equations and the 3 coordinate plane - mater …...2012/05/23 · rate of change relation slope...

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Unit

115

3

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Equations and the Coordinate Plane

Essential Questions

Why is it important to consider slope, domain, and range in problem situations?

How can graphs be used to interpret solutions of real world problems?

Unit OverviewIn this unit you will compare and contrast linear and non-linear patterns and write expressions to represent these patterns. You will study functions, domain, range, slope, and forms of linear equations. You will model and solve problems involving systems of equations and you will collect and analyze bivariate data.

Academic VocabularyAdd these words to your vocabulary notebook.

bivariate data continuous data discrete data domain function linear data range rate of change relation

slope solution to a system of linear equations

system of linear equations

trend line x-intercept y-intercept

This unit has three Embedded Assessments—after Activities 3.2, 3. 5, and 3.7. These embedded assessments allow you to demonstrate your understanding of linear relations and linear equations, systems of equations, and applications of bivariate data.

Embedded Assessment 1

Linear Relationships and Functions p. 135


Slopes and Intercepts p. 161


Bivariate Data and Systems p. 175

EMBEDDED ASSESSMENTS

??

115-116_SB_MS3_3-0_SE.indd 115115-116_SB_MS3_3-0_SE.indd 115 4/10/09 7:25:02 PM4/10/09 7:25:02 PM

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116 SpringBoardTM Mathematics with MeaningTM Level 3

Write your answers on notebook paper. Show your work.

1. On the grid below, draw a fi gure that illustrates the meaning of linear.

2. Name fi ve ordered pairs that would be on a graph made from the following table.

input output 5 110 215 3

3. Complete the table below so that the data is linear.

input output12 63 10 4

4. Name 3 ordered pairs that satisfy the equation y = 2x + 1.

5. A line contains the points (2, 5) and (4, 6): a. Where does it cross the x-axis?b. Where does it cross the y-axis?

6. Use the graph below to:a. Plot and label the points R(3, 5) and S(6, 0).b. Give the coordinates of point T.

7. Draw a horizontal line that contains the point (2, 3) and a vertical line that contains (1, 4).

8. Write a ratio that compares the shaded region in the fi gure below to all the regions.

UNIT 3

Getting Ready

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Unit 3 • Equations and the Coordinate Plane 117

My Notes

ACTIVITY

3.1Linear and Non-Linear Patterns Fill It UpSUGGESTED LEARNING STRATEGIES: Predict and Confirm, Use Manipulatives, Create Representations, Look for a Pattern, Quickwrite

Using the template that your teacher gave you, create the cube and cone, which you will use in an experiment. You will be fi lling the cube and the cone by adding 20 beans at a time. Before you begin the experiment, make the following predictions.

1. Predict how many groups of 20 beans you can add to the cube until it is full.

2. Predict how many groups of 20 beans you can add to the cone until it is full.

3. Are your predictions diff erent? Explain how the shape of the fi gure aff ected your predictions.

Do the following experiment for the cube. Each stage of the experiment consists of 3 steps. Each time you complete the three steps you complete a stage.

Step 1: Add 20 beans to the cube.

Step 2: Shake the fi gure gently to allow the beans to settle.

Step 3: Measure the height of the beans in centimeters.

4. Complete the experiment for the cube.

a. Enter the data in the table.

b. Do you see a pattern in the data in the table?

c. Plot the values from the table on the grid.

d. Looking at the graph, what do you notice about the relationship between the stage number and the height of the beans?

Stage # Height of Beans (cm)1234567

Stage Number

Hei

ght o

f Bea

ns

2 41 3 5 76

1

2

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3

4

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118 SpringBoard® Mathematics with MeaningTM Level 3

My Notes

ACTIVITY 3.1continued

Linear and Non-Linear Patterns Fill It UpFill It Up

5. Complete the experiment for the cone.

a. Fill in the table for each stage and plot the points on the grid.

Stage # Height of Beans (cm)1234567

b. Look at the data in the table. What, if any, patterns do you notice?

c. Look at the graph. What patterns do you notice about the relationship between the stage number and the height of the beans?

SUGGESTED LEARNING STRATEGIES: Use Manipulatives, Create Representations, Look for a Pattern, Quickwrite

Stage Number

Hei

ght o

f Bea

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2 41 3 5 76

1

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3

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My NotesTh e following data was collected as beans were added to a cylinder.

Stage # Height of Beans (cm)1 32 63 94 125 156 187 21

6. Plot the points on the grid for the cylinder.

a. What patterns do you notice for the data in the table?

b. Looking at the graph, what patterns do you notice about the relationship between the stage number and the height of the beans?

c. What conjecture can you make about the rate of change of the height of the beans as the stages increase?

SUGGESTED LEARNING STRATEGIES: Create Representations, Look for a Pattern, Quickwrite



Stage Number

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4 821–1 3 5 7 96 10

5

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20

ACADEMIC VOCABULARY

The rate of change in a relationship represents the fraction of vertical change in the output to the horizontal change in the input. The output is often represented by the variable y. The input is often represented by the variable x.


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My NotesMy Notes



Th e following data was collected as beans were added to an irregular polyhedron.

Stage # Height of Beans (cm)1 12 23 54 105 156 187 19

7. Plot the points on the grid for the irregular polyhedron.

a. What patterns do you notice for the data in the table?

b. Looking at the graph, what patterns do you notice about the relationship between the stage number and the height of the beans?

c. What conjecture can you make about the rate of change of the height of the beans as the stage numbers increase?

