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Slope as a Rate of Change Tick Tock Toothpick

SUGGESTED LEARNING STRATEGIES: Interactive Word Wall, Use Manipulatives, Create Representations, Debriefing, Look for a Pattern, Think/Pair/Share

ACTIVITY

2.3

My Notes

ACTIVITY 2.3

Slope as a Rate of Change

Activity Focus

· · · · · Slope as a rate of change Pattern recognition Equivalent ratios Linear models Linear models to make predictions · Multiple representations

It takes four identical toothpicks to build a square.

1. How many identical toothpicks does it take to build a line of two adjacent squares, as in this drawing?

Materials

· Toothpicks

It takes seven toothpicks.

Chunking the Activity

#12 #35 #67 #810 #1113 #1417 #1819 #2021 #2223 #24 #25 #2628 #2930

2. Use toothpicks to build lines of three, four, and five adjacent squares. Write the number of toothpicks needed and make drawings of your work in the My Notes space. a. 3 squares 10 toothpicks b. 4 squares 13 toothpicks

© 2010 College Board. All rights reserved.

c. 5 squares 16 toothpicks 3. The number of toothpicks needed, y, depends on the number of squares, x. Use your results from Items 12 to complete this table. Squares (x) Toothpicks (y) 4 1 7 2 10 3 13 4 16 5 4. Describe any patterns you see in the table in Item 3.

Answers may vary. Sample answer: for each additional square, the number of toothpicks increases by three. Unit 2 · Equations, Inequalities, and Linear Relationships

1 Interactive Word Wall The

word adjacent should be added to the Interactive Word Wall at this point. The definition of adjacent is essential to the problem situation so that all students build the same design and use the same data.

2 Use Manipulatives, Create

Representations Make sure that students sketch the designs so that they have them for further reference in the activity.

3 Create Representations,

Debriefing

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4 Look for a Pattern, Think/

Pair/Share, Debriefing Have students share the patterns they observe to enhance everyone's understanding of this situation.

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© 2010 College Board. All rights reserved.

TEACHER TO TEACHER

A As students work on Questions 12, monitor each group. S Students should use toothpicks to build the physical models T models. This will help them to recognize that they create each new square by adding on three toothpicks to the current number of squares. This experience will prove valuable later in the activity when students discuss slope as a rate of change and write an equation to represent this situation.

Unit 2 · Equations, Inequalities, and Linear Relationships

Sample taken from: SpringBoard Middle School Mathematics 2 Annotated Teacher Edition.

109

ACTIVITY 2.3 Continued

5 Debriefing Students should

recognize the pattern in the table from Item 3 as a constant rate of change. Some students may question that the first square is built with four toothpicks. Have them focus on the additional squares being created after this first one. Some students may note that the first square is actually three toothpicks being added to an initial side made using one toothpick.

ACTIVITY 2.3 continued

Slope as a Rate of Change

Tick Tock Toothpick

SUGGESTED LEARNING STRATEGIES: Debriefing, Create Representations, Think/Pair/Share, Group Presentation, Shared Reading

My Notes

5. Use numbers to complete this statement: Each time the number of squares in the line increases by 1 , the number of toothpicks increases by 3 . 6. Use this grid to make a scatter plot of the data from Question 3. Label the axes and title the graph.

y 20 E D 10 B 5 A C

Toothpick Squares

Number of Toothpicks

15

67 Create Representations,

Think/Pair/Share, Group Presentation Students may want to connect the points on their graph when they notice that the points lie on a line. Discussion is needed to point out that this is discrete data. It does not make sense to address part of a square in this situation so the data should be the counting or natural numbers. The points on the graph should not be connected. Having students label the graph and create the scale serves as formative assessment regarding student knowledge of these skills. Note that many students will reverse the independent and dependent variables. This question provides an opportunity for discussion of proper labeling and units. Group presentations may be easier if students use chart paper for their graphs so that they are large enough to see. Suggested Assignment CHECK YOUR UNDERSTANDING p. 118, #1 UNIT 2 PRACTICE pp.144, #17

TEACHER TO TEACHER

0

5 10 Number of Squares

15

x

7. Label the leftmost point on the graph point A. Label the other points, from left to right, points B, C, D, and E. Explain what you notice about the points in your scatter plot.

The points form a line.

8. Describe how to move along the grid between each pair of points. a. From A to B: Go Up b. From B to C: Go Up

3 3

and Go Right and Go Right

1 1

. .

