HCPS Course 2 Curriculum Guide - Henrico County Public Schools [PDF]

Course 2 Pacing Guide and Curriculum Reference Based on the 2009 Virginia Standards of Learning

2016-2017 Henrico County Public Schools

Course 2 Curriculum Guide

HENRICO COUNTY PUBLIC SCHOOLS

Introduction The Mathematics Curriculum Guide serves as a guide for teachers when planning instruction and assessment. It defines the content knowledge, skills, and understandings that are measured by the Standards of Learning assessment. It provides additional guidance to teachers as they develop an instructional program appropriate for their students. It also assists teachers in their lesson planning by identifying essential understandings, defining essential content knowledge, and describing the intellectual skills students need to use. This Guide delineates in greater specificity the content that all teachers should teach and all students should learn. The format of the Curriculum Guide facilitates teacher planning by identifying the key concepts, knowledge, and skills that should be the focus of instruction for each objective. The Curriculum Guide is divided into sections: Vertical Articulation, Curriculum Information, Essential Knowledge and Skills, Key Vocabulary, Essential Questions and Understandings, and Teacher Notes and Elaborations. The purpose of each section is explained below. Vertical Articulation: This section includes the foundational objectives and the future objectives correlated to each SOL. Curriculum Information: This section includes the objective and SOL Reporting Category, focus or topic, pacing guidelines, and links to VDOE’s Enhanced Scope and Sequence lessons. Essential Knowledge and Skills: Each objective is expanded in this section. What each student should know and be able to do in each objective is outlined. This is not meant to be an exhaustive list nor is a list that limits what taught in the classroom. This section is helpful to teachers when planning classroom assessments as it is a guide to the knowledge and skills that define the objective. (Taken from the Curriculum Framework) Key Vocabulary: This section includes vocabulary that is key to the objective and many times the first introduction for the student to new concepts and skills. Essential Questions and Understandings: This section delineates the key concepts, ideas and mathematical relationships that all students should grasp to demonstrate an understanding of the objectives. (Taken from the Curriculum Framework) Teacher Notes and Elaborations: This section includes background information for the teacher. It contains content that is necessary for teaching this objective and may extend the teachers’ knowledge of the objective beyond the current grade level. Special thanks to Prince William County Public Schools for allowing information from their curriculum documents to be included in this document.

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Course 2 Pacing and Curriculum Guide Course Outline First Marking Period

Second Marking Period

Third Marking Period

Fourth Marking Period

7.1d* - Square Roots

7.1a - Negative Exponents

7.9 - Probability

7.5 - Volume and Surface Area

7.1e - Absolute Value

7.1bc* - Scientific Notation, Fractions, Decimals, Percents

7.10 - Probability of Compound

7.8 - Transformations

7.14 - Equations

7.4 - Proportional Reasoning

7.13 - Expressions

7.15 - Inequalities

7.6 - Similar Figures

7.2 - Sequences

7.12 - Representations

7.11 - Statistics

7.3* - Integer Operations

7.16 - Properties

Events

7.7 - Quadrilaterals

SOL Review * SOL test items measuring Objectives 7.1b-d and 7.3b will be completed without the use of a calculator.

View the online HCPS Pacing Guide for more details GRADE 7 SOL TEST BLUEPRINT (45 QUESTIONS TOTAL) *The Grade 7 SOL is a computer adaptive test (CAT)* Number and Number Sense, Computation and Estimation Measurement and Geometry Probability, Statistics, Patterns, Functions, and Algebra

14 Questions 12 Questions 19 Questions

31 % of the Test 27 % of the Test 42 % of the Test

Henrico County Course 2 website Grade 7 Mathematics Formula Sheet VDOE Middle School Math Vocabulary Cards

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Vertical Articulation Grade 5

Grade 7

Grade 8

Algebra 1

6.5 investigate/describe positive exponents, perfect squares

7.1 a) investigate/describe negative 8.5 a) determine if a number is a exponents; d) determine square perfect square; b) find two roots consecutive whole numbers between which a square root lies

A.3 express square roots/cube roots of whole numbers/the square root of monomial algebraic expression (simplest radical form)

5.2 a) recognize/name fractions in their equivalent decimal form and vice versa; b) compare/order fracts and decimals

6.2 a) frac/dec/% - a) describe as ratios; b) ID from representation; c) equiv relationships; d) compare/order

7.1 b) determine scientific notation 8.1 b) compare/order fract/dec/%, for numbers > zero; c) and scientific notation compare/order fract/dec/%, and scientific notation e) ID/describe absolute value for rational numbers

A.1 represent verbal quantitative situations algebraically/evaluate expressions for given replacement values of variables

5.18 c) model one-step linear equations using add/sub

6.3 a) ID/represent integers; b) order/compare integers; c) ID/describe absolute value of integers

7.3 a) model operations (add/sub/mult/div) w/ integers

5.17 describe/express the relationship in a number pattern

6.17 ID/extend geometric/arithmetic 7.2 describe/represent sequences arithmetic/geometric sequences using variable expressions

5.5 a) find sum/diff/product/quotient of two decimals through thousandths

6.6 a) mult/div fractions

7.3 b) add/sub/mult/div integers

6.1 describe/compare data using ratios

7.4 single and multistep practical problems with proportional reasoning

Exponents/ Squares/ Square Roots Modeling/ Comparing/Ordering Operations/ Alg Patt/ Recall Seq Ratios/ Proportions

Grade 6

8.3 a) solve practical problems involving rational numbers, percent, ratios, and prop

6.2 frac/dec/% - a) describe as ratios; 7.6 determine similarity of plane b) ID from representation; c) equiv figures and write proportions to relationships; express relationships between similar quads and triangles

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Plane and Solid Figures

Measurement Apps - Geom Figures

Grade 5

Probability

Grade 7

Grade 8

5.8 a) find perimeter/area/volume; 6.9 make ballpark comparisons b) differentiate among between U.S. Cust/metric system perimeter/area/ volume, ID which measure is appropriate; c) ID equiv measurements within metric system; d) estimate/measure U.S. Cust/metric; e) choose appropriate unit of measure w/ U.S. Cust/ metric

7.5 a) describe volume/surface area of cylinders; b) solve practical problems involving volume/surface area of rect. prims and cylinders; c) describe how changes in measured attribute affects volume/surface area

8.7 a) investigate/solve practical problems involving volume/surface area of prisms, cylinders, cones, pyramids; b) describe how changes in measured attribute affects volume/surface area

5.13 a) using plane figures will 6.12 determine congruence of develop definitions of plane figures; segments/angles/polygons b) investigate/describe results of combining/subdividing plane figures

7.6 determine similarity of plane figures and write proportions to express relationships between similar quads and triangles

8.10 a) verify the Pythagorean Theorem; b) apply the Pythagorean Theorem

5.12 a) classify angles as right/ 6.13 ID/describe properties of acute/ obtuse/straight; b) triangles quadrilaterals as right/ acute/obtuse/equilateral/scalene/is osceles.

7.7 compare/contrast quadrilaterals 8.6 a) verify/describe relationships based on properties among vertical/adjacent/supplementary/co mplementary angles; b) measure angles < 360°

5.11 measure right/acute/obtuse/straight angles

Collect/Represent Data

HENRICO COUNTY PUBLIC SCHOOLS Grade 6

Algebra 1

6.11 a) ID coordinates of a point in a 7.8 represent transformations of 8.8 a) apply transformations to plane coordinate plane; b) graph ordered polygons in the coordinate plane by figures; b) ID applications of pairs in coordinate plane graphing transformations

5.14 make predictions/determine 6.16 a) compare/contrast dep/indep 7.9 investigate/describe the 8.12 determine probability of probability by constructing a sample events; b) determine probabilities difference between the indep/dep events with and without space for dep/indep events experimental/theoretical probability replacement 7.10 determine the probability of compound events, Basic Counting Principle 5.15 collect/organize/interpret data, 6.14 a) construct circle graphs; b) 7.11 a) construct/analyze using stem-and-leaf plots/line draw conclusions/make predictions, histograms; b) compare/contrast graphs using circle graphs; c) histograms compare/contrast graphs

8.13 a) make A.10 compare/contrast multiple comparisons/predictions/inferences, univariate data sets with box-andusing information displayed in whisker plots graphs; b) construct/analyze scatterplots

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Course 2 Curriculum Guide Grade 5

HENRICO COUNTY PUBLIC SCHOOLS Grade 6

Grade 7

Equations and Inequalities Collect/Represent Data

7.12 represent relationships with tables, graphs, rules, and words

5.18 a) investigate/describe concept 6.18 solve one-step linear equations 7.14 a) solve one- and two-step of variable; b) write open sentence in one variable linear equations; b) solve practical using variable; c) model one-step problems in one variable linear equations using add/sub; d) create problems based on open sentence

6.20 graph inequalities on number line

7.15 a) solve one-step inequalities; b) graph solutions on number line

Grade 8

Algebra 1

8.14 make connections between any A.7 investigate/analyze functions two representations (tables, graphs, (linear/quadratic) families and words, rules) characteristics (algebraically/graphically) - a) determine relation is function; b) domain/range; c) zeros; d) x- and yintercepts; e) find values of function for elements in domain; f) make connect between/among multiple representation of functions (concrete/verbal/numeric/graphic/al gebraic)

8.15 a) solve multistep linear equations in one variable (variable on one and two sides of equations); b) solve two-step linear inequalities and graph results on number line; c) ID properties of operations used to solve

A.4 solve multistep linear/quad equation (in 2 variables) - a) solve literal equation; b) justify steps used in simplifying expressions and solving equations; c) solve quad equations (algebraically/graphically); d) solve multistep linear equations (algebraically/graphically); e) solve systems of two linear equation (2 variable-algebraically/graphically); f) solve real-world problems involving equations and systems of equations

8.16 graph linear equation in two variables

A.5 solve multistep linear inequalities (2 variables) - a) solve multistep linear inequalities (algebraically/graphically); b) justify steps used in solving inequalities; c) solve real-world problems involving inequalities; d) solve systems of inequalities

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Properties

Expressions/ Equations and Inequalities Collect/Represent Data Operations

Grade 5

Grade 6

Grade 7

Grade 8

Algebra 1 A.6 graph linear equations/linear inequalities (in 2 variables) - a) determine slope of line given equation of line/graph of line or two points on line - slope as rate of change; b) write equation of line given graph of line, two points on line or slope & point on line

5.7 evaluate whole number 6.8 evaluate whole number numerical expressions using order of expressions using order of operations operations

7.13 a) write verbal expressions as algebraic expressions and sentences as equations and vice versa; b) evaluate algebraic expressions

5.19 distributive property of mult over addition

7.16 a) apply properties w/ real 8.15 c) ID properties of operations numbers: commutative and used to solve equations associative properties for add/mult; b) distributive property; c) additive/ multiplicative identity properties; d) additive/ multiplicative inverse properties; e) multiplicative property of zero

6.19 a) investigate/recognize identity properties for add/mult; b) multiplicative property of zero; c) inverse preperty for mult

Virginia Department of Education - Fall 2010

8.1 a) simplify numerical expressions involving positive exponents, using rational numbers, order of operations, properties

A.1 represent verbal quantitative situations algebraically/evaluate expressions for given replacement values of variables A.2 perform operations on polynomials - a) apply laws of exponents to perform ops on expressions; b) add/subtract/multiply/divide polynomials; c) factor first and second degree binomials/trinomials (1 or 2 variables)

DRAFT -Vertical Articulation of the 2009 Mathematics Standards of Learning

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Course 2 Curriculum Guide Curriculum Information SOL 7.1 SOL Reporting Category Number and Number Sense, Computation and Estimation Focus Proportional Reasoning Virginia SOL 7.1 The student will a. investigate and describe the concept of negative exponents for powers of ten; b. determine scientific notation for numbers greater than zero;* c. compare and order fractions, decimals, percents and numbers written in scientific notation;* d. determine square roots;* and e. identify and describe absolute value for rational numbers. *SOL test items measuring Objective 7.1b-d will be completed without the use of a calculator. HCPS Website SOL 7.1a – Negative Exponents SOL 7.1bc – Compare & Order SOL 7.1d – Square Roots SOL 7.1e – Absolute Value DOE Lesson Plans  Powers of Ten (PDF) - Describing the concept of negative exponents for powers of ten (Word)  Scientific Notation (PDF) Ordering numbers written in scientific notation (Word)  Ordering Fractions, Decimals, and Percents (PDF) - Ordering fractions, decimals, and percents (Word) Return to Course Outline (continued)

Essential Knowledge and Skills Key Vocabulary The student will use problem solving, mathematical communication, mathematical reasoning, connections and representations to:  Recognize powers of 10 with negative exponents by examining patterns.  Write a power of 10 with a negative exponent in fraction and decimal form.  Recognize a number greater than zero in scientific notation.  Write a number greater than zero in scientific notation.  Compare and determine equivalent relationships between numbers larger than zero, written in scientific notation.  Order no more than three numbers greater than zero written in scientific notation.  Represent a number in fraction, decimal, and percent forms.  Compare, order, and determine equivalent relationships among fractions, decimals, and percents. Decimals are limited to the thousandths place, and percents are limited to the tenths place. Ordering is limited to no more than four numbers.  Compare and order fractions, decimals, percents, and numbers written in scientific notation.  Determine the square root of a perfect square less than or equal to 400 without the use of a calculator.  Demonstrate absolute value using a number line.  Determine the absolute value of a rational number. (continued)

HENRICO COUNTY PUBLIC SCHOOLS Essential Questions and Understandings Teacher Notes and Elaborations Essential Questions and Understandings  When should scientific notation be used? Scientific notation should be used whenever the situation calls for use of very large or very small numbers.  How are fractions, decimals and percents related? Any rational number can be represented in fraction, decimal and percent form.  What does a negative exponent mean when the base is 10? A base of 10 raised to a negative exponent represents a number between 0 and 1.  How is taking a square root different from squaring a number? Squaring a number and taking a square root are inverse operations.  Why is the absolute value of a number positive? The absolute value of a number represents distance from zero on a number line regardless of direction. Distance is positive. Teacher Notes and Elaborations  Recognize powers of 10 with negative exponents by examining patterns. o An exponent tells how many times the base is used as a factor. In the expression 32, 3 is the base and 2 is the exponent. o Negative exponents for powers of 10 can be investigated through patterns such as: 102  100 101  10 100  1 1 1   0.1 101 10 1 1  2   0.01 100 10

101  102

 Write a power of 10 with a negative exponent in fraction and decimal form. o Negative exponents for powers of 10 are used to represent numbers between 0 1 1 and 1 (e.g., 103  3 and 3  0.001 ). 10 10

(continued) Page 7 of 71

Course 2 Curriculum Guide Curriculum Information SOL 7.1 (continued from previous page) SOL Reporting Category Number and Number Sense, Computation and Estimation Focus Proportional Reasoning Virginia SOL 7.1 The student will a. investigate and describe the concept of negative exponents for powers of ten; b. determine scientific notation for numbers greater than zero;* c. compare and order fractions, decimals, percents and numbers written in scientific notation;* d. determine square roots;* and e. identify and describe absolute value for rational numbers. *SOL test items measuring Objective 7.1b-d will be completed without the use of a calculator. DOE Lesson Plans (continued)  Absolute Value (PDF) - Identifying and describing absolute value for rational numbers (Word)  Square Roots (PDF) - Determining square roots (Word)

Essential Knowledge and Skills Key Vocabulary (continued from previous page)

HENRICO COUNTY PUBLIC SCHOOLS Essential Questions and Understandings Teacher Notes and Elaborations (continued from previous page)

The student will use problem solving, mathematical communication, mathematical reasoning, connections and representations to:  Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle to solve practical problems.

