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### Transcript of LESSON 1: PRINT JOB EXERCISES

Grade 8 Unit 4: Linear Relationships

EXERCISES

EXERCISES

1. Write three things you already know about linear relationships. Share your work with a classmate. Does your classmate understand what you wrote?

3. Consider everyday life situations. Think of a situation in your life that can be modeled by a linear relationship. Describe this situation and explain why it is a linear relationship.

4. Write a goal stating what you plan to accomplish in this unit.

LESSON 1: PRINT JOB

Grade 8 Unit 4: Linear Relationships

EXERCISES

EXERCISES

This graph represents four runners competing in the 100 m dash. Use this to answer questions 1–5.

20

40

60

80

100

Dis

tanc

e (m

)

Time (sec)

Runners Competing in the 100 m Dash

050 15 2010

A B C D

x

y

1. Who ran the 100 m dash the fastest?

A Runner A B Runner B C Runner C D Runner D

2. At what time does runner D cross the 80 m mark?

A 8 sec B 11 sec C 13 sec D 16 sec

3. What is the rate (in distance over time) for runner C?

A 5 m/sec B 5.25 m/sec C 6 m/sec D 6.25 m/sec

4. Write an equation to represent each runner’s graph.

5. Compare the times of each runner at 20 m, 40 m, 60 m, and 80 m. You may want to create a table to organize your data.

LESSON 2: MODELING RUNNING SPEEDS

Grade 8 Unit 4: Linear Relationships

EXERCISES

6. Consider the “hoops” in this coordinate plane. Write an equation for a proportional relationship line that goes through as many hoops as possible. First graph the line to check how many hoops it goes through, and then write the corresponding equation for that line.

x

y

2

–2

–4

–6

4

6

8

10

5 10 15 20

7. A house has very old plumbing that has many leaks. This graph shows how much water is wasted by the leaks over time. What is the unit rate in gallons per hour?

Wa

sted

Wa

ter

(gal

)

Time (hr)

Amount of Wasted Water Over Time

2

1

0

14

12

34

10 x

y

A 12 gal/hr B x gal/hr C y gal/hr D 2 gal/hr

LESSON 2: MODELING RUNNING SPEEDS

Grade 8 Unit 4: Linear Relationships

EXERCISES

8. The flow of traffic is monitored on a bridge during 1 hour. A total of 3,180 cars cross the bridge during the first half hour.

500

0

1,000

1,500

2,000

2,500

3,000

3,500

4,000

4,500

5,000

5,500

Num

ber

of C

ars

Time (min)

Number of Cars Crossing the Bridge

6,000

6,500

20 30 40 50 60100 x

y Which rates show the number of cars that cross the bridge per number of minutes? There may be more than one correct rate.

A 3,180 cars per 30 minutes

B 53 cars per minute

C 106 cars per minute

D 30 minutes per 3,180 cars

E 6,360 cars per 60 minutes

Use this information to answer questions 9–10.

Erin babysits on weekends. She earns \$15 for each hour she works.

9. Erin says that her pay is a proportional relationship. Do you agree? Explain why or why not.

10. Write a formula that shows the relationship.

11. Jacob is paid according to the rate in this table. How much money is Jacob paid for working 8 hours?

Hours Worked Pay

1 \$5

2 \$10

3 \$15

4 \$20

A \$25 B \$28

C \$35 D \$40

LESSON 2: MODELING RUNNING SPEEDS

Grade 8 Unit 4: Linear Relationships

EXERCISES

12. This graph shows two proportional relationships. Which line has the smaller slope?

–10 –5 5 10

–10

–5

5

10

y

x

Line A

Line B

A Line A has a smaller slope.

B Line B has a smaller slope.

C Both lines have the same slope.

D This graph does not show the slopes of the lines, so you cannot tell.

Challenge Problem

13. Proportional relationships can be represented by the general equation y = kx.

Write the equations of four proportional relationships. For two of them, make the values of k a positive number. For the other two, make the values of k a negative number.

Draw the graphs of your four proportional relationships in the same coordinate plane.

Draw some conclusions about the graphs of proportional relationships, depending on whether k is a positive or negative number. Explain your observations.

