Equations, Tables, and Graphs - EWMA Middle School

phm07c3_te_0305.fm Page 130 Thursday, May 25, 2006 3:56 PM

3-5

3-5

1. Plan Objective

What You’ll Learn

To use tables, equations, and graphs to solve problems

Examples 1 2

Making Tables and Writing Equations Graphing Linear Equations

Math Understandings: p. 104C

Equations, Tables, and Graphs

1. Vocabulary Review What do you call a symbol that stands for one or more numbers? variable Evaluate for a 4. 2. 6a 21 3 3. 13 2a 21 4. 5a 8 28

Lesson 1-1

To use tables, equations, and graphs to solve problems New Vocabulary solution, linear equation

Why Learn This? You can use equations, tables, and graphs to represent the same data. For example, you can use a table of values for plant growth to write an equation or make a graph. Given a word problem, you can sometimes make a table of data. Then you can write an equation to model the situation.

Math Background The solutions of an equation in two variables, such as y x 5, can be graphed on a coordinate plane as a line formed by an infinite set of ordered pairs. Tables can be used to represent a set of x- and y-coordinates that satisfy the equation. The set of x-values is known as the domain and the set of corresponding y-values is known as the range. In real-world linear situations, the domain and range must be reasonable values.

Making Tables and Writing Equations Suppose you save $3 each week. Make a table and write an equation to represent your total savings after a given number of weeks.

For help making a table of data, go to Lesson 1-1, Example 4.

More Math Background: p. 104C

Number of Weeks

Total Savings (dollars)

Expression

0

0

3(0)

1

3

3(1)

2

6

3(2)

3

9

3(3)

w

t

3(w)

Let w represent the number of weeks.

Lesson Planning and Resources

Look for a pattern in the table. Your total savings for a given week is 3 times the number of weeks you have been saving.

Let t represent your total savings.

The equation t 3w models your total savings.

See p. 104E for a list of the resources that support this lesson. 1. You buy CDs from a music store. Each CD costs $15. Make a table

and write an equation to represent the total cost of buying a given number of CDs. See back of book.

Bell Ringer Practice Check Skills You’ll Need Use student page, transparency, or PowerPoint. For intervention, direct students to:

Algebraic Expressions and Order of Operations Lesson 1-1 Extra Skills and Word Problems Practice, Ch. 1

130

130

Chapter 3 Real Numbers and the Coordinate Plane

Special Needs

Below Level

L1

Students work in pairs. Match those who have difficulty graphing equations on the grids with those who can do so more easily. The first student can make the data tables while the partner can graph the equations. Then partners can check each other’s work.

learning style: visual

L2

To help students remember the order of a coordinate pair, have them write x over the first value and y over the second value.

learning style: visual


Any ordered pair that makes an equation true is a solution of the equation. For example, (2, 6) is a solution of y 3 x because 6 3( 2). An equation with two variables can have many solutions. You can show these solutions on a graph. An equation is a linear equation if all of its solutions lie on a line.

Vocabulary Tip The term linear means “relating to a line.”

Graphing Linear Equations Jerrod keeps track of how much dry food is in his cat’s feeder. Graph the linear equation y 21 x 12, where y represents the cups of food left and x represents the number of days since he filled the twelve-cup feeder.

x

y 12 x 12

0

12 (0) 12 12

4

12 (4) 12 10

9

12 (9) 12 7 12

16

12 (16) 12 4

Step 2 Graph the ordered pairs and draw a line through the points. Cups of Food

Step 1 Make a table.

8 4 0

4

Activity Lab Use before the lesson. Student Edition Activity Lab, Data Analysis 3-5a, Tables and Graphs, p. 129 Teaching Resources

Activity Lab 3-5: Hidden Equation

Additional Examples Suppose you buy a bag of food for your pet dog every week. Dog food costs $4 per bag. Make a table and write an equation to represent the total cost of buying dog food for any number of weeks. c 4w

y

12

2. Teach

x 8 12 16 Days Passed

Each point (x, y) on the graph represents a solution of the equation. For example, the point (4, 10) means that after 4 days, 10 cups are left.

Number of Weeks

1

Cost of 4 Dog Food

2

3

4

w

8 12 16 c

Graph the linear equation y x 3, where y represents the pressure inside a deflating balloon after x seconds.

2. Graph the linear equation y 5 x 50, where y represents the

temperature in F of a chemical solution after x minutes.

See back of book.

A plant is 4 cm tall and grows 2 cm per day. Predict how tall the plant will be after 8 days.