Essential Questions and Understandings  Scientific notation is used to represent very large and very small numbers. o A number is in scientific notation when it is written in the form: a ∙ 10n where 1  a  10 and n is an integer. o A number written in scientific notation is the product of two factors, a decimal greater than or equal to one but less than 10, and a power of 10 (e.g., 3.1 ∙ 10 5 = 310,000 and 2.85 ∙ 10-4 = 0.000285).

Key Vocabulary absolute value exponent percent perfect square rational number scientific notation square root

 Recognize a number greater than zero in scientific notation.

 When should scientific notation be used? Scientific notation should be used whenever the situation calls for use of very large or very small numbers.

 Write a number greater than zero in scientific notation.  Compare and determine equivalent relationships between numbers larger than zero, written in scientific notation.  Order no more than three numbers greater than zero written in scientific notation.  Compare and order fractions, decimals, percents, and numbers written in scientific notation.  How are fractions, decimals and percents related? Any rational number can be represented in fraction, decimal and percent form. a  A rational number is any number that can be expressed in the form , where a and b b are integers and b ≠ 0.  Percent means “per hundred”. A number followed by a percent symbol (%) is 3 60 60 equivalent to that number with a denominator of 100 (e.g.,  ,  0.60 , 0.60 5 100 100 = 60%). (continued)

Return to Course Outline

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Course 2 Curriculum Guide Curriculum Information SOL 7.1 SOL Reporting Category Number and Number Sense, Computation and Estimation

HENRICO COUNTY PUBLIC SCHOOLS Essential Questions and Understandings Teacher Notes and Elaborations (continued)  Equivalent relationships among fractions, decimals, and percents can be determined by using manipulatives (e.g., fraction bars, Base10 blocks, fraction circles, graph paper, number lines and calculators).

Focus Proportional Reasoning

 Compare, order, and determine equivalent relationships among fractions, decimals, and percents. Decimals are limited to the thousandths place, and percents are limited to the tenths place. Ordering is limited to no more than four numbers.

 Represent a number in fraction, decimal, and percent forms.

 How is taking a square root different from squaring a number? Squaring a number and taking a square root are inverse operations. Virginia SOL 7.1 The student will a. investigate and describe the concept of negative exponents for powers of ten; b. determine scientific notation for numbers greater than zero;* c. compare and order fractions, decimals, percents and numbers written in scientific notation;* d. determine square roots;* and e. identify and describe absolute value for rational numbers. *SOL test items measuring Objective 7.1b-d will be completed without the use of a calculator.

 A square root of a number is a number which, when multiplied by itself, produces the given number (e.g., 121 is 11 since 11 ∙ 11 = 121). A whole number that can be named as a product of a number with itself is a perfect square (e.g., 81 = 9 ∙ 9, where 81 is a perfect square).  The square root of a number can be represented geometrically as the length of a side of the square.  Determine the square root of a perfect square less than or equal to 400 without the use of a calculator.  Why is the absolute value of a number positive? The absolute value of a number represents distance from zero on a number line regardless of direction. Distance is always positive.  The absolute value of a number is the distance from 0 on the number line regardless of direction (e.g.,  1  1 , 1  1 , 1  1 , 2

2

2

2

2

2

and 1  1 ). 2 2  The distance between two rational numbers on the number line is the absolute value of their difference. o

Example 1: The distance between 5 and 2 is 5  2  3 or 2  5  3 .

o

Example 2: The distance between 3.5 and ( 7.4 ) is 3.5   7.4   10.9 or

o

Example 3: The distance between ( 4 ) and ( 1 ) is

 4   1

 3 or

 7.4  3.5  10.9 .  1   4  3 .

1 2 and 4 is 1 2  4 1  2 8 or 4 1  1 2  2 8 . 3 5 3 5 15 5 3 15  Demonstrate absolute value using a number line.

o

Example 4: The distance between 1

 Determine the absolute value of a rational number.  Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle to solve practical problems.


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Course 2 Curriculum Guide Curriculum Information SOL 7.2 SOL Reporting Category Number and Number Sense, Computation and Estimation Focus Proportional Reasoning Virginia SOL 7.2 The student will describe and represent, arithmetic and geometric sequences using variable expressions. HCPS Website SOL 7.2 - Sequences ESS Lesson Arithmetic and Geometric Sequences (PDF) - Describing arithmetic and geometric sequences (Word)

Essential Knowledge and Skills Key Vocabulary The student will use problem solving, mathematical communication, mathematical reasoning, connections and representations to:  Analyze arithmetic and geometric sequences to discover a variety of patterns.  Identify the common difference in an arithmetic sequence.  Identify the common ratio in a geometric sequence.  Given an arithmetic or geometric sequence, write a variable expression to describe the relationship between two consecutive terms in the sequence. Key Vocabulary arithmetic sequence common difference common ratio consecutive terms geometric sequence variable expression

HENRICO COUNTY PUBLIC SCHOOLS Essential Questions and Understandings Teacher Notes and Elaborations Essential Questions and Understandings  What are arithmetic sequences? In an arithmetic sequence, the numbers are found by using a common difference.  What are geometric sequences? In a geometric sequence, the numbers are found by using a common ratio.  When are variable expressions used? Variable expressions can express the relationship between two consecutive terms in a sequence. Teacher Notes and Elaborations  Analyze arithmetic and geometric sequences to discover a variety of patterns.  In the numeric pattern of an arithmetic sequence, students must determine the difference, called the common difference, between each succeeding number in order to determine what is added to each previous number to obtain the next number. o Sample arithmetic sequences include: 4, 7, 10, 13, … (The common difference is 3) 10, 3, 4 , 11 , … (The common difference is 7 ) (The common difference is 5) 6 , 1 , 4, 9, …  Identify the common difference in an arithmetic sequence.  What are geometric sequences? In a geometric sequence, the numbers are found by using a common ratio.  In geometric sequences, students must determine what each number is multiplied by in order to obtain the next number in the geometric sequence. This multiplier is called the common ratio. o Sample geometric sequences include: 2, 4, 8, 16, 32,… (The common ratio is 2) 1, 5, 25, 125, 625,… (The common ratio is 5) 1 80, 20, 5, 1.25,…. (The common ratio is ) 4  Identify the common ratio in a geometric sequence.  When are variable expressions used? Variable expressions can express the relationship between two consecutive terms in a sequence.

Return to Course Outline (continued) Page 10 of 71

Course 2 Curriculum Guide Curriculum Information SOL 7.2 SOL Reporting Category Number and Number Sense, Computation and Estimation Focus Proportional Reasoning Virginia SOL 7.2 The student will describe and represent, arithmetic and geometric sequences using variable expressions.

HENRICO COUNTY PUBLIC SCHOOLS Essential Questions and Understandings Teacher Notes and Elaborations (continued)  A variable expression can be written to express the relationship between two consecutive terms of a sequence o If n represents a number in the sequence 3, 6, 9, 12…, the next term in the sequence can be determined using the variable expression n + 3. o If n represents a number in the sequence 1, 5, 25, 125…, the next term in the sequence can be determined by using the variable expression 5n.  Given an arithmetic or geometric sequence, write a variable expression to describe the relationship between two consecutive terms in the sequence.  Consecutive terms immediately follow each other in some order. For example 5 and 6 are consecutive whole numbers, 2 and 4 are consecutive even numbers.


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Course 2 Curriculum Guide Curriculum Information SOL 7.3 SOL Reporting Category Number and Number Sense, Computation and Estimation Focus Integer Operations and Proportional Reasoning Virginia SOL 7.3 The student will a. model addition, subtraction, multiplication and division of integers; and b. add, subtract, multiply, and divide integers.* *SOL test items measuring Objective 7.3b will be completed without the use of a calculator. HCPS Website SOL 7.3a – Modeling Integer Operations SOL 7.3b –Integers DOE Lesson Plans  Integers: Multiplication and Division (PDF) - Solve multiplication and division problems that contain integers (Word)  Integers: Addition and Subtraction (PDF) - Modeling addition, subtraction, multiplication, and division of integers (Word)


Essential Knowledge and Skills Key Vocabulary The student will use problem solving, mathematical communication, mathematical reasoning, connections and representations to:  Model addition, subtraction, multiplication and division of integers using pictorial representations of concrete manipulatives.  Formulate rules for addition, subtraction, multiplication, and division of integers.  Add, subtract, multiply and divide integers.  Simplify numerical expressions involving addition, subtraction, multiplication and division of integers using order of operations.  Solve practical problems involving addition, subtraction, multiplication, and division with integers.

Key Vocabulary absolute value integers opposites

HENRICO COUNTY PUBLIC SCHOOLS Essential Questions and Understandings Teacher Notes and Elaborations Essential Questions and Understandings  The sums, differences, products and quotients of integers are either positive, zero, or negative. How can this be demonstrated? This can be demonstrated through the use of patterns and models. Teacher Notes and Elaborations  The set of integers is the set of whole numbers and their opposites o (…-3, -2, -1, 0, 1, 2, 3…) o Integers are used in practical situations such as temperature changes (above/below zero), balance a checking account (deposits/withdrawals), and changes in altitude (above /below sea level).  The absolute value of an integer is the distance on a number line that a number is from zero. o It is always written as a positive number. o Students should recognize and be able to read the symbol for absolute value (e.g., 7  7 is read as “The absolute value of negative seven equals seven.”).  The order of operations is a convention that defines the computation order to follow in simplifying an expression. In grades 5 and 6, students simplify expressions by using the order of operations in a demonstrated step-by-step approach.  Modeling using concrete materials should be explored BEFORE any other strategies are used.  Concrete experiences must be used to formulate rules for adding, subtracting, multiplying, and dividing integers o Use number lines to model addition, subtraction, multiplication and division of integers o Use manipulatives, such as two-color counters or algebra tiles to model addition, subtraction, multiplication and division of integers o Use pictorial representations to model addition, subtraction, multiplication and division of integers o Examine patterns using calculators  Use real-life examples such as weather maps to demonstrate positive and negative temperatures, the stock market to illustrate gains and losses, banking examples involving credits and debits, and problems involving sea level to understand ways in which positives and negatives are used. Page 12 of 71

Course 2 Curriculum Guide Curriculum Information SOL 7.4 SOL Reporting Category Number and Number Sense, Computation and Estimation Focus Integer Operations and Proportional Reasoning Virginia SOL 7.4 The student will solve single-step and multi-step practical problems, using proportional reasoning. HCPS Website SOL 7.4 – Proportions ESS Lessons  Proportions (PDF) - Solving a proportion to find a missing term (Word)  Sales Tax & Tip (PDF) - Solving problems, using proportional reasoning (Word)


Essential Knowledge and Skills Key Vocabulary The student will use problem solving, mathematical communication, mathematical reasoning, connections and representations to:  Write proportions that represent equivalent relationships between two sets.  Solve a proportion to find a missing term.  Apply proportions to convert units of measurement between the U.S. Customary System and the metric system. Calculators may be used.  Apply proportions to solve problems that involve percents.  Apply proportions to solve practical problems, including scale drawings. Scale factors shall have denominators no greater than 12 and decimals no less than tenths. Calculators may be used.  Using 10% as a benchmark, mentally compute 5%, 10%, 15%, or 20% in a practical situation such as tips, tax and discounts.  Solve problems involving tips, tax, and discounts. Limit problems to only one percent computation per problem. Key Vocabulary discount (amount of discount) equivalent extremes means percent proportion rate (discount rate, tax rate, unit rate) ratio sale price (discount price) scale factor tax tip

HENRICO COUNTY PUBLIC SCHOOLS Essential Questions and Understandings Teacher Notes and Elaborations Essential Questions and Understandings  What makes two quantities proportional? Two quantities are proportional, when one quantity is a constant multiple of the other. Teacher Notes and Elaborations  A proportion is a statement of equality between two ratios.  A ratio is a comparison of two numbers or measures using division. Both numbers in a ratio have the same unit of measure. A ratio may be written three ways: as a fraction a , using the notation a:b, or in words a to b. b a c  A proportion can be written as = , a:b = c:d, or a is to b as c is to d. b d  A proportion can be solved by finding the product of the means and the product of the extremes. o In the proportion a:b = c:d, a and d are the extremes and b and c are the means. If values are substituted for a, b, c, and d such as 5:12 = 10:24, then the product of extremes (5  24) is equal to the product of the means (12  10).  In a proportional situation, both quantities increase or decrease together.  In a proportional situation, two quantities increase multiplicatively. Both are multiplied by the same factor.  A proportion can be solved by finding equivalent fractions.  A rate is a ratio that compares two quantities measured in different units.  A unit rate is a rate with a denominator of 1. Examples of rates include miles/hour and revolutions/minute.  A percent is a special ratio in which the denominator is 100. o Proportions can be used to represent percent problems as follows: percent part  100 whole


Course 2 Curriculum Guide Curriculum Information SOL 7.4 SOL Reporting Category Number and Number Sense, Computation and Estimation

HENRICO COUNTY PUBLIC SCHOOLS Essential Questions and Understandings Teacher Notes and Elaborations (continued)  A discount rate is the percent off an item (e.g., If an item is reduced in price by 20%, 20% is the discount rate.) o The amount of discount (discount) is how much is subtracted from the original amount. o The sale price (discount price) is the result of subtracting the discount from the original price.