Change in x Change in y

9. Each movement you described in Question 8 can be written units up as a ratio in the form _________. Describe each movement units right units up by writing a ratio in the form _________. units right a. A to B b. B to C

3 __ 1 3 __ 1

You can think of units up as a change in the y direction and of units right as a change in the x direction. A movement up or to the right is positive. 110 SpringBoard® Mathematics with MeaningTM Level 2

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80 Shared Reading, Think/Pair/Share, Debriefing These

questions help students realize that there is a constant rate of change between any two points in a set of linear data, and to recognize slope as the ratio of the "units up (or down)" and the "units right (or left)" and that this ratio is also referred to as rise over run. Students connect the terms "up (or down)" with rise and "right (or left)" with run.

Students must use the S graphs created in Item 6 g to answer Items 810.

110 SpringBoard® Mathematics with MeaningTM Level 2

Sample taken from: SpringBoard Middle School Mathematics 2 Annotated Teacher Edition.

© 2010 College Board. All rights reserved.

© 2010 College Board. All rights reserved.

Slope as a Rate of Change

Tick Tock Toothpick

SUGGESTED LEARNING STRATEGIES: Debriefing, Look for a Pattern, Think/Pair/Share, Visualize, Create Representations

ACTIVITY 2.3

continued

ACTIVITY 2.3 Continued

Differentiating Instruction

My Notes

10. Describe moves along the grid between each pair of points. a. From A to C: units up = Ratio form: 2

6 3 __ = __ 1 9 6

and units right =

2

b. From B to E: units up = Ratio form: 3

9 3 __ = __ 1

and units right =

3

c. From A to E: units up = Ratio form: 4

12 3 ___ = __ 1

12

and units right =

4

11. What do you notice about these ratios?

Answers may vary. Sample answer: The ratios are equivalent.

Some students may need more support in finding the rise and run and writing the ratio for slope. Providing extra practice for these students outside this investigation will help them understand this concept. Look for other situations that have a constant rate of change and have students graph the data and look for the ratio of rise over run that exists on these graphs. Pre-made linear graphs can also be used to find rise, run, and slope.

a Look for a Pattern, Think/

Pair/Share, Debriefing Students should notice that the ratios are equivalent.

WRITING MATH

Some mathematicians call the change in y the rise and the change in x the run. You may see the ratio for slope written rise as ____. run

12. How do the ratios relate to the number of squares and toothpicks? The ratios represent the number of additional

toothpicks needed to add a given number of squares. For 3 example, __ represents three additional toothpicks for one 1 additional square.

© 2010 College Board. All rights reserved.

13. When points on a scatter plot lie on a line, a ratio such change in y units up as _________ or __________ is the slope of that line. units right change in x a. What is the slope of the line in the scatter plot you made for Question 6?

3 The slope of this line is __ or 3. 1

b Visualization, Think/Pair/

Share, Debriefing Students should share their ideas so that the class can learn about the relationships that have been noticed. Some students may find it easier to explain a relationship if they can draw it or use a sketch.

b. What do you think is true about the slope ratios between any two points on a line?

Answers may vary. Sample answer: They are all the same.

14. Use the grid to move from B to A and from E to B. a. Describe the movement from B to A and express the movement from B to A as a ratio.

Answers may vary. Sample answer: Down 3, left 1; -3 slope is ___. -1

c Think/Pair/Share, Debriefing

Students should use the ratios that they wrote in Item 10 to find the slope of the line. Make sure that students notice that no matter what two points are chosen, the ratios are equivalent, meaning the slope is the same between all points on the line.

111

b. Describe the movement from E to B and express the movement from E to B as a ratio.

Answers may vary. Sample answer: Change in y is 9, -3 -9 change in x is -3; slope is ___, or ___. -3 -1 Unit 2 · Equations, Inequalities, and Linear Relationships

d Create Representations,

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© 2010 College Board. All rights reserved.

Look for a Pattern Students may use various ways of completing the table, such as extending the table, using the patterns that they noticed earlier in this activity, or using an equation that they have developed.

Unit 2 · Equations, Inequalities, and Linear Relationships

Sample taken from: SpringBoard Middle School Mathematics 2 Annotated Teacher Edition.

111

ACTIVITY 2.3 Continued

e Think/Pair/Share, Group

Presentation Students should share their work so that they can hear how others found the number of toothpicks needed for 10 and 20 squares.