Seconds

0

1

2

3

Pressure

3

2

1

0

Pressure

Roberto’s Method I can make a table of data. Height of Plant Days Passed

0

1

2

3

4

5

6

7

8

Height

4

6

8

10

12

14

16

18

20

3 2 1 0

1 2 3 Seconds

After 8 days, the plant will be 20 cm tall.

3-5 Equations, Tables, and Graphs

Advanced Learners

L4

The equation y 6 3x represents the height of water in a bottle pouring out over time. Do all of its solutions apply to the graph of the equation? no Which of the solutions to the equation should be shown on the graphed line? Sample: the positive

values, including zero

learning style: verbal

131

English Language Learners Students may be comfortable mentally calculating the answer for the Choose a Method example since the numbers are small integers. Stress that the tables and graphs also allow them to make predictions, whether the numbers are easy or hard to mentally calculate.

learning style: verbal

131


Guided Instruction Example 1

Jasmine’s Method

Ask: • How do the x-values change as you move down the table? increase by 1 • How do the y-values change? increase by 3 • If this pattern continues, what would be the next ordered pair in the table? (4, 12)

I can make a graph. Let x represent the number of days that have passed. Let y represent the height of the plant. I can make a table of solutions. Three points on the graph are (1, 6), (2, 8), and (3, 10).

Height of Plant Days Passed (x)

1

2

3

Height (y)

6

8

10

Height (cm)

Error Prevention! In a real-world situation, students need to consider not only coordinates that are solutions to the equation, but values that are reasonable for the situation. Ask: Why would you not choose negative values for x in Example 2? There cannot be a negative number of days.

I can draw a line through the points. Then I can use the graph to find the height y when x 8.

y

24 16 8

x 0

2

After 8 days, the plant will be 20 cm tall.

4 6 8 Days Passed

Choose a Method A bag of rice weighs 80 oz. If a serving of rice is 2 oz, how much rice will be left after you prepare 10 servings? Explain why you chose the method you used. 60 oz; check students’ methods.

Teaching Resources • Daily Notetaking Guide 3-5 L3 • Adapted Notetaking 3-5 L1

Closure • How do you graph a linear

Check Your Understanding

equation in two variables? Sample: Make a table of values by choosing several x-values and substituting them into the equation and simplifying. Graph the resulting ordered pairs and connect them in a straight line. • In many cases when a linear equation represents a realworld situation, such as height vs. age, which values cannot apply to the situation? In which quadrant will the graph be shown? negative nonzero xand y-values; first quadrant

1. Vocabulary Which statement about linear equations is not true? C

The graph of a linear equation is a line. Every point on the graph of a linear equation is a solution. A point that does not lie on the graph of a linear equation may still be a solution of the equation. You can write solutions of a linear equation as ordered pairs.

2. A leaky pipe loses 0.75 gallons of water every minute. Complete the data table below. See margin. Water Loss Number of Minutes (t)

1

2

3

4

Gallons of Water Lost (g)

■

■

■

■

3. Use the table from Exercise 2. Write a linear equation to represent the amount of water lost from the leaky pipe. g 0.75 t 4. Suppose you wanted to graph the equation in Exercise 3. Use the

4. (1, 0.75), (2, 1.5), (3, 2.25), (4, 3)

132 2.

132

table from Exercise 2 to name four points that lie on the graph.


Water Loss Number of Minutes (t)

1

2

3

4

Gallons of Water Lost (g)

0.75

1.5

2.25

3


3. Practice

Homework Exercises For more exercises, see Extra Skills and Word Problems.

Assignment Guide

5. In 2000, about four babies were born world-wide every second. For Exercises See Examples 5–6

1

7–8

2

Make a table and write an equation to represent the total number of babies born over time. See margin.

Check Your Understanding Go over Exercises 1–4 in class before assigning the Homework Exercises.

6. The temperature drops 2F every hour. Make a table and write an

equation to represent the total temperature drop over time.

Homework Exercises

See margin. 7. For a certain repair, an auto shop charges a $20 fee for materials

plus $40 per hour for labor. Graph the linear equation y 40x 20, where y represents the total cost and x represents the hours of labor. 7–8. See back of book. 8. On a 100-point test, each question is worth 5 points. Partial answers

A Practice by Example B Apply Your Skills C Challenge

5–8 9–17 18

Test Prep and Mixed Review

19–24

Homework Quick Check

receive partial credit. Graph the linear equation y 100 5x, where y represents your score and x represents the number of incorrect answers.

To check students’ understanding of key skills and concepts, go over Exercises 6, 7, 13, 15, and 16.