Focus Integer Operations and Proportional Reasoning

 A sales tax rate is the percent of tax (e.g., Virginia has a 5% tax rate on most items purchased.) Sales tax is the amount added to the price of an item based on the tax rate.

Virginia SOL 7.4 The student will solve single-step and multi-step practical problems, using proportional reasoning.

 A tip is a small sum of money given as acknowledgment of services rendered, (a gratuity). It is often times computed as a percent of the bill or service.  How are proportions used? o Proportions are used in everyday contexts, such as speed, recipe conversions, scale drawings, map reading, reducing and enlarging, comparison shopping, and monetary conversions. o Apply proportions to solve practical problems, including scale drawings. Scale factors shall have denominators no greater than 12 and decimals no less than tenths. Calculators may be used. o Solve problems involving tips, tax, and discounts. Limit problems to only one percent computation per problem. o Using 10% as a benchmark, mentally compute 5%, 10%, 15%, or 20% in a practical situation such as tips, tax and discounts. o Apply proportions to convert units of measurement between the U.S. Customary System and the metric system. Calculators may be used. o Proportions can be used to convert between measurement systems. For example: if 2 inches is about 5 cm, how many inches are 2inches 5cm in 16 cm?  x 16cm o Write proportions that represent equivalent relationships between two sets. o Solve a proportion to find a missing term.


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Course 2 Curriculum Guide Curriculum Information SOL 7.5 SOL Reporting Category Measurement and Geometry Focus Proportional Reasoning Virginia SOL 7.5 The student will a. describe volume and surface area of cylinders; b. solve practical problems involving the volume and surface area of rectangular prisms and cylinders; and c. describe how changing one measured attribute of a rectangular prism affects its volume and surface area. HCPS Website SOL 7.5 – Volume and Surface Area ESS Lessons  S.A. & Vol of a Cylinder (PDF) Determining the surface area and volume of a cylinder (Word)  Vol of a Rectangular Prism (PDF) Determining the volume of a rectangular prism (Word)  S.A. of a Rectangular Prism (PDF) - Determining the surface area of a rectangular prism (Word)  Attributes of a Rectangular Prism (PDF) - Exploring how changing an attribute of a rectangular prism affects its volume and surface area (Word)

Essential Knowledge and Skills Key Vocabulary The student will use problem solving, mathematical communication, mathematical reasoning, connections and representations to:  Determine if a practical problem involving a rectangular prism or cylinder represents the application of volume or surface area.  Find the surface area of a rectangular prism.  Solve practical problems that require finding the surface area of a rectangular prism.  Develop a procedure and formula for finding the surface area of a cylinder.  Find the surface area of a cylinder.  Solve practical problems that require finding the surface area of a cylinder.  Find the volume of a rectangular prism.  Solve practical problems that require finding the volume of a rectangular prism.  Develop a procedure and formula for finding the volume of a cylinder.  Find the volume of a cylinder.  Solve practical problems that require finding the volume of a cylinder.  Describe how the volume of a rectangular prism is affected when one measured attribute is multiplied by a scale factor. Problems will be limited to changing attributes by scale factors 1 (e.g., , 2, 3, 5, and 10) only. 2

HENRICO COUNTY PUBLIC SCHOOLS Essential Questions and Understandings Teacher Notes and Elaborations Essential Questions and Understandings  How are volume and surface area related? Volume is a measure of the amount a container holds while surface area is the sum of the areas of the surfaces on the container.  How does the volume of a rectangular prism change when one of the attributes is increased? There is a direct relationship between the volume of a rectangular prism increasing when the length of one of the attributes of the prism is changed by a scale factor. Teacher Notes and Elaborations  The ratio of the circumference of any circle to the length of its diameter is  (pi).  is a nonterminating nonrepeating decimal. The most commonly used rational number 22 approximations for  are 3.14 and . 7  The area of a rectangle is computed by multiplying the lengths of two adjacent sides.  The area of a circle is computed by squaring the radius and multiplying that product 22 by π (A = πr2, where π ≈ 3.14 or ). 7  The radius of a circle is a segment connecting the center of the circle to a point on the circle.  The diameter of a circle is a segment connecting two points on the circle and passing through the center.  Nets are two-dimensional drawings (e.g., a drawing of a figure that has length and width) of three-dimensional figures (e.g., a figure that has length, width, and height) that can be used to help students find surface area.  A net of a solid is a two dimensional figure that can be folded into a three dimensional shape.  Determine if a practical problem involving a rectangular prism or cylinder represents the application of volume or surface area.

Return to Course Outline (continued)


Course 2 Curriculum Guide Curriculum Information SOL 7.5 (continued from previous page) SOL Reporting Category Measurement and Geometry Focus Proportional Reasoning Virginia SOL 7.5 The student will a. describe volume and surface area of cylinders; b. solve practical problems involving the volume and surface area of rectangular prisms and cylinders; and c. describe how changing one measured attribute of a rectangular prism affects its volume and surface area.

Essential Knowledge and Skills Key Vocabulary (continued from previous page) The student will use problem solving, mathematical communication, mathematical reasoning, connections and representations to:  Describe how the surface area of a rectangular prism is affected when one measured attribute is multiplied by a scale factor. Problems will be limited to changing attributes by scale factors 1 (e.g., , 2, 3, 5, and 10) only. 2 Key Vocabulary base cube cylinder diameter face formula height length net pi (  ) radius rectangular prism scale factor surface area volume width

HENRICO COUNTY PUBLIC SCHOOLS Essential Questions and Understandings Teacher Notes and Elaborations (continued from previous page) Teacher Notes and Elaborations  Surface area of any solid figure is the total area of the surface of the solid.  The volume of a solid is the total amount of space inside a three-dimensional object. A unit for measuring volume is the cubic unit.  Find the surface area of a rectangular prism.  Find the volume of a rectangular prism.  Solve practical problems that require finding the surface area of a rectangular prism.  Solve practical problems that require finding the volume of a rectangular prism.  A rectangular prism can be represented on a flat surface as a net that contains six rectangles – two that have measures of the length and width of the base, two others that have measures of the length and height, and two others that have measures of the width and height. o The surface area of a rectangular prism is the sum of the areas of all six faces (SA = 2lw + 2lh + 2wh). o The volume of a rectangular prism is computed by multiplying the area of the base, B, (length times width) by the height of the prism (V = lwh or V = Bh).

 Find the surface area of a cylinder.  Find the volume of a cylinder.  Solve practical problems that require finding the surface area of a cylinder.



Course 2 Curriculum Guide Curriculum Information SOL 7.5 SOL Reporting Category Measurement and Geometry Focus Proportional Reasoning Virginia SOL 7.5 The student will a. describe volume and surface area of cylinders; b. solve practical problems involving the volume and surface area of rectangular prisms and cylinders; and c. describe how changing one measured attribute of a rectangular prism affects its volume and surface area.

HENRICO COUNTY PUBLIC SCHOOLS Essential Questions and Understandings Teacher Notes and Elaborations (continued)  Solve practical problems that require finding the volume of a cylinder.  A cylinder can be represented on a flat surface as a net that contains two circles (bases for the cylinder) and one rectangular region whose length is the circumference of the circular base and whose width is the height of the cylinder. o The surface area of the cylinder is the area of the two circles and the rectangle (SA = 2πr2 + 2πrh). o The volume of a cylinder is computed by multiplying the area of the circular base, B, (πr2) by the height of the cylinder (V = πr2h or V = Bh).

 Describe how the volume of a rectangular prism is affected when one measured attribute is multiplied by a scale factor. Problems will be limited to changing attributes by scale factors only.  Describe how the surface area of a rectangular prism is affected when one measured attribute is multiplied by a scale factor. Problems will be limited to changing attributes by scale factors only.



Course 2 Curriculum Guide Curriculum Information SOL 7.5 SOL Reporting Category Measurement and Geometry Focus Proportional Reasoning Virginia SOL 7.5 The student will a. describe volume and surface area of cylinders; b. solve practical problems involving the volume and surface area of rectangular prisms and cylinders; and c. describe how changing one measured attribute of a rectangular prism affects its volume and surface area.

HENRICO COUNTY PUBLIC SCHOOLS Essential Questions and Understandings Teacher Notes and Elaborations (continued)  There is a direct relationship between changing one measured attribute of a rectangular prism by a scale factor and its volume. For example, doubling the length of a prism will double its volume. o This direct relationship does not hold true for surface area. For example, doubling the length will only double the area of the affected sides. It will not double the total surface area. o Example: Given a rectangular prism with the following dimensions: l = 5 meters, w = 4 meters and h = 3 meters. Students should describe how the volume and surface area of a rectangular prism is affected when one attribute is multiplied by a scale factor. A scale factor is a ratio that compares the sizes of the parts of the scale drawing of an object with the actual sizes of the corresponding parts of the object (e.g., If the scale drawing is ten times the size of the actual object, the scale factor is 10). Length

Width

Height

Volume

Surface Area

5

4

3

60 m3

94 m2

Multiply length by 2

10

4

3

120 m3

164 m2

Multiply width by 2

5

8

3

120 m3

158 m2

Multiply height by 2

5

4

6

120 m3

148 m2

1 2

4

3

30 m3

59 m2

3

30 m3

62 m2

1 2

30 m3

67 m2

Original Figure Using the original figure:

Multiply length by

1 2

Multiply width by

1 2

5

2

Multiply height by

1 2

5

4

2

1


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Course 2 Curriculum Guide Curriculum Information SOL 7.6 SOL Reporting Category Measurement and Geometry Focus Proportional Reasoning Virginia SOL 7.6 The student will determine whether plane figures (quadrilaterals and triangles) are similar and write proportions to express the relationships between corresponding sides of similar figures. HCPS Website SOL 7.6 – Similar Figures ESS Lesson Similar Figures (PDF) - Determining whether two plane figures are similar, identifying corresponding sides (Word)

Essential Knowledge and Skills Key Vocabulary The student will use problem solving, mathematical communication, mathematical reasoning, connections and representations to:  Identify corresponding sides and corresponding and congruent angles of similar figures using the traditional notation of curved lines for the angles.  Write proportions to express the relationships between the lengths of corresponding sides of similar figures.  Determine if quadrilaterals or triangles are similar by examining congruence of corresponding angles and proportionality of corresponding sides.  Given two similar figures, write similarity statements using symbols such as ΔABC ~ ΔDEF, ∠A corresponds to ∠D, and AB corresponds to DE . Key Vocabulary corresponding parts congruent hatch mark polygon proportion ratio similar figures


HENRICO COUNTY PUBLIC SCHOOLS Essential Questions and Understandings Teacher Notes and Elaborations Essential Questions and Understandings  How do polygons that are similar compare to polygons that are congruent? Congruent polygons have the same size and shape. Similar polygons have the same shape, and corresponding angles between the similar figures are congruent. However, the lengths of the corresponding sides are proportional. All congruent polygons are considered similar with the ratio of the corresponding sides being 1:1. Teacher Notes and Elaborations  Two polygons are similar if corresponding (matching) angles are congruent and the lengths of corresponding sides are proportional. o The symbol ~ is used to indicate that two polygons (a closed plane figure constructed with three or more straight-line segments that intersect only at their vertices) are similar.  Congruent polygons have the same size and shape.  Congruent polygons are similar polygons for which the ratio of the corresponding sides is 1:1.  Given two similar figures, write similarity statements.  Identify corresponding sides and corresponding and congruent angles of similar figures using the traditional notation of curved lines for the angles.  Similarity statements can be used to determine corresponding parts of similar figures such as: ABC ~ DEF

 A corresponds to  D AB corresponds to DE


Course 2 Curriculum Guide Curriculum Information SOL 7.6 SOL Reporting Category Measurement and Geometry Focus Proportional Reasoning Virginia SOL 7.6 The student will determine whether plane figures (quadrilaterals and triangles) are similar and write proportions to express the relationships between corresponding sides of similar figures.

HENRICO COUNTY PUBLIC SCHOOLS Essential Questions and Understandings Teacher Notes and Elaborations (continued)  The traditional notation for marking congruent angles is to use a curve on each angle. Denote which angles are congruent with the same number of curved lines. For example, if  A congruent to  B, then both angles will be marked with the same number of curved lines.  Write proportions to express the relationships between the lengths of corresponding sides of similar figures.  Congruent sides are denoted with the same number of hatch marks on each congruent side. For example, a side on a polygon with 2 hatch marks is congruent to the side with 2 hatch marks on a congruent polygon.

 Determine if quadrilaterals or triangles are similar by examining congruence of corresponding angles and proportionality of corresponding sides.  Corresponding parts is a one-to-one mapping between two figures. Similar figures are the same shape, but not always the same size. o ΔABC ~ ΔDEF. Therefore: A corresponds to D and A  D B corresponds to E and B  E C corresponds to F and C  F AB BC AC = = DE EF DF o

Two polygons are similar if corresponding (matching) angles are congruent and the lengths of corresponding sides are proportional.


