ, π) is not on the graph of the equation π β π = π. Therefore, (βπ, π) cannot be a solution to the system of equations.
Discussion (10 minutes) The formal proof shown below is optional. The discussion can be modified by first asking students the questions in the first two bullet points, then having them make conjectures about why there is just one solution to the system of equations. ο§
How many points of intersection of the two lines were there? οΊ
ο§
Was your answer to part (d) in Exercises 1β5 ever βyesβ? Explain. οΊ
ο§
There was only one point of intersection for each pair of lines. No, the answer was always βno.β Each time, the point given was either a solution to just one of the equations, or it was not a solution to either equation.
So far, what you have observed strongly suggests that the solution of a system of linear equations is the point of intersection of two distinct lines. Letβs show that this observation is true. Theorem. Let οΏ½
π1 π₯ + π1 π¦ = π1 π2 π₯ + π2 π¦ = π2
be a system of linear equations, where π1 , π1 , π1 , π2 , π2 , and π2 are constants. At least one of π1 , π1 is not equal to zero, and at least one of π2 , π2 is not equal to zero. Let π1 and π2 be the lines defined by π1 π₯ + π1 π¦ = π1 and π2 π₯ + π2 π¦ = π2 , respectively. Then, the solution of the system is exactly the point of intersection of π1 and π2 .
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ο§
8β’4
To prove the theorem, we have to show two things. (1) Any point that lies in the intersection of the two lines is a solution of the system.
MP.6
(2) Any solution of the system lies in the intersection of the two lines. ο§
We begin by showing that (1) is true. To show that (1) is true, use the definition of a solution. What does it mean to be a solution to an equation? A solution is the ordered pair of numbers that makes an equation true. A solution is also a point on the graph of a linear equation.
οΊ ο§
ο§
ο§
Suppose (π₯0 , π¦0 ) is the point of intersection of π1 and π2 . Since the point (π₯0 , π¦0 ) is on π1 , it means that it is a solution to π1 π₯ + π1 π¦ = π1 . Similarly, since the point (π₯0 , π¦0 ) is on π2 , it means that it is a solution to π2 π₯ + π2 π¦ = π2 . Therefore, since the point (π₯0 , π¦0 ) is a solution to each linear equation, it is a solution to the system of linear equations, and the proof of (1) is complete.
To prove (2), use the definition of a solution again. Suppose (π₯0 , π¦0 ) is a solution to the system. Then, the point (π₯0 , π¦0 ) is a solution to π1 π₯ + π1 π¦ = π1 , and therefore, lies on π1 . Similarly, the point (π₯0 , π¦0 ) is a solution to π2 π₯ + π2 π¦ = π2 , and therefore, lies on π2 . Since there can be only one point shared by two distinct lines, then (π₯0 , π¦0 ) must be the point of intersection of π1 and π2 . This completes the proof of (2).
Therefore, given a system of two distinct linear equations, there can be only one point of intersection, which means there is just one solution to the system.
Exercise 6 (3 minutes) This exercise is optional. It requires students to write two different systems for a given solution. The exercise can be completed independently or in pairs. 6.
Write two different systems of equations with (π, βπ) as the solution.
Answers will vary. Two sample solutions are provided: οΏ½
ππ + π = π ππ β ππ = π
and οΏ½
βπ + ππ = βπ ππ β ππ = ππ
Closing (5 minutes) Summarize, or ask students to summarize, the main points from the lesson: ο§
We know how to graph a system of linear equations and find the point of intersection of the lines.
ο§
We know that the point of intersection represents the solution to the system of linear equations.
ο§
We know that two distinct lines will intersect at only one point; therefore, the point of intersection is an ordered pair of numbers that makes both equations in the system true.
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8β’4
Lesson Summary When a system of linear equations is graphed, the point of intersection of the lines of the graph represents the solution to the system. Two distinct lines intersect at most at one point. The coordinates of that point (π, π) represent values that make both equations of the system true. Example: The system οΏ½
π+π =π graphs as shown below. πβπ =π
The graphs of the lines intersect at (π, βπ). That means the equations in the system are true when π = π and π = βπ: π+π =π π + (βπ) = π π=π
πβπ =π π β (βπ) = π π=π
Exit Ticket (5 minutes)
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387
Lesson 25
NYS COMMON CORE MATHEMATICS CURRICULUM
Name
8β’4
Date
Lesson 25: Geometric Interpretation of the Solutions of a Linear System Exit Ticket 1.
Graph the linear system on a coordinate plane: οΏ½
2π₯ β π¦ = β1 . π¦ = 5π₯ β 5
a.
Name the ordered pair where the graphs of the two linear equations intersect.
b.
Verify that the ordered pair named in (a) is a solution to 2π₯ β π¦ = β1.
c.
Verify that the ordered pair named in (a) is a solution to π¦ = 5π₯ β 5.
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Geometric Interpretation of the Solutions of a Linear System 11/8/13 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
388
Lesson 25
NYS COMMON CORE MATHEMATICS CURRICULUM
8β’4
Exit Ticket Sample Solutions 1.
Graph the linear system on a coordinate plane: οΏ½
a.
ππ β π = βπ π = ππ β π
Name the ordered pair where the graphs of the two linear equations intersect. (π, π)
b.
Verify that the ordered pair named in (a) is a solution to ππ± β π² = βπ. π(π) β π = βπ π β π = βπ βπ = βπ
The left and right sides of the equation are equal. c.
Verify that the ordered pair named in (a) is a solution to π = ππ β π. π = π(π) β π π = ππ β π π=π
The left and right sides of the equation are equal.
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389
Lesson 25
NYS COMMON CORE MATHEMATICS CURRICULUM
8β’4
Problem Set Sample Solutions 1.
Graph the linear system on a coordinate plane: π π = π+π οΏ½ π π = βππ + ππ π π
For the equation π = π + π: π
The slope is , and the π-intercept is (π, π). π
For the equation π = βππ + ππ: π π
The slope is β , and the π-intercept is (π, ππ). a.
Name the ordered pair where the graphs of the two linear equations intersect. (π, π)
b.
Verify that the ordered pair named in (a) is π π
a solution to π = π + π. π (π) + π π π=π+π π=π π=
The left and right sides of the equation are equal. c.
Verify that the ordered pair named in (a) is a solution to π = βππ + ππ. π = βπ(π) + ππ π = βπ + ππ π=π
The left and right sides of the equation are equal. 2.
Graph the linear system on a π
π= π+π π coordinate plane: οΏ½ π + ππ = π π π
For the equation π = π + π: π
The slope is , and the π-intercept is
(π, π).
π
For the equation π + ππ = π: π + ππ = π ππ = π π=π
The π-intercept is (π, π). π + π(π) = π π=π
The π-intercept is (π, π). Lesson 25: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
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390
Lesson 25
NYS COMMON CORE MATHEMATICS CURRICULUM
a.
8β’4
Name the ordered pair where the graphs of the two linear equations intersect. (βπ, π)
b.
π π
Verify that the ordered pair named in (a) is a solution to π = π + π. π (βπ) + π π π = βπ + π π=π π=
The left and right sides of the equation are equal. c.
Verify that the ordered pair named in (a) is a solution to π + ππ = π. βπ + π(π) = π βπ + π = π π=π
The left and right sides of the equation are equal. 3.
Graph the linear system on a π=π coordinate plane: οΏ½ π + ππ = ππ
For the equation π + ππ = ππ: π + ππ = ππ ππ = ππ π=π
The π-intercept is (π, π). π + π(π) = ππ π = ππ
The π-intercept is (ππ, π). a.
b.
Name the ordered pair where the graphs of the two linear equations intersect. (π, π)
Verify that the ordered pair named in (a) is a solution to π = π. π=π
The left and right sides of the equation are equal. c.
Verify that the ordered pair named in (a) is a solution to π + ππ = ππ. π + π(π) = ππ π + π = ππ ππ = ππ
The left and right sides of the equation are equal.
Lesson 25: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
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391
Lesson 25
NYS COMMON CORE MATHEMATICS CURRICULUM
4.
Graph the linear system on a coordinate plane: οΏ½
For the equation βππ + ππ = ππ: The π-intercept is (π, π). The π-intercept is (βπ, π).
For the equation ππ + ππ = π: The π-intercept is (π, π). The π-intercept is (π, π).
a.
8β’4
βππ + ππ = ππ ππ + ππ = π
βπ(π) + ππ = ππ ππ = ππ π=π
βππ + π(π) = ππ βππ = ππ π = βπ π(π) + ππ = π ππ = π π=π
ππ + π(π) = π ππ = π π=π
Name the ordered pair where the graphs of the two linear equations intersect. (βπ, π)
b.
Verify that the ordered pair named in (a) is a solution to βππ + ππ = ππ. βπ(βπ) + π(π) = ππ π + ππ = ππ ππ = ππ
The left and right sides of the equation are equal. c.
Verify that the ordered pair named in (a) is a solution to ππ + ππ = π. π(βπ) + π(π) = π βπ + ππ = π π=π
The left and right sides of the equation are equal.
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392
Lesson 25
NYS COMMON CORE MATHEMATICS CURRICULUM
5.
8β’4
π + ππ = π Graph the linear system on a coordinate plane: οΏ½ π π = πβπ π
For the equation π + ππ = π:
π + ππ = π ππ = π π=π
The π-intercept is (π, π).
π + π(π) = π π=π
The π-intercept is (π, π). π π
For the equation π = π β π: π
The slope is , and the π-intercept is (π, βπ). π
a.
Name the ordered pair where the graphs of the two linear equations intersect. (π, βπ)
b.
Verify that the ordered pair named in (a) is a solution to π + ππ = π. π + π(βπ) = π πβπ =π π=π
The left and right sides of the equation are equal.
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393
Lesson 25
NYS COMMON CORE MATHEMATICS CURRICULUM
c.
8β’4
π π
Verify that the ordered pair named in (a) is a solution to π = π β π. π (π) β π π βπ = π β π βπ = βπ βπ =
The left and right sides of the equation are equal. 6.
Without graphing, name the ordered pair where the graphs of the two linear equations intersect. π=π οΏ½ π = βπ (π, βπ)
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394
Lesson 26
NYS COMMON CORE MATHEMATICS CURRICULUM
8β’4
Lesson 26: Characterization of Parallel Lines Student Outcomes ο§
Students know that when a system of linear equations has no solution, i.e., no point of intersection of the lines, then the lines are parallel.
Lesson Notes The discussion that begins on page 398 is an optional proof of the theorem about parallel lines. You can choose to discuss the proof with students or have students complete Exercises 4β10.
Classwork Exercises 1β3 (10 minutes) Students complete Exercises 1β3 independently. Once students are finished, debrief their work using the questions in the discussion that follows the exercises. Exercises 1.
Graph the system: οΏ½ a.
π
π= π+π π . π π= πβπ π
Identify the slope of each equation. What do you notice? π
The slope of the first equation is , and the slope π
π
of the second equation is . The slopes are equivalent. b.
π
Identify the π-intercept of each equation. Are the π-intercepts the same or different?
The π-intercepts are (π, π) and (π, βπ). The πintercepts are different.
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Lesson 26
NYS COMMON CORE MATHEMATICS CURRICULUM
2.
Graph the system: οΏ½ a.
8β’4
π
π= β π+π π . π π= β π+π π
Identify the slope of each equation. What do you notice? π π
The slope of both equations is β . The slopes are equal.
b.
Identify the π-intercept of each equation. Are the π-intercepts the same or different?
The π-intercepts are (π, π) and (π, π). The π-intercepts are different.
3.
Graph the system: οΏ½ a.
π = ππ β π . π = ππ β π
Identify the slope of each equation. What do you notice? The slope of both equations is π. The slopes are equal.
b.
Identify the π-intercept of each equation. Are the π-intercepts the same or different?
The π-intercepts are (π, βπ) and (π, βπ). The π-intercepts are different.
Discussion (10 minutes) ο§
What did you notice about each of the systems you graphed in Exercises 1β3? οΊ
MP.3 & MP.8
ο§
What did you notice about the slopes in each system? οΊ
ο§
For each exercise, the lines graphed as parallel lines. Each time, the linear equations of the system had the same slope.
What did you notice about the π¦-intercepts of the equations in each system? οΊ
In each case, the π¦-intercepts were different. Lesson 26: Date:
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Lesson 26
NYS COMMON CORE MATHEMATICS CURRICULUM
ο§
If the equations had the same π¦-intercept and the same slope, what would we know about the graphs of the lines? οΊ
ο§
8β’4
There is only one line that can go through a given point with a given slope. If the equations had the same slope and π¦-intercept, they would graph as the same line.
For that reason, when we discuss lines with the same slope, we must make sure to identify them as distinct lines.
ο§ Write a summary of the conclusions you have reached by completing Exercises 1β3. MP.3 & Provide time for students to write their conclusions. Share the theorem with them, and have students compare the MP.8 conclusions that they reached to the statements in the theorem. ο§
What you observed in Exercises 1β3 can be captured in the following theorem. Theorem. (1) Two distinct, non-vertical lines in the plane are parallel if they have the same slope. (2) If two distinct, non-vertical lines have the same slope, then they are parallel.
ο§
Suppose you have a pair of parallel lines on a coordinate plane. In how many places will those lines intersect? οΊ
ο§
Suppose you are given a system of linear equations that graph as parallel lines. How many solutions will the system have? οΊ
ο§ ο§
ο§
By definition, parallel lines never intersect.
Based on work in the previous lesson, students learned that the solution to a system of equations is represented by the ordered pair that is the location where the lines intersect. Since parallel lines do not intersect, then a system containing linear equations that graph as parallel lines will have no solution.
What we want to find out is how to recognize when the lines defined by the equations are parallel. Then, we would know immediately that we have a system with no solution as long as the lines are unique. π¦=6 π₯=2 or οΏ½ . Why A system can easily be recognized as having no solutions when it is in the form of οΏ½ π¦ = 15 π₯ = β7 is that so? π₯=2 graphs as two vertical lines. All vertical lines are parallel to the π¦-axis, οΊ Because the system οΏ½ π₯ = β7 π¦=6 and therefore, are parallel to one another. Similarly, the system οΏ½ graphs as two horizontal π¦ = 15 lines. All horizontal lines are parallel to the π₯-axis, and therefore, are parallel to one another.
We want to be able to recognize when we have a system of parallel lines that are not vertical or horizontal. What characteristics did each pair of equations have? οΊ
Each pair of equations had the same slope, but different π¦-intercepts.
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NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 26
8β’4
Discussion (15 minutes) The following discussion is proof of the theorem about parallel lines. Having this discussion with students is optional. Instead, you may have students complete Exercises 4β10 beginning on page 401. ο§
We begin by proving (1). Recall that when we were shown that the slope between any two points would be equal to the same constant, π, we used what we knew about similar triangles. We will do something similar to prove (1).
ο§
Suppose we have two non-vertical and non-horizontal parallel lines π1 and π2 in the coordinate plane. Assume that the π¦-intercept of π1 is greater than the π¦-intercept of π2 . (We could assume the opposite is true; it doesnβt make a difference with respect to the proof. We just want to say something clearly so our diagram will make sense.) Letβs also assume that these lines are left-to-right inclining, i.e., positive slope.
ο§
Pick a point π = (π1 , π2 ) on π1, and draw a vertical line from π so that it intersects π2 at point π and the π₯-axis at point π
. From points π and π
, draw horizontal lines so that they intersect lines π1 and π2 at points π and π, respectively.
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Characterization of Parallel Lines 11/8/13
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Lesson 26
NYS COMMON CORE MATHEMATICS CURRICULUM
ο§
By construction, β πππ and β ππ
π are right angles. How do we know for sure? οΊ
ο§
We drew a vertical line from point π. Therefore, the vertical line is parallel to the π¦-axis and perpendicular to the π₯-axis. Therefore, β ππ
π = 90Λ. Since we also drew horizontal lines, we know they are parallel. The vertical line through π is then a transversal that intersects parallel lines, which means corresponding angles are congruent. Since β ππ
π corresponds to β πππ, then β πππ = 90Λ.