ACTIVITY 2.3 continued

Slope as a Rate of Change

Tick Tock Toothpick

SUGGESTED LEARNING STRATEGIES: Create Representations, Look for a Pattern, Think/Pair/Share, Group Presentation

My Notes

c. What kind of numbers did you use when you wrote the ratios in Parts a and b?

negative numbers

f Create Representations,

Think/Pair/Share, Group Presentation Have students share the equations that they developed using the graph and table. If students struggle, have them look again at the sketches that they made earlier in the activity. The sketches will help them to see that the equation is 1 plus 3 added on for each additional square.

d. How do the slope ratios compare to the slope ratios you wrote in question 10?

Answers may vary. Sample answer: They are the same.

e. Use this slope ratio to add another point to the graph and explain what the point represents.

Answers may vary. Sample answer: I put a point at (6, 19); it shows that 19 toothpicks are needed to make 6 squares.

15. Suppose that you wanted to find the number of toothpicks needed to build 50 squares. a. Would your graph be helpful for finding that number? Explain your answer.

Explanations may vary. Sample answer: No, I would have to graph 50 points to find the answer.

b. Recall that the variable x represents the number of squares in a line and the variable y represents the number of toothpicks used to build the squares. Complete this table that was made by adding more rows for more squares to the table from Question 3.

c. Explain how you found the number of toothpicks needed for 10 squares and for 20 squares.

Answers may vary. Sample answer: I looked at the table and noticed that y was always 1 more than 3 times x.

16. Use the information from the graph and the table to write an equation that represents this situation.

y = 3x + 1

112 SpringBoard® Mathematics with MeaningTM Level 2

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112 SpringBoard® Mathematics with MeaningTM Level 2

Sample taken from: SpringBoard Middle School Mathematics 2 Annotated Teacher Edition.

© 2010 College Board. All rights reserved.

© 2010 College Board. All rights reserved.

Squares (x) Toothpicks (y) 4 1 7 2 10 3 13 4 16 5 19 6 22 7 25 8 31 10 61 20

Slope as a Rate of Change

Tick Tock Toothpick

SUGGESTED LEARNING STRATEGIES: Work Backward, Think/ Pair/Share, Debriefing, Create Representations, Visualize

ACTIVITY 2.3

continued

ACTIVITY 2.3 Continued

g Work Backward, Think/Pair/

Share, Debriefing Students should share their work with the class, perhaps on a whiteboard, so that they can compare strategies. Students may not catch the change in the format of the problem between part a and parts b and c. Be sure to monitor student work and watch for this. Suggested Assignment

My Notes

17. Use your equation from Item 16 to help answer each part. Show your work or explain how you arrived at your answers.

Methods and explanations may vary. Sample answers given:

a. How many toothpicks are needed to build 50 squares?

y = 3(50) + 1, so 151 toothpicks are needed.

b. How many squares can be built from a total of 94 toothpicks?

94 = 3x + 1, so 31 squares can be built.

c. How many squares can be built from a total of 62 toothpicks?

62 = 3x + 1, so 20 squares can be built and there is one toothpick left but that is not enough to make another square.

CHECK YOUR UNDERSTANDING p. 118, #23 UNIT 2 PRACTICE p. 144, #1820 hi Create Representations, Visualization, Debriefing Students should use the Math Tip as a help in labeling the quadrants. After the quadrants are labeled, students need to plot the points given on the graph. Both of these skills have been taught in previous math courses. This question provides an opportunity to review these skills and also provides you with feedback regarding what students recall about these processes. It may be necessary to take the time now to give additional practice.

Now examine how slope is a part of other linear relationships. Recall that the coordinate plane is divided into four quadrants. 18. Label the quadrants I, II, III, and IV on the coordinate grid. Then plot and label each ordered pair. A(3, -5) B(5, 0) C(-2, 4)

y

10

© 2010 College Board. All rights reserved.

D(-3, -4)

E(0, 5)

Thinking about how you would write the letter "c" might help you to remember how to label the quadrants correctly.

y 5 4 3

II

8 6

I

2

E

1 5 4 3 2 1 1 2 3 4 5 1 2 3 4 5 x

C

4 2 (0, 0)

B

4 6 8 10

10 8

6

4

2 2

2

x

D

4 6

A

IV

III

8 10

19. What ordered pair represents the origin? Plot the origin on the grid.

(0, 0), see above.