LESSON 2: MODELING RUNNING SPEEDS

Grade 8 Unit 4: Linear Relationships

EXERCISES

EXERCISES

1. What is the slope of this line?

A Slope = 1

B Slope = 2

C Slope = 3

D Slope = 4

2. Which graph shows a proportional relationship?

A

2

3

1

–2

–3

–1

–4

4

–2–3–4 2 31 4

y

x

B y

x

2

3

1

4

–2–3–4

–2

–3

–1

–4

2 31 4

C

2

3

1

–2

–3

–1

–4

4

–2–3–4 2 31 4

y

x

D

2

3

1

–2

–3

–1

–4

4

–2–3–4 2 31 4

y

x

2

4

6

8

10

642–2

y

x

LESSON 3: INVESTIGATING GRAPHS

Grade 8 Unit 4: Linear Relationships

EXERCISES

3. Pedra says, “In this proportional relationship, when x is 3, y is 9.” What is the slope of Pedra’s proportional relationship?

A Slope = 13

B Slope = −13

C Slope = 3 D Slope = –3

4. Proportional relationships always contain what point?

A (1, 1) B (0, 0)

C (1, 0) D (0, 1)

5. Find the slope of line A and line B.

2

–2

–4

–6

4

6

8

5–5–10 10

Line B

Line A

y

x

–8

6. This table shows a proportional relationship. What is the rate of change?

x 0 3 6 9 12

y 0 6 12 18 24

A 2 B 3

C 6 D 12

LESSON 3: INVESTIGATING GRAPHS

Grade 8 Unit 4: Linear Relationships

EXERCISES

7. Graph the following four proportional relationships with the slopes given.

Line a: slope = 12

Line b: slope = –4

Line c: slope = −35

Line d: slope = 3

8. Consider this graph.

–10 –5 5 10

–15

–10

–5

5

10

15

20

y

x

a. Complete the table that goes with the graph.

x –4 –2 0 2 6y –16 0 8 16 24

Slope

b. Explain whether the graph represents a proportional relationship.

LESSON 3: INVESTIGATING GRAPHS

Grade 8 Unit 4: Linear Relationships

EXERCISES

9. a. Draw a linear graph that intersects the y-axis at (0, 0) and has a slope of –3.

b. Write an equation that represents this linear graph.

Challenge Problem

10. Marshall draws a line representing a proportional relationship. His line, line A, is the graph of a proportional relationship that passes through point W(x1, y1).

The slope of line A is yx

1

1

.

Marshall draws another line—line B. This line is the same as line A, but it is shifted vertically so that it that passes through point (0, 1) instead of the origin, (0, 0). Line B does not pass through point W. Instead, it passes through point Z(x2, y2). Point Z has the same x-value as point W (x2 = x1), but point Z has a different y-value.

Marshall says that the slope of line B is yx

2

1

.

Do you agree with Marshall? Explain why or why not.

LESSON 3: INVESTIGATING GRAPHS

Grade 8 Unit 4: Linear Relationships

EXERCISES

EXERCISES

1. When a line representing a proportional relationship is translated, which attribute of the new line does not change?

A The x-intercept

B The y-intercept

C The slope

D The equation

2. This table shows the coordinates and ratios yx

of four points from the proportional

relationship represented by the equation y = 3x.

x 1 2 3 4

y 3 6 9 12

yx 3 3 3 3

Suppose you translate this line so that it shifts up 2 units along the y-axis and now passes through point (0, 2) instead of the origin, (0, 0).

a. What are the missing values in the table for this new line? Round any decimals to the nearest tenth.

x 1 2 3 4

y

yx

b. How does the table show that the new line no longer represents a proportional relationship?

3. After a line representing a proportional relationship is translated, does the new line still represent a proportional relationship? Explain your reasoning.

LESSON 4: TRANSLATING GRAPHS

Grade 8 Unit 4: Linear Relationships

EXERCISES

4. This set of data shows the distance traveled by a train over 5 hr on a stretch of flat land. The train does not make any stops on that part of the journey.

120

4080

240

160200

280320

0543210 x

y

Dis

tanc

e (m

i)

Time (hr)

What is the slope of the linear equation that models this data? Slope = _____

5. This graph shows the line y x=12

, which represents a proportional relationship,

translated to a new line represented with the equation y x= +12

4 .