We want to show that β³ πππ~ β³ ππ
π, which means we need another pair of equal angles in order to use the AA criterion. Do we have another pair of equal angles? Explain. οΊ
ο§
8β’4
Yes. We know that lines π1 and π2 are parallel. Still using the vertical line through π as the transversal, corresponding angles β πππ
= β πππ. Therefore, β³ πππ~ β³ ππ
π.
To better see what we are doing, we will translate β³ πππ along vector οΏ½οΏ½οΏ½οΏ½οΏ½β ππ
as shown.
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Lesson 26
NYS COMMON CORE MATHEMATICS CURRICULUM
ο§
8β’4
By definition of dilation, we know that: |πβ²πβ²| |πβ²πβ²| = . |ππ
| |π
π|
Equivalently, by the multiplication property of equality:
|πβ²πβ²| |ππ
| = . |πβ²πβ²| |π
π|
Because translation preserves lengths of segments, we know that |πβ² πβ² | = |ππ| and |πβ² π β² | = |ππ|, so we have |ππ| |ππ
| = . |ππ| |π
π| By definition of slope,
|ππ| |ππ|
equal and (1) is proved. ο§
is the slope of π1 and
|ππ
| |π
π|
is the slope of π2 . Therefore, the slopes of π1 and π2 are
To prove (2), use the same construction as we did for (1). The difference this time is that we know we have side lengths that are equal in ratio because we are given that the slopes are the same, so we are trying to prove that the lines π1 and π2 are parallel. Since we donβt know the lines are parallel, we also donβt know that β πππ
= β πππ, but we do know that β πππ and β ππ
π are right angles.
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Lesson 26
NYS COMMON CORE MATHEMATICS CURRICULUM
ο§
Then again, we translate β³ πππ along vector οΏ½οΏ½οΏ½οΏ½οΏ½β ππ
as shown.
ο§
Since the corresponding sides are equal in ratio to the scale factor
|πβ²πβ²| |ππ
|
=
|πβ²πβ²| |π
π|
8β’4
and share a common angle
β πβ²π
πβ², by the Fundamental Theorem of Similarity, we know that the lines containing πβ²πβ² and ππ are parallel. Since the line containing πβ²πβ² is a translation of line ππ, and translations preserve angle measures, then we know that line ππ is parallel to line ππ. Since the line containing ππ is π1 and the line containing ππis line π2 , then we can conclude that π1 β₯ π2 . This finishes the proof of the theorem.
Exercises 4β10 (15 minutes) Students complete Exercises 4β10 independently. Once students are finished, debrief their work using the questions in the discussion that follows the exercises. 4.
Write a system of equations that has no solution. Answers will vary. Verify that the system that has been written has equations that have the same slope and unique π-intercepts. Sample student solution: οΏ½
5.
π
π= π+π π . π π= πβπ π
Write a system of equations that has (π, π) as a solution.
Answers will vary. Verify that students have written a system where (π, π) is a solution to each equation. Sample ππ + π = ππ . student solution: οΏ½ π π= π π
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Characterization of Parallel Lines 11/8/13
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Lesson 26
NYS COMMON CORE MATHEMATICS CURRICULUM
6.
8β’4
How can you tell if a system of equations has a solution or not? If the slopes of the equations are different, the lines will intersect at some point, and there will be a solution to the system. If the slopes of the equations are the same, and the π-intercepts are different, then the equations will graph as parallel lines, which means the system will not have a solution.
7.
Does the system of linear equations shown below have a solution? Explain. οΏ½
ππ β ππ = π ππ β ππ = π
π
Yes, this system does have a solution. The slope of the first equation is π, and the slope of the second equation is . Since the slopes are different, these equations will graph as non-parallel lines, which means they will intersect at some point.
8.
π
Does the system of linear equations shown below have a solution? Explain. οΏ½
βππ + ππ = ππ π = ππ + π
No, this system does not have a solution. The slope of the first equation is π
π π
π
= , and the slope of the second π
equation is . Since the slopes are the same, but the lines are distinct, these equations will graph as parallel lines. π
Parallel lines never intersect, which means this system has no solution. 9.
Does the system of linear equations shown below have a solution? Explain. οΏ½
πππ + ππ = βπ ππ + π = π
No, this system does not have a solution. The slope of the first equation is β
ππ = βπ, and the slope of the second π
equation is βπ. Since the slopes are the same but the lines are distinct, these equations will graph as parallel lines. Parallel lines never intersect, which means this system has no solution. 10. Genny babysits for two different families. One family pays her $π each hour and a bonus of $ππ at the end of the night. The other family pays her $π every half hour and a bonus of $ππ dollars at the end of the night. Write and solve the system of equations that represents this situation. At what number of hours do the two families pay the same for babysitting service from Genny? Let π represent the total amount Genny is paid for babysitting π hours. The first family pays π = ππ + ππ. Since the other family pays by the half hour, π β π would represent the amount Genny is paid each hour. So, the other family pays π = (π β π)π + ππ, which is the same as π = ππ + ππ. The system is οΏ½
π = ππ + ππ π = ππ + ππ
Since the equations in the system have the same slope and different π-intercepts, there will not be a point of intersection. That means that there will not be a number of hours for when Genny is paid the same amount by both families. The second family will always pay her $π more than the first family.
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Lesson 26
NYS COMMON CORE MATHEMATICS CURRICULUM
8β’4
Closing (5 minutes) Summarize, or ask students to summarize, the main points from the lesson: ο§
ο§
We know that systems of vertical lines will have no solution because all vertical lines are parallel to the π¦-axis, and therefore, are parallel to one another. Similarly, systems of horizontal lines will have no solution because all horizontal lines are parallel to the π₯-axis, and therefore, are parallel to one another.
We know that if a system contains linear equations with graphs of distinct lines that have the same slope, then the graphs of the lines are parallel and, therefore, the system has no solution.
Lesson Summary By definition, parallel lines do not intersect; therefore, a system of linear equations that graph as parallel lines will have no solution. Parallel lines have the same slope, but no common point. Verify that lines are parallel by comparing their slopes and their π-intercepts.
Exit Ticket (5 minutes)
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Characterization of Parallel Lines 11/8/13
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Lesson 26
NYS COMMON CORE MATHEMATICS CURRICULUM
Name
8β’4
Date
Lesson 26: Characterization of Parallel Lines Exit Ticket Does each system of linear equations have a solution? Explain your answer. 1.
οΏ½
2.
οΏ½
3.
5 4
π¦= π₯β3 5 4
π¦+2= π₯
2 3
π¦= π₯β5
4π₯ β 8π¦ = 11
1
π₯+π¦=8 οΏ½3 π₯ + 3π¦ = 12
Lesson 26: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
Characterization of Parallel Lines 11/8/13
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Lesson 26
NYS COMMON CORE MATHEMATICS CURRICULUM
8β’4
Exit Ticket Sample Solutions Does each system of linear equations have a solution? Explain your answer. 1.
οΏ½
π π
π= πβπ π π
π+π= π
π
No, this system does not have a solution. The slope of the first equation is , and the slope of the second equation is π π
π
. Since the slopes are the same, and they are distinct lines, these equations will graph as parallel lines; since
parallel lines never intersect, this system has no solution.
2.
π
π= πβπ π οΏ½ ππ β ππ = ππ
π
π
Yes, this system does have a solution. The slope of the first equation is , and the slope of the second equation is . π
Since the slopes are different, these equations will graph as non-parallel lines, which means they will intersect at some point.
3.
π
π
π+π= π οΏ½π π + ππ = ππ
π π
No, this system does not have a solution. The slope of the first equation is β , and the slope of the second equation π π
is β . Since the slopes are the same, and they are distinct lines, these equations will graph as parallel lines; since parallel lines never intersect, this system has no solution.
Problem Set Sample Solutions Answer problems 1β5 without graphing the equations. 1.
Does the system of linear equations shown below have a solution? Explain. οΏ½
ππ + ππ = π βππ β πππ = π
π π
No, this system does not have a solution. The slope of the first equation is β , and the slope of the second equation is β
π π , which is equivalent to β . Since the slopes are the same, but the lines are distinct, these equations will ππ π
graph as parallel lines. Parallel lines never intersect, which means this system has no solution.
2.
Does the system of linear equations shown below have a solution? Explain. π πβπ =π οΏ½π ππ β ππ = π
π
π
Yes, this system does have a solution. The slope of the first equation is , and the slope of the second equation is . π
Since the slopes are different, these equations will graph as non-parallel lines, which means they will intersect at some point.
Lesson 26: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
π
Characterization of Parallel Lines 11/8/13
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Lesson 26
NYS COMMON CORE MATHEMATICS CURRICULUM
3.
8β’4
Does the system of linear equations shown below have a solution? Explain. οΏ½
π + ππ = π ππ β π = βπ
π π
Yes, this system does have a solution. The slope of the first equation is β , and the slope of the second equation is
π. Since the slopes are different, these equations will graph as non-parallel lines, which means they will intersect at some point.
4.
Does the system of linear equations shown below have a solution? Explain. οΏ½
π = ππ + ππ πππ β ππ = π
No, this system does not have a solution. The slope of the first equation is π, and the slope of the second equation is ππ π
, which is equivalent to π. Since the slopes are the same but the lines are distinct, these equations will graph as
parallel lines. Parallel lines never intersect, which means this system has no solution.
5.
Does the system of linear equations shown below have a solution? Explain. οΏ½
π π + ππ π ππ β ππ = π
π=
π
No, this system does not have a solution. The slope of the first equation is , and the slope of the second equation is π π
π
. Since the slopes are the same, but the lines are distinct, these equations will graph as parallel lines. Parallel lines
never intersect, which means this system has no solution. 6.
Given the graph of a system of linear equations below, is there a solution to the system that we cannot see on this portion of the coordinate plane? That is, will the lines intersect somewhere on the plane not represented in the picture? Explain.
π
π
The slope of ππ is , and the slope of ππ is . Since the slopes are different, these equations will graph as non-parallel π
π
lines, which means they will intersect at some point. Therefore, the system of linear equations that represents the graph will have a solution.
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7.
Lesson 26
8β’4
Given the graph of a system of linear equations below, is there a solution to the system that we cannot see on this portion of the coordinate plane? That is, will the lines intersect somewhere on the plane not represented in the picture? Explain.
π π
π π
The slope of ππ is β , and the slope of ππ is β . Since the slopes are different, these equations will graph as nonparallel lines, which means they will intersect at some point. Therefore, the system of linear equations that represents the graph will have a solution.
8.
Given the graph of a system of linear equations below, is there a solution to the system that we cannot see on this portion of the coordinate plane? That is, will the lines intersect somewhere on the plane not represented in the picture? Explain.
The slope of ππ is βπ, and the slope of ππ is βπ. Since the slopes are the same, these equations will graph as parallel lines, which means they will not intersect, and the system of linear equations that represents the graph will have no solution.
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9.
Lesson 26
8β’4
Given the graph of a system of linear equations below, is there a solution to the system that we cannot see on this portion of the coordinate plane? That is, will the lines intersect somewhere on the plane not represented in the picture? Explain.
π
The slope of ππ is , and the slope of ππ is π
π
ππ
. Since the slopes are different, these equations will graph as non-
parallel lines, which means they will intersect at some point. Therefore, the system of linear equations that represents the graph will have a solution.
10. Given the graph of a system of linear equations below, is there a solution to the system that we cannot see on this portion of the coordinate plane? That is, will the lines intersect somewhere on the plane not represented in the picture? Explain.
Lines ππ and ππ are horizontal lines. That means that they are both parallel to the π-axis, and therefore, are parallel to one another. Therefore, the system of linear equations that represents the graph will have no solution.
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Lesson 27
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8β’4
Lesson 27: Nature of Solutions of a System of Linear Equations Student Outcomes ο§
Students know that since two equations in the form ππ₯ + ππ¦ = π and πβ²π₯ + πβ²π¦ = πβ² graph as the same line when
ο§
πβ² π
=
πβ² π
=
πβ² π
, then the system of linear equations has infinitely many solutions.
Students know a strategy for solving a system of linear equations algebraically.
Classwork Exercises 1β3 (5 minutes) Students complete Exercises 1β3 independently in preparation for the discussion that follows about infinitely many solutions. Exercises 1β3 Determine the nature of the solution to each system of linear equations. 1.
ππ + ππ = π οΏ½ π π =β π+π π
The slopes of these two distinct equations are the same, which means they graph as parallel lines. Therefore, this system will have no solutions.
2.
οΏ½
ππ + ππ = βπ πβπ=π
οΏ½
ππ + ππ = π ππ + ππ = π
The slopes of these two equations are unique. That means they graph as distinct lines and will intersect at one point. Therefore, this system has one solution.
3.
These equations graph as the same line because they have the same slope and the same π-intercept.
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Discussion (7 minutes) Ask students to summarize the nature of the solutions for each of the Exercises 1β3. Students should be able to state clearly what they observed in Exercise 1 and Exercise 2 as stated in (1) and (2) below. ο§
So far, our work with systems of linear equations has taught us that: (1) If the lines defined by the equations in the system are parallel, then the system has no solutions. (2) If the lines defined by the equations in the system are not parallel, then the system has exactly one solution, and it is the point of intersection of the two lines.
Have students discuss their βsolutionβ to Exercise 3. Ask students what they noticed, if anything, about the equations in Exercise 3. If necessary, show that the first equation can be obtained from the second equation by multiplying each term by 3. Then, the equations are exactly the same. Proceed with the second example below, along with the discussion points that follow. ο§
We now know there is a third possibility with respect to systems of equations. Consider the following system of linear equations: 3π₯ + 2π¦ = 5 οΏ½ 6π₯ + 4π¦ = 10
What do you notice about the constants π, π, and π of the first equation, compared to the constants πβ², πβ and πβ² of the second equation?
MP.2
οΊ
ο§
π
=
πβ² π
=
πβ² π
, they are equal to the same constant 2.
Sample student work:
2(3π₯ + 2π¦ = 5) 6π₯ + 4π¦ = 10
When you multiply each term of the first equation by the constant 2, the result is the second equation.
What does this mean about the graphs of the equations in the system? οΊ
ο§
πβ²
If you multiplied each term of the first equation by the constant 2, what is the result? οΊ
ο§
When you compare
It means that the equations define the same line. That is, they graph as the same line.
When two equations define the same line, it means that every solution to one equation is also a solution to the other equation. Since we can find an infinite number of solutions to any linear equation in two variables, systems that are comprised of equations that define the same line have infinitely many solutions.
3π₯ + 2π¦ = 5 , has infinitely many solutions. Instruct 6π₯ + 4π¦ = 10 them to find at least two solutions to the first equation, and then show that the same ordered pairs satisfy the second equation as well. Provide students with time to verify the fact that the system, οΏ½ οΊ
Sample student work: 3π₯ + 2π¦ = 5 οΏ½ 6π₯ + 4π¦ = 10
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Two solutions to 3π₯ + 2π¦ = 5 are (1,1) and (3, β2). Replacing the solutions in the second equation results in the true equations: 6(1) + 4(1) = 10 6 + 4 = 10 10 = 10
and
ο§
6(3) + 4(β2) = 10 18 β 8 = 10 10 = 10
Therefore, the nature of the solutions of a system of linear equations is one of three possibilities: one solution, no solution, or infinitely many solutions. A system will have one solution when the equations graph as distinct lines with different slopes. A system will have no solution when the equations have the same slope and different π¦-intercepts, which produces a graph of parallel lines. A system will have infinitely many solutions when the equations define the same line; these equations will have the same slope and same π¦intercept, producing the same line.