Unit 2 · Equations, Inequalities, and Linear Relationships

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© 2010 College Board. All rights reserved.

Unit 2 · Equations, Inequalities, and Linear Relationships

Sample taken from: SpringBoard Middle School Mathematics 2 Annotated Teacher Edition.

113

ACTIVITY 2.3 Continued

Paragraph Shared Reading, Interactive Word Wall The term function needs to be added to the Interactive Word Wall. Create Representations, Think/Pair/Share, Group Presentation Have students work together to complete the table and graph the points. When the groups are done, they should share their tables and graphs with the class as a check for accuracy and to share methods that they used for the problem.

ACTIVITY 2.3 continued

Slope as a Rate of Change

Tick Tock Toothpick

SUGGESTED LEARNING STRATEGIES: Shared Reading, Interactive Word Wall, Create Representations, Think/Pair/ Share, Group Presentation, Visualize, Debriefing

My Notes

ACADEMIC VOCABULARY function

Linear functions can be graphed in the coordinate plane. A function pairs each input value with exactly one output value and can be represented by a table, a graph, or an equation. 1 a. Complete this input/output table for the equation y = __x - 2. 2 Input, Output, Ordered pair x y (x, y) -2 -1 0 2

-3 1 -2 __ 2 -2 -1 1 - __ 2 (-2, -3) 1 (-1, -2 __) 2 (0, -2) (2, -1) 1 (3, -__) 2

j Visualization, Think/

Pair/Share, Debriefing This question is a review of the process of finding rise, run and slope that the students used in Items 10 and 13. This can serve as formative assessment on this skill.

TEACHER TO TEACHER

3

b. Then plot each ordered pair on the coordinate plane. Finally, connect the points with a line.

y

8 6 4

Students may not have S le learned the term y intercep in previous math y-intercept courses. At this point, you may have them use this term or you may say "the point where the line crosses the y-axis."

2 8 6 4 2 2 2 4 6 8 4 6 8

x

A

B

20. Select two points on the graph of the linear equation in Item 20. Label the point on the left A and the point on the right B.

2 units 4 units

Answers may vary. Sample answers given: A(-2, -3); B(2, -1)

a. To move from point A to point B, what is the change in y? b. To move from point A to point B, what is the change in x? c. What is the slope between point A and point B? 114 SpringBoard Mathematics with Meaning Level 2

® TM

1 2 __, or __ 4 2

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114 SpringBoard® Mathematics with MeaningTM Level 2

Sample taken from: SpringBoard Middle School Mathematics 2 Annotated Teacher Edition.

© 2010 College Board. All rights reserved.

© 2010 College Board. All rights reserved.

Slope as a Rate of Change

Tick Tock Toothpick

SUGGESTED LEARNING STRATEGIES: Create Representations, Think/Pair/Share, Debriefing, Quickwrite

ACTIVITY 2.3

continued

ACTIVITY 2.3 Continued

kl Create Representations, Think/Pair/Share, Debriefing These questions allow the students to practice the skills that they have been learning in the activity so far. It also allows you to see what skills the students understand well and what skills they need more practice on. Be sure the students read the Math Tip. Discuss with them what it means specifically and why it may be important to see what is happening around the origin. Showing the students a graph with negative slope, one with positive slope and perhaps a couple of nonlinear graphs will help them to see why choosing a variety of numbers is important in graphing a function. Depending on your district or state standards, you may need to spend more time on domain and range. For now, students may note that the x-values are the domain and the y-values are the range.

My Notes

21. Make an input/output table for the linear equation y = 3x + 1. Choose your own values for x. Include some positive values, negative values, and zero. Then calculate the values of y and graph the equation.

Answers may vary. Sample answer:

x

-2 -1 0 1 2

y

8 6 4 2 8 6 4 2 2 4

© 2010 College Board. All rights reserved.

y

-5 -2 1 4 7

When you choose input values, it is a good idea to choose a negative number that is close to (0, 0) and a small positive number so that you can see what is happening around the origin.

2

4

6

8

x

6 8

22. Find the slope of the linear equation in Item 22.

3 - __, or -3 1

m Quickwrite, Think/Pair/

23. Look back at the previous two equations, tables, and graphs. How do the slopes in these problems relate to the original equations?