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Course 2 Curriculum Guide Curriculum Information SOL 7.7 SOL Reporting Category Measurement and Geometry Focus Relationships between Figures Virginia SOL 7.7 The student will compare and contrast the following quadrilaterals based on properties: parallelogram, rectangle, square, rhombus, and trapezoid. HCPS Website SOL 7.7 – Quadrilaterals ESS Lesson Quadrilateral Sort (PDF) - Comparing and contrasting the properties of quadrilaterals (Word)


Essential Knowledge and Skills Key Vocabulary The student will use problem solving, mathematical communication, mathematical reasoning, connections and representations to:  Identify the classification(s) to which a quadrilateral belongs, using deductive reasoning and inference.  Compare and contrast attributes of the following quadrilaterals: parallelogram, rectangle, square, rhombus, and trapezoid. Key Vocabulary congruent decagon diagonal hatch marks heptagon hexagon isosceles trapezoid kite nonagon octagon parallel

parallelogram pentagon plane figure polygon quadrilateral rectangle regular polygon rhombus square trapezoid vertex

HENRICO COUNTY PUBLIC SCHOOLS Essential Questions and Understandings Teacher Notes and Elaborations Essential Questions and Understandings  Why can some quadrilaterals be classified in more than one category? Every quadrilateral in a subset has all of the defining attributes of the subset. For example, if a quadrilateral is a rhombus, it has all the attributes of a rhombus. However, if that rhombus also has the additional property of 4 right angles, then that rhombus is also a square. Teacher Notes and Elaborations  A polygon is a simple closed plane figure whose sides are line segments that intersect only at their endpoints.  Two lines in the same plane are parallel if they do not intersect. They are always the same distance from each other.  Two geometric figures that are the same shape and size are congruent. Two angles are congruent if they have the same measure. Two line segments are congruent if they are the same length.  A quadrilateral is a closed plane figure (two-dimensional) with four sides that are line segments.  A parallelogram is a quadrilateral whose opposite sides are parallel and congruent. Opposite angles are congruent. o A diagonal divides the parallelogram into two congruent triangles. The diagonals of a parallelogram bisect each other. o Denote which angles are congruent with the same number of curved lines. Congruent sides are denoted with the same number of hatch marks on each congruent side. o Arrows are used in diagrams to indicate that lines are parallel.

(continued)

Page 21 of 71

Course 2 Curriculum Guide Curriculum Information SOL 7.7 SOL Reporting Category Measurement and Geometry Focus Relationships between Figures Virginia SOL 7.7 The student will compare and contrast the following quadrilaterals based on properties: parallelogram, rectangle, square, rhombus, and trapezoid.

HENRICO COUNTY PUBLIC SCHOOLS Essential Questions and Understandings Teacher Notes and Elaborations (continued)  A rectangle is a parallelogram with four right angles. The diagonals of a rectangle are the same length (congruent) and bisect each other. Since a rectangle is a parallelogram, a rectangle has the same properties as those of a parallelogram.  A square is a rectangle with four congruent sides and a rhombus with four right angles. Squares have special characteristics that are true for all squares, such as diagonals are perpendicular bisectors and diagonals bisect opposite angles. Since a square is a rectangle, a square has all the properties of a rectangle and of a parallelogram.  A rhombus is a parallelogram with four congruent sides whose diagonals bisect each other and intersect at right angles. Opposite angles are congruent.  A trapezoid is a quadrilateral with exactly one pair of parallel sides. A trapezoid may have none or two right angles. A trapezoid with congruent, non-parallel sides is called an isosceles trapezoid.  A kite is a quadrilateral with two pairs of adjacent congruent sides. One pair of opposite angles is congruent.  The number of sides determines the name of the polygon. A pentagon has 5 sides; a hexagon, 6 sides; a heptagon, 7 sides; an octagon, 8 sides; a nonagon, 9 sides; and a decagon, 10 sides.  Prefixes in the names of polygons tell the number of sides: penta = 5, hexa = 6, hepta = 7, octa = 8, nona = 9, and deca = 10.  In regular polygons all angles are congruent and all sides are congruent.


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Course 2 Curriculum Guide Curriculum Information SOL 7.8 SOL Reporting Category Measurement and Geometry Focus Relationships between Figures Virginia SOL 7.8 The student, given a polygon in the coordinate plane, will represent transformations (reflections, dilations, rotations, and translations) by graphing in the coordinate plane. HCPS Website SOL 7.8 - Transformations ESS Lessons  Rotation (PDF) - Rotating a polygon on the coordinate plane (Word)  Dilation (PDF) - Dilate a polygon (Word)  Translation & Reflection (PDF) Translating and reflecting polygons on the coordinate plane (Word)

Essential Knowledge and Skills Key Vocabulary The student will use problem solving, mathematical communication, mathematical reasoning, connections and representations to:  Identify the coordinates of the image of a right triangle or rectangle that has been translated either vertically, horizontally or a combination of a vertical and horizontal translation.  Identify the coordinates of the image of a right triangle or rectangle that has been rotated 90° or 180° about the origin.  Identify the coordinates of the image of a right triangle or a rectangle that has been reflected over the x- or y-axis.  Identify the coordinates of a right triangle or rectangle that has been dilated. The center of the dilation will be the origin.  Sketch the image of a right triangle or rectangle translated vertically or horizontally.  Sketch the image of a right triangle or rectangle that has been rotated 90° or 180° about the origin.  Sketch the image of a right triangle or rectangle that has been reflected over the x- or y-axis.  Sketch the image of a dilation of a right triangle or rectangle limited to a scale 1 1 factor of , , 2, 3, or 4. 4 2

HENRICO COUNTY PUBLIC SCHOOLS Essential Questions and Understandings Teacher Notes and Elaborations Essential Questions and Understandings  How does the transformation of a figure affect the size, shape and position of that figure? Translations, rotations and reflections do not change the size or shape of a figure. A dilation of a figure and the original figure are similar. Reflections, translations and rotations usually change the position of the figure. Teacher Notes and Elaborations  A rotation of a geometric figure is a turn of the figure around a fixed point. The point may or may not be on the figure. The fixed point is called the center of rotation.  Identify the coordinates of the image of a right triangle or rectangle that has been rotated 90° or 180° about the origin.  Sketch the image of a right triangle or rectangle that has been rotated 90° or 180° about the origin.  A translation of a geometric figure is a slide of the figure in which all the points on the figure move the same distance in the same direction.  Identify the coordinates of the image of a right triangle or rectangle that has been translated either vertically, horizontally, or a combination of a vertical and horizontal translation.  Sketch the image of a right triangle or rectangle translated vertically or horizontally.  A reflection is a transformation that reflects a figure across a line in the plane.  Identify the coordinates of the image of a right triangle or a rectangle that has been reflected over the x- or y-axis.  Sketch the image of a right triangle or rectangle that has been reflected over the x- or y-axis.  A dilation of a geometric figure is a transformation that changes the size of a figure by scale factor to create a similar figure.


(continued)


Course 2 Curriculum Guide Curriculum Information SOL 7.8 (continued from previous page) SOL Reporting Category Measurement and Geometry Focus Relationships between Figures Virginia SOL 7.8 The student, given a polygon in the coordinate plane, will represent transformations (reflections, dilations, rotations, and translations) by graphing in the coordinate plane.

Essential Knowledge and Skills Key Vocabulary (continued from previous page) Key Vocabulary center of rotation coordinate plane coordinates (ordered pair) dilation horizontal axis (x-axis) image origin

pre-image quadrant reflection rotation scale factor transformation translation vertical axis (y-axis)

HENRICO COUNTY PUBLIC SCHOOLS Essential Questions and Understandings Teacher Notes and Elaborations (continued from previous page) Teacher Notes and Elaborations  Identify the coordinates of a right triangle or rectangle that has been dilated. The center of the dilation will be the origin.  Sketch the image of a dilation of a right triangle or rectangle limited to a scale factor 1 1 of , , 2, 3 or 4. 4 2  The image of a polygon is the resulting polygon after the transformation. The preimage is the polygon before the transformation.  A transformation of preimage point A can be denoted as the image A (read as “A prime”).  When a geometric figure is translated on a coordinate plane, the new vertices are labeled as follows: point A corresponds to A , point B corresponds to B , and so on. Sometimes double prime ( A ) and triple prime ( A ) notations are used.


Page 24 of 71

Course 2 Curriculum Guide Curriculum Information SOL 7.9 SOL Reporting Category Probability, Statistics, Patterns, Functions, and Algebra Focus Applications of Statistics and Probability Virginia SOL 7.9 The student will investigate and describe the difference between the experimental probability and theoretical probability of an event. HCPS Website SOL 7.9 - Probability ESS Lesson What are the chances? (PDF) Investigating and describing the theoretical and experimental probabilities (Word)

Essential Knowledge and Skills Key Vocabulary The student will use problem solving, mathematical communication, mathematical reasoning, connections and representations to:  Determine the theoretical probability of an event.  Determine the experimental probability of an event.  Describe changes in the experimental probability as the number of trials increases.  Investigate and describe the difference between the probability of an event found through experiment or simulation versus the theoretical probability of that same event. Key Vocabulary event experimental probability Law of Large Numbers outcome probability sample space sampling simulation theoretical probability

HENRICO COUNTY PUBLIC SCHOOLS Essential Questions and Understandings Teacher Notes and Elaborations Essential Questions and Understandings  What is the difference between the theoretical and experimental probability of an event? Theoretical probability of an event is the expected probability and can be found with a formula. The experimental probability of an event is determined by carrying out a simulation or an experiment. In experimental probability, as the number of trials increases, the experimental probability gets closer to the theoretical probability. Teacher Notes and Elaborations  The probability of an event occurring is a ratio expressing the chance or likelihood that a certain event will occur, given the number of possible outcomes (results) of an experiment. An event is a subset of a sample space. The sample space is the set of all possible outcomes of an experiment.  Determine the theoretical probability of an event.  The theoretical probability of an event is the expected probability and can be found with a formula. number of possible favorable outcomes Theoretical probability of an event  total number of possible outcomes  Determine the experimental probability of an event. Describe changes in the experimental probability as the number of trials increases.  The experimental probability of an event is determined by carrying out a simulation or an experiment. The experimental probability is found by repeating an experiment many times and using the ratio. o Experimental probability is not exact since the results may vary if the experiment is repeated. o In experimental probability, as the number of trials increases, the experimental probability gets closer to the theoretical probability (Law of Large Numbers). o An important use of experimental probability is to make predictions about a large group of people based on the results of a poll or survey. This technique, called sampling, is used when it is impossible to question every member of a group. Experimental probability 


number of times desired outcomes occur total number of trials in the experiment

 Investigate and describe the difference between the probability of an event found through experiment or, simulation versus the theoretical probability of that same event.

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Course 2 Curriculum Guide Curriculum Information SOL 7.10 SOL Reporting Category Probability, Statistics, Patterns, Functions, and Algebra Focus Applications of Statistics and Probability Virginia SOL 7.10 The student will determine the probability of compound events using the Fundamental (Basic) Counting Principle. HCPS Website SOL 7.10 - Probability ESS Lesson The Real Meal Deal (PDF) Computing the number of outcomes, using the Fundamental (Basic) Counting Principle (Word)

Essential Knowledge and Skills Key Vocabulary The student will use problem solving, mathematical communication, mathematical reasoning, connections and representations to:  Compute the number of possible outcomes by using the Fundamental (Basic) Counting Principle.  Determine the probability of a compound event containing no more than two events. Key Vocabulary compound event dependent event Fundamental Counting Principle independent event outcomes probability sample space tree diagram

HENRICO COUNTY PUBLIC SCHOOLS Essential Questions and Understandings Teacher Notes and Elaborations Essential Questions and Understandings  What is the Fundamental (Basic) Counting Principle? The Fundamental (Basic) Counting Principle is a computational procedure used to determine the number of possible outcomes of several events.  What is the role of the Fundamental (Basic) Counting Principle in determining the probability of compound events? It is the product of the number of outcomes for each event that can be chosen individually (e.g., the possible outcomes or outfits of four shirts, two pants, and three shoes is 4 · 2 · 3 or 24). Teacher Notes and Elaborations  Compute the number of possible outcomes by using the Fundamental (Basic) Counting Principle.  Tree diagrams are used to illustrate possible outcomes of events. They can be used to support the Fundamental (Basic) Counting Principle. o The following tree diagram illustrates the possible outcomes (results). Using the Fundamental (Basic) Counting Principle the possible outcomes can be found by multiplying the number of pant choices times the shirt choices (2 ∙ 3 = 6)

.  Events are independent when the outcome of one has no effect on the outcome of the other. For example, rolling a number cube and flipping a coin are independent events.

Return to Course Outline (continued) Page 26 of 71

Course 2 Curriculum Guide Curriculum Information SOL 7.10 SOL Reporting Category Probability, Statistics, Patterns, Functions, and Algebra Focus Applications of Statistics and Probability Virginia SOL 7.10 The student will determine the probability of compound events using the Fundamental (Basic) Counting Principle.

HENRICO COUNTY PUBLIC SCHOOLS Essential Questions and Understandings Teacher Notes and Elaborations (continued)  Events are dependent when the outcome of one event is influenced by the outcome of the other. For example, when drawing two marbles from a bag, not replacing the first after it is drawn affects the outcome of the second draw.  A compound event combines two or more simple events (independent or dependent). For example, a bag contains 4 red, 3 green and 2 blue marbles. What is the probability of selecting a green and then a blue marble (with or without replacement)? 3 2 6 2 o With replacement (independent) the probability is:   which can be simplified to . 9 9 81 27 1 3 2 6 o Without replacement (dependent) the probability is: which can be simplified to .   12 9 8 72


Page 27 of 71

Course 2 Curriculum Guide Curriculum Information SOL 7.11 SOL Reporting Category Probability, Statistics, Patterns, Functions, and Algebra Focus Applications of Statistics and Probability Virginia SOL 7.11 The student, given data in a practical situation, will a. construct and analyze histograms; and b. compare and contrast histograms with other types of graphs presenting information from the same data set. HCPS Website SOL 7.11 – Statistical Graphs ESS Lesson Numbers in a Name (PDF) Constructing, analyzing, comparing, and contrasting histograms (Word)


Essential Knowledge and Skills Key Vocabulary The student will use problem solving, mathematical communication, mathematical reasoning, connections and representations to:  Collect, analyze, display, and interpret a data set using histograms. For collection and display of raw data, limit the data to 20 items.  Determine patterns and relationships within data sets (e.g., trends).  Make inferences, conjectures, and predictions based on analysis of a set of data.  Compare and contrast histograms with line plots, circle graphs, and stem and leaf plots presenting information from the same data set. Key Vocabulary circle graph conjecture frequency distribution histogram inference intervals line plot prediction stem-and-leaf plot tally trends

HENRICO COUNTY PUBLIC SCHOOLS Essential Questions and Understandings Teacher Notes and Elaborations Essential Questions and Understandings  What type of data is most appropriate to display in a histogram? Numerical data that can be characterized using consecutive intervals are best displayed in a histogram. Teacher Notes and Elaborations  All graphs tell a story and include a title and labels that describe the data.  Collect, analyze, display, and interpret a data set using histograms. For collection and display of raw data, limit the data to 20 items.  Determine patterns and relationships within data sets (e.g., trends).  Make inferences, conjectures, and predictions based on analysis of a set of data.  Compare and contrast histograms with line plots, circle graphs, and stem-and-leaf plots presenting information from the same data set.  A line plot shows the frequency of data on a number line. Line plots are used to show the spread of the data and quickly identify the range, mode, and any outliers.