Example 1 (7 minutes) In this example, students realize that graphing a system of equations will yield a solution, but the precise coordinates of the solution cannot be determined from the graph. π¦ = 3π₯ + 5 ο§ The following is the graph of the system οΏ½ . π¦ = 8π₯ + 3
ο§
Though the equations were easy to graph, it is not easy to identify the solution to the system because the intersection of the lines does not have integer coordinates. Estimate the solution, based on the graph. οΊ
ο§
1 3
The solution looks like it could be οΏ½ , 6οΏ½.
When we need a precise answer, we must use an alternative strategy for solving systems of linear equations.
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Understanding the substitution method requires an understanding of the transitive property. One accessible way to introduce this to students at this level is through symbolic puzzles; one possible example follows. If ! = $$ and $$ = &, is it true that ! = &? Why or why not? Allow students time to think about the symbol puzzle and share their thoughts with the class. Then, continue with the points below. ο§
When two linear expressions are equal to the same number, then the expressions are equal to each other, just like ! = & because both ! and & are equal to $$. Look at the system of equations given in this example: π¦ = 3π₯ + 5 οΏ½ π¦ = 8π₯ + 3
How is this like our puzzle about !, $$, and &? οΊ
ο§
The equation 3π₯ + 5 = 8π₯ + 3 is a linear equation in one variable. Solve it for π₯. οΊ
ο§
Both of the equations in the system are equal to π¦; therefore, 3π₯ + 5 must be equal to 8π₯ + 3.
Sample student work:
3π₯ + 5 = 8π₯ + 3 2 = 5π₯ 2 =π₯ 5
Keep in mind that we are trying to identify the point of intersection of the lines, that is, the solution that is 2
common to both equations. What we have just found is the π₯-coordinate of that solution, , and we must now 5
ο§
determine the π¦-coordinate. To do so, we can substitute the value of π₯ into either of the two equations of the system to determine the value for π¦. 2 π¦ = 3οΏ½ οΏ½+ 5 5 6 π¦= +5 5 31 π¦= 5 Verify that we would get the same value for π¦ using the second equation. οΊ
ο§
2 π¦ = 8οΏ½ οΏ½+ 3 5 16 π¦= +3 5 31 π¦= 5
2 31 οΏ½. Look at the graph. Does it look like this could be the solution? 5 5
The solution to the system is οΏ½ , οΊ
ο§
Sample student work:
Yes, the coordinates look correct, and they are close to our estimated solution.
Our estimation was close to the actual answer, but not as precise as when we solved the system algebraically.
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Example 2 (4 minutes) ο§
π¦ = 7π₯ β 2 Does the system οΏ½ have a solution? 2π¦ β 4π₯ = 10 οΊ
Yes, the slopes are different, which means they are not parallel and not the same line.
ο§
Now that we know that the system has a solution, we will solve it without graphing.
ο§
Notice that in this example we do not have two linear expressions equal to the same number. However, we do know what π¦ is; it is equal to the expressions 7π₯ β 2. Therefore, we can substitute the expression that π¦ is equal to in the second equation. Since π¦ = 7π₯ β 2, then:
ο§
Since we know the π₯-coordinate of the solution, we can substitute the value of π₯ into either equation to determine the value of the π¦-coordinate.
Determine the π¦-coordinate of the solution to the system. οΊ
ο§
It represents the π₯-coordinate of the point of intersection of the graphs of the lines, or the solution to the system.
How can we determine the π¦-coordinate of the solution to the system? οΊ
ο§
7 5
What does π₯ = represent? οΊ
ο§
2π¦ β 4π₯ = 10 2(7π₯ β 2) β 4π₯ = 10 14π₯ β 4 β 4π₯ = 10 10π₯ β 4 = 10 10π₯ = 14 14 π₯= 10 7 π₯= 5
Sample student work:
7 39 οΏ½. What does the solution represent? 5 5
The solution to this system is οΏ½ , οΊ
7 π¦ = 7οΏ½ οΏ½β 2 5 49 π¦= β2 5 39 π¦= 5
The solution is the point of intersection of the graphs of the lines of the system. It is a solution that makes both equations of the system true.
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Example 3 (8 minutes) ο§
4π¦ = 26π₯ + 4 Does the system οΏ½ have a solution? π¦ = 11π₯ β 1 οΊ
ο§
Yes, the slopes are different, which means they are not parallel and not the same line.
Solve this system using substitution; since we know what π¦ is equal to, we can replace that value with the expression 11π₯ β 1 in the first equation. οΊ
Sample student work:
4π¦ = 26π₯ + 4 οΏ½ π¦ = 11π₯ β 1 4(11π₯ β 1) = 26π₯ + 4 44π₯ β 4 = 26π₯ + 4 44π₯ β 26π₯ β 4 = 26π₯ β 26π₯ + 4 18π₯ β 4 = 4 18π₯ β 4 + 4 = 4 + 4 18π₯ = 8 8 π₯= 18 4 π₯= 9
4 35 οΏ½. 9 9
The solution to the system is οΏ½ , ο§
4 π¦ = 11 οΏ½ οΏ½ β 1 9 44 π¦= β1 9 35 π¦= 9
There is another option for solving this equation. We could multiply the second equation by 4. Then, we would have two linear equations equal to 4π¦ that could then be written as a single equation, as in Example 1. 1
Or, we could multiply the first equation by . Then, we would have two linear equations equal to π¦ that could 4
be written as a single equation, as in Example 1, again.
ο§
For this system, letβs multiply the second equation by 4. Then, we have: 4(π¦ = 11π₯ β 1) 4π¦ = 44π₯ β 4
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ο§
4π¦ = 26π₯ + 4 , and we can write the two linear expressions 26π₯ + 4 and 4π¦ = 44π₯ β 4 44π₯ β 4 as equal to one another. Solve the system. Now, the system can be written as οΏ½ οΊ
Sample student work:
26π₯ + 4 = 44π₯ β 4 4 = 18π₯ β 4 8 = 18π₯ 8 =π₯ 18 4 =π₯ 9 4 π¦ = 11 οΏ½ οΏ½ β 1 9 44 π¦= β1 9 35 π¦= 9
4 35 οΏ½. 9 9
The solution to the system is οΏ½ , ο§
8β’4
Compare the solution we got algebraically to the graph of the system of linear equations:
4 35 οΏ½. 9 9
We can see that our answer is correct because it looks like the lines intersect at οΏ½ ,
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Exercises 4β7 (7 minutes) Students complete Exercises 4β7 independently. Exercises 4β7 Determine the nature of the solution to each system of linear equations. If the system has a solution, find it algebraically; then, verify that your solution is correct by graphing. 4.
οΏ½
ππ + ππ = βππ π + π = βπ
These equations define the same line. Therefore, this system will have infinitely many solutions.
5.
π
π = πβπ π οΏ½ ππ = π + π
The slopes of these two equations are unique. That means they graph as distinct lines and will intersect at one point. Therefore, this system has one solution. π οΏ½π =
π π β ποΏ½ π
π πβπ π π π+π = πβπ π π π = πβπ π π π= π π ππ =π π ππ =
π ππ οΏ½ οΏ½βπ π π ππ π= βπ π π π= π π=
The solution is οΏ½
ππ π , οΏ½. π π
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6.
οΏ½
8β’4
π = πππ β π π = ππ + π
The slopes of these two equations are unique. That means they graph as distinct lines and will intersect at one point. Therefore, this system has one solution. πππ β π = ππ + π ππ β π = π ππ = ππ ππ π= π
The solution is οΏ½ππ,
7.
ππ οΏ½. π
ππ π = ποΏ½ οΏ½+ π π π = ππ + π π = ππ
Write a system of equations with (π, βπ) as its solution.
Answers will vary. Verify that students have written a system of equations where (π, βπ) is a solution to each π =πβπ equation in the system. Sample solution: οΏ½ π + π = βπ
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Closing (3 minutes) Summarize, or ask students to summarize, the main points from the lesson: ο§
A system can have one solution, no solution, or infinitely many solutions. It will have one solution when the lines are distinct and their slopes are different; it will have no solution when the equations graph as distinct lines with the same slope; it will have infinitely many solutions when the equations define the same line.
ο§
We learned a method for solving a system of linear equations algebraically. It requires us to write linear expressions equal to one another and substitution.
Lesson Summary A system of linear equations can have a unique solution, no solutions, or infinitely many solutions. Systems with a unique solution will be comprised of linear equations that have different slopes that graph as distinct lines, intersecting at only one point. Systems with no solution will be comprised of linear equations that have the same slope that graph as parallel lines (no intersection). Systems with infinitely many solutions will be comprised of linear equations that have the same slope and πintercept that graph as the same line. When equations graph as the same line, every solution to one equation will also be a solution to the other equation. A system of linear equations can be solved using a substitution method. That is, if two expressions are equal to the same value, then they can be written equal to one another. Example: οΏ½
π = ππ β π π = ππ + π
Since both equations in the system are equal to π, we can write the equation, ππ β π = ππ + π, and use it to solve for π and then the system. Example:
οΏ½
ππ = ππ + π π =π+π
In this example, you can multiply each term of the equation π = π + π by π to produce the equivalent equation ππ = ππ + ππ. Now, as is the previous example, since both equations equal ππ we can write ππ + π = ππ + ππ, and use this equation to solve for π and then the system.
Exit Ticket (4 minutes)
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Name
8β’4
Date
Lesson 27: Nature of Solutions of a System of Linear Equations Exit Ticket Determine the nature of the solution to each system of linear equations. If the system has a solution, then find it without graphing. 1.
2.
3.
1
5
π¦= π₯+ 2 2 οΏ½ π₯ β 2π¦ = 7
οΏ½
2 3 1 2π¦ + π₯ = 2 2
π¦= π₯+4
π¦ = 3π₯ β 2 οΏ½ β3π₯ + π¦ = β2
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Exit Ticket Sample Solutions Determine the nature of the solution to each system of linear equations. If the system has a solution, then find it without graphing. 1.
π
π
π = π+ π π οΏ½ π β ππ = π
The slopes of these two equations are the same, and the π-intercepts are different, which means they graph as parallel lines. Therefore, this system will have no solutions.
2.
οΏ½
π π π ππ + π = π π
π= π+π
The slopes of these two equations are unique. That means they graph as distinct lines and will intersect at one point. Therefore, this system has one solution.
The solution is οΏ½β 3.
οΏ½
ππ ππ , οΏ½. ππ ππ
π π π οΏ½ π + ποΏ½ + π = π π π π π π+π+ π =π π π ππ π+π =π π ππ π = βπ π ππ π=β ππ ππ π π = οΏ½β οΏ½ + π ππ π ππ +π π=β ππ ππ π= ππ
π = ππ β π βππ + π = βπ
These equations define the same line. Therefore, this system will have infinitely many solutions.
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Problem Set Sample Solutions Students practice determining the nature of solutions of a system of linear equations and finding the solution for systems that have one. Determine the nature of the solution to each system of linear equations. If the system has a solution, find it algebraically; then, verify that your solution is correct by graphing. 1.
π
π = πβπ π οΏ½ ππ β ππ = π
The slopes of these two equations are the same, which means they graph as parallel lines. Therefore, this system will have no solutions.
2.
οΏ½
ππ β π = π βππ β π = ππ
(ππ β π = π)π ππ β ππ = ππ
οΏ½
ππ β ππ = ππ βππ β π = ππ
ππ β ππ = βππ β π
ππ β ππ = βπ ππ = π π=
π π
π π = ποΏ½ οΏ½ β π π ππ π= βπ π ππ π=β π π π
The solution is οΏ½ , β
ππ οΏ½. π
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3.
οΏ½
π = ππ + π π = πππ + π
The solution is οΏ½
4.
οΏ½
ππ π , οΏ½. π π
8β’4
ππ + π = πππ + π π = ππ + π π = ππ π =π π π π = ποΏ½ οΏ½+ π π ππ π= +π π ππ π= π
ππ π + ππ π π π= π+π π
ππ =
These equations define the same line. Therefore, this system will have infinitely many solutions.
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5.
οΏ½
π+π=π π = ππ β π
ππ β π + π = π ππ + π = π π = βππ βπ = π π + π = βπ π = βππ
The solution is (βππ, βπ).
6.
οΏ½
8β’4
ππ = ππ β ππ ππ = πππ β π
ππ β ππ = πππ β π βππ = ππ β π βππ = ππ ππ β =π π ππ ππ = π οΏ½β οΏ½ β ππ π ππ ππ = β β ππ π πππ ππ = β π πππ π=β ππ
The solution is οΏ½β
ππ πππ ,β οΏ½. π ππ
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7.
8β’4
π
ππ β ππ = π οΏ½ πππ β πππ = π
These equations define the same line. Therefore, this system will have infinitely many solutions.
8.
οΏ½
ππ β ππ = π βπππ + ππ = βππ
The slopes of these two equations are the same, which means they graph as parallel lines. Therefore, this system will have no solutions.
9.
π
π= πβπ π οΏ½ ππ = π β ππ
π π β ποΏ½ π ππ = ππ β ππ ππ = ππ β ππ οΏ½ ππ = π β ππ ππ β ππ = π β ππ ππ β ππ = π ππ = ππ ππ π= π π ππ π = οΏ½ οΏ½βπ π π ππ π= βπ ππ ππ π=β ππ π οΏ½π =
The solution is οΏ½ 10. οΏ½
ππ ππ , β οΏ½. π ππ
ππ β ππ = π π = ππ + ππ
ππ β ππ = ππ + ππ ππ β ππ = ππ ππ = ππ π = ππ π = π(ππ) + ππ π = ππ + ππ π = ππ
The solution is (ππ, ππ).
11. Write a system of linear equations with (βπ, π) as its solution.
Answers will vary. Verify that students have written a system of equations where (βπ, π) is a solution to each π = π + ππ equation in the system. Sample solution: οΏ½ π+π =π Lesson 27: Date:
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Lesson 28: Another Computational Method of Solving a Linear System Student Outcomes ο§
Students learn the elimination method for solving a system of linear equations.
ο§
Students use properties of rational numbers to find a solution to a system, if it exists, through computation using substitution and elimination methods.
Lesson Notes Throughout the lesson, students are asked to verify that their solution to a system is correct by graphing the system and comparing the point of intersection to their solution. For that reason, provide graph paper for student use for both Exercises and the Problem Set. Graphs should be provided during the presentation of the Examples so that you can discuss with students whether or not their estimated point of intersection verifies their solution.
Classwork Discussion (5 minutes) ο§ ο§
In the last lesson, we saw that if a system of linear equations has a solution, it can be found without graphing. In each case, the first step was to eliminate one of the variables. π¦ = 3π₯ + 5 Describe how you would solve this system algebraically: οΏ½ π¦ = 8π₯ + 3
Since both equations were equal to π¦, we could write 3π₯ + 5 = 8π₯ + 3, thereby eliminating the π¦ from the system. π¦ = 7π₯ β 2 Describe how you would solve this system algebraically: οΏ½ 2π¦ β 4π₯ = 10 οΊ
ο§
We can substitute 7π₯ β 2 for π¦ in the second equation, i.e., 2(7π₯ β 2) β 4π₯ = 10, thereby eliminating the π¦ again. π₯ = 6π¦ + 7 Describe how you would solve this system algebraically: οΏ½ π₯ = 10π¦ + 2 οΊ
ο§
οΊ
ο§
Since both equations are equal to π₯, we could write 6π¦ + 7 = 10π¦ + 2, thereby eliminating the π₯.
In this lesson, we will learn a method for solving systems that requires us to eliminate one of the variables, but in a different way from the last lesson.