9. Guided Problem Solving In the design at

the right, 12 squares surround a row of 3 circles. Predict the number of squares needed to surround a row of 10 circles. • Make a Plan Make a table of values. 26 squares Graph the ordered pairs from the table and draw a line through the points. Then use the graph to find the answer. • Carry Out the Plan Complete the table below.

Adapted Practice 3-5

Equations, Tables, and Graphs

Use the equation y 2x 1. Complete each solution. 1. (0, 9)

2. (5, 9)

1

1

2

3

4

5

6

Number of Squares (y)

■

■

■

■

■

■

3. (20, 9)

11 yes

a. (0, 8)

b. (6, 10)

no

no

a. (3, 4)

b. (0, 19)

yes

3 5

11. y x 2

nline

-6

Visit: PHSchool.com Web Code: ase-0305

-4

2

4

6

-6

-4

Total Cost

1

2

3

4

5

13

21

29

37

45

4

6

-6

y 6 3x 1 4

2

x -6

-4

-2 O -2

2

x 2

4

6

-6

-4

y 2x-4 7

-6

6

y

y

6

4

-6

4

4

2

12. y 3x 1

6

2

x

-2 O -2

-4

y 12 x-4 3

-6

-4

-2 O -2

x 2

4

6

-4

-6

-6

13. Jan wants to buy maps for her trip. The maps cost $2 each and she has $25. Make a table and write an equation to represent the amount she will have left if she buys m maps.

5 maps

L3

3-5 • Guided Problem Solving GPS

Number of Mugs

-2 O -2

2

11. y 2x + 7

2 -4

2

x

-2 O -2

y

-6

4

-4

4

12. y 1 . 5 x 4

y 6

2

x

-2 O -2

6 y 12 x 12

14. The table below shows the cost of buying class mugs online. Write an equation to model the data. See left.

14. Let m the number of mugs. Let t the total cost. t 8m 5

yes

9. y 5 212 x 1 3 y

10. y 5 12 x 2 12

same linear equation. Which one is not? Explain. See left. B( 0, 4) C(1, 2) D(4, 4) A(2, 1) E(3, 2)

yes

d. (4, 39)

6 y 52 x 5 4

4

13. Writing in Math Four of the five points below are solutions of the

d. (4, 4)

yes

8. y 5 52 x 2 5

y 4-6x 6

2 3

c. (2, 9)

y

-4

10. y x 3

c. (2, 2)

Graph each linear equation.

6

Graph each linear equation. 10–12. See back of book.

137 no

6. Determine whether each ordered pair is a solution of y 5x + 19.

2

13. A; A is not on the line passing through the other points.

4. (68, 9)

–39

5. Determine whether each ordered pair is a solution of y 3x 8.

7. y 4x + 6

Number of Circles (x)

L1 L3

Practice 3-5

Student Page 133, Exercise 15:

Engraving a key chain costs $10 plus $1.50 for each engraved letter. You can only spend $20. What is the maximum number of letters you can engrave? Solve by making a table and writing an equation.

Understand 1. What are you being asked to do?

15. Engraving a key chain costs $10 plus $1.50 for each engraved letter.

You can only spend $20. What is the maximum number of letters you can engrave? Solve by making a table and writing an equation. See back of book.

find the maximum number of letters that can be engraved for $20 Plan and Carry Out 2. Write an expression for the cost l per letters in a key chain.

3.

Number Total Cost of Letters Expression (dollars)

1.5l

1

10 1.5(1)

$11.50

What is the flat fee charged per key chain? 10

2

10 1.5(2)

$13.00

3

10 1.5(3)

$14.50

4

10 1.5(4)

$16.00

5

10 1.5(5)

$17.50

6

10 1.5(6)

$19.00

7

10 1.5(7)

$20.50

4. Write an equation for the cost c of engraving a key chain. Be sure to include the flat fee and the cost per letter.

c 10 1.5l 5. Use the equation in Step 4 to complete the table. 6. What is the maximum number of letters you can engrave for $20

lesson quiz, PHSchool.com, Web Code: asa-0305

3-5 Equations, Tables, and Graphs

5–6. See back of book.

133

or less? 6

letters

Check 7. Compare the cost of your answer with the cost of having one more letter engraved. How should these two amounts compare

to $20?

the cost for 6 letters should be less than or equal

to $20, and the cost for 7 letters should be more Solve Another Problem 8. Suzy is going bowling at EZ Lanes. The cost to rent shoes is $4, and each bowling game costs $3.50. If Suzy needs to rent shoes and has $16, how many games can she play? Solve by drawing a table and writing an equation.