 A stem-and-leaf plot displays data from least to greatest using the digits of the greatest place value to group data.

(continued)

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Course 2 Curriculum Guide Curriculum Information SOL 7.11 SOL Reporting Category Probability, Statistics, Patterns, Functions, and Algebra

HENRICO COUNTY PUBLIC SCHOOLS Essential Questions and Understandings Teacher Notes and Elaborations (continued)  A frequency distribution shows how often an item, a number, or range of numbers occurs. It can be used to construct a histogram. A tally is a mark used to keep count in each interval.

Focus Applications of Statistics and Probability Virginia SOL 7.11 The student, given data in a practical situation, will a. construct and analyze histograms; and b. compare and contrast histograms with other types of graphs presenting information from the same data set.

 Bar graphs are utilized to compare counts of different categories both categorical and discrete data. A bar graph uses parallel bars; either horizontal or vertical, to represent counts for several categories. One bar is used for each category with the length of the bar representing the count for that category. There is space before, between, and after the bars. The axis displaying the scale representing the count for the categories should extend one increment above the greatest recorded piece of data. The values should represent equal increments. Each axis should be labeled, and the graph should have a title.

 A histogram is a form of bar graph in which the categories are consecutive and equal intervals. If no data exists in an interval, that interval must still be labeled in the graph. A histogram uses numerical instead of categorical data. A histogram is constructed from a frequency table. The intervals are shown on the x-axis and the number of elements in each interval is represented by the height of a bar located above the interval. The length or height of each bar is determined by the number of data elements (frequency) falling into a particular interval. Histograms summarize data but do not provide information about specific data points.


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Course 2 Curriculum Guide Curriculum Information SOL 7.12 SOL Reporting Category Probability, Statistics, Patterns, Functions, and Algebra Focus Linear Equations Virginia SOL 7.12 The student will represent relationships with tables, graphs, rules, and words. HCPS Website SOL 7.12 – Functions & Relations ESS Lesson 7.12 - Relationships Round Robin (PDF) - Representing relationships with tables, graphs, rules, and words (Word)

Essential Knowledge and Skills Key Vocabulary The student will use problem solving, mathematical communication, mathematical reasoning, connections and representations to:  Describe and represent relations and functions, using tables, graphs, rules, and words. Given one representation, students will be able to represent the relation in another form. Key Vocabulary function relation table of values vertical horizontal input output dependent variable independent variable

HENRICO COUNTY PUBLIC SCHOOLS Essential Questions and Understandings Teacher Notes and Elaborations Essential Questions and Understandings  What are the different ways to represent the relationship between two sets of numbers? Rules that relate elements in two sets can be represented by word sentences, equations, tables of values, graphs or illustrated pictorially. Teacher Notes and Elaborations  Rules that relate elements in two sets can be represented by word sentences, equations, tables of values, graphs, or illustrated pictorially.  A relation is any set of ordered pairs. For each first member, there may be many second members.  A function is a relation in which there is one and only one second member for each first member. o The function that relates earnings to time worked is earnings = rate of pay × hours worked. o The function that relates distance traveled to the rate of travel and the time is distance = rate × time; for example, a student traveling at 30 miles per hour on a motor bike, would produce the following table: TIME (t) DISTANCE (d)

1 hour

2 hours

3 hours

4 hours

30 miles

60 miles

90 miles

120 miles

The equation that represents this function is d = 30t. o

A person makes $30 an hour. A function representing this is e = 30h where e represents the earnings and h is the number of hours worked. The following represents a table of values for this function. TIME (t) EARNINGS (e)

1 hour

2 hours

3 hours

4 hours

$30

$60

$90

$120

 As a table of values, a function has a unique value assigned to the second variable for each value of the first variable. Return to Course Outline


Course 2 Curriculum Guide Curriculum Information SOL 7.12 SOL Reporting Category Probability, Statistics, Patterns, Functions, and Algebra Focus Linear Equations

HENRICO COUNTY PUBLIC SCHOOLS Essential Questions and Understandings Teacher Notes and Elaborations (continued)  A table of values is the data used to make a graph in the coordinate system. The values are used to graph points.  As a graph, a function is any curve (including straight lines) such that any vertical line would pass through the curve only once. o

Graphs may be constructed from ordered pairs represented in a table. The ordered pairs in the following table are (2,0), (1,1), (0, 2), (1,3), (2, 4) . The equation represented in this table and graph is y  x  2 .

Virginia SOL 7.12 The student will represent relationships with tables, graphs, rules, and words.

10

y

9 8 7 6 5 4

x+2

3

2

0

1

1

0

2

1

3

2

4

2 1 -9 -8 -7 -6 -5 -4 -3 -2 -1 -1

x 1

2

3

4

5

6

7

8

9

-2 -3 -4 -5 -6 -7 -8 -9 -10

 Some relations are functions; all functions are relations.  Describe and represent relations and functions, using tables, graphs, rules, and words. Given one representation, students will be able to represent the relation in another form.


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Course 2 Curriculum Guide Curriculum Information SOL 7.13 SOL Reporting Category Probability, Statistics, Patterns, Functions, and Algebra Focus Linear Equations Virginia SOL 7.13 The student will a. write verbal expressions as algebraic expressions and sentences as equations and vice versa; and b. evaluate algebraic expressions for given replacement values of the variables. HCPS Website SOL 7.13 - Expressions ESS Lessons Translate and Evaluate (PDF) - Writing and evaluating algebraic expressions (Word)

Essential Knowledge and Skills Key Vocabulary The student will use problem solving, mathematical communication, mathematical reasoning, connections and representations to:  Write verbal expressions as algebraic expressions. Expressions will be limited to no more than two operations.  Write verbal sentences as algebraic equations. Equations will contain no more than one variable term.  Translate algebraic expressions and equations to verbal expressions and sentences. Expressions will be limited to no more than two operations.  Identify examples of expressions and equations.  Apply the order of operations to evaluate expressions for given replacement values (integers, fractions, and decimals) of the variables. Limit the number of replacements to no more than three per expression. Key Vocabulary algebraic equation algebraic expression coefficient constant expression grouping symbols order of operations substitution term variable variable expression verbal expression verbal sentence


HENRICO COUNTY PUBLIC SCHOOLS Essential Questions and Understandings Teacher Notes and Elaborations Essential Questions and Understandings  How can algebraic expressions and equations be written? Word phrases and sentences can be used to represent algebraic expressions and equations. Teacher Notes and Elaborations  An expression is a name for a number.  An expression that contains a variable is a variable expression.  An expression that contains only numbers is a numerical expression.  A verbal expression is a word phrase (e.g., “the sum of two consecutive integers”).  A verbal sentence is a complete word statement (e.g., “The sum of two consecutive integers is five.”).  An algebraic expression is a variable expression that contains at least one variable (e.g., 2x – 5).  An algebraic equation is a mathematical statement that says that two expressions are equal (e.g., 2x + 1 = 5).  A term is a number, variable, product, or quotient in an expression of sums and/or differences. The expression 3x + 4y – 7 contains 3 terms (3x, 4y, 7 ).  A coefficient is the numerical factor of a variable in a term. In the term 2x, 2 is the coefficient of x.  A constant is a numerical expression that is part of an algebraic expression. In the expression 4x + 9, 9 is the constant.  Write verbal expressions as algebraic expressions. Expressions will be limited to no more than 2 operations.



HENRICO COUNTY PUBLIC SCHOOLS Essential Questions and Understandings Teacher Notes and Elaborations (continued)  Write verbal sentences as algebraic equations. Equations will contain no more than 1 variable term.  Translate algebraic expressions and equations to verbal expressions and sentences. Expressions will be limited to no more than 2 operations.  Identify examples of expressions and equations.

Virginia SOL 7.13 The student will a. write verbal expressions as algebraic expressions and sentences as equations and vice versa; and b. evaluate algebraic expressions for given replacement values of the variables.

 Apply the order of operations to evaluate expressions for given replacement values of the variables. Limit the number of replacements to no more than 3 per expression. o To evaluate an algebraic expression, substitute a given replacement value for a variable and apply the order of operations. For example, if a = 3 and b = -2 then 5a + b can be evaluated as: 5(3) + (-2) 15 + (-2) 13


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Course 2 Curriculum Guide Curriculum Information SOL 7.14 SOL Reporting Category Probability, Statistics, Patterns, Functions, and Algebra

HCPS Website SOL 7.14 - Equations

Essential Knowledge and Skills Key Vocabulary The student will use problem solving, mathematical communication, mathematical reasoning, connections and representations to:  Represent and demonstrate steps for solving one- and two-step equations in one variable using concrete materials, pictorial representations, and algebraic sentences.  Translate word problems/practical problems into algebraic equations and solve them.  Solve one- and two-step linear equations in one variable.  Solve practical problems that require the solution of a one- or two-step linear equation.

ESS Lesson Equations (PDF) - Solving one and two step linear equations (Word)

Key Vocabulary inverse operations coefficient

Focus Linear Equations Virginia SOL 7.14 The student will a. solve one- and two-step linear equations in one variable; and b. solve practical problems requiring the solution of one- and two-step linear equations.

HENRICO COUNTY PUBLIC SCHOOLS Essential Questions and Understandings Teacher Notes and Elaborations Essential Questions and Understandings  When solving an equation, why is it important to perform identical operations on each side of the equal sign? An operation that is performed on one side of an equation must be performed on the other side to maintain equality. Teacher Notes and Elaborations  An equation is a mathematical sentence that states that two expressions are equal.  A one-step equation is defined as an equation that requires the use of one operation to solve (e.g., x + 3 = – 4 ).  Inverse operations undo each other. The inverse operation for addition is subtraction, and the inverse operation for multiplication is division.  A two-step equation is defined as an equation that requires the use of two operations to x7 solve (e.g., 2x + 1 = -5; -5 = 2x + 1;  4 ). 3  Represent and demonstrate steps for solving one- and two-step equations in one variable. It is important to use the concrete and pictorial representations to create the rules for solving the equations. The concrete and pictorial representations should be explored before moving to the abstract concepts. o Use concrete materials such as Algeblocks or Algebra tiles o Use pictorial representations such as drawing out the concrete representations o Use algebraic sentences  Solve one- and two-step linear equations in one variable. o The following demonstrates steps for solving a two-step equation algebraically. 2( x  2)  14 2( x  2) 14  2 2 x27 -2 -2 x+0=5 x=5


divide or multiply by the reciprocal (multiplicative inverse) subtract or add the opposite (additive inverse)

 Solve practical problems that require the solution of a one- or two-step linear equation. o Translate word problems/practical problems into algebraic equations then solve the problem Page 34 of 71

Course 2 Curriculum Guide Curriculum Information SOL 7.15 SOL Reporting Category Probability, Statistics, Patterns, Functions, and Algebra Focus Linear Equations Virginia SOL 7.15 The student will a. solve one-step inequalities in one variable and b. graph solutions to inequalities on the number line. HCPS Website SOL 7.15 - Inequalities ESS Lesson Inequalities (PDF) - Solving one and two step linear equations (Word)

Essential Knowledge and Skills Key Vocabulary The student will use problem solving, mathematical communication, mathematical reasoning, connections and representations to:  Represent and demonstrate steps in solving inequalities in one variable, using concrete materials, pictorial representations, and algebraic sentences.  Graph solutions to inequalities on the number line.  Identify a numerical value that satisfies the inequality. Key Vocabulary inequality inverse operations coefficient

HENRICO COUNTY PUBLIC SCHOOLS Essential Questions and Understandings Teacher Notes and Elaborations Essential Questions and Understandings  How are the procedures for solving equations and inequalities the same? The procedures are the same except for the case when an inequality is multiplied or divided on both sides by a negative number. Then the inequality sign is changed from less than to greater than, or greater than to less than.  How is the solution to an inequality different from that of a linear equation? In an inequality, there can be more than one value for the variable that makes the inequality true. Teacher Notes and Elaborations  An inequality is a mathematical sentence that states that one quantity is less than (or greater than) another quantity. An inequality is a mathematical sentence that compares two expressions using one of the symbols , , , or .  A one-step inequality is defined as an inequality that requires the use of one operation to solve (e.g., x – 4 > 9).  Inverse operations undo each other. The inverse operation for addition is subtraction, and the inverse operation for multiplication is division.  When both expressions of an inequality are multiplied or divided by a negative number, the inequality symbol reverses (e.g., –3x < 15 is equivalent to x > –5).  Solutions to inequalities can be represented using a number line. o Inequalities using the < or > symbols are represented on a number line with an open circle on the number and a shaded line over the solution set. o Inequalities using the ≤ or ≥ symbols are represented on a number line with a closed circle on the number and shaded line in the direction of the solution set.  Represent and demonstrate steps for solving one- and two-step equations in one variable. It is important to use the concrete and pictorial representations to create the rules for solving the equations. The concrete and pictorial representations should be explored before moving to the abstract concepts. o Use concrete materials such as Algeblocks or Algebra tiles o Use pictorial representations such as drawing out the concrete representations o Use algebraic sentences  Identify a numerical value that satisfies the inequality.