Example 1 (8 minutes) Example 1 Use what you noticed about adding equivalent expressions to solve the following system by elimination: ππ β ππ = ππ οΏ½ ππ + ππ = βπ Lesson 28: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
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Show students the three examples of adding integer equations together. Ask students to verbalize what they notice in the examples and to generalize what they observe. The goal is for students to see that we can add equivalent expressions and still have an equivalence. Example 1: If 2 + 5 = 7 and 1 + 9 = 10, does 2 + 5 + 1 + 9 = 7 + 10?
Example 2: If 1 + 5 = 6 and 7 β 2 = 5, does 1 + 5 + 7 β 2 = 6 + 5?
Example 3: If β3 + 11 = 8 and 2 + 1 = 3, does β3 + 11 + 2 + 1 = 8 + 3? ο§
MP.1
Use what you noticed about adding equivalent expressions to solve the following system by elimination: 6π₯ β 5π¦ = 21 οΏ½ 2π₯ + 5π¦ = β5
Provide students with time to attempt to solve the system by adding the equations together. Have students share their work with the class. If necessary, use the points below to support students. ο§
ο§
ο§
Notice that terms β5π¦ and 5π¦ are opposites; that is, they have a sum of zero when added. If we were to add the equations in the system, the π¦ would be eliminated. 6π₯ β 5π¦ + 2π₯ + 5π¦ = 21 + (β5) 6π₯ + 2π₯ β 5π¦ + 5π¦ = 16 8π₯ = 16 π₯=2
Just as before, now that we know what π₯ is, we can substitute it into either equation to determine the value of π¦.
9 5
The solution to the system is οΏ½2, β οΏ½.
2(2) + 5π¦ = β5 4 + 5π¦ = β5 5π¦ = β9 9 π¦=β 5
We can verify our solution by graphing the system:
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Example 2 (5 minutes) Example 2 Solve the following system by elimination: οΏ½
ο§
ο§
βππ + ππ = π ππ β ππ = ππ
We will solve the following system by elimination: β2π₯ + 7π¦ = 5 οΏ½ 4π₯ β 2π¦ = 14
In this example it is not as obvious which variable to eliminate. It will become obvious as soon as we multiply the first equation by 2: 2(β2π₯ + 7π¦ = 5) β4π₯ + 14π¦ = 10
β4π₯ + 14π¦ = 10 Now we have the system οΏ½ , and it is clear that when we add β4π₯ + 4π₯, the π₯ will be 4π₯ β 2π¦ = 14 eliminated. Add the equations of this system together and determine the solution to the system. οΊ
Sample student work:
β4π₯ + 14π¦ + 4π₯ β 2π¦ = 10 + 12 14π¦ β 2π¦ = 24 12π¦ = 24 π¦=2
ο§
9 2
The solution to the system is οΏ½ , 2οΏ½.
4π₯ β 2(2) = 14 4π₯ β 4 = 14 4π₯ = 18 18 π₯= 4 9 π₯= 2
We can verify our solution by graphing the system:
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Example 3 (5 minutes) Example 3 Solve the following system by elimination: οΏ½
ο§
ππ β ππ = βπ ππ β ππ = π
We will solve the following system by elimination: 7π₯ β 5π¦ = β2 οΏ½ 3π₯ β 3π¦ = 7
Provide time for students to solve this system on their own before discussing it as a class. ο§
In this case, it is even less obvious which variable to eliminate. On these occasions, we need to rewrite both equations. We multiply the first equation by β3 and the second equation by 7. β3(7π₯ β 5π¦ = β2)
MP.1
ο§
β21π₯ + 15π¦ = 6 7(3π₯ β 3π¦ = 7) 21π₯ β 21π¦ = 49
β21π₯ + 15π¦ = 6 Now we have the system οΏ½ , and it is obvious that the π₯ can be eliminated. 21π₯ β 21π¦ = 49
Look at the system again:
7π₯ β 5π¦ = β2 οΏ½ 3π₯ β 3π¦ = 7
What would we do if we wanted to eliminate the π¦ from the system?
We could multiply the first equation by 3 and the second equation by β5.
οΊ
Students may say to multiply the first equation by β3 and the second equation by 5. Whichever answer is given first, ask if the second is also a possibility. Students should answer, βyes.β
Exercises 1β3 (8 minutes) Students complete Exercises 1β3 independently. Exercises Each of the following systems has a solution. Determine the solution to the system by eliminating one of the variables. Verify the solution using the graph of the system. 1.
οΏ½
ππ β ππ = βππ ππ + ππ = βπ
ππ β ππ + ππ + ππ = βππ + (βπ) ππ = βππ π = βπ π(βπ) + ππ = βπ βπ + ππ = βπ ππ = βπ π π=β π π The solution is οΏ½βπ, β οΏ½. π
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2.
οΏ½
8β’4
π β ππ = π ππ + ππ = π
βπ(π β ππ = π) βππ + πππ = βππ
οΏ½
βππ + πππ = βππ ππ + ππ = π
βππ + πππ + ππ + ππ = βππ + π πππ = βππ π = βπ π β π(βπ) = π π+π= π π=π
The solution is (π, βπ). 3.
οΏ½
ππ β ππ = βπ ππ + ππ = π
βπ(ππ β ππ = βπ) βππ + ππ = ππ
π(ππ + ππ = π) οΏ½
ππ + πππ = π
βππ + ππ = ππ ππ + πππ = π
βππ + ππ + ππ + πππ = ππ + π πππ = ππ π=
ππ ππ
ππ = βπ +
ππ ππ
ππ ππ β π οΏ½ οΏ½ = βπ ππ ππ ππ β = βπ ππ
ππ ππ ππ π=β ππ ππ π=β ππ
ππ = β
The solution is οΏ½β
ππ ππ , οΏ½. ππ ππ
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Discussion (6 minutes) ο§
Systems of linear equations can be solved by graphing, substitution (as was shown in the last lesson), or elimination (as was shown in this lesson). Some systems can be solved more efficiently by elimination, while others by substitution. Which method do you think would be most efficient for the following system? Explain. π¦ = 5π₯ β 19 οΏ½ 3π₯ + 11 = π¦ οΊ
ο§
What method would you use for the following system? Explain. 2π₯ β 9π¦ = 7 οΏ½ π₯ + 9π¦ = 5 οΊ
ο§
Elimination would be the most efficient method because the terms β9π¦ + 9π¦, when added, would eliminate the π¦ from the equation.
What method would you use for the following system? Explain. 4π₯ β 3π¦ = β8 οΏ½ π₯ + 7π¦ = 4 οΊ
ο§
Substitution would be the most efficient method. Since each equation is equal to π¦, it would be easiest to write the expressions 5π₯ β 19 and 3π₯ + 11 equal to one another; then, solve for π¦.
Elimination would likely be the most efficient method because we could multiply the second equation by β4 to eliminate the π₯ from the equation.
What method would you use for the following system? Explain. π₯ + π¦ = β3 οΏ½ 6π₯ + 6π¦ = 6 οΊ
Accept any reasonable answer students provide; then, remind them that the most efficient use of time is to check to see if the system has a solution at all. Since the slopes of the graphs of these lines are parallel, this system has no solution.
Closing (4 minutes) Summarize, or ask students to summarize, the main points from the lesson: ο§
We know how to solve a system by eliminating one of the variables. In some cases, you will have to multiply one or both of the given equations by a constant in order to eliminate a variable.
ο§
We know that some systems are solved more efficiently by elimination than by other methods.
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Lesson Summary Systems of linear equations can be solved by eliminating one of the variables from the system. One way to eliminate a variable is by setting both equations equal to the same variable, then writing the expressions equal to one another. Example: Solve the system οΏ½
π = ππ β π π = ππ + π
Since both equation of the system are equal to π, then we can write and solve the equation: ππ β π = ππ + π
Another way to eliminate a variable is by multiplying each term of an equation by the same constant to make an equivalent equation. Then use the equivalent equation to eliminate one of the variables and solve the system. Example: Solve the system οΏ½
ππ + π = π π + π = ππ
Multiply the second equation by βπ to eliminate the π: Now we have the system οΏ½
ππ + π = π βππ β ππ = βππ
βπ(π + π = ππ) βππ β ππ = βππ
When the equations are added together, the π is eliminated:
ππ + π β ππ β ππ = π + (βππ) π β ππ = π + (βππ)
Once a solution has been found, verify the solution graphically or by substitution.
Exit Ticket (4 minutes) The graphs have been provided in the Exit Ticket in order to allow students to check their solution within the four minutes allotted.
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8β’4
Date
Lesson 28: Another Computational Method of Solving a Linear System Exit Ticket Determine the solution, if it exists, for each system of linear equations. Verify your solution on the coordinate plane. 1.
2.
π¦ = 3π₯ β 5 οΏ½ π¦ = β3π₯ + 7
π¦ = β4π₯ + 6 οΏ½ 2π₯ β π¦ = 11
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Exit Ticket Sample Solutions Determine the solution, if it exists, for each system of linear equations. Verify your solution on the coordinate plane. 1.
οΏ½
π = ππ β π π = βππ + π
ππ β π = βππ + π ππ = ππ π=π π = π(π) β π π=πβπ π=π
The solution is (π, π).
2.
οΏ½
π = βππ + π ππ β π = ππ ππ β (βππ + π) = ππ ππ + ππ β π = ππ ππ = ππ ππ π= π
The solution is οΏ½
ππ π = βπ οΏ½ οΏ½ + π π ππ π=β +π π ππ π=β π
ππ ππ , β οΏ½. π π
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Problem Set Sample Solutions Determine the solution, if it exists, for each system of linear equations. Verify your solution on the coordinate plane. 1.
π
π+π= π οΏ½π ππ + π = π
π ππ + π + π = π π π π+π =π π π π = βπ π π π=β π π π οΏ½β οΏ½ + π = π π ππ β +π= π π ππ π= π
π ππ οΏ½. π π
The solution is οΏ½β , 2.
οΏ½
ππ + ππ = π βππ + π = π
π(βππ + π = π) βππ + ππ = π οΏ½
ππ + ππ = π βππ + ππ = π
ππ + ππ β ππ + ππ = ππ ππ = ππ π=π
βππ + π = π
βππ = βπ
π π
π=
The solution is οΏ½ , ποΏ½.
π π
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3.
οΏ½
π = ππ β π ππ = ππ β π
οΏ½
ππ + ππ = ππ βππ + π = βπ
8β’4
These equations define the same line. Therefore, this system will have infinitely many solutions.
4.
ππ + ππ β ππ + π = ππ β π ππ + π = ππ ππ = ππ π=π ππ + π(π) = ππ ππ + ππ = ππ ππ = π π π= π π π
The solution is οΏ½ , ποΏ½.
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5.
οΏ½
8β’4
π+π= π ππ + ππ = π
ππ + π(π + π) = π ππ + ππ + ππ = π
ππ + ππ = π
ππ = βπ π=β
π β +π =π π ππ =π π
π π
π ππ οΏ½. π π
The solution is οΏ½β ,
6.
οΏ½
π = ππ + π ππ = ππ + πππ
The equations graph as distinct lines. The slopes of these two equations are the same, which means they graph as parallel lines. Therefore, this system will have no solutions.
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7.
οΏ½
8β’4
ππ β ππ = ππ βππ + ππ = βπ
π(βππ + ππ = βπ) οΏ½
βππ + ππ = βπ
ππ β ππ = ππ βππ + ππ = βπ
ππ β ππ β ππ + ππ = ππ β π βππ + ππ = ππ
ππ = ππ π=
ππ π
ππ ππ β π οΏ½ οΏ½ = ππ π ππ ππ β = ππ π πππ ππ = π ππ π= π
The solution is οΏ½ 8.
οΏ½
ππ ππ , οΏ½. π π
ππ + ππ = π ππ β ππ = ππ
The equations graph as distinct lines. The slopes of these two equations are the same, which means they graph as parallel lines. Therefore, this system will have no solutions.
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9.
οΏ½
8β’4
π = βππ + π ππ = π β π
π(π = βππ + π)
ππ = βππ + ππ
ππ = βππ + ππ οΏ½ ππ = π β π
βππ + ππ = π β π
ππ = ππ β π ππ = ππ ππ =π π
π=π
π = βπ(π) + π
π = βπ + π π=π
The solution is (π, π). 10. οΏ½
π = ππ β π πππ = ππ + π
These equations define the same line. Therefore, this system will have infinitely many solutions.
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11. οΏ½
8β’4
ππ β ππ = ππ ππ + ππ = ππ
ππ β ππ + ππ + ππ = ππ + ππ ππ = ππ
π(π) β ππ = ππ π β ππ = ππ βππ = π π π=β π
π=π
π π
The solution is οΏ½π, οΏ½.
π
π = πβπ π 12. οΏ½ π =π+π
π πβπ =π+π π π πβπ =π π π π = ππ π π = ππ
π = ππ + π
π = ππ
The solution is (ππ, ππ).
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13.
οΏ½
8β’4
ππ β ππ = ππ π + ππ = ππ
βπ(π + ππ = ππ) βππ β ππ = βππ ππ β ππ β ππ β ππ = ππ β ππ βπππ = βππ ππ π= ππ
ππ π + π οΏ½ οΏ½ = ππ ππ ππ = ππ π+ ππ ππ π= ππ
The solution is οΏ½ 14. οΏ½
ππ ππ , οΏ½. ππ ππ
πππ + πππ = π πππ + ππ = ππ
The slopes of these two equations are the same and different π-intercepts, which means they graph as parallel lines. Therefore, this system will have no solution.
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Lesson 29: Word Problems Student Outcomes ο§
Students write word problems into systems of linear equations.
ο§
Students solve systems of linear equations using elimination and substitution methods.
Lesson Notes In this lesson, students will use many of the skills learned in this module. For example, they begin defining the variables as described in a word problem. Next, students must write a system of linear equations to represent the situation. MP.4 Students will then have to decide which method for solving the system is most efficient. Finally, they solve the system and check to make sure their answer is correct. For each of the examples in the lesson, pose the questions and provide students time to answer them on their own. Then select students to share their thoughts and solutions with the class.
Classwork Example 1 (5 minutes) Example 1 The sum of two numbers is πππ and the difference between the two numbers is πππ. What are the two numbers?
ο§
Together we will read a word problem and work towards finding the solution.
ο§
The sum of two numbers is 361 and the difference between the two numbers is 173. What are the two numbers?
Provide students time to work independently or in pairs to solve this problem. Have students share their solutions and explain how they arrived at their answers. Then show how the problem can be solved using a system of linear equations. ο§
What do we need to do first? οΊ
ο§
If we define our variables we can better represent the situation we have been given. What should the variables be for this problem? οΊ
ο§
Let π₯ represent one number and let π¦ represent the other number.
Now that we know the numbers are π₯ and π¦, what do we need to do now? οΊ
ο§
We need to define our variables.
We need to write equations to represent the information in the word problem.
Using π₯ and π¦, write equations to represent the information we are provided. οΊ
The sum of two numbers is 361 can be written as π₯ + π¦ = 361. The difference between the two numbers is 173 can be written as π₯ β π¦ = 173.
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Word Problems 11/8/13
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ο§
We have two equations to represent this problem. What is it called when we have more than one linear equation for a problem, and how is it represented symbolically? οΊ
ο§
8β’4
We have a system of linear equations: π₯ + π¦ = 361 οΏ½ π₯ β π¦ = 173
We know several methods for solving systems of linear equations. Which method do you think will be the most efficient and why? We should add the equations together to eliminate the variable π¦ because we can do that in one step. π₯ + π¦ = 361 Solve the system: οΏ½ . π₯ β π¦ = 173 οΊ
ο§
οΊ
Sample student work:
π₯ + π¦ = 361 οΏ½ π₯ β π¦ = 173 π₯ + π₯ + π¦ β π¦ = 361 + 173 2π₯ = 534 π₯ = 267
ο§
The solution is (267, 94).