Suzy can bowl 3 games.

c 4 35(g) Number Total Cost of Games Expression (dollars) 1

4 3.50(1)

$7.50

2

4 3.50(2)

$11.00

3

4 3.50(3)

$14.50

4

4 3.50(4)

$18.00

5

4 3.50(5)

$21.50

133


4. Assess & Reteach

16. Choose a Method You start an exercise routine by lifting 3 lb and

increase the weight by 2 lb per month. Predict how much weight you will lift after 5 months. Explain why you chose the method you used.

1–3. See back of book.

13 lb; check students’ methods. 17. Error Analysis Gina owes her father $200. During each week, she

Lesson Quiz

pays $20. They both draw graphs to represent the money she owes. Who is correct? Explain. See left.

2. Membership at a video store costs $5 per month, plus $1.50 to rent each movie. Graph the linear equation y 5 1.50x, where y represents the total cost in a month and x represents the number of movies rented each month.

17. Gina; the graph drawn by Gina’s father shows the amount owed after Gina gives him $40 each week, not $20.

3. Suppose you rent 6 movies in a month, in the situation above. Make a table that represents your total costs.

Gina

y

2

Money Gina Owes Me

200 160 120 80 40 0

Money I Owe

1. Suppose you make $8 per hour at an after-school job. Make a table and write an equation to represent your total pay after 6 hours of work.

200 160 120 80 40 0

x

4 6 Week

Gina’s Dad

y

2

4 6 Week

x

18. Challenge A club sells calendars for $4 each. It spends $2 to make

each calendar and $20 on film. Write and graph two equations to represent income and expenses. Where do the graphs intersect? See back of book.

Test Prep and Mixed Review

4. The air pressure in a tire is 32 pounds per square inch. Every hour, air is leaking out at the rate of 3 pounds per square inch. Write an equation that describes this situation. Sample: p 32 3h

Multiple Choice

Practice

1

19. The graph of y 2 x 1 is shown on

y

the coordinate grid at the right. Which table of ordered pairs contains only points on this line? B

y

x

x

2 2 O

2

4

x

x

y

x

y

0

0

2

3

1.5

y

4

1

2

2

2

1

1.5

1

0

0

2

3

2.5

4

3

2

2

5

3.5

20. Audrey bought a box of cereal and some bananas for $4.69. If the Enrichment 3-5

cereal cost $3.99 and the bananas were on sale for $0.28 per pound, how many pounds of bananas did Audrey buy? H 0.42 lb 2.2 lb 2.5 lb 4.2 lb

L4 Equations, Tables, and Graphs L2

Reteaching 3-5 You can use equations, tables, and graphs to represent the same data.

Suppose you spend $8 each day on food. Make a table and write an equation to represent the total amount of money spent after a given number of days. Then graph your equation. Let x represent the number of days.

2

Let y represent the total money spent.

3

Find at least three solutions. Days Passed

Total Spending Expression (dollars)

0

0

8(0)

1

8

8(1)

2

16

3

24

8(3)

x

y

8(x)

8(2)

4

The equation y 8x models your total spending.

5

Show your solutions on a graph. Graph the ordered pairs and draw a line through the points.

30 20 10 2

4

6

8

x

Days Passed

2

6 8 10 12

3 4 5

2. Mary buys bottled water for her family. Graph the linear equation y x 10, where x represents the number of days since she bought 10 bottles and y represents the number of bottles left.

Height (ft)

4

22–24

1-6

134

15

Solve each equation. 22. b 6 10 4

23. k 1 24 25

24. 4 n 40 44


10 5 O

2

4

6

8

x

Test Prep

Alternative Assessment

Resources

Students write a paragraph explaining how to graph the following situation in the coordinate plane: A scientist is releasing 20 drops of liquid from a tube every 10 seconds. Students write an equation that describes the situation and draw the graph.

Years

y 8 6 4 2 O

2

4

6

Days Passed

134

See Lesson

20

Bottles of Water

2

1

For Exercises

y

1. A tree is 2 feet tall and grows 2 feet per year. Complete the table and graph each (x, y) solution. How tall will the tree be in 5 years?

0

should Sam find the number of miles per hour that equals 1 knot? B Divide 20 by 23. Divide 3 by 20. Divide 23 by 20. Divide 3 by 23.

40

O

Years Passed (x) Height in Feet (y)

21. A ferry travels at 20 knots, which is about 23 miles per hour. How

y Total Spending ($)

1

8

x

For additional practice with a variety of test item formats: • Test-Taking Strategies, p. 151 • Test Prep, p. 155 • Test-Taking Strategies with Transparencies