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Course 2 Curriculum Guide Curriculum Information SOL 7.16 SOL Reporting Category Probability, Statistics, Patterns, Functions, and Algebra Focus Linear Equations Virginia SOL 7.16 The student will apply the following properties of operations with real numbers: a. the commutative and associative properties for addition and multiplication; b. the distributive property; c. the additive and multiplicative identity properties; d. the additive and multiplicative inverse properties; and e. the multiplicative property of zero. HCPS Website SOL 7.16 - Properties ESS Lessons Properties (PDF) - Identifying and applying properties (Word)


Essential Knowledge and Skills Key Vocabulary The student will use problem solving, mathematical communication, mathematical reasoning, connections and representations to:  Identify properties of operations used in simplifying expressions.  Apply the properties of operations to simplify expressions. Key Vocabulary additive identity property additive inverse property associative property of addition associative property of multiplication commutative property of addition commutative property of multiplication distributive property identity elements inverses multiplicative identity property multiplicative inverse property multiplicative property of zero reciprocal

HENRICO COUNTY PUBLIC SCHOOLS Essential Questions and Understandings Teacher Notes and Elaborations Essential Questions and Understandings  Why is it important to apply properties of operations when simplifying expressions? Using the properties of operations with real numbers helps with understanding mathematical relationships. Teacher Notes and Elaborations  The commutative property for addition states that changing the order of the addends does not change the sum (e.g., 5 + 4 = 4 + 5). 

The commutative property for multiplication states that changing the order of the factors does not change the product (e.g., 5 · 4 = 4 · 5).



The associative property of addition states that regrouping the addends does not change the sum [e.g., 5 + (4 + 3) = (5 + 4) + 3].



The associative property of multiplication states that regrouping the factors does not change the product [e.g., 5 · (4 · 3) = (5 · 4) · 3].



Subtraction and division are neither commutative nor associative.



The distributive property states that the product of a number and the sum (or difference) of two other numbers equals the sum (or difference) of the products of the number and each other number [e.g., 5 · (3 + 7) = (5 · 3) + (5 · 7), or 5 · (3 – 7) = (5 · 3) – (5 · 7)].



Identity elements are numbers that combine with other numbers without changing the other numbers. The additive identity is zero (0). The multiplicative identity is one (1). There are no identity elements for subtraction and division.



The additive identity property states that the sum of any real number and zero is equal to the given real number (e.g., 5 + 0 = 5).



The multiplicative identity property states that the product of any real number and one is equal to the given real number (e.g., 8 · 1 = 8).



Inverses are numbers that combine with other numbers and result in identity elements 1 [e.g., 5 + (–5) = 0; · 5 = 1]. 5 (continued) Page 36 of 71


HENRICO COUNTY PUBLIC SCHOOLS Essential Questions and Understandings Teacher Notes and Elaborations (continued)  The additive inverse property states that the sum of a number and its additive inverse always equals zero [e.g., 5 + (–5) = 0].  The multiplicative inverse property states that the product of a number and its multiplicative inverse (or reciprocal) always equals one 1 (e.g., 4 · = 1). 4  Zero has no multiplicative inverse.

Virginia SOL 7.16 The student will apply the following properties of operations with real numbers: a. the commutative and associative properties for addition and multiplication; b. the distributive property; c. the additive and multiplicative identity properties; d. the additive and multiplicative inverse properties; and e. the multiplicative property of zero.

 The multiplicative property of zero states that the product of any real number and zero is zero.  Division by zero is not a possible arithmetic operation. Division by zero is undefined.  Identify properties of operations used in simplifying expressions. o -25(7)( -4) o 7( -25)( -4) Commutative property of multiplication o 7[( -25)( -4)] Associative property of multiplication o 7(100) o 700  Apply the properties of operations to simplify expressions.


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Course 2 Curriculum Guide HENRICO COUNTY PUBLIC SCHOOLS

http://www.doe.virginia.gov/testing/test_administration/ancilliary_materials/mathematics/2009/2009_sol_formula_sheet_math_7.pdf

Page 38 of 71

Grade 7

Copyright © 2009 by the Virginia Department of Education P.O. Box 2120 Richmond, Virginia 23218-2120 http://www.doe.virginia.gov All rights reserved. Reproduction of these materials for instructional purposes in public school classrooms in Virginia is permitted. Superintendent of Public Instruction Patricia I. Wright, Ed.D. Assistant Superintendent for Instruction Linda M. Wallinger, Ph.D. Office of Elementary Instruction Mark R. Allan, Ph.D., Director Deborah P. Wickham, Ph.D., Mathematics Specialist Office of Middle and High School Instruction Michael F. Bolling, Mathematics Coordinator Acknowledgements The Virginia Department of Education wishes to express sincere thanks to Deborah Kiger Bliss, Lois A. Williams, Ed.D., and Felicia Dyke, Ph.D. who assisted in the development of the 2009 Mathematics Standards of Learning Curriculum Framework. NOTICE

The Virginia Department of Education does not unlawfully discriminate on the basis of race, color, sex, national origin, age, or disability in employment or in its educational programs or services. The 2009 Mathematics Curriculum Framework can be found in PDF and Microsoft Word file formats on the Virginia Department of Education’s Web site at http://www.doe.virginia.gov.

Virginia Mathematics Standards of Learning Curriculum Framework 2009 Introduction

The 2009 Mathematics Standards of Learning Curriculum Framework is a companion document to the 2009 Mathematics Standards of Learning and amplifies the Mathematics Standards of Learning by defining the content knowledge, skills, and understandings that are measured by the Standards of Learning assessments. The Curriculum Framework provides additional guidance to school divisions and their teachers as they develop an instructional program appropriate for their students. It assists teachers in their lesson planning by identifying essential understandings, defining essential content knowledge, and describing the intellectual skills students need to use. This supplemental framework delineates in greater specificity the content that all teachers should teach and all students should learn. Each topic in the Mathematics Standards of Learning Curriculum Framework is developed around the Standards of Learning. The format of the Curriculum Framework facilitates teacher planning by identifying the key concepts, knowledge and skills that should be the focus of instruction for each standard. The Curriculum Framework is divided into three columns: Understanding the Standard; Essential Understandings; and Essential Knowledge and Skills. The purpose of each column is explained below. Understanding the Standard This section includes background information for the teacher (K-8). It contains content that may extend the teachers’ knowledge of the standard beyond the current grade level. This section may also contain suggestions and resources that will help teachers plan lessons focusing on the standard. Essential Understandings This section delineates the key concepts, ideas and mathematical relationships that all students should grasp to demonstrate an understanding of the Standards of Learning. In Grades 6-8, these essential understandings are presented as questions to facilitate teacher planning. Essential Knowledge and Skills Each standard is expanded in the Essential Knowledge and Skills column. What each student should know and be able to do in each standard is outlined. This is not meant to be an exhaustive list nor a list that limits what is taught in the classroom. It is meant to be the key knowledge and skills that define the standard. The Curriculum Framework serves as a guide for Standards of Learning assessment development. Assessment items may not and should not be a verbatim reflection of the information presented in the Curriculum Framework. Students are expected to continue to apply knowledge and skills from Standards of Learning presented in previous grades as they build mathematical expertise.

FOCUS 6–8

STRAND: NUMBER AND NUMBER SENSE

GRADE LEVEL 7

In the middle grades, the focus of mathematics learning is to  build on students’ concrete reasoning experiences developed in the elementary grades;  construct a more advanced understanding of mathematics through active learning experiences;  develop deep mathematical understandings required for success in abstract learning experiences; and  apply mathematics as a tool in solving practical problems. Students in the middle grades use problem solving, mathematical communication, mathematical reasoning, connections, and representations to integrate understanding within this strand and across all the strands. 

Students in the middle grades focus on mastering rational numbers. Rational numbers play a critical role in the development of proportional reasoning and advanced mathematical thinking. The study of rational numbers builds on the understanding of whole numbers, fractions, and decimals developed by students in the elementary grades. Proportional reasoning is the key to making connections to most middle school mathematics topics.



Students develop an understanding of integers and rational numbers by using concrete, pictorial, and abstract representations. They learn how to use equivalent representations of fractions, decimals, and percents and recognize the advantages and disadvantages of each type of representation. Flexible thinking about rational number representations is encouraged when students solve problems.



Students develop an understanding of the properties of operations on real numbers through experiences with rational numbers and by applying the order of operations.



Students use a variety of concrete, pictorial, and abstract representations to develop proportional reasoning skills. Ratios and proportions are a major focus of mathematics learning in the middle grades.

Mathematics Standards of Learning Curriculum Framework 2009: Grade 7

1

STANDARD 7.1 7.1


The student will a) investigate and describe the concept of negative exponents for powers of ten; b) determine scientific notation for numbers greater than zero; c) compare and order fractions, decimals, percents and numbers written in scientific notation; d) determine square roots; and e) identify and describe absolute value for rational numbers. UNDERSTANDING THE STANDARD

ESSENTIAL UNDERSTANDINGS

(Background Information for Instructor Use Only) 



Negative exponents for powers of 10 are used to represent numbers between 0 and 1. 1 3 (e.g., 10 = 3 = 0.001). 10 Negative exponents for powers of 10 can be investigated through patterns such as:





2



1

10 = 10 0

10 = 1 1 1 1 10 = 1  = 0.1 10 10 A number followed by a percent symbol (%) is equivalent to that number with a denominator of 100 3 60 (e.g., = = 0.60 = 60%). 5 100



Scientific notation is used to represent very large or very small numbers.



A number written in scientific notation is the product of two factors — a decimal greater than or equal to 1 but less than 10, and a power of 10 (e.g., 3.1  105= 310,000 and 2.85 x 10 0.000285).

4





ESSENTIAL KNOWLEDGE AND SKILLS

When should scientific notation be used? Scientific notation should be used whenever the situation calls for use of very large or very small numbers.

The student will use problem solving,

How are fractions, decimals and percents related? Any rational number can be represented in fraction, decimal and percent form.

reasoning, connections, and representations

10 =100



GRADE LEVEL 7

What does a negative exponent mean when the base is 10? A base of 10 raised to a negative exponent represents a number between 0 and 1. How is taking a square root different from squaring a number? Squaring a number and taking a square root are inverse operations. Why is the absolute value of a number positive? The absolute value of a number represents distance from zero on a number line regardless of direction. Distance is positive.

=


mathematical communication, mathematical

to  Recognize powers of 10 with negative exponents by examining patterns.  Write a power of 10 with a negative exponent in fraction and decimal form.  Write a number greater than 0 in scientific notation.  Recognize a number greater than 0 in scientific notation.  Compare and determine equivalent relationships between numbers larger than 0 written in scientific notation.  Represent a number in fraction, decimal, and percent forms.  Compare, order, and determine equivalent relationships among fractions, decimals, and percents. Decimals are limited to the thousandths place, and percents are limited to the tenths place. Ordering is limited to no more than 4 numbers. 2

STANDARD 7.1 7.1


The student will a) investigate and describe the concept of negative exponents for powers of ten; b) determine scientific notation for numbers greater than zero; c) compare and order fractions, decimals, percents and numbers written in scientific notation; d) determine square roots; and e) identify and describe absolute value for rational numbers. UNDERSTANDING THE STANDARD


GRADE LEVEL 7



Equivalent relationships among fractions, decimals, and percents can be determined by using manipulatives (e.g., fraction bars, Base-10 blocks, fraction circles, graph paper, number lines and calculators).

 Order no more than 3 numbers greater than 0 written in scientific notation.

A square root of a number is a number which, when multiplied by itself, produces the given number (e.g., 121 is 11 since 11 x 11 = 121).

 Demonstrate absolute value using a number line.



The square root of a number can be represented geometrically as the length of a side of the square.



The absolute value of a number is the distance from 0 on the number line regardless of direction. 1 1 (e.g.,  ). 2 2

 Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle to solve practical problems.†



 Determine the square root of a perfect square less than or equal to 400.

 Determine the absolute value of a rational number.

†


Revised March 2011

3

STANDARD 7.2 7.2


The student will describe and represent arithmetic and geometric sequences using variable expressions. UNDERSTANDING THE STANDARD





GRADE LEVEL 7

In the numeric pattern of an arithmetic sequence, students must determine the difference, called the common difference, between each succeeding number in order to determine what is added to each previous number to obtain the next number.



2, 4, 8, 16, 32, …; 1, 5, 25, 125, 625, …; and 80, 20, 5, 1.25, ….

 A variable expression can be written to express the relationship between two consecutive terms of a sequence -

If n represents a number in the sequence 3, 6, 9, 12…, the next term in the sequence can be determined using the variable expression n + 3.

-

If n represents a number in the sequence 1, 5, 25, 125…, the next term in the sequence can be determined by using the variable expression 5n.


When are variable expressions used? Variable expressions can express the relationship between two consecutive terms in a sequence.

In geometric sequences, students must determine what each number is multiplied by in order to obtain the next number in the geometric sequence. This multiplier is called the common ratio. Sample geometric sequences include –



mathematical communication, mathematical reasoning, connections, and representations to 

Analyze arithmetic and geometric sequences to discover a variety of patterns.



Identify the common difference in an arithmetic sequence.



Identify the common ratio in a geometric sequence.



Given an arithmetic or geometric sequence, write a variable expression to describe the relationship between two consecutive terms in the sequence.

4

FOCUS 6–8

STRAND: COMPUTATION AND ESTIMATION

GRADE LEVEL 7


Students develop conceptual and algorithmic understanding of operations with integers and rational numbers through concrete activities and discussions that bring meaning to why procedures work and make sense.



Students develop and refine estimation strategies and develop an understanding of when to use algorithms and when to use calculators. Students learn when exact answers are appropriate and when, as in many life experiences, estimates are equally appropriate.



Students learn to make sense of the mathematical tools they use by making valid judgments of the reasonableness of answers.