2

–2

–4

–6

4

6

8

10

5–5–10 10

y = x + 412

y = x12

x

y

A The line was translated 4 units up.

B The line was translated 4 units down.

C The line was translated 8 units to the left.

D The line was translated 8 units to the right.

E Both lines have the same slope.

LESSON 4: TRANSLATING GRAPHS

Grade 8 Unit 4: Linear Relationships

EXERCISES

6. This graph shows the line y = 3x, which represents a proportional relationship, and a translation of the line. What is the equation of the translated line?

2

–2

–4

–6

4

6

5–5

y = 3x ?

y

x

–8

A y = 3x – 2 B y = 3x + 2

C y = 3x – 6 D y = 3x + 6

7. Translate the line y x=13

down 4 units.

What is the equation of the new line?

8. Translate the line y = –2x + 4 up 3 units.

What is the equation of the new line?

9. This is a translated line. The original line represents a proportional relationship. How was this line translated from the original?

2–2–4–6 4 6 8

2

–2

4

6

8y

x

LESSON 4: TRANSLATING GRAPHS

Grade 8 Unit 4: Linear Relationships

EXERCISES

Challenge Problem

10. Graph five different lines in the same coordinate plane that could that could have the

equation y x b= +12

, where b is any real number.

LESSON 4: TRANSLATING GRAPHS

Grade 8 Unit 4: Linear Relationships

EXERCISES

EXERCISES

1. Here is the graph of a line.

2

–2

–4

–6

–8

5–5 x

y

What are the slope and y-intercept of the line?

A Slope = −34

, y-intercept = 4

B Slope = 34

, y-intercept = –3

C Slope = 43

, y-intercept = 4

D Slope = −43

, y-intercept = –3

LESSON 5: LINEAR EQUATIONS

Grade 8 Unit 4: Linear Relationships

EXERCISES

2. Which equation represents this line?

2

–2

–4

6

8

5–5

4

x

y

A y = –5x + 2

B y = 5x + 2

C y = –2x + 5

D y = 2x + 5

3. Which equation represents this line?

2

–2

–4

–6

–8

5–5 x

y

A y x= −53

3

B y x= −35

3

C y x= −335

D y x= − +35

3

LESSON 5: LINEAR EQUATIONS

Grade 8 Unit 4: Linear Relationships

EXERCISES

4. Which line represents the equation y x= − +12

4

2

–2

6

–6

5–5 x

y

–4

4

line a

line b

line c

line d

A Line a

B Line b

C Line c

D Line d

5. What properties do you know about a line that has an equation in slope-intercept form y = mx + b with a negative value of m and a positive value of b? There may be more than one correct property.

A The slope is positive (going up to the right).

B The slope is negative (going down to the right).

C The y-intercept is positive.

D The y-intercept is negative.

E The line goes through the origin point (0, 0).

6. Graph the equation and find the slope and y-intercept. y x= +13

4

7. Graph the equation and find the slope and y-intercept. y = –4x + 1

8. Graph the equation and find the slope and y-intercept. y x= − −34

2

LESSON 5: LINEAR EQUATIONS

Grade 8 Unit 4: Linear Relationships

EXERCISES

9. A line goes through the points (–2, 2) and (1, 8).

a. Graph this line.b. Find the slope and y-intercept.c. Write the equation of the line in slope-intercept form.

10. A door-to-door salesman gets paid \$1,000 each month to cover his travel expenses and \$10 for each item he sells. The relationship between his monthly income and the number of items he sells can be represented as this equation.

monthly income = 1,000 + (10 • number of items sold)

Explain why this situation does not represent a proportional relationship.

Challenge Problem

11. There is a relationship between two quantities.

• The relationship is linear, but it is not proportional. • The line representing the relationship intersects the y-axis at (0, 3). • The line of this non-proportional relationship is parallel to the line of a proportional relationship that intersects (2, 5).

Find the equation of this linear relationship based on this description.

Graph the equation.

LESSON 5: LINEAR EQUATIONS

Grade 8 Unit 4: Linear Relationships

EXERCISES

EXERCISES

1. Based on what your classmates shared about their projects, revise and make improvements on your project proposal.

2. Make a plan for how you will complete your project by the end of the unit.

3. If you have a partner or are working in a group, make a list of assignments for each person. Be sure that each assignment has a due date.