Based on our work, we believe the two numbers are 267 and 94. Check to make sure your answer is correct by substituting the numbers in to both equations. If it makes a true statement, then we know we are correct. If it makes a false statement, then we need to go back and check our work. οΊ
Sample student work: 267 + 94 = 361 361 = 361
and
ο§
267 + π¦ = 361 π¦ = 94
267 β 94 = 173 173 = 173
Now we are sure that the numbers are 267 and 94. Does it matter which number is π₯ and which number is π¦? οΊ
Not necessarily, but we need their difference to be positive, so π₯ must be the larger of the two numbers to make sense of our equation π₯ β π¦ = 173.
Example 2 (7 minutes) Example 2 There are πππ Grade 8 students at Euclidβs Middle School. Thirty-four more than four times the number of girls is equal to half the number of boys. How many boys are in Grade 8 at Euclidβs Middle School? How many girls?
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Word Problems 11/8/13
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8β’4
ο§
Again we will work together to solve the following word problem.
ο§
There are 356 Grade 8 students at Euclidβs Middle School. Thirty-four more than four times the number of girls is equal to half the number of boys. How many boys are in Grade 8 at Euclidβs Middle School? How many girls? What do we need to do first? οΊ
ο§
We need to define our variables.
If we define our variables we can better represent the situation we have been given. What should the variables be for this problem? οΊ
Let π₯ represent the number of girls and let π¦ represent the number of boys.
Whichever way students define the variables, ask them if it could be done the opposite way. For example, if the students respond as stated above, ask them if we could let π₯ represent the number of boys and π¦ represent the number of girls. They should say that at this stage it doesnβt matter if π₯ represents girls or boys, but once the variable is defined, it does matter. ο§
Now that we know that π₯ is the number of girls and π¦ is the number of boys, what do we need to do now? οΊ
ο§
Using π₯ and π¦, write equations to represent the information we are provided. οΊ
ο§
There are 356 Grade 8 students can be represented as π₯ + π¦ = 356. Thirty-four more than four times 1 2
the number of girls is equal to half the number of boys can be represented as 4π₯ + 34 = π¦.
We have two equations to represent this problem. What is it called when we have more than one linear equation for a problem, and how is it represented symbolically? οΊ
ο§
We need to write equations to represent the information in the word problem.
We have a system of linear equations:
π₯ + π¦ = 356 1 4π₯ + 34 = π¦ 2 We know several methods for solving systems of linear equations. Which method do you think will be the most efficient and why? οΏ½
Answers will vary. There is no obvious βmost efficientβ method. Accept any reasonable responses as long as they are justified. π₯ + π¦ = 356 Solve the system: οΏ½ 1 . 4π₯ + 34 = π¦ οΊ
ο§
οΊ
Sample student work:
2
οΏ½
π₯ + π¦ = 356 1 4π₯ + 34 = π¦ 2
2 οΏ½4π₯ + 34 =
8π₯ + 68 = π¦ οΏ½
Lesson 29: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
1 π¦οΏ½ 2
π₯ + π¦ = 356 8π₯ + 68 = π¦
Word Problems 11/8/13
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8β’4
π₯ + 8π₯ + 68 = 356 9π₯ + 68 = 356 9π₯ = 288 π₯ = 32
ο§
What does the solution mean in context? οΊ
ο§
The solution is (32, 324).
32 + π¦ = 356 π¦ = 324
Since we let π₯ represent the number of girls and π¦ represent the number of boys, it means that there are 32 girls and 324 boys at Euclidβs Middle School in Grade 8.
Based on our work, we believe there are 32 girls and 324 boys. How can we be sure we are correct? οΊ
We need to substitute the values into both equations of the system to see if it makes a true statement. Then 32 + 324 = 356 356 = 356
and
1 4(32) + 34 = (324) 2 128 + 34 = 162 162 = 162
Example 3 (5 minutes) Example 3 A family member has some five-dollar bills and one-dollar bills in their wallet. Altogether she has ππ bills and a total of $ππ. How many of each bill does she have?
ο§
Again we will work together to solve the following word problem.
ο§
A family member has some five-dollar bills and one-dollar bills in their wallet. Altogether she has 18 bills and a total of $62. How many of each bill does she have? What do we do first? οΊ
ο§
We need to define our variables.
If we define our variables we can better represent the situation we have been given. What should the variables be for this problem? οΊ
Let π₯ represent the number of $5 bills and let π¦ represent the number of $1 bills.
Again, whichever way students define the variables, ask them if it could be done the opposite way. ο§
Now that we know that π₯ is the number of $5 bills and π¦ is the number of $1 bills, what do we need to do now? οΊ
We need to write equations to represent the information in the word problem.
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Lesson 29
NYS COMMON CORE MATHEMATICS CURRICULUM
ο§
Using π₯ and π¦, write equations to represent the information we are provided. οΊ
ο§
Altogether she has 18 bills and a total of $62 must be represented with two equations. The first being π₯ + π¦ = 18 to represent the total of 18 bills and the second being 5π₯ + π¦ = 62 to represent the total amount of money she has.
We have two equations to represent this problem. What is it called when we have more than one linear equation for a problem, and how is it represented symbolically? οΊ
ο§
8β’4
We have a system of linear equations:
π₯ + π¦ = 18 οΏ½ 5π₯ + π¦ = 62
We know several methods for solving systems of linear equations. Which method do you think will be the most efficient and why? Answers will vary. Students will either say they could multiply one of the equations by β1, then they would be able to eliminate the variable π¦ when they add the equations together. Other students may say it would be easiest to solve for π¦ in the first equation and then substitute the value of π¦ into the second equation. After they have justified their method, allow them to solve the system in any manner they choose. π₯ + π¦ = 18 Solve the system: οΏ½ . 5π₯ + π¦ = 62 οΊ
ο§
οΊ
Sample student work:
οΏ½
π₯ + π¦ = 18 5π₯ + π¦ = 62
π₯ + π¦ = 18 π¦ = βπ₯ + 18 π¦ = βπ₯ + 18 οΏ½ 5π₯ + π¦ = 62
5π₯ + (βπ₯) + 18 = 62 4π₯ + 18 = 62 4π₯ = 44 π₯ = 11
ο§
What does the solution mean in context? οΊ
ο§
The solution is (11, 7).
11 + π¦ = 18 π¦=7
Since we let π₯ represent the number of $5 bills and π¦ represent the number of $1 bills, it means that our family member has 11 $5 bills, and 7 $1 bills.
The next step is to check our work. οΊ
It is obvious that 11 + 7 = 18, so we know our family member has 18 bills. It makes more sense to check our work again the actual value of those 18 bills in this case. Now we check the second equation 5(11) + 1(7) = 62 55 + 7 = 62 62 = 62
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Lesson 29
NYS COMMON CORE MATHEMATICS CURRICULUM
8β’4
Example 4 (9 minutes) Example 4 A friend bought π boxes of pencils and π notebooks for school and it cost him $ππ. He went back to the store the same day to buy school supplies for his younger brother. He spent $ππ. ππ on π boxes of pencils and π notebooks. How much would π notebooks cost?
ο§
Again we will work together to solve the following word problem.
ο§
A friend bought 2 boxes of pencils and 8 notebooks for school and it cost him $11. He went back to the store the same day to buy school supplies for his younger brother. He spent $11.25 on 3 boxes of pencils and 5 notebooks. How much would 7 notebooks cost? What do we do first? οΊ
ο§
We need to define our variables.
If we define our variables we can better represent the situation we have been given. What should the variables be for this problem? οΊ
Let π₯ represent the cost of a box of pencils and let π¦ represent the cost of a notebook.
Again, whichever way students define the variables, ask them if it could be done the opposite way. ο§
Now that we know that π₯ is the cost of a box of pencils and π¦ is the cost of a notebook, what do we need to do now? οΊ
ο§
Using π₯ and π¦, write equations to represent the information we are provided. οΊ
ο§
A friend bought 2 boxes of pencils and 8 notebooks for school and it cost him $11 is represented by the equation 2π₯ + 8π¦ = 11. . He spent $11.25 on 3 boxes of pencils and 5 notebooks is represented by the equation 3π₯ + 5π¦ = 11.25.
We have two equations to represent this problem. What is it called when we have more than one linear equation for a problem, and how is it represented symbolically? οΊ
ο§
We need to write equations to represent the information in the word problem.
We have a system of linear equations: 2π₯ + 8π¦ = 11 οΏ½ 3π₯ + 5π¦ = 11.25
We know several methods for solving systems of linear equations. Which method do you think will be the most efficient and why? οΊ
Answers will vary. Ask several students how what they believe the most efficient method is and have them share with the class a brief description of their plan. For example, a student may decide to multiply the first equation by 3 and the second equation by β2 to eliminate π₯ from the system after adding the equations together. After several students have shared their plan, allow students to solve in any manner they choose.
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Word Problems 11/8/13
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Lesson 29
NYS COMMON CORE MATHEMATICS CURRICULUM
ο§
2π₯ + 8π¦ = 11 Solve the system: οΏ½ . 3π₯ + 5π¦ = 11.25 οΊ
8β’4
Sample student work:
2π₯ + 8π¦ = 11 3π₯ + 5π¦ = 11.25 3(2π₯ + 8π¦ = 11) 6π₯ + 24π¦ = 33 β2(3π₯ + 5π¦ = 11.25) β6π₯ β 10π¦ = β22.50 οΏ½
6π₯ + 24π¦ = 33 οΏ½ β6π₯ β 10π¦ = β22.50
6π₯ + 24π¦ β 6π₯ β 10π¦ = 33 β 22.50 24π¦ β 10π¦ = 10.50 14π¦ = 10.50 10.50 π¦= 14 π¦ = 0.75
ο§
The solution is (2.50, 0.75).
What does the solution mean in context? οΊ
ο§
It means that a box of pencils costs $2.50 and a notebook costs $0.75.
Before we answer the question that this word problem asked, check to make sure the solution is correct. οΊ
Sample student work: 2(2.50) + 8(0.75) = 11 5 + 6 = 11 11 = 11
and
ο§
3(2.50) + 5(0.75) = 11.25 7.50 + 3.75 = 11.25 11.25 = 11.25
Now that we know we have the correct costs for the box of pencils and notebooks we can answer the original question: How much would 7 notebooks cost? οΊ
ο§
2π₯ + 8(0.75) = 11 2π₯ + 6 = 11 2π₯ = 5 π₯ = 2.50
The cost of 7 notebooks is 7(0.75) = 5.25. Therefore, 7 notebooks cost $7.25.
Keep in mind that some word problems require us to solve the system in order to answer a specific question, like this example about the cost of 7 notebooks. Other problems may just require the solution to the system to answer the word problem, like the first example about the two numbers and their sums and difference. It is always a good practice to reread the word problem to make sure you know what you are being asked to do.
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Lesson 29
NYS COMMON CORE MATHEMATICS CURRICULUM
8β’4
Exercises 1β3 (9 minutes) Students complete Exercises 1β3 independently or in pairs. Exercises 1.
A farm raises cows and chickens. The farmer has a total of ππ animals. One day he counts the legs of all of his animals and realizes he has a total of πππ. How many cows does the farmer have? How many chickens?
Let π represent the number of cows and π represent the number of chickens. Then, οΏ½
π + π = ππ ππ + ππ = πππ
βπ(π + π = ππ)
βππ β ππ = βππ οΏ½
βππ β ππ = βππ ππ + ππ = πππ
βππ β ππ + ππ + ππ = βππ + πππ βππ + ππ = ππ
ππ = ππ
π = ππ
ππ + π = ππ
π = ππ
The solution is (ππ, ππ).
π(ππ) + π(ππ) = πππ ππ + ππ = πππ
πππ = πππ
The farmer has ππ cows and ππ chickens. 2.
The length of a rectangle is π times the width. The perimeter of the rectangle is ππ inches. What is the area of the rectangle? Let π represent the length and π represent the width. Then, οΏ½
π = ππ ππ + ππ = ππ
π(ππ) + ππ = ππ ππ + ππ = ππ
πππ = ππ
π = π. π
π = π(π. π)
The solution is (ππ, π. π).
π = ππ
π(ππ) + π(π. π) = ππ
ππ + π = ππ
The area of the rectangle is ππ Γ π. π = ππ in2. Lesson 29: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
ππ = ππ
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Lesson 29
NYS COMMON CORE MATHEMATICS CURRICULUM
3.
8β’4
The sum of the measures of angles π and π is πππΛ. If the measure of β π is ππΛ more than half the measure of β π, what is the measure of each angle?
Let π represent the measure of β π and π represent the measure of β π. Then, π + π = πππ π οΏ½ π = ππ + π π π ππ + π + π = πππ π π ππ + π = πππ π π π = ππ π π = ππ
π + ππ = πππ π = ππ
The solution is (ππ, ππ).
π ππ = ππ + (ππ) π ππ = ππ + ππ ππ = ππ
The measure of β π = ππΛ and the measure of β π = ππΛ.
Closing (5 minutes) Summarize, or ask students to summarize, the main points from the lesson: ο§
We know how to write information from word problems into a system of linear equations.
ο§
We can solve systems of linear equations using the elimination and substitution methods.
ο§
When we solve a system we must clearly define the variables we intend to use, consider which method for solving the system would be most efficient, check our answer and think about what it means in context. Finally, we should check to make sure we have answered the question posed in the problem.
Exit Ticket (5 minutes)
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Lesson 29
NYS COMMON CORE MATHEMATICS CURRICULUM
Name
8β’4
Date
Lesson 29: Word Problems Exit Ticket 1.
Small boxes contain DVDs and large boxes contain one gaming machine. Three boxes of gaming machines and a box of DVDs weigh 48 pounds. Three boxes of gaming machines and five boxes of DVDs weigh 72 pounds. How much does each box weigh?
2.
A language arts test is worth 100 points. There are a total of 26 questions. There are spelling word questions that are worth 2 points each and vocabulary word questions worth 5 points each. How many of each type of question is there?
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Lesson 29
NYS COMMON CORE MATHEMATICS CURRICULUM
8β’4
Exit Ticket Sample Solutions 1.
Small boxes contain DVDs and large boxes contain one gaming machine. Three boxes of gaming machines and a box of DVDs weigh ππ pounds. Three boxes of gaming machines and five boxes of DVDs weigh ππ pounds. How much does each box weigh?
Let π represent the weight of the gaming machine box and let π represent the weight of the DVD box. Then, ππ + π = ππ ππ + ππ = ππ
οΏ½
βπ(ππ + π = ππ) οΏ½
βππ β π = βππ
βππ β π = βππ ππ + ππ = ππ
ππ β ππ + ππ β π = ππ β ππ ππ = ππ π=π
ππ + π = ππ ππ = ππ
π = ππ
The solution is (ππ, π)
π(ππ) + π = ππ
ππ = ππ
The box with one gaming machine weighs ππ pounds and the box containing DVDs weighs π pounds. 2.
A language arts test is worth πππ points. There are a total of ππ questions. There are spelling word questions that are worth π points each and vocabulary word questions worth π points each. How many of each type of question is there?
Let π represent the number of spelling word questions and let π represent the number of vocabulary word questions. Then, οΏ½
π + π = ππ ππ + ππ = πππ
βπ(π + π = ππ)
οΏ½
βππ β ππ = βππ
βππ β ππ = βππ ππ + ππ = πππ
ππ β ππ + ππ β ππ = πππ β ππ ππ = ππ
π = ππ
π + ππ = ππ π = ππ
The solution is (ππ, ππ)
ππ + ππ = ππ
ππ = ππ
There are ππ spelling word questions and ππ vocabulary word questions.
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Lesson 29
NYS COMMON CORE MATHEMATICS CURRICULUM
8β’4
Problem Set Sample Solutions 1.
Two numbers have a sum of π, πππ and a difference of πππ. What are the two numbers?