Students reinforce skills with operations with whole numbers, fractions, and decimals through problem solving and application activities.


5

STANDARD 7.3 7.3


The student will a) model addition, subtraction, multiplication and division of integers; and b) add, subtract, multiply, and divide integers. UNDERSTANDING THE STANDARD









GRADE LEVEL 7

The set of integers is the set of whole numbers and their opposites (e.g., … –3, –2, –1, 0, 1, 2, 3, …). Integers are used in practical situations, such as temperature changes (above/below zero), balance in a checking account (deposits/withdrawals), and changes in altitude (above/below sea level).



ESSENTIAL KNOWLEDGE AND SKILLS The student will use problem solving,

The sums, differences, products and quotients of integers are either positive, zero, or negative. How can this be demonstrated? This can be demonstrated through the use of patterns and models.

Concrete experiences in formulating rules for adding and subtracting integers should be explored by examining patterns using calculators, along a number line and using manipulatives, such as twocolor counters, or by using algebra tiles. Concrete experiences in formulating rules for multiplying and dividing integers should be explored by examining patterns with calculators, along a number line and using manipulatives, such as two-color counters, or by using algebra tiles.



Model addition, subtraction, multiplication and division of integers using pictorial representations of concrete manipulatives.



Add, subtract, multiply, and divide integers.



Simplify numerical expressions involving addition, subtraction, multiplication and division of integers using order of operations.



Solve practical problems involving addition, subtraction, multiplication, and division with integers.

6

STANDARD 7.4 7.4


The student will solve single-step and multistep practical problems, using proportional reasoning. UNDERSTANDING THE STANDARD



A proportion is a statement of equality between two ratios.



A proportion can be written as

a c = , a:b = c:d, or b d




What makes two quantities proportional? Two quantities are proportional when one quantity is a constant multiple of the other.

mathematical communication, mathematical reasoning, connections, and representations

a is to b as c is to d. 

A proportion can be solved by finding the product of the means and the product of the extremes. For example, in the proportion a:b = c:d, a and d are the extremes and b and c are the means. If values are substituted for a, b, c, and d such as 5:12 = 10:24, then the product of extremes (5  24) is equal to the product of the means (12  10).



In a proportional situation, both quantities increase or decrease together.



In a proportional situation, two quantities increase multiplicatively. Both are multiplied by the same factor.



A proportion can be solved by finding equivalent fractions.



A rate is a ratio that compares two quantities measured in different units. A unit rate is a rate with a denominator of 1. Examples of rates include miles/hour and revolutions/minute.



GRADE LEVEL 7

Proportions are used in everyday contexts, such as speed, recipe conversions, scale drawings, map reading, reducing and enlarging, comparison shopping, and monetary conversions.


to 

Write proportions that represent equivalent relationships between two sets.



Solve a proportion to find a missing term.



Apply proportions to convert units of measurement between the U.S. Customary System and the metric system. Calculators may be used.



Apply proportions to solve practical problems, including scale drawings. Scale factors shall have denominators no greater than 12 and decimals no less than tenths. Calculators may be used.



Using 10% as a benchmark, mentally compute 5%, 10%, 15%, or 20% in a practical situation such as tips, tax and discounts.



Solve problems involving tips, tax, and discounts. Limit problems to only one percent computation per problem.

7

STANDARD 7.4 7.4


GRADE LEVEL 7

The student will solve single-step and multistep practical problems, using proportional reasoning. UNDERSTANDING THE STANDARD


Proportions can be used to convert between measurement systems. For example: if 2 inches is about 5 cm, how many inches are in 16 cm? 2inches 5cm  – x 16cm



A percent is a special ratio in which the denominator is 100.



Proportions can be used to represent percent problems as follows: percent part  – 100 whole




8

FOCUS 6–8

STRAND: MEASUREMENT

GRADE LEVEL 7


Students develop the measurement skills that provide a natural context and connection among many mathematics concepts. Estimation skills are developed in determining length, weight/mass, liquid volume/capacity, and angle measure. Measurement is an essential part of mathematical explorations throughout the school year.



Students continue to focus on experiences in which they measure objects physically and develop a deep understanding of the concepts and processes of measurement. Physical experiences in measuring various objects and quantities promote the long-term retention and understanding of measurement. Actual measurement activities are used to determine length, weight/mass, and liquid volume/capacity.



Students examine perimeter, area, and volume, using concrete materials and practical situations. Students focus their study of surface area and volume on rectangular prisms, cylinders, pyramids, and cones.


9

STANDARD 7.5 7.5

UNDERSTANDING THE STANDARD 

The area of a rectangle is computed by multiplying the lengths of two adjacent sides.



The area of a circle is computed by squaring the radius and multiplying that product by  (A = r2 , 22 where   3.14 or ). 7







GRADE LEVEL 7

The student will a) describe volume and surface area of cylinders; b) solve practical problems involving the volume and surface area of rectangular prisms and cylinders; and c) describe how changing one measured attribute of a rectangular prism affects its volume and surface area.

(Background Information for Instructor Use Only)



STRAND: MEASUREMENT

A rectangular prism can be represented on a flat surface as a net that contains six rectangles — two that have measures of the length and width of the base, two others that have measures of the length and height, and two others that have measures of the width and height. The surface area of a rectangular prism is the sum of the areas of all six faces ( SA  2lw  2lh  2wh ).

ESSENTIAL UNDERSTANDINGS  How are volume and surface area related? Volume is a measure of the amount a container holds while surface area is the sum of the areas of the surfaces on the container.  How does the volume of a rectangular prism change when one of the attributes is increased? There is a direct relationship between the volume of a rectangular prism increasing when the length of one of the attributes of the prism is changed by a scale factor.

A cylinder can be represented on a flat surface as a net that contains two circles (bases for the cylinder) and one rectangular region whose length is the circumference of the circular base and whose width is the height of the cylinder. The surface area of the cylinder is the area of the two circles and the rectangle (SA = 2r2 + 2rh). The volume of a rectangular prism is computed by multiplying the area of the base, B, (length times width) by the height of the prism (V = lwh = Bh). The volume of a cylinder is computed by multiplying the area of the base, B, (r2) by the height of the cylinder (V = r2h = Bh).


ESSENTIAL KNOWLEDGE AND SKILLS The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to  Determine if a practical problem involving a rectangular prism or cylinder represents the application of volume or surface area.  Find the surface area of a rectangular prism.  Solve practical problems that require finding the surface area of a rectangular prism.  Find the surface area of a cylinder.  Solve practical problems that require finding the surface area of a cylinder.  Find the volume of a rectangular prism.  Solve practical problems that require finding the volume of a rectangular prism.  Find the volume of a cylinder.  Solve practical problems that require finding the volume of a cylinder.  Describe how the volume of a rectangular prism is affected when one measured attribute is multiplied by a scale factor. Problems will be limited to 10

STANDARD 7.5 7.5

GRADE LEVEL 7

The student will a) describe volume and surface area of cylinders; b) solve practical problems involving the volume and surface area of rectangular prisms and cylinders; and c) describe how changing one measured attribute of a rectangular prism affects its volume and surface area. UNDERSTANDING THE STANDARD


STRAND: MEASUREMENT


There is a direct relationship between changing one measured attribute of a rectangular prism by a scale factor and its volume. For example, doubling the length of a prism will double its volume. This direct relationship does not hold true for surface area.


ESSENTIAL KNOWLEDGE AND SKILLS changing attributes by scale factors only.  Describe how the surface area of a rectangular prism is affected when one measured attribute is multiplied by a scale factor. Problems will be limited to changing attributes by scale factors only.

11

STANDARD 7.6 7.6

STRAND: MEASUREMENT

The student will determine whether plane figures – quadrilaterals and triangles – are similar and write proportions to express the relationships between corresponding sides of similar figures. UNDERSTANDING THE STANDARD



Two polygons are similar if corresponding (matching) angles are congruent and the lengths of corresponding sides are proportional.



Congruent polygons have the same size and shape.



Congruent polygons are similar polygons for which the ratio of the corresponding sides is 1:1.



Similarity statements can be used to determine corresponding parts of similar figures such as: ABC ~ DEF  A corresponds to  D AB corresponds to DE





GRADE LEVEL 7




How do polygons that are similar compare to polygons that are congruent? Congruent polygons have the same size and shape. Similar polygons have the same shape, and corresponding angles between the similar figures are congruent. However, the lengths of the corresponding sides are proportional. All congruent polygons are considered similar with the ratio of the corresponding sides being 1:1.

The traditional notation for marking congruent angles is to use a curve on each angle. Denote which angles are congruent with the same number of curved lines. For example, if  A congruent to  B, then both angles will be marked with the same number of curved lines. Congruent sides are denoted with the same number of hatch marks on each congruent side. For example, a side on a polygon with 2 hatch marks is congruent to the side with 2 hatch marks on a congruent polygon.



Identify corresponding sides and corresponding and congruent angles of similar figures using the traditional notation of curved lines for the angles.



Write proportions to express the relationships between the lengths of corresponding sides of similar figures.



Determine if quadrilaterals or triangles are similar by examining congruence of corresponding angles and proportionality of corresponding sides.



Given two similar figures, write similarity statements using symbols such as ABC ~ DEF , A

corresponds to  D, and AB corresponds to DE .

12

FOCUS 6–8

STRAND: GEOMETRY

GRADE LEVEL 7


Students expand the informal experiences they have had with geometry in the elementary grades and develop a solid foundation for the exploration of geometry in high school. Spatial reasoning skills are essential to the formal inductive and deductive reasoning skills required in subsequent mathematics learning.



Students learn geometric relationships by visualizing, comparing, constructing, sketching, measuring, transforming, and classifying geometric figures. A variety of tools such as geoboards, pattern blocks, dot paper, patty paper, miras, and geometry software provides experiences that help students discover geometric concepts. Students describe, classify, and compare plane and solid figures according to their attributes. They develop and extend understanding of geometric transformations in the coordinate plane.



Students apply their understanding of perimeter and area from the elementary grades in order to build conceptual understanding of the surface area and volume of prisms, cylinders, pyramids, and cones. They use visualization, measurement, and proportional reasoning skills to develop an understanding of the effect of scale change on distance, area, and volume. They develop and reinforce proportional reasoning skills through the study of similar figures.



Students explore and develop an understanding of the Pythagorean Theorem. Mastery of the use of the Pythagorean Theorem has far-reaching impact on subsequent mathematics learning and life experiences.

The van Hiele theory of geometric understanding describes how students learn geometry and provides a framework for structuring student experiences that should lead to conceptual growth and understanding. 

Level 0: Pre-recognition. Geometric figures are not recognized. For example, students cannot differentiate between three-sided and four-sided polygons.



Level 1: Visualization. Geometric figures are recognized as entities, without any awareness of parts of figures or relationships between components of a figure. Students should recognize and name figures and distinguish a given figure from others that look somewhat the same. (This is the expected level of student performance during grades K and 1.)



Level 2: Analysis. Properties are perceived but are isolated and unrelated. Students should recognize and name properties of geometric figures. (Students are expected to transition to this level during grades 2 and 3.)


13

FOCUS 6–8

STRAND: GEOMETRY

GRADE LEVEL 7



Level 3: Abstraction. Definitions are meaningful, with relationships being perceived between properties and between figures. Logical implications and class inclusions are understood, but the role and significance of deduction is not understood. (Students should transition to this level during grades 5 and 6 and fully attain it before taking algebra.)



Level 4: Deduction. Students can construct proofs, understand the role of axioms and definitions, and know the meaning of necessary and sufficient conditions. Students should be able to supply reasons for steps in a proof. (Students should transition to this level before taking geometry.)


14

STANDARD 7.7 7.7

STRAND: GEOMETRY

The student will compare and contrast the following quadrilaterals based on properties: parallelogram, rectangle, square, rhombus, and trapezoid. UNDERSTANDING THE STANDARD



A quadrilateral is a closed plane (two-dimensional) figure with four sides that are line segments.



A parallelogram is a quadrilateral whose opposite sides are parallel and opposite angles are congruent.



A rectangle is a parallelogram with four right angles. The diagonals of a rectangle are the same length and bisect each other.





GRADE LEVEL 7




Why can some quadrilaterals be classified in more than one category? Every quadrilateral in a subset has all of the defining attributes of the subset. For example, if a quadrilateral is a rhombus, it has all the attributes of a rhombus. However, if that rhombus also has the additional property of 4 right angles, then that rhombus is also a square.

mathematical communication, mathematical reasoning, connections, and representations to

A square is a rectangle with four congruent sides whose diagonals are perpendicular. A square is a rhombus with four right angles.



A rhombus is a parallelogram with four congruent sides whose diagonals bisect each other and intersect at right angles.

Compare and contrast attributes of the following quadrilaterals: parallelogram, rectangle, square, rhombus, and trapezoid.



Identify the classification(s) to which a quadrilateral belongs, using deductive reasoning and inference.



A trapezoid is a quadrilateral with exactly one pair of parallel sides.



A trapezoid with congruent, nonparallel sides is called an isosceles trapezoid.



Quadrilaterals can be sorted according to common attributes, using a variety of materials.



A chart, graphic organizer, or Venn diagram can be made to organize quadrilaterals according to attributes such as sides and/or angles.


15

STANDARD 7.8 7.8

STRAND: GEOMETRY

GRADE LEVEL 7

The student, given a polygon in the coordinate plane, will represent transformations (reflections, dilations, rotations, and translations) by graphing in the coordinate plane. UNDERSTANDING THE STANDARD


(Background Information for Instructor Use Only)  A rotation of a geometric figure is a turn of the figure around a fixed point. The point may or may not be on the figure. The fixed point is called the center of rotation.  A translation of a geometric figure is a slide of the figure in which all the points on the figure move the same distance in the same direction.




How does the transformation of a figure affect the size, shape and position of that figure? Translations, rotations and reflections do not change the size or shape of a figure. A dilation of a figure and the original figure are similar. Reflections, translations and rotations usually change the position of the figure.

 A reflection is a transformation that reflects a figure across a line in the plane.


Identify the coordinates of the image of a right triangle or rectangle that has been translated either vertically, horizontally, or a combination of a vertical and horizontal translation.