4. Complete any exercises from this unit that you have not finished.

LESSON 6: LINEAR RELATIONSHIPS PROJECT

Grade 8 Unit 4: Linear Relationships

EXERCISES

EXERCISES

1. For each equation, find the matching table and graph.

y = 2x + 3 Table: _____ A I B II C III D IVGraph: _____ A A B B C C D D

y = 12

x Table: _____ A I B II C III D IVGraph: _____ A A B B C C D D

y = 35

x – 2 Table: _____ A I B II C III D IVGraph: _____ A A B B C C D D

y = –32

x + 1 Table: _____ A I B II C III D IVGraph: _____ A A B B C C D D

Tables I II

x –2 –1 0 1 2 x –2 –1 0 1 2y 4 2.5 1 –0.5 –2 y –3.2 –2.6 –2 –1.4 –0.8

III IVx –2 –1 0 1 2 x –2 –1 0 1 2y –1 –0.5 0 0.5 1 y –1 1 3 5 7

Graphs

A

–2

–4

4

–4 –2 2

2

–2

–4

4

–4

–2

–4

4

–4

2

–4

4

–4 –2

2

x x

x x

2

y y

y y

–2 2 44

–2 2 4

–2

2 4

B

–2

–4

4

–4 –2 2

2

–2

–4

4

–4

–2

–4

4

–4

2

–4

4

–4 –2

2

x x

x x

2

y y

y y

–2 2 44

–2 2 4

–2

2 4

C

–2

–4

4

–4 –2 2

2

–2

–4

4

–4

–2

–4

4

–4

2

–4

4

–4 –2

2

x x

x x

2

y y

y y

–2 2 44

–2 2 4

–2

2 4

D

–2

–4

4

–4 –2 2

2

–2

–4

4

–4

–2

–4

4

–4

2

–4

4

–4 –2

2

x x

x x

2

y y

y y

–2 2 44

–2 2 4

–2

2 4

LESSON 7: REPRESENTATIONS

Grade 8 Unit 4: Linear Relationships

EXERCISES

2. Consider this graph of proportional relationship A and this table of proportional relationship B.

2

4

6

8

10

02 4 6 80 10 x

yProportional Relationship A

x 0 1 2 3 4y 0 2 4 6 8

Proportional Relationship B

Which statement about the rates of change of these two proportional relationships is true?

A Proportional relationship A has a smaller rate of change than B.

B Proportional relationship B has a smaller rate of change than A.

C Both proportional relationships A and B have the same rate of change.

D You cannot tell the rate of change from the data given.

3. Three linear relationships are each described in a different way.

• Relationship A is described by two points: (1, 2) and (3, 6) • Relationship B is described by this table:

x 3 6 9 12 15 18y 7 14 21 28 35 42

• Relationship C is described by this formula: y x=12

a. What is the constant of proportionality for relationship A?b. Using the table, explain how you know that relationship B is a proportional

relationship.c. Which of the three relationships has the greatest unit rate? Explain how you

know.

LESSON 7: REPRESENTATIONS

Grade 8 Unit 4: Linear Relationships

EXERCISES

4. Select all the proportional relationships.

A Relationship A

x –2 0 2 4 6y 5 0 –5 –10 –15

B Relationship B A runner ran a half marathon at a constant rate of 5 miles per hour. He started the race right on the starting line.

C Relationship C

2

–2

–4

4

6

–6

642–6 –4 –2 x

y

D Relationship D A line is represented by the equation y = mx + b, where m = –3 and b = 0.

E Relationship E

2

–2

–4

4

6

–6

642–6 –4 –2 x

y

LESSON 7: REPRESENTATIONS

Grade 8 Unit 4: Linear Relationships

5. Three runners take part in a 10 km race. The three runners run the entire race at constant speeds. Measurements are taken of their respective speeds throughout the race, but the measurements are recorded differently for each runner.

Runner ADistance (km) 0 2 4 6 8 10

Time (min) 0 12 24 36 48 60

Runner B This runner runs the half marathon at a constant rate of 0.2 km per minute. He started the race right on the starting line.

Runner C

2

4

6

8

10

010 20 30 400 50 60 x

y

Time (min)

Dis

tanc

e (k

m)

Who is the fastest runner?