Let π represent one number and π represent the other number. Then, οΏ½
π + π = ππππ π β π = πππ
π + π + π β π = ππππ β πππ ππ = πππ
π = πππ
πππ + π = ππππ π = πππ
The solution is (πππ, πππ).
πππ + πππ = ππππ
ππππ = ππππ
The two numbers are πππ and πππ. 2.
The sum of the ages of two brothers is ππ. The younger brother is ππ more than a third of the older brothers age. How old is the younger brother? Let π represent the age of the younger brother and π represent the age of the older brother. Then, οΏ½
π + π = ππ π π = ππ + π π
π ππ + π + π = ππ π π ππ + π = ππ π π π = ππ π π = ππ
π + ππ = ππ π = ππ
The solution is (ππ, ππ).
π ππ = ππ + (ππ) π ππ = ππ + π
The younger brother is ππ years old.
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ππ = ππ
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Lesson 29
NYS COMMON CORE MATHEMATICS CURRICULUM
3.
8β’4
One angle measures ππ more than π times another angle. The angles are supplementary. What are their measures?
Let π represent the measure of one angle and π represent the measure of the other angle. Then, οΏ½
π = ππ + ππ π + π = πππ
ππ + ππ + π = πππ ππ + ππ = πππ
ππ = πππ
π = ππ. π
π = π(ππ. π) + ππ
π = ππ. π + ππ The solution is (ππ. π, πππ. π).
π = πππ. π
πππ. π + ππ. π = πππ
πππ = πππ
One angle measures πππ. πΛ and the other measures ππ. πΛ. 4.
Some friends went to the local movie theatre and bought four buckets of large popcorn and six boxes of candy. The total for the snacks was $ππ. ππ. The last time you were at the theatre you bought a large popcorn and a box of candy and the total was $π. ππ. How much would π large buckets of popcorn and π boxes of candy cost? Let π represent the cost of a large bucket of popcorn and π represent the cost of a box of candy. Then, οΏ½
ππ + ππ = ππ. ππ π + π = π. ππ
βπ(π + π = π. ππ)
βππ β ππ = βππ οΏ½
ππ + ππ = ππ. ππ βππ β ππ = βππ
ππ + ππ β ππ β ππ = ππ. ππ β ππ ππ β ππ = π. ππ
ππ = π. ππ
π = π. ππ
π + π. ππ = π. ππ The solution is (π, π. ππ).
π=π
π(π) + π(π. ππ) = ππ. ππ ππ + ππ. ππ = ππ. ππ
ππ. ππ = ππ. ππ
Two large buckets of popcorn and three boxes of candy will cost π(π) + π(π. ππ) = ππ + ππ. ππ = ππ. ππ dollars.
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Lesson 29
NYS COMMON CORE MATHEMATICS CURRICULUM
5.
8β’4
You have ππ total coins for a total of $ππ. ππ. You only have quarters and dimes. How many of each coin do you have?
Let π represent the number of quarters and π represent the number of dimes. Then, οΏ½
π + π = ππ π. πππ + π. ππ = ππ. ππ
βπ(π. πππ + π. ππ = ππ. ππ) οΏ½
βπ β π. ππ = βππ. ππ
π + π = ππ βπ β π. ππ = βππ. ππ
π + π β π β π. ππ = ππ β ππ. ππ π β π. ππ = ππ. ππ
π. ππ = ππ. ππ π=
ππ. ππ π. π
π = ππ
π + ππ = ππ π = ππ
The solution is (ππ, ππ).
π. ππ(ππ) + π. π(ππ) = ππ. ππ ππ. ππ + π. ππ = ππ. ππ
ππ. ππ = ππ. ππ
I have ππ quarters and ππ dimes. 6.
A piece of string is πππ inches long. Isabel wants to cut it into π pieces so that one piece is three times as long as the other. How long is each piece? Let π represent one piece and π represent the other, then
π + π = πππ οΏ½ ππ = π
ππ + π = πππ ππ = πππ π = ππ
π + ππ = πππ π = ππ
The solution is (ππ, ππ).
ππ + ππ = πππ
πππ = πππ
One piece should be ππ inches long and the other should be ππ inches long.
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Lesson 30
NYS COMMON CORE MATHEMATICS CURRICULUM
8β’4
Lesson 30: Conversion between Celsius and Fahrenheit Student Outcomes ο§
Students learn a real-world application of linear equations with respect to the conversion of temperatures from Celsius to Fahrenheit and Fahrenheit to Celsius.
Classwork Discussion (20 minutes) ο§
ο§
There are two methods for measuring temperature: (a) Fahrenheit, which assigns the number 32 to the temperature of water freezing, and the number 212 to the temperature of water boiling; and (b) Celsius, which assigns the numbers 0 and 100, respectively, to the same temperatures. These numbers will be denoted by 32ΛπΉ, 212ΛπΉ, 0ΛπΆ, 100ΛπΆ, respectively. Our goal is to address the following two questions: (1)
If π is a number, what is the degree in Fahrenheit that corresponds to πΛπͺ?
If π is a number, what is the degree in Fahrenheit that corresponds to (βπ)Λπͺ?
(2)
ο§ ο§
ο§ MP.3
Instead of trying to answer these questions directly, letβs try something simpler. With this in mind, can we find out what degree in Fahrenheit corresponds to 1ΛπΆ? Explain. We can use the following diagram to organize our thinking:
At this point, the only information we have is that 0ΛπΆ = 32ΛπΉ and 100ΛπΆ = 212ΛπΉ. We want to figure out what degree of Fahrenheit corresponds to 1ΛπΆ. Where on the diagram would 1ΛπΆ be located? Be specific.
Provide students time to talk to their partners about a plan and then have them share. Ask them to make conjectures about what degree in Fahrenheit corresponds to 1ΛπΆ, have them explain their rationale for the numbers they chose. Consider recording the information and have the class vote on which answer they think is closest to correct. οΊ
We need to divide the Celsius number line from 0 to 100, into 100 equal parts. The first division to the right of zero will be the location of 1ΛπΆ. Lesson 30: Date:
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Lesson 30
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8β’4
Now that we know where to locate 1ΛπΆ on the lower number line, we need to figure out what number it corresponds to on the upper number line, representing Fahrenheit. Like we did with Celsius, we divide the number line from 32 to 212 into 100 equal parts. The number line from 32 to 212 is actually a length of 180 units (212 β 32 = 180). Now, how would we determine the precise number in Fahrenheit that corresponds to 1ΛπΆ? Provide students time to talk to their partners and compute the answer. οΊ
We need to take the length 180 and divide it into 100 equal parts. That is, 180 9 4 = = 1 = 1.8. 100 5 5
ο§
If we look at a magnified version of the number line with this division, we have:
ο§
Based on your computation, what number falls at the intersection of the Fahrenheit number line and the red line that corresponds to 1ΛπΆ? Explain. οΊ
Since we know that each division on the Fahrenheit number line has a length of 1.8, then when we start from 32 and add 1.8 we get 33.8. Therefore 1ΛπΆ is equal to 33.8ΛπΉ.
Revisit the conjecture made at the beginning of the activity and note which student came closest to guessing 33.8ΛπΉ. Ask the student to explain how they arrived at such a close answer. ο§
Eventually, we want to revisit the original two questions. But first, letβs look at a few more concrete questions. What is 37ΛπΆ in Fahrenheit? Explain.
Provide students time to talk to their partners about how to answer the question. Ask students to share their ideas and explain their thinking. οΊ
Since the unit length on the Celsius scale is equal to the unit length on the Fahrenheit scale, then 37ΛπΆ means we need to multiply (37 Γ 1.8) to determine the corresponding location on the Fahrenheit scale. But, because 0 on the Celsius scale is 32 on the Fahrenheit scale, we will need to add 32 to our answer. In other words, 37ΛπΆ = (32 + 37 Γ 1.8)ΛπΉ = (32 + 66.6)ΛπΉ = 98.6ΛπΉ.
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456
Lesson 30
NYS COMMON CORE MATHEMATICS CURRICULUM
8β’4
Exercises 1β5 (8 minutes) Have students work in pairs or small groups to determine the corresponding Fahrenheit temperature for each given Celsius temperature. The goal is for students to be consistent in their use of repeated reasoning to lead them to the general equation for the conversion between Celsius and Fahrenheit. Exercises 1β5 Determine the corresponding Fahrenheit temperate for the given Celsius temperatures in Exercises 1β5. 1.
How many degrees Fahrenheit is ππΛπͺ?
ππΛπͺ = (ππ + ππ Γ π. π)Λπ = (ππ + ππ)Λπ = ππΛπ. 2.
How many degrees Fahrenheit is ππΛπͺ?
ππΛπͺ = (ππ + ππ Γ π. π)Λπ = (ππ + ππ. π)Λπ = πππ. πΛπ.
MP.7
3.
How many degrees Fahrenheit is ππΛπͺ?
ππΛπͺ = (ππ + ππ Γ π. π)Λπ = (ππ + πππ. π)Λπ = πππ. πΛπ. 4.
How many degrees Fahrenheit is ππΛπͺ?
ππΛπͺ = (ππ + ππ Γ π. π)Λπ = (ππ + πππ. π)Λπ = πππ. πΛπ. 5.
How many degrees Fahrenheit is πΛπͺ?
πΛπͺ = (ππ + π. ππ)Λπ
Discussion (10 minutes) Have students share their answers from Exercise 5. Select several students to explain how they derived the equation to convert between Celsius and Fahrenheit. Close that part of the discussion by letting them know that they answered Question 1 that was posed at the beginning of the lesson: (1) If π‘ is a number, what is the degree in Fahrenheit that corresponds to π‘ΛπΆ?
The following discussion will answer question (2):
ο§
(2) If π‘ is a number, what is the degree in Fahrenheit that corresponds to (βπ‘)ΛπΆ?
Now that Question 1 has been answered, letβs beginning thinking about Question 2. Where on the number line would we find a negative Celsius temperature? οΊ
A negative Celsius temperature will be to the left of zero on the number line.
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Lesson 30
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ο§
8β’4
Again, we will start simply. How can we determine the Fahrenheit temperature that corresponds to β1ΛπΆ?
Provide students time to think, confirm with a partner, and then share with the class. οΊ
ο§
We know that each unit on the Celsius scale is equal to 1.8ΛπΉ. Then (β1)ΛπΆ will equal (32 β 1.8)ΛπΉ = 30.2ΛπΉ.
How many degrees Fahrenheit corresponds to (β15)ΛπΆ?
Provide students time to think, confirm with a partner, and then share with the class. οΊ
ο§
(β15)ΛπΆ = (32 β 15 Γ 1.8)ΛπΉ = (32 β 27)ΛπΉ = 5ΛπΉ.
How many degrees Fahrenheit corresponds to (β36)ΛπΆ?
Provide students time to think, confirm with a partner, and then share with the class. οΊ
ο§
(β36)ΛπΆ = (32 β 36 Γ 1.8)ΛπΉ = (32 β 64.8)ΛπΉ = β32.8ΛπΉ.
How many degrees Fahrenheit corresponds to (βπ‘)ΛπΆ?
Provide students time to think, confirm with a partner, and then share with the class. οΊ
ο§
Each of the previous four temperatures was negative. Then (32 β 1.8π‘)ΛπΉ can be rewritten as οΏ½32 + 1.8(βπ‘)οΏ½ΛπΉ. Where the second equation looks a lot like the one we wrote for π‘ΛπΆ, i.e., π‘ΛπΆ = (32 + 1.8π‘)ΛπΉ. Are they the same equation? In other words, given any number π‘, positive or negative, would the result be the correct answer? We already know that the equation works for positive Celsius temperatures, so now letβs focus on negative Celsius temperatures. Use π‘ΛπΆ = (32 + 1.8π‘)ΛπΉ where π‘ = β15. We expect the same answer as before: (β15)ΛπΆ = (32 β 15 Γ 1.8)ΛπΉ = (32 β 27)ΛπΉ = 5ΛπΉ. Show that it is true. οΊ
ο§ ο§
(βπ‘)ΛπΆ = (32 β 1.8π‘)ΛπΉ
(β15)ΛπΆ = οΏ½32 + 1.8(β15)οΏ½ΛπΉ = οΏ½32 + (β27)οΏ½ = 5ΛπΉ.
Therefore, the equation π‘ΛπΆ = (32 + 1.8π‘)ΛπΉ will work for any π‘.
On a coordinate plane, if we let π₯ be the given temperature and π¦ be the temperature in Celsius, then we have the equation π¦ = π₯. That is, there is no change in temperature. But, when we let π₯ be the given temperature and π¦ be the temperature in Fahrenheit, we have the equation π¦ = 32 + 1.8π₯.
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ο§
Will these lines intersect? Explain? οΊ
ο§
Yes. They have different slopes so at some point they will intersect.
What will that point of intersection represent? οΊ
ο§
8β’4
π¦=π₯ It represents the solution to the system οΏ½π¦ = 1.8π₯ + 32. That point will be when the given temperature
in Celsius is the same in Celsius, π¦ = π₯, and when the given temperature in Celsius is the same temperature in Fahrenheit, π¦ = 1.8π₯ + 32.
Solve the system of equations algebraically to determine at what number π‘ΛπΆ = π‘ΛπΉ. οΊ
Sample student work:
π¦=π₯ οΏ½π¦ = 1.8π₯ + 32
π₯ = 1.8π₯ + 32 β0.8π₯ = 32 π₯ = β40
At β40 degrees, the temperatures will be equal in both units. In other words, at β40 degrees Celcius, the temperature in Fahrenheit will also be β40 degrees.
Closing (3 minutes) Summarize, or ask students to summarize, the main points from the lesson: ο§
We know how to use a linear equation in a real-world situation like converting between Celsius and Fahrenheit.
ο§
We can use the computations we make for specific numbers to help us determine a general linear equation for a situation.
Exit Ticket (4 minutes)
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Name
8β’4
Date
Lesson 30: Conversion between Celsius and Fahrenheit Exit Ticket Use the equation developed in class to answer the following questions: 1.
How many degrees Fahrenheit is 11ΛπΆ?
2.
How many degrees Fahrenheit is β3ΛC?
3.
Graph the equation developed in class and use it to confirm your results from Questions 1 and 2.
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Exit Ticket Sample Solutions Use the equation developed in class to answer the following questions: 1.
2.
3.
How many degrees Fahrenheit is ππΛπͺ?
How many degrees Fahrenheit is βπΛπͺ?
ππΛπͺ = (ππ + ππ Γ π. π)Λπ ππΛπͺ = (ππ + ππ. π)Λπ ππΛπͺ = ππ. πΛπ. βπΛπͺ = (ππ + (βπ) Γ π. π)Λπ βπΛπͺ = (ππ β π. π)Λπ βπΛπͺ = ππ. πΛπ.
Graph the equation developed in class and use it to confirm your results from Questions 1 and 2.
When I graph the equation developed in class, πΛπͺ = (ππ + π. ππ)Λπ , the results from questions 1 and 2 are on the line; therefore, confirming they are solutions to the equation.
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Problem Set Sample Solutions 1.
π π
Does the equation, πΛπͺ = (ππ + π. ππ)Λπ, work for any rational number π? Check that is does with π = π and π π
π = βπ .
π π οΏ½π οΏ½ Λπͺ = οΏ½ππ + π Γ π. ποΏ½ Λπ = (ππ + ππ. π)Λπ = ππ. πΛπ π π
π π οΏ½βπ οΏ½ Λπͺ = οΏ½ππ + οΏ½βπ οΏ½ Γ π. ποΏ½ Λπ = (ππ β ππ. π)Λπ = ππ. πΛπ π π 2.