 The image of a polygon is the resulting polygon after the transformation. The preimage is the polygon before the transformation.



Identify the coordinates of the image of a right triangle or rectangle that has been rotated 90° or 180° about the origin.

 A transformation of preimage point A can be denoted as the image A (read as “A prime”).



Identify the coordinates of the image of a right triangle or a rectangle that has been reflected over the x- or y-axis.



Identify the coordinates of a right triangle or rectangle that has been dilated. The center of the dilation will be the origin.



Sketch the image of a right triangle or rectangle translated vertically or horizontally.



Sketch the image of a right triangle or rectangle that has been rotated 90° or 180° about the origin.



Sketch the image of a right triangle or rectangle that has been reflected over the x- or y-axis.



Sketch the image of a dilation of a right triangle or

 A dilation of a geometric figure is a transformation that changes the size of a figure by scale factor to create a similar figure.


16

STANDARD 7.8 7.8

STRAND: GEOMETRY

GRADE LEVEL 7

The student, given a polygon in the coordinate plane, will represent transformations (reflections, dilations, rotations, and translations) by graphing in the coordinate plane. UNDERSTANDING THE STANDARD

(Background Information for Instructor Use Only)


ESSENTIAL KNOWLEDGE AND SKILLS 1 1 rectangle limited to a scale factor of , , 2, 3 or 4. 4 2


17

FOCUS 6–8

STRAND: PROBABILITY AND STATISTICS

GRADE LEVEL 7


Students develop an awareness of the power of data analysis and probability by building on their natural curiosity about data and making predictions.



Students explore methods of data collection and use technology to represent data with various types of graphs. They learn that different types of graphs represent different types of data effectively. They use measures of center and dispersion to analyze and interpret data.



Students integrate their understanding of rational numbers and proportional reasoning into the study of statistics and probability.



Students explore experimental and theoretical probability through experiments and simulations by using concrete, active learning activities.


18

STANDARD 7.9 7.9


The student will investigate and describe the difference between the experimental probability and theoretical probability of an event. UNDERSTANDING THE STANDARD



Theoretical probability of an event is the expected probability and can be found with a formula.



Theoretical probability of an event =



number of possible favorable outcomes total number of possible outcomes 



The experimental probability of an event is determined by carrying out a simulation or an experiment.


What is the difference between the theoretical and experimental probability of an event? Theoretical probability of an event is the expected probability and can be found with a formula. The experimental probability of an event is determined by carrying out a simulation or an experiment. In experimental probability, as the number of trials increases, the experimental probability gets closer to the theoretical probability.



The experimental probability = number of times desired outcomes occur number of trials in the experiment



GRADE LEVEL 7

In experimental probability, as the number of trials increases, the experimental probability gets closer to the theoretical probability (Law of Large Numbers).



Determine the theoretical probability of an event.



Determine the experimental probability of an event.



Describe changes in the experimental probability as the number of trials increases.



Investigate and describe the difference between the probability of an event found through experiment or simulation versus the theoretical probability of that same event.

19

STANDARD 7.10 7.10


The student will determine the probability of compound events, using the Fundamental (Basic) Counting Principle.

UNDERSTANDING THE STANDARD


(Teacher Notes) 





GRADE LEVEL 7

The Fundamental (Basic) Counting Principle is a computational procedure to determine the number of possible outcomes of several events. It is the product of the number of outcomes for each event that can be chosen individually (e.g., the possible outcomes or outfits of four shirts, two pants, and three shoes is 4 · 2 · 3 or 24). Tree diagrams are used to illustrate possible outcomes of events. They can be used to support the Fundamental (Basic) Counting Principle. A compound event combines two or more simple events. For example, a bag contains 4 red, 3 green and 2 blue marbles. What is the probability of selecting a green and then a blue marble?





What is the Fundamental (Basic) Counting Principle? The Fundamental (Basic) Counting Principle is a computational procedure used to determine the number of possible outcomes of several events. What is the role of the Fundamental (Basic) Counting Principle in determining the probability of compound events? The Fundamental (Basic) Counting Principle is used to determine the number of outcomes of several events. It is the product of the number of outcomes for each event that can be chosen individually.


ESSENTIAL KNOWLEDGE AND SKILLS The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to  Compute the number of possible outcomes by using the Fundamental (Basic) Counting Principle.  Determine the probability of a compound event containing no more than 2 events.

20

STANDARD 7.11 7.11


The student, given data in a practical situation, will a) construct and analyze histograms; and b) compare and contrast histograms with other types of graphs presenting information from the same data set.




All graphs tell a story and include a title and labels that describe the data.



A histogram is a form of bar graph in which the categories are consecutive and equal intervals. The length or height of each bar is determined by the number of data elements frequency falling into a particular interval.



GRADE LEVEL 7




What types of data are most appropriate to display in a histogram?


Numerical data that can be characterized using consecutive intervals are best displayed in a histogram.

reasoning, connections, and representations to 

Collect, analyze, display, and interpret a data set using histograms. For collection and display of raw data, limit the data to 20 items.



Determine patterns and relationships within data sets (e.g., trends).



Make inferences, conjectures, and predictions based on analysis of a set of data.



Compare and contrast histograms with line plots, circle graphs, and stem-and-leaf plots presenting information from the same data set.

A frequency distribution shows how often an item, a number, or range of numbers occurs. It can be used to construct a histogram.


21

STANDARD 7.11 7.11


GRADE LEVEL 7

The student, given data in a practical situation, will a) construct and analyze histograms; and b) compare and contrast histograms with other types of graphs presenting information from the same data set.

UNDERSTANDING THE STANDARD (Background Information for Instructor Use Only) 

Comparisons, predictions and inferences are made by examining characteristics of a data set displayed in a variety of graphical representations to draw conclusions.



The information displayed in different graphs may be examined to determine how data are or are not related, ascertaining differences between characteristics (comparisons), trends that suggest what new data might be like (predictions), and/or “what could happen if” (inference).




22

FOCUS 6–8

STRAND: PATTERNS, FUNCTIONS, AND ALGEBRA

GRADE LEVEL 7

In the middle grades, the focus of mathematics learning is to  build on students’ concrete reasoning experiences developed in the elementary grades;  construct through active learning experiences a more advanced understanding of mathematics;  develop deep mathematical understandings required for success in abstract learning experiences; and  apply mathematics as a tool in solving practical problems. Students in the middle grades use problem solving, mathematical communication, mathematical reasoning, connections, and representations to integrate understanding within this strand and across all the strands. 

Students extend their knowledge of patterns developed in the elementary grades and through practical experiences by investigating and describing functional relationships.



Students learn to use algebraic concepts and terms appropriately. These concepts and terms include variable, term, coefficient, exponent, expression, equation, inequality, domain, and range. Developing a beginning knowledge of algebra is a major focus of mathematics learning in the middle grades.



Students learn to solve equations by using concrete materials. They expand their skills from one-step to two-step equations and inequalities.



Students learn to represent relations by using ordered pairs, tables, rules, and graphs. Graphing in the coordinate plane linear equations in two variables is a focus of the study of functions.


23

STANDARD 7.12 7.12


The student will represent relationships with tables, graphs, rules, and words.




GRADE LEVEL 7

Rules that relate elements in two sets can be represented by word sentences, equations, tables of values, graphs, or illustrated pictorially.



A relation is any set of ordered pairs. For each first member, there may be many second members.



A function is a relation in which there is one and only one second member for each first member.



As a table of values, a function has a unique value assigned to the second variable for each value of the first variable.



As a graph, a function is any curve (including straight lines) such that any vertical line would pass through the curve only once.



Some relations are functions; all functions are relations.




What are the different ways to represent the relationship between two sets of numbers? Rules that relate elements in two sets can be represented by word sentences, equations, tables of values, graphs or illustrated pictorially.



Describe and represent relations and functions, using tables, graphs, rules, and words. Given one representation, students will be able to represent the relation in another form.

24

STANDARD 7.13 7.13


GRADE LEVEL 7

The student will a) write verbal expressions as algebraic expressions and sentences as equations and vice versa; and b) evaluate algebraic expressions for given replacement values of the variables.



(Teacher Notes) 




An expression is a name for a number.



An expression that contains a variable is a variable expression.



An expression that contains only numbers is a numerical expression.

reasoning, connections, and representations



A verbal expression is a word phrase (e.g., “the sum of two consecutive integers”).

to



A verbal sentence is a complete word statement (e.g., “The sum of two consecutive integers is five.”).



Write verbal expressions as algebraic expressions. Expressions will be limited to no more than 2 operations.



An algebraic expression is a variable expression that contains at least one variable (e.g., 2x – 5).



Write verbal sentences as algebraic equations. Equations will contain no more than 1 variable term.



An algebraic equation is a mathematical statement that says that two expressions are equal (e.g., 2x + 1 = 5).



Translate algebraic expressions and equations to verbal expressions and sentences. Expressions will be limited to no more than 2 operations.



To evaluate an algebraic expression, substitute a given replacement value for a variable and apply the order of operations. For example, if a = 3 and b = -2 then 5a + b can be evaluated as: 5(3) + (-2) = 15 + (-2) = 13.



Identify examples of expressions and equations.



Apply the order of operations to evaluate expressions for given replacement values of the variables. Limit the number of replacements to no more than 3 per expression.

How can algebraic expressions and equations be written? Word phrases and sentences can be used to represent algebraic expressions and equations.



25

STANDARD 7.14 7.14


GRADE LEVEL 7

The student will a) solve one- and two-step linear equations in one variable; and b) solve practical problems requiring the solution of one- and two-step linear equations.




An equation is a mathematical sentence that states that two expressions are equal.



A one-step equation is defined as an equation that requires the use of one operation to solve (e.g., x + 3 = – 4 ).



The inverse operation for addition is subtraction, and the inverse operation for multiplication is division.



A two-step equation is defined as an equation that requires the use of two operations to solve x7  4 ). (e.g., 2x + 1 = -5; -5 = 2x + 1; 3




When solving an equation, why is it important to perform identical operations on each side of the equal sign? An operation that is performed on one side of an equation must be performed on the other side to maintain equality.





Represent and demonstrate steps for solving one- and two-step equations in one variable using concrete materials, pictorial representations and algebraic sentences.



Solve one- and two-step linear equations in one variable.



Solve practical problems that require the solution of a one- or two-step linear equation.

26

STANDARD 7.15 7.15


The student will a) solve one-step inequalities in one variable; and b) graph solutions to inequalities on the number line.




A one-step inequality is defined as an inequality that requires the use of one operation to solve (e.g., x – 4 > 9).



The inverse operation for addition is subtraction, and the inverse operation for multiplication is division.



When both expressions of an inequality are multiplied or divided by a negative number, the inequality symbol reverses (e.g., –3x < 15 is equivalent to x > –5).



GRADE LEVEL 7






How are the procedures for solving equations and inequalities the same? The procedures are the same except for the case when an inequality is multiplied or divided on both sides by a negative number. Then the inequality sign is changed from less than to greater than, or greater than to less than. How is the solution to an inequality different from that of a linear equation? In an inequality, there can be more than one value for the variable that makes the inequality true.


Represent and demonstrate steps in solving inequalities in one variable, using concrete materials, pictorial representations, and algebraic sentences.



Graph solutions to inequalities on the number line.



Identify a numerical value that satisfies the inequality.

Solutions to inequalities can be represented using a number line.


27

STANDARD 7.16 7.16


The student will apply the following properties of operations with real numbers: a) the commutative and associative properties for addition and multiplication; b) the distributive property; c) the additive and multiplicative identity properties; d) the additive and multiplicative inverse properties; and e) the multiplicative property of zero.




The commutative property for addition states that changing the order of the addends does not change the sum (e.g., 5 + 4 = 4 + 5).



The commutative property for multiplication states that changing the order of the factors does not change the product (e.g., 5 · 4 = 4 · 5).



The associative property of addition states that regrouping the addends does not change the sum [e.g., 5 + (4 + 3) = (5 + 4) + 3].



GRADE LEVEL 7



Subtraction and division are neither commutative nor associative.



The distributive property states that the product of a number and the sum (or difference) of two other numbers equals the sum (or difference) of the products of the number and each other number [e.g., 5 · (3 + 7) = (5 · 3) + (5 · 7), or 5 · (3 – 7) = (5 · 3) – (5 · 7)].



Identity elements are numbers that combine with other numbers without changing the other numbers. The additive identity is zero (0). The multiplicative identity is one (1). There are no identity elements for subtraction and division.


Why is it important to apply properties of operations when simplifying expressions? Using the properties of operations with real numbers helps with understanding mathematical relationships.

The associative property of multiplication states that regrouping the factors does not change the product [e.g., 5 · (4 · 3) = (5 · 4) · 3].






Identify properties of operations used in simplifying expressions.



Apply the properties of operations to simplify expressions.

28

STANDARD 7.16 7.16


GRADE LEVEL 7

The student will apply the following properties of operations with real numbers: a) the commutative and associative properties for addition and multiplication; b) the distributive property; c) the additive and multiplicative identity properties; d) the additive and multiplicative inverse properties; and e) the multiplicative property of zero.

UNDERSTANDING THE STANDARD (Background Information for Instructor Use Only) 

The additive identity property states that the sum of any real number and zero is equal to the given real number (e.g., 5 + 0 = 5).



The multiplicative identity property states that the product of any real number and one is equal to the given real number (e.g., 8 · 1 = 8).



Inverses are numbers that combine with other numbers and result in identity elements 1 [e.g., 5 + (–5) = 0; · 5 = 1]. 5



The additive inverse property states that the sum of a number and its additive inverse always equals zero [e.g., 5 + (–5) = 0].



The multiplicative inverse property states that the product of a number and its multiplicative inverse 1 (or reciprocal) always equals one (e.g., 4 · = 1). 4



Zero has no multiplicative inverse.



The multiplicative property of zero states that the product of any real number and zero is zero.



Division by zero is not a possible arithmetic operation. Division by zero is undefined.




29

STANDARD 7.16



GRADE LEVEL 7

30