A Runner A

B Runner B

C Runner C

6. This table represents a linear relationship. Create a graph and write an equation that represent the linear relationship.

x –4 –2 0 2 4y –7 –4 –1 2 5

7. Marshall is tracking the height of a tomato plant in his garden. It starts at a height of 5 cm. Each week, it grows another 2 cm. Create a table, a graph, and an equation that represent this situation.

LESSON 7: REPRESENTATIONS

Grade 8 Unit 4: Linear Relationships

EXERCISES

8. Create a table and an equation that represent the linear relationship in this graph.

2

–2

–4

4

6

8

–6

–8

–2–4–6 2 4 6 8 x

y

Challenge Problem

9. Consider these two equations. y = 32 – 4x y = 8(6 – x) Describe a method for finding the intersection point of these equations.

LESSON 7: REPRESENTATIONS

Grade 8 Unit 4: Linear Relationships

EXERCISES

EXERCISES

1. Select all the graphs that have a negative slope.

A

–2

–4

–6

6

2 4 6–2–4–6

4

x

y

2

B

–2

–4

–6

6

2 4 6–2–4–6

4

x

y

2

C

–2

–4

–6

6

2 4 6–2–4–6

4

x

y

2

D

–2

–4

–6

6

2 4 6–2–4–6

4

x

y

2

E

–2

–4

–6

6

2 4 6–2–4–6

4

x

y

2

A If m is positive and b is negative, the line will slope down from left to right.

B If m is negative and b is negative, the line will slope up from left to right.

C If m is positive and b is positive, the line will slope up from left to right.

D If m is negative and b is positive, the line will slope down from left to right.

E If m is positive and b is negative, the line will slope up from left to right.

LESSON 8: NEGATIVE SLOPE

Grade 8 Unit 4: Linear Relationships

EXERCISES

3.

–2

–4

–6

6

2 4 6–2–4–6

4

x

y

2

What is the slope of this graph? Slope = ______

4. Explain why the relationship represented by the graph is not proportional.

–2

–4

–6

6

2 4 6–2–4–6

4

x

y

2

LESSON 8: NEGATIVE SLOPE

Grade 8 Unit 4: Linear Relationships

EXERCISES

5. What are the slope and y-intercept of line a? Explain how you determined these values.

5

10

15

20

05 100 15

y

x

line bline c

line dline a

6.

5

10

15

20

05 100 15

y

x

line bline c

line dline a

What are the slope and y-intercept of line b? Slope = _____ y-intercept = _____

LESSON 8: NEGATIVE SLOPE

Grade 8 Unit 4: Linear Relationships

EXERCISES

7.

5

10

15

20

05 100 15

y

x

line bline c

line dline a

What are the slope and y-intercept of line c? Slope = _____ y-intercept = _____

8.

5

10

15

20

05 100 15

y

x

line bline c

line dline a

What are the slope and y-intercept of line d? Slope = _____ y-intercept = _____

LESSON 8: NEGATIVE SLOPE

Grade 8 Unit 4: Linear Relationships

EXERCISES

9. How can you tell when a slope is negative? Use a right triangle diagram on a line to show what a negative slope looks like.

10. Without graphing, determine which of these equations has a negative slope. Explain how you can find the answers just by looking at the equation.

Equation A: y x= − +34

5

Equation B: y = –4x – 2 Equation C: y = 6x – 3

11. This formula approximates the maximum heart rate for male patients.

y x= −20012

, where y represents the maximum heart rate (beats per minute or

bpm) and x the age of the patient (years).

a. Graph the equation.b. What are the coordinates of the y-intercept?c. What is the slope? What unit of measurement is the slope, knowing that y

represents the maximum heart rate and x represents the age of the patient?d. A man is 40 years old. Calculate his maximum heart rate using this formula.

12. The maximum heart rate for a good athlete under the age of 30 can be determined using a combination of scientific results.

Max heart rate = 217 – age Subtract 3 beats when training for running. Subtract 3 beats when training for rowing. Subtract 5 beats when training for bicycling.

Calculate your max heart rate for running, rowing, and biking.