π π
π π
Knowing that πΛπͺ = οΏ½ππ + ποΏ½ Λπ for any rational π, show that for any rational number π
, π
Λπ = οΏ½ (π
β ππ)οΏ½ Λπͺ. π π
π π
π π
Since π
Λπ can be found by οΏ½ππ + ποΏ½, then π
= οΏ½ππ + ποΏ½, and π
Λπ = πΛπͺ. Substituting π
= οΏ½ππ + ποΏ½ into π
Λπ we get:
π π
Λπ = οΏ½ππ + ποΏ½ Λπ π π π
= ππ + π π π π
β ππ = π π
π (π
β ππ) = π π
π π
π π
Now that we know π = (π
β ππ), then π
Λπ = οΏ½ (π
β ππ)οΏ½ Λπͺ. 3.
Drake was trying to write an equation to help him predict the cost of his monthly phone bill. He is charged $ππ just for having a phone, and his only additional expense comes from the number of texts that he sends. He is charged $π. ππ for each text. Help Drake out by completing parts (a)β(f). a.
How much was his phone bill in July when he sent πππ texts? ππ + πππ(π. ππ) = ππ + ππ. π = ππ. π His bill in July was $ππ. ππ.
b.
How much was his phone bill in August when he sent πππ texts?
ππ + πππ(π. ππ) = ππ + ππ. ππ = ππ. ππ His bill in August was $ππ. ππ.
c.
How much was his phone bill in September when he sent πππ texts? ππ + πππ(π. ππ) = ππ + ππ. ππ = ππ. ππ His bill in September was $ππ. ππ.
d.
Let π represent the total cost of Drakeβs phone bill. Write an equation that represents the total cost of his phone bill in October if he sends π texts. π = ππ + π(π. ππ)
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e.
8β’4
Another phone plan charges $ππ for having a phone and $π. ππ per text. Let π represent the total cost of the phone bill for sending π texts. Write an equation to represent his total bill.
π = ππ + π(π. ππ) f.
Write your equations in parts (d) and (e) as a system of linear equations and solve. Interpret the meaning of the solution in terms of the phone bill. οΏ½
π = ππ + π(π. ππ) π = ππ + π(π. ππ)
ππ + (π. ππ)π = ππ + (π. ππ)π ππ + (π. ππ)π = (π. ππ)π ππ = π. πππ πππ = π π = ππ + πππ(π. ππ) π = ππ The solution is (πππ, ππ), meaning that when Drake sends πππ texts, the cost of his bill will be $ππ using his current phone plan or the new one.
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New York State Common Core
8
Mathematics Curriculum
GRADE
GRADE 8 β’ MODULE 4
Optional Topic E:
Pythagorean Theorem 8.G.B.7, 8.EE.C.8 Focus Standard:
Instructional Days:
8.G.B.7
Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.
8.EE.C.8
Analyze and solve pairs of simultaneous linear equations. a.
Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously.
b.
Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. For example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6.
c.
Solve real-world and mathematical problems leading to two linear equations in two variables. For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair.
1
Lesson 31: System of Equations Leading to Pythagorean Triples (S)
1
Lesson 31 shows students how to apply what they learned about systems of linear equations to find a Pythagorean triple (8.G.B.7, 8.EE.C.8b). The Babylonian method of generating Pythagorean triples, described in Lesson 31, uses a system of linear equations.
1
Lesson Structure Key: P-Problem Set Lesson, M-Modeling Cycle Lesson, E-Exploration Lesson, S-Socratic Lesson
Topic E: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
Pythagorean Theorem 11/6/13
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Lesson 31
NYS COMMON CORE MATHEMATICS CURRICULUM
8β’4
Lesson 31: System of Equations Leading to Pythagorean Triples Student Outcomes ο§
Students know that a Pythagorean triple can be obtained by multiplying any known triple by a common whole number. Students use this method to generate Pythagorean triples.
ο§
Students use a system of equations to find three numbers, π, π, and π, so that π2 + π 2 = π 2 .
Lesson Notes This lesson is optional as it includes content related to the Pythagorean Theorem. The purpose of this lesson is to demonstrate an application of systems of linear equations to other content in the curriculum. Though Pythagorean triples are not part of the standard for the grade, it is an interesting topic and should be shared with students if time permits.
Classwork Discussion (10 minutes) ο§
A New York publicist, George Arthur Plimpton, bought a clay tablet from an archaeological dealer for $10 in 1922. This tablet was donated to Columbia University in 1936 and became known by its catalog number, Plimpton 322. What made this tablet so special was not just that it was 4,000 years old, but that it showed a method for finding Pythagorean triples. It was excavated near old Babylonia (which is now Iraq).
ο§
Any three numbers, π, π, π, that satisfy π2 + π 2 = π 2 are considered a triple, but when the three numbers are positive integers, then they are known as Pythagorean triples. It is worth mentioning that one of the Pythagorean triples found on the tablet was 12709, 13500, 18541. Lesson 31: Date:
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ο§
8β’4
An easy-to-remember Pythagorean Triple is 3, 4, 5. (Quickly verify for students that 3, 4, 5 is a triple). To generate another Pythagorean triple, we need only to multiply each of the numbers 3, 4, 5 by the same whole number. For example, the numbers 3, 4, 5 when each is multiplied by 2, the result is the triple 6, 8, 10. (Again, quickly verify that 6, 8, 10 is a triple). Letβs think about why this is true in a geometric context. Shown below are the two right triangles:
ο§
Discuss with your partners how the method for finding Pythagorean triples can be explained mathematically. οΊ
ο§
Triangle β³ π΄β²π΅β²πΆβ² can be obtained by dilating β³ π΄π΅πΆ by a scale factor of 2. Each triangle has a right angle with corresponding sides that are equal in ratio to the same constant, 2. That is how we know that these triangles are similar. The method for finding Pythagorean triples can be directly tied to our understanding of dilation and similarity. Each triple is just a set of numbers that represent a dilation of β³ π΄π΅πΆ by a whole-number scale factor.
Of course, we can also find triples by using a scale factor 0 < π < 1, but since it produces a set of numbers
that are not whole numbers, they are not considered to be Pythagorean triples. For example, if π = triple using side lengths 3, 4, 5 is 0.3, 0.4, 0.5.
1 , then a 10
Exercises 1β3 (5 minutes) Students complete Exercises 1β3 independently. Allow students to use a calculator to verify that they are identifying triples. Exercises 1β3 1.
Identify two Pythagorean triples using the known triple π, π, π (other than π, π, ππ). Answers will vary. Accept any triple that is a whole number multiple of π, π, π.
2.
Identify two Pythagorean triples using the known triple π, ππ, ππ.
Answers will vary. Accept any triple that is a whole number multiple of π, ππ, ππ.
3.
Identify two triples using either π, π, π or π, ππ, ππ. Answers will vary.
Lesson 31: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
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Discussion (10 minutes) ο§
Pythagorean triples can also be explained algebraically. For example, assume π, π, π represent a triple. Let π be a positive integer. Then by the Pythagorean Theorem, π2 + π 2 = π 2 :
(ππ)2 + (ππ)2 = π2 π2 + π2 π 2 = π2 (π2 + π 2 ) = π2 π 2
By the second law of exponents By the Distributive Property By substitution (π2 + π 2 = π 2 )
= (ππ)2
ο§
Our learning of systems of linear equations leads us to another method for finding Pythagorean triples, and it is actually the method that was discovered on the tablet Plimpton 322. Consider the system of linear equations:
π‘
π₯+π¦ = π οΏ½ π π₯βπ¦ =
MP.6
π‘
where π and π‘ are positive integers and π‘ > π . Incredibly, the solution to this system results in a Pythagorean
ο§
π, π, π are a Pythagorean triple.
ο§
Which method should we use to solve this system? Explain.
π
π π
To make this simpler, letβs replace π and π‘ with 1 and 2, respectively. Then we have: 2 π₯+π¦= 1 οΏ½ 1 π₯βπ¦ = 2 οΊ
ο§
π
triple. When the solution is written as fractions with the same denominator, οΏ½ , οΏ½ for example, the numbers
We should add the equations together to eliminate the variable π¦.
By the elimination method we have:
π₯+π¦+π₯βπ¦ =2+
5 2 5 π₯= 4 Now, we can substitute π₯ into one of the equations to find π¦: 5 +π¦=2 4 2π₯ =
π¦=2β
5 3 4 4
π¦=
3 4
1 2
5 4
Then, the solution to the system is οΏ½ , οΏ½. Since the solution, when written as fractions with the same π π π π
denominator,οΏ½ , οΏ½ for example, represents the Pythagorean triple, π, π, π, then our solution yields the triple 3,
4, 5.
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The remaining time can be used to complete Exercises 4β7 where students practice finding triples using the system of linear equations just described, or with the discussion below which shows the solution to the general system (without using concrete numbers for π and π‘).
Exercises 4β7 (10 minutes) These exercises are to be completed in place of the discussion below. Have students complete Exercises 4β7 independently. Exercises 4β7 π
π+π = π Use the system οΏ½ π to find Pythagorean triples for the given values of π and π. Recall that the solution, in the πβπ = π
π π π π
form of οΏ½ , οΏ½ is the triple, π, π, π.
4.
π = π, π = π
π π + π π π π ππ = + π π ππ ππ = ππ ππ π= ππ
π π+π = π οΏ½ π πβπ = π
π+π+πβπ=
Then the solution is οΏ½ 5.
ππ π +π= ππ π π ππ π= β π ππ π π= ππ
ππ π , οΏ½ and the triple is π, ππ, ππ. ππ ππ
π = π, π = ππ
ππ π + π ππ πππ ππ = ππ πππ π= πππ
ππ π+π = π οΏ½ π πβπ = ππ
π+π+πβπ=
Then the solution is οΏ½
πππ
,
ππ
πππ πππ
Lesson 31: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
πππ ππ +π= πππ π ππ πππ π= β π πππ ππ π= πππ
οΏ½ and the triple is ππ, πππ, πππ.
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6.
π = π, π = π
π+π+πβπ=π+ ππ π ππ π= π
ππ =
Then the solution is οΏ½ 7.
ππ ππ π
,
π
π π
π π+π = π οΏ½ π πβπ = π
ππ π +π = π π
π=πβ π=
ππ π
8β’4
ππ π
οΏ½ and the triple is ππ, π, ππ.
Use a calculator to verify that you found a Pythagorean triple in each of the Exercises 4β6. Show your work below. For the triple π, ππ, ππ:
ππ + πππ = πππ
ππ + π, πππ = π, πππ
π, πππ = π, πππ
For the triple ππ, πππ, πππ:
πππ + ππππ = ππππ
ππππ + ππ, πππ = ππ, πππ
ππ, πππ = ππ, πππ
For the triple ππ, π, ππ:
πππ + ππ = πππ
πππ + ππ = πππ
πππ = πππ
Discussion (10 minutes) This discussion is optional and replaces Exercises 4β7 above. ο§
ο§
Now we solve the system generally. π‘ π₯+π¦ = π οΏ½ π π₯βπ¦ = π‘
Which method should we use to solve this system? Explain. οΊ
We should add the equations together to eliminate the variable π¦.
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ο§
8β’4
By the elimination method we have: π‘ π π₯+π¦+π₯βπ¦= + π π‘ π‘ π 2π₯ = + π π‘
To add the fractions we will need the denominators to be the same. So, we use what we know about equivalent fractions, and multiply the first fraction by
π‘ π‘
π‘ π‘ π π 2π₯ = οΏ½ οΏ½ + οΏ½ οΏ½ π π‘ π‘ π 2π₯ =
2π₯ = 1
Now, we multiply both sides of the equation by : 2
π
and the second fraction by : π
π‘ 2 π 2 + π π‘ π π‘
π‘ 2 + π 2 π π‘
1 1 π‘ 2 + π 2 (2π₯) = οΏ½ οΏ½ π π‘ 2 2 π₯=
π‘ 2 + π 2 2π π‘
Now that we have a value for π₯ we can solve for π¦ as usual, but it is simpler to go back to the system: π‘ π₯+π¦= π οΏ½ π π₯βπ¦= π‘ Is equivalent to the system: π‘ π₯ = βπ¦ π οΏ½ π π₯ = +π¦ π‘ π‘ π βπ¦= +π¦ π π‘ π‘ π = + 2π¦ π π‘ π‘ π β = 2π¦ π π‘ Which is very similar to what we have done before when we solved for π₯. Therefore, π¦=
π‘2 +π 2 π‘2 βπ 2 , 2π π‘ οΏ½. Since the solution, when written as fractions with the same 2π π‘
The solution to the system is οΏ½
denominator, such as οΏ½
π‘ 2 β π 2 2π π‘
π π , οΏ½ for example, represent the Pythagorean triple, π, π, π, then our solution yields the π π
triple π‘ 2 β π 2 , 2π π‘, π‘ 2 + π 2 . Lesson 31: Date:
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Closing (5 minutes) Summarize, or ask students to summarize, the main points from the lesson: ο§
We know how to find an infinite number of Pythagorean triples: multiply a known triple by a whole number.
ο§
We know that if the numbers π, π, π are not whole numbers they can still be considered a triple, just not a Pythagorean triple.
ο§
We know how to use a system of linear equations, just like the Babylonians did 4,000 years ago, to find Pythagorean triples.
Lesson Summary A Pythagorean triple is a set of three positive integers that satisfies the equation ππ + ππ = ππ .
An infinite number of Pythagorean triples can be found by multiplying the numbers of a known triple by a whole number. For example, π, π, π is a Pythagorean triple. Multiply each number by π, then you have ππ, ππ, ππ which is also a Pythagorean triple. The system of linear equations, οΏ½ π, πππ years ago.
π π can be used to find Pythagorean triples, just like the Babylonians did π πβπ= π
π+π=
Exit Ticket (5 minutes)
Lesson 31: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
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Name
8β’4
Date
Lesson 31: System of Equations Leading to Pythagorean Triples Exit Ticket Use a calculator to complete problems 1β3. 15 17
1.
Is 7, 20, 21 a Pythagorean triple? Is 1, ,
2.
Identify two Pythagorean triples using the known triple 9, 40, 41.
3.
8 8
a Pythagorean triple? Explain.
π‘
π₯+π¦ = π Use the system οΏ½ π to find Pythagorean triples for the given values of π = 2 and π‘ = 3. Recall that the π₯βπ¦ = π‘ π π π π
solution, in the form of οΏ½ , οΏ½, is the triple, π, π, π. Verify your results.
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Exit Ticket Sample Solutions Use a calculator to complete problems 1β3. 1.
ππ ππ
Is π, ππ, ππ a Pythagorean triple? Is π,
,
π π
a Pythagorean triple? Explain.
The set of numbers π, ππ, ππ is not a Pythagorean triple because ππ + πππ β πππ . ππ ππ
The set of numbers π,
,
π π
is not a Pythagorean triple because the numbers
But, they are a triple because ππ + οΏ½ 2.
ππ π ππ π οΏ½ =οΏ½ οΏ½ . π π
ππ π
and
ππ π
are not whole numbers.
Identify two Pythagorean triples using the known triple π, ππ, ππ.
Answers will vary. Accept any triple that is a whole number multiple of π, ππ, ππ.
3.
π
π+π = π Use the system οΏ½ π to find Pythagorean triples for the given values of π = π and π = π. Recall that the πβπ = π π π π π
solution, in the form of οΏ½ , οΏ½, is the triple, π, π, π. Verify your results.
π π + π π ππ ππ = π ππ π= ππ
π+π+πβπ=
Then the solution is οΏ½
π π+π = π οΏ½ π πβπ = π
ππ π , οΏ½ and the triple is π, ππ, ππ. ππ ππ
ππ π +π= ππ π π ππ π= β π ππ π π= ππ
ππ + πππ = πππ
ππ + πππ = πππ
πππ = πππ
Lesson 31: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
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Lesson 31
NYS COMMON CORE MATHEMATICS CURRICULUM
8β’4
Problem Set Sample Solutions Students practice finding triples using both methods discussed in this lesson. 1.