Answers may vary. Sample answer: The slope is the same as the coefficient of the variable and the value of y where the line crosses the y-axis is equal to the constant in the equation. This number is called the y-intercept.

Unit 2 · Equations, Inequalities, and Linear Relationships

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© 2010 College Board. All rights reserved.

Share, Debriefing Students need to notice that the slope of the line is the same as the coefficient of the variable in the equation and the y-intercept is the same as the constant in the equation. If students are struggling with this, have them make a graphic organizer that includes the equation from each problem along with its slope and y-intercept. When isolated like this, the pattern may become more apparent. Suggested Assignment CHECK YOUR UNDERSTANDING p. 118, #4 UNIT 2 PRACTICE p. 144, #19

Unit 2 · Equations, Inequalities, and Linear Relationships

Sample taken from: SpringBoard Middle School Mathematics 2 Annotated Teacher Edition.

115

ACTIVITY 2.3 Continued

Paragraph Students will now mirror the first part of this activity with data that has a negative slope. Let them try to complete this section with less guidance than before. If safety issues can be addressed, you may choose to burn a candle in the classroom and collect actual data, before students use the data given in the unit. It is recommended to record the height of a burning birthday candle, measured every 20 seconds, to obtain accurate data.

ACTIVITY 2.3 continued

Slope as a Rate of Change

Tick Tock Toothpick

SUGGESTED LEARNING STRATEGIES: Shared Reading, Marking the Text, Create Representations, Debriefing, Quickwrite

My Notes

Before the invention of clocks, candles were sometimes used to measure time. The height of a burning candle depends on the amount of time that has elapsed since the candle was lighted. In this table, t represents the time in hours that a candle has been burning and h represents the height of the candle in inches. t 3 4 5 6 h 7.5 6 4.5 3

n Shared Reading, Mark the

Text, Create Representations, Debriefing Students should recall the labeling that they did on the scatter plot earlier in the unit and use a similar format for this graph. If the concepts of domain and range are important in your curriculum, you might note that every point in time has an associated height, so in this case, unlike with the toothpicks, the data is continuous.

24. Use this grid to make a scatter plot of the data for the rate at which the candle burns. Label the axes and title the graph.

y Candle Burning 15 Height of Candle 10 5

A

B

C

D x

© 2010 College Board. All rights reserved.

0

5 15 10 Time in Hours

25. From left to right, label the points on your scatter plot A, B, C, and D. Describe moves along the grid between each pair of points and express the moves as a sloop ratio in lowest terms. a. From B to D: change in y =

2 Slope ratio form: _______ -3 ___ -3

o Quickwrite, Debriefing

Encourage students to think about the movement between points in terms of rise and run. If students give answers in the form of units up, down, left or right, question further so that they rephrase the movements using the terms rise and run.

and change in x = 2

b. From C to D: change in y = -1.5 and change in x =

-1.5 _____ 1

1

Slope ratio form:

c. From C to A: change in y =

-2 Slope ratio form: _______ 3 ___

3

and change in x =

-2

116 SpringBoard® Mathematics with MeaningTM Level 2

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116 SpringBoard® Mathematics with MeaningTM Level 2

Sample taken from: SpringBoard Middle School Mathematics 2 Annotated Teacher Edition.

© 2010 College Board. All rights reserved.

Slope as a Rate of Change

Tick Tock Toothpick

SUGGESTED LEARNING STRATEGIES: Quickwrite, Think/Pair/Share, Debriefing, Self/Peer Revision, Visualize, Look for a Pattern, Group Presentation, Create Representations, Work Backward

ACTIVITY 2.3

continued

ACTIVITY 2.3 Continued

pq Quickwrite, Think/Pair/ Share, Debriefing, Self/Peer Edit Students need to use the information they found regarding movement on the grid in Item 26 to write the slope ratio of this graph. They will then relate this slope to the situation being modeled. Encourage students to consider "rate of change."

My Notes

26. What do you notice about the slope ratios in Item 26?

-1.5 They are all the same: _____, or -1.5. 1

27. Interpret the slope for this graph. What does it tell you about the rate at which the candle burns?

Answers may vary. Sample answer: For each hour the candle burns, the height decreases by 1.5 inches.

28. Use your scatter plot from Item 25 and the slope to answer these questions. Explain how you arrived at your answers. a. How tall was the candle before it was lighted?