Challenge Problem

13. Find the two equations that match these rules. Rule 1: Both equations have negative slopes. Rule 2: The lines of these equations intersect at (1, –1). Rule 3: The y-intercept of one of the equations is 3. Rule 4: One of the equations represents a proportional relationship.

LESSON 8: NEGATIVE SLOPE

Grade 8 Unit 4: Linear Relationships

EXERCISES

EXERCISES

1. Look back at your reflection from today’s lesson regarding the next steps in your project. Continue to work on your project focusing on these next steps.

2. Check that you have addressed all the rubric criteria for your project.

If you have not addressed one or more of the rubric criteria, write your plans to update your project to address those aspects of the rubric.

Describe one thing about your project that needs more attention or that has not gone as well as you had expected.

4. Complete any exercises from the unit that you have not finished.

LESSON 9: PROJECT WORK DAY

Grade 8 Unit 4: Linear Relationships

EXERCISES

EXERCISES

1. What is the slope of this line?

A Slope = 15

B Slope = −15

C Slope = –1

D Slope = –10

2. What is the slope of this line?

A Slope = 1

20

B Slope = 1

10

C Slope = 15

D Slope = 1

1

2

–1

3

10 155–5 x

y

5

15

25

30

10

010 2 3 5

20

x

y

4

LESSON 10: THE EFFECT OF SCALE

Grade 8 Unit 4: Linear Relationships

EXERCISES

3. What is the slope-intercept equation for this line?

A y = 6x

B y = x + 3

C y = 6x + 3

D y = 3x

4. What is the slope-intercept equation for this line?

A y = –0.1x + 0.5

B y = –x + 0.5

C y = –x

D y = –0.5x

1

–1–1

–2

3

5

2

1

4

x

y

–0.2

0.4

0.8

1

5–5

0.2

0.6

x

y

LESSON 10: THE EFFECT OF SCALE

Grade 8 Unit 4: Linear Relationships

EXERCISES

5. Look at these two graphs.

–2

4

6

8

10

12

Graph A

Graph B14

5–5

2

4

–2

6

8

2 4–4 –2

x

y

2 x

y

Explain the similarities and differences between the two graphs.

6. The line in this graph seems to make an angle of about 45º with the x-axis, which usually indicates a slope of –1.

Find the equation for this graph. What is the slope? Explain why the slope looks like it should be –1 but is not.

0.5

1

–0.5

–1

–1.5

–2

0.5 1 1.5 x

y

LESSON 10: THE EFFECT OF SCALE

Grade 8 Unit 4: Linear Relationships

EXERCISES

7. Select the graphs that represent the equation y = 2x + 1. There may be more than one correct graph.

A

–10 –5 5 10

–10

–5

5

10y

x

B

–4

–2

–2 2

4

2

y

x

C

–1 –0.5 0.5 1

–1

1y

x

D

–20 –10 10 20

–10

–5

5

10y

x

E

–10

–5

–2 2

10

5

y

x

LESSON 10: THE EFFECT OF SCALE

Grade 8 Unit 4: Linear Relationships

EXERCISES

8. This graph consists of five linear parts. Each part of the graph has a “duration” of 6 sec. There are two different slopes in the graph.

20

15

10

5

Val

ue

Time

AB

C

DE

5 100 15 20 25 300

x

y

Which formula matches each part of the graph?

y x= − 7 y x= − 14 y x= y x= − +16

14 y x= − +16

7

A B C D E

Challenge Problem

9. Line a is a line that intersects points (0, 40) and (30, 60). Line b is a line that intersects points (0, 50) and (30, 60).

Line c is represented by the equation y x= − +13

10

a. Draw the three lines in the same coordinate plane.

b. Determine the slope-intercept equations of lines a and b.

c. Complete this table.

x –60 –30 0 30y (line a)

y (line b)

y (line c)

d. What is the relationship among the three graphs? Explain your thinking.