Explain in terms of similar triangles why it is that when you multiply the known Pythagorean triple π, π,π by ππ, it generates a Pythagorean triple.
The triangle with lengths π, π, π is similar to the triangle with lengths ππ, ππ, ππ. They are both right triangles whose corresponding side lengths are equal to the same constant: ππ ππ ππ = = = ππ. π π π
Therefore, the triangles are similar. Thus, we can say that there is a dilation from some center with scale factor π = ππ that makes the triangles congruent. 2.
Identify three Pythagorean triples using the known triple π, ππ, ππ.
Answers will vary. Accept any triple that is a whole number multiple of π, ππ, ππ.
3.
Identify three triples (numbers that satisfy ππ + ππ = ππ, but π, π, π are not whole numbers) using the triple π, ππ, ππ. Answers will vary. Accept any triple that is not a set of whole numbers.
π
π+π = π Use the system οΏ½ π to find Pythagorean triples for the given values of π and π. Recall that the solution, in the πβπ = form of οΏ½
4.
π π π , οΏ½, is the triple, π, π, π. π π
π = π, π = π
π π + π π ππ ππ = ππ ππ π= ππ
π π+π = π οΏ½ π πβπ = π
π+π+πβπ =
Then the solution is οΏ½
ππ ππ , οΏ½ and the triple is ππ, ππ, ππ. ππ ππ
Lesson 31: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
ππ π +π= ππ π π ππ π= β π ππ ππ π= ππ
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Lesson 31
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5.
π = π, π = π
π π + π π ππ ππ = ππ ππ π= ππ
π π+π = π οΏ½ π πβπ = π
ππ π +π = ππ π π ππ π= β π ππ ππ π= ππ
π+π+πβπ=
Then the solution is οΏ½ 6.
π = π, π = π
ππ ππ , οΏ½ and the triple is ππ, ππ, ππ. ππ ππ
π π + π π ππ ππ = ππ ππ π= ππ
π π+π = π οΏ½ π πβπ = π
ππ π +π= ππ π π ππ π= β π ππ π π= ππ
π+π+πβπ=
Then the solution is οΏ½ 7.
8β’4
ππ π , οΏ½ and the triple is π, ππ, ππ. ππ ππ
Use a calculator to verify that you found a Pythagorean triple in each of the problems 4β6. Show your work below. For the triple ππ, ππ, ππ:
For the triple ππ, ππ, ππ:
For the triple π, ππ, ππ:
Lesson 31: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
πππ + πππ = πππ ππππ + π, πππ = π, πππ π, πππ = π, πππ πππ + πππ = πππ πππ + π, πππ = π, πππ π, πππ = π, πππ ππ + πππ = πππ ππ + πππ = πππ πππ = πππ
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End-of-Module Assessment Task
Name
8β’4
Date
1. Use the graph below to answer parts (a)β(c).
a.
Use any pair of points to calculate the slope of the line.
b.
Use a different pair of points to calculate the slope of the line.
c.
Explain why the slopes you calculated in parts (a) and (b) are equal.
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NYS COMMON CORE MATHEMATICS CURRICULUM
End-of-Module Assessment Task
8β’4
2. Jeremy rides his bike at a rate of 12 miles per hour. Below is a table that represents the number of hours and miles Kevin rides. Assume both bikers ride at a constant rate. Time in hours (π₯)
Distance in miles (π¦)
2
23
1.5 3.5 4
17.25 40.25 46
a.
Which biker rides at a greater speed? Explain your reasoning.
b.
Write an equation for a third biker, Lauren, who rides twice as fast as Kevin. Use π¦ to represent the number of miles Lauren travels in π₯ hours. Explain your reasoning.
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End-of-Module Assessment Task
c.
Create a graph of the equation in part (b).
d.
Calculate the slope of the line in part (c) and interpret its meaning in this situation.
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8β’4
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End-of-Module Assessment Task
8β’4
3. The cost of five protractors is $14.95 at Store A. The graph below compares the cost of protractors at Store A with the cost at Store B.
Estimate the cost of one protractor at Store B. Use evidence from the graph to justify your answer.
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End-of-Module Assessment Task
8β’4
4. Given the equation, 3π₯ + 9π¦ = β8, write a second linear equation to create a system that: a.
Has exactly one solution. Explain your reasoning.
b.
Has no solution. Explain your reasoning.
c.
Has infinitely many solutions. Explain your reasoning.
d.
Interpret the meaning of the solution, if it exists, in the context of the graph of the following system of equations. β5π₯ + 2π¦ = 10 οΏ½ 10π₯ β 4π¦ = β20
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End-of-Module Assessment Task
8β’4
5. Students sold 275 tickets for a fundraiser at school. Some tickets are for children and cost $3, while the rest are adult tickets that cost $5. If the total value of all tickets sold was $1,025, how many of each type of ticket was sold?
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6.
End-of-Module Assessment Task
8β’4
a.
Determine the equation of the line connecting the points (0, β1) and (2,3).
b.
Will the line described by the equation in (a) intersect the line passing through the points (β2,4) and (β3,3)? Explain why or why not.
Module 4: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
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NYS COMMON CORE MATHEMATICS CURRICULUM
End-of-Module Assessment Task
8β’4
7. Line π1 and line π2 are shown on the graph below. Use the graph to answer parts (a)β(d).
a.
What is the π¦-intercept of π1 ?
b.
What is the π¦-intercept of π2 ?
c.
Write a system of linear equations representing lines π1 and π2 .
d.
Use the graph to estimate the solution to the system.
e.
Solve the system of linear equations algebraically.
f.
Show that your solution from part (e) satisfies both equations.
Module 4: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
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NYS COMMON CORE MATHEMATICS CURRICULUM
End-of-Module Assessment Task
8β’4
A Progression Toward Mastery Assessment Task Item
1
aβb 8.EE.B.5
c 8.EE.B.6
2
a 8.EE.B.5
STEP 1 Missing or incorrect answer and little evidence of reasoning or application of mathematics to solve the problem.
STEP 2 Missing or incorrect answer but evidence of some reasoning or application of mathematics to solve the problem.
STEP 3 A correct answer with some evidence of reasoning or application of mathematics to solve the problem, OR an incorrect answer with substantial evidence of solid reasoning or application of mathematics to solve the problem.
STEP 4 A correct answer supported by substantial evidence of solid reasoning or application of mathematics to solve the problem.
Student made no attempt to find the slope in part (a) and/or part (b).
Student computed the slope in parts (a) and (b), but made computational errors leading to slopes that were not equal. Student may have used the same two points for both parts (a) and (b).
Student computed slope, both times, but may have forgotten to include the negative sign, or made another simple computational error. Student found the slopes in both parts (a) and (b) to be equal.
Student correctly computed the slope, both times, and found
β ). Student found the 2 slopes in both parts (a) and (b) to be equal.
Student made no attempt to answer the question.
Student stated that the slopes in parts (a) and (b) were not equal.
Student made a weak argument by stating that the slopes are equal because the fractions are equal, or that the fractions representing the slope are equivalent.
Student made a convincing argument and referenced similar triangles to explain why the slopes between any two points on a line are equal.
Student made no attempt to answer the question or wrote βKevinβ or βJeremyβ with no evidence of an application of mathematics to solve the problem.
Student wrote an incorrect answer but showed some evidence of reasoning in their explanation.
Student wrote the correct answer. Studentsβ explanation lacked precision or was incorrect. For example, student may have written that Jeremy travels a farther distance in 2 hours, instead of referencing the rates of each biker.
Student wrote the correct answer. Student provided a strong mathematical explanation as to which biker rides at a greater speed by referencing a graph (slopes of each line where one slope is steeper) or a numerical comparison of their rates.
Module 4: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
3
π = β (or an 2 equivalent fraction to 3
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bβd 8.EE.B.5
3 8.EE.B.5
4
aβc 8.EE.C.7a 8.EE.C.8
End-of-Module Assessment Task
8β’4
Student made little or no attempt to complete parts (b)β(d). Student may have plotted points on the graph with no relevance to the problem.
Student wrote an incorrect equation in part (b) and/or graphed the equation incorrectly and/or calculated the slope incorrectly. Student did not connect the slope of the line to Laurenβs rate.
Student may have correctly identified the equation, graphed, and calculated slope and identified it as Laurenβs rate but may not have explained their reasoning in part (b). OR Student may have made a mistake in writing the equation for part (b) which led to an incorrect graph and slope in parts (c)β(d).
Student correctly wrote the equation, π¦ = 23π₯ in part (b) or an equivalent equation. Student explained βtwice as fastβ in terms of distance traveled for a given time interval compared to the data for Kevin given in the table. Student correctly graphed the situation in (c) and correctly identified the slope of the line, 23, as the rate that Lauren rides for part (d).
Student made no attempt to answer the question OR wrote a dollar amount with no explanation.
Student may or may not have correctly calculated the unit rate of protractors for Store A. Student wrote an estimate for Store B, but did not justify their estimate using evidence from the graph.
Student used the information provided about Store A to determine the unit rate of protractors buy may have made a computation error leading to a poor estimate. Student may or may not have used evidence from the graph to justify their estimate or made a weak connection between their estimate and the graph.
Student used the information provided about Store A to determine the unit rate of protractors and referenced the unit rate at Store A in the justification of their estimate. Student wrote an estimate that made sense (e.g., less than Store A, about half as much) and used evidence from the graph in their justification (e.g., comparison of slopes, size of angles).
Student made no attempt to answer any parts of (a)β(c) OR just rewrote the given equation as an answer.
Student answered at least one part of (a)β(c) correctly. Student may have left two parts blank. Student may or may not have explained their reasoning.
Student answered at least two parts of (a)β(c) correctly. Student may have left one part blank. Student explained their reasoning in at least two parts of (a)β(c) noting the characteristics required to achieve the desired number of solutions.
Student provided a correct equation and explanation for each of the parts of (a)β(c). Specifically, for part (a) an equation that represents a distinct line from the given equation will have a slope 1 different than β , for 3 part (b) an equation that represents a line parallel to the given equation will have the same slope, and for part (c) an equation that represents the same line as the given equation whose graphs coincide.
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NYS COMMON CORE MATHEMATICS CURRICULUM
d 8.EE.C.8
Student made little or no attempt to answer the question. Student did not provide a mathematical explanation or apply any mathematical reasoning to support their answer.
End-of-Module Assessment Task
Student gave an incorrect answer. Student may have said that the point of intersection of the lines is the solution to the system, or that there is no solution because the lines are parallel.
Student may have tried to find the solution algebraically. Student stated that there are infinitely many solutions to the system, but may not have referenced what the graph would look like (i.e., each equation produced the same line on the graph). Student supplied weak mathematical reasoning to support their answer.
8β’4
Student correctly stated that the graphs of the equations produce the same line. Student explained that one equation can be obtained by the other by multiplying the first equation by β2, or the 1
second equation by β . 2 OR Student explained that both lines had the 5 2
same slope of and the same π¦-intercept of (0,5). Student supplied strong mathematical reasoning to support their answer.
5
6
8.EE.C.8
a 8.EE.C.8
b 8.EE.C.8
Student made little or no attempt to write and solve a system of linear equations.
Student may have written an incorrect system of equations to represent the situation. Student may or may not have defined their variables. Student may have used another strategy to determine the numbers of tickets of each type were sold. There is some evidence of mathematical reasoning.
Student correctly wrote a system of linear equations to represent the situation, but made a computational error leading to an incorrect solution OR student correctly wrote and solved a system but did not define their variables.
Student correctly wrote and solved a system of linear equations to solve the problem. Student defined the variables used in the system. Student stated clearly that 175 childrenβs tickets and 100 adultβs tickets were sold.
Student made little or no attempt to write the equation.
Student incorrectly computed the slope of the line as something other than 2. Student did not write the correct equation of the line.
Student used the points to correctly determine the slope of the line as 2, but may have written an incorrect equation.
Student made little or no attempt to answer the question. OR Student responded with βyesβ or βnoβ only.
Student incorrectly computed the slope of the line as something other than 1 and may or may not have drew an incorrect conclusion about whether or not the lines would intersect.
Student used the points to correctly determine the slope of the line as 1, but made an incorrect conclusion about whether or not the lines would intersect. OR Student made a computational error for the slope and drew the
Student used the points to determine the slope of the line, then wrote the equation of the line passing through those two points as π¦ = 2π₯ β 1 or equivalent.
Module 4: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
Students used the points to determine the slope as 1 AND correctly concluded that the lines will intersect because the slopes are different.
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NYS COMMON CORE MATHEMATICS CURRICULUM
End-of-Module Assessment Task
8β’4
wrong conclusion about the lines. OR Student said the lines would intersect, but did not provide an explanation.
7
aβb 8.EE.C.8
cβd 8.EE.C.8
eβf 8.EE.C.8
Student left both parts (a) and (b) blank. OR Student identified coordinates that do not fall on either the π₯ or π¦ axis.
Student identified one of the two π¦-intercepts, but may have inversed the coordinates. Student may have left one of (a) or (b) blank.
Student left the item blank. OR Student only wrote one equation that may or may not have represented one of the lines on the graph. Student may or may not have written an estimate or wrote and estimate where the π₯-value was not between 1 and 2 and the π¦-value was not between 3 and 4.
Student wrote two equations to represent the system but the equations did not represent the lines on the graph. Student may or may not have written an estimate or wrote and estimate where the π₯-value was not between 1 and 2 and the π¦-value was not between 3 and 4.
Student was unable to solve the system algebraically.
Module 4: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
Student solved the system algebraically, but made serious computational errors leading to an incorrect solution. Student was unable to complete part (f) or may have realized their error and did not correct it.
Student identified the π¦intercepts of π1 and π2 , but may have inversed the coordinates, i.e., (4, 0) and (2, 0), or identified the slope of π1 as (0, 2), and π2 as (0, 4).
Student correctly identified the π¦intercepts of π1 and π2 , (0, 4) and (0, 2), respectively.
Student wrote a system of equations but one of the equations may have been written incorrectly. Student wrote an estimate where the π₯value was between 1 and 2 and the π¦-value was between 3 and 4.
Student correctly wrote the system as π₯ β π¦ = β2 οΏ½ or a system π₯ + 2π¦ = 8 equivalent to this given one. Student wrote an estimate where the π₯value was between 1 and 2 and the π¦-value was between 3 and 4.
Student solved the system, but may have made a computational error leading to an incorrect π₯ or π¦ coordinate. Student verified their solution in part (f). Student may have made a computational error leading them to believe their solution was correct.
Student correctly solved the system and identified the solution as 4 10 οΏ½. Student 3 3
οΏ½ ,
verified their solution in part (f).
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NYS COMMON CORE MATHEMATICS CURRICULUM
Name
End-of-Module Assessment Task
8β’4
Date
Module 4: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
Linear Equations 11/8/13
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NYS COMMON CORE MATHEMATICS CURRICULUM
End-of-Module Assessment Task
8β’4
2. Jeremy rides his bike at a rate of 12 miles per hour. Below is a table that represents the number of hours and miles Kevin rides. Assume both bikers ride at a constant rate. Time in hours (π₯)
Distance in miles (π¦)
2
23
1.5 3.5 4
Module 4: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
17.25 40.25 46
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Module 4: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
End-of-Module Assessment Task
8β’4
Linear Equations 11/8/13
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Module 4: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
End-of-Module Assessment Task
8β’4
Linear Equations 11/8/13
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Module 4: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
End-of-Module Assessment Task
8β’4
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Module 4: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
End-of-Module Assessment Task
8β’4
Linear Equations 11/8/13
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Module 4: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
End-of-Module Assessment Task
8β’4
Linear Equations 11/8/13
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Module 4: Date: Β© 2013 Common Core, Inc. Some rights reserved. commoncore.org
End-of-Module Assessment Task
8β’4
Linear Equations 11/8/13
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