Explanations may vary. Sample answer: The height of the candle before it was lighted was 12 inches. To determine this height, I used 7.5 + 4.5 = 12.

r Visualization, Look for a

Pattern, Think/Pair/Share, Group Presentation Students should work together using the scatterplot they made and the slope that they found to determine the height of the candle before it was lit and the time it takes the candle to melt completely from the time it is lit. Encourage students to share with the class how they solved these problems.

b. How much time elapses from the time the candle is lighted until it is completely melted?

Explanations may vary. Sample answer: 8 hours will have elapsed from the time the candle was lit until it 1 completely melted. I divided 12 by 1__. 2

© 2010 College Board. All rights reserved.

29. Recall that t represents the time in hours that a candle has been burning and h represents the height of the candle in inches. Write an equation for h in terms of t.

h = -1.5t + 12 or h = 12 - 1.5t

TECHNOLOGY

These equations can be graphed using a graphing calculator.

s Create Representations,

Think/Pair/Share, Group Presentation Have students share the equations that they develop from this situation. Students may struggle with developing this equation as they need to consider the candle height as 12 inches before burning, then decreasing in height at a constant rate of 1.5 inches per minute. Students are more familiar with the idea of situations with an increasing quantity. Doing at least part of this as a group may prove helpful.

30. Use the equation you wrote in Item 30 to answer these questions. Show your work. a. How tall was the candle after it had been burning for 3.5 hours?

The candle was 6.75 inches tall after burning for 3.5 hours.

b. After how many hours was the burning candle 4 inches tall?

The candle was 4 inches tall after burning for about 5.3 hours.

Unit 2 · Equations, Inequalities, and Linear Relationships

117

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© 2010 College Board. All rights reserved.

t Work Backward, Think/Pair/

Share, Group Presentation Students should share their work to check for accuracy and to note different methods of solving. Suggested Assignment CHECK YOUR UNDERSTANDING p. 118, #56 UNIT 2 PRACTICE p. 144, #21

Unit 2 · Equations, Inequalities, and Linear Relationships

Sample taken from: SpringBoard Middle School Mathematics 2 Annotated Teacher Edition.

117

ACTIVITY 2.3 Continued

ACTIVITY 2.3 continued

Slope as a Rate of Change

Tick Tock Toothpick

1.

Input (x) 1 2 3 4 5

Output (y) 2 4 6 8 10

y 10 8 6 4 2 x

CHECK YOUR UNDERSTANDING

Write your answers on notebook paper Show your work. Copy and complete the input/output table Write your answers on notebook paper. or grid 4. 2 paper. Show your work. for the equation y = __x + 1. Then graph 3 the equation. 1. Make an input/output table for y = 2x. Use 1, 2, 3, 4, and 5 as the input, or x, values. Then plot the ordered pairs on a coordinate grid. Input, x Output, y -3 0 3 6 5. Graph y = -3x - 6. Use this input/output table. Input, x -2 -1 0 1 Output, y 0 -3 -6 -9

2. Plot (1, 2) and (4, 3) on a coordinate grid. Then determine the change in y, change in x, and slope of the line between the two points. 3. Find the change in y, change in x, and slope of this graph.

y 10 8 6 4 2 10 8 6 4 2 2 4 6 8 10 2 4 6 8 10 x

10 8 6 4 2

2 4 6 8

2

4

6

8

10

10

3. The rise is 2, the run is 1, and 2 the slope is __ or 2. 1 4. Output values are -1, 1, 3 and 5.

y 10 8 6 4 2 10 8 6 4 2 2 4 6 8 10 y 10 8 6 4 2 10 8 6 4 2 2 4 6 8 10 2 4 6 8 10 x 2 4 6 8 10 x

118 SpringBoard® Mathematics with MeaningTM Level 2

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5.

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6. Answers may vary. Sample answer: Slope might be used in building ramps, stairs, and roofs; in finding elevation of landforms; and in analyzing data to find a pattern in the data points and then write an equation that describes the data.

118 SpringBoard® Mathematics with MeaningTM Level 2

Sample taken from: SpringBoard Middle School Mathematics 2 Annotated Teacher Edition.

© 2010 College Board. All rights reserved.

© 2010 College Board. All rights reserved.

2. The rise is 1, the run is 3, and 1 the slope is __. 3

6. MATHEMATICAL Give some examples of R E F L E C T I O N how slope might be used in real-world applications.

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