LESSON 10: THE EFFECT OF SCALE

Grade 8 Unit 4: Linear Relationships

EXERCISESLESSON 11: PUTTING IT TOGETHER 1

EXERCISES

1. Match each graph with the appropriate description.

12345

y

x1 2 3 4 5 6 7 8

678

0

0

12345

y

x1 2 3 4 5 6 7 8

678

0

0

123

y

x1 2 3 4 5 6 7 80

0

123

y

x1 2 3 4 5 6 7 80

0

Linear Linear and Proportional Non-Linear

2. Match each graph with the appropriate description.

12345

y

x1 2 3 4 5 6 7 8

678

0

0

12345

y

x1 2 3 4 5 6 7 8

678

0

0

123

y

x1 2 3 4 5 6 7 80

0

123

y

x1 2 3 4 5 6 7 80

0

Graphs With a Positive Constant Slope

Graphs With a Negative Constant Slope

Graphs With a Slope That is Not Constant

3. Find the slope of the line represented in this table. Write your answer as a decimal.

x –16 –6 2 6 10

y –16 –3.5 6.5 11.5 16.5

Slope = ____

4. Identify which graph represents each of the relationships described.

Grade 8 Unit 4: Linear Relationships

EXERCISES

0.5

0

1.0

y

x0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

1.5

2.0

2.5

3.0

Amount of Rice (lb)

Cost of Three Types of Rice

Cos

t (\$

)

(1.0, 1.29)

(2.0, 2.19)

(3.0, 2.89)

CBA

a. The price of bulk short grain rice is \$0.89 per pound.

A Graph A

B Graph B

C Graph C

b. The price of bulk long grain rice is \$1.09 per pound.

A Graph A

B Graph B

C Graph C

c. The price of packaged rice is \$1.29 for a 1-pound bag, \$2.19 for a 2-pound bag, and \$2.89 for a 3-pound bag.

A Graph A

B Graph B

C Graph C

LESSON 11: PUTTING IT TOGETHER 1

Grade 8 Unit 4: Linear Relationships

EXERCISES

Use this Graph to answer questions 5–7.

0.5

0

1.0

y

x0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

1.5

2.0

2.5

3.0

Amount of Rice (lb)

Cost of Three Types of Rice

Cos

t (\$

)

(1.0, 1.29)

(2.0, 2.19)

(3.0, 2.89)

CBA

5. In which relationship is there a unit price? There may be more than one relationship.

A Graph A B Graph B

C Graph C D None of the graphs

6. Write a formula in the form y = kx to represent each relationship, if possible.

7. The price of bulk short grain rice is \$0.89 per pound.

a. How much would 3 pounds of bulk short grain rice cost? 3 pounds = \$_____

b. How much bulk short grain rice could you buy for \$2.89? Round your answer to the nearest hundredth. \$2.89 = _____ pounds

Challenge Problem

8. Suppose you wanted to show the graph of the conversion from feet to miles. If there are 5,280 feet in 1 mile, what scale would you use for each axis? Sketch the graph.

LESSON 11: PUTTING IT TOGETHER 1

Grade 8 Unit 4: Linear Relationships

EXERCISESLESSON 12: PUTTING IT TOGETHER 2

EXERCISES

Share your work with a classmate. Does your classmate understand what you wrote?

Include the vocabulary word or phrase, a definition, and one or more examples. When appropriate, your example should include a diagram, a picture, or a step-by-step problem-solving approach.

Word or Phrase Definition Examples

proportional linear relationship

A relationship between two variables in which the ratio of the two variables is a constant value

This value is called the constant of proportionality and also describes the slope of the line representing the relationship. The general equation for a proportional relationship can be written as y = kx, where k is the constant of proportionality. A line representing such a relationship always goes through the origin, (0, 0).

2

–2–2–4

–4

2 4 6

4

6

x

y

y = 2x

• non-proportional linear relationship • slope • y-intercept

Grade 8 Unit 4: Linear Relationships

EXERCISES

3. In this unit, you learned about linear relationships. You explored word problems involving real-world situations that could be represented using linear relationships. For example, you looked at modeling running speeds during races.

Reflect back on the everyday life situation you described in problem 3 of the Lesson 1 exercises. Do you still think this situation can be described by a linear relationship?

If so, consider adding more information to your description if necessary. Indicate whether the relationship is proportional, write an equation to model the situation, and draw the corresponding graph.

If not, describe another situation from your everyday life that you could represent with a linear relationship.

Indicate whether the relationship is proportional, write an equation to model this situation, and draw the corresponding graph.

4. Complete any exercises from this unit that you have not finished.

LESSON 12: PUTTING IT TOGETHER 2