SUGGESTED LEARNING STRATEGIES: Create Representations, Look for a Pattern, Quickwrite

Stage Number

Hei

ght o

f Bea

ns

4 821–1 3 5 7 96 10

5

10

15

20


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My Notes

SUGGESTED LEARNING STRATEGIES: Look for a Pattern, Quickwrite, Group Discussion, Group Presentation, Create Representations, Simplify a Problem



8. How does the rate of change from the cylinder experiment diff er from the rate of change for the irregular polyhedron experiment?

9. Explain how the shape of the object aff ects the rate of change.

10. If the height of the cylinder and the irregular polyhedron were extended indefi nitely, explain how the height of the beans would change as the stage number increased.

11. Th e graphs and tables below show what happened when the bean experiment was performed with each of the vases shown. Match each vase to a graph and a table. Explain the reasoning behind your choices.

a. b. c.

x y0 01 12 23 2.54 35 3.256 47 6

x y0 01 22 43 64 75 86 8.57 8.75

x y0 01 42 83 124 165 206 247 28


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My Notes




Data is linear if it has a constant rate of change. When you plot the points of linear data on a coordinate plane, they lie on a straight line.

12. Compare and contrast the graphs and tables of the three fi gures in Item 10. Which of the fi gures appeared to encourage a linear relationship? Explain your reasoning.

A person is drinking water from a cylindrical cup using a straw. Th e following graph gives the height of the water at diff erent time intervals.

13. Using the data from the graph fi ll in the table below.

Time (sec) Height (cm) 01020304050

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

Time

Hei

ght o

f Wat

er

20 4010 30 50

21

43

65

87

15

91011121314

ACADEMIC VOCABULARY

linear data


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My Notes



14. Using the table, describe any patterns you see in the height of the water over time.

15. Is the relationship between time and the height of the water linear?

a. Explain using the graph.

b. Explain using the table.

16. What is the rate of change in the water level from 20 seconds to 30 seconds?

17. Connect the data points, and determine what the rate of change in the water level is from 20 seconds to 21 seconds.

18. Predict the height of the water at 25 seconds. How did you make your prediction? If you wanted to look at many diff erent times, would your method still be eff ective?

19. Create an expression that gives the height of the water in terms of the time (t).

20. How long will it take for the water to completely empty out of the cup? Explain using multiple representations.

SUGGESTED LEARNING STRATEGIES: Quickwrite, Create Representations, Group Presentation, Identify a Subtask, Discussion Group


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CHECK YOUR UNDERSTANDING


1. Find the rate of change for the table.

2. Find the rate of change for the table.

3. Determine which of the following tables displays linear data. Explain your reasoning.a. b.

4. Which equation matches the data in the table?a. y = x + 5b. x = y - 5c. y = 7x - 1d. y = 2x + 4

5. Graph the following points and determine if the data is linear.

{(5, -3), (7, -1), (9, 0), (11, 2)} 6. Determine which of the following

expressions displays a linear relationship. Use multiple representations to explain your reasoning.a. 2xb. -2x + 2c. x(4x) d. 4 - 3x

7. MATHEMATICAL R E F L E C T I O N

In this activity, you explored three ways to

represent linear data: in a table, graphically, and with an expression. Which representation of linear data do you understand most easily and why?

x y0 51 92 133 174 215 256 297 33

x y0 45

10 4020 3530 3040 3550 4060 4570 50

x y-5 -2.5-3 -5.5-1 -8.5

1 -11.53 14.55 17.57 20.59 23.5

x y1 62 83 104 12

x y-2 6-1 4 0 2 1 0 2 -2 3 -4 4 -6 5 -8


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My Notes

ACTIVITY

3.2FunctionsWho Am I?SUGGESTED LEARNING STRATEGIES: Create Representations, Quickwrite

Relationships can exist between diff erent sets of information. For example, the pairing of the names of students in your class and their heights is one such relationship.

1. Collect the following information for 15 members of your class.

Student Number First Name Height (cm)

Length of Index Finger (cm)

1

2

3

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6

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15

2. Write the student numbers of 5 students in the class and their height in the following form: (Number, Height).


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My NotesTh e number and height that you wrote for the fi ve students in your class is called an ordered pair. Given the input (x) of a student’s number, you can get an output (y) of that student’s height. In a cartesian coordinate plane, ordered pairs are represented by (x,y).

3. Graph and label the coordinates of a point for each of the fi ve ordered pairs you wrote in Question 2.

1512963

180

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90

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4. Using the information from the table, what other relationships can you create?

5. Using one of the relationships you described in Question 4 that contains numeric values only, create fi ve ordered pairs of students in your class.

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

An ordered pair is two numbers written in a certain order. Most often, the term “ordered pair” will refer to the x and y coordinates of a point on the coordinate plane, which are always written (x, y). The term can also refer to any values paired together according to a specifi c order.

MATH TERMS

FunctionsWho Am I?Who Am I?



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My Notes

A relation is a set of ordered pairs. For example, the pairing of students’ index fi nger length with their height is a relation.Th e set of all the starting values or inputs is called the domain. In a relation, all domain values must be matched with an output value. Th e set of all output values is called the range.

EXAMPLE 1

Find the domain and range of the following set:

{(1,2), (2,4), (4,5), (8,3)}

Step 1: Look at the fi rst number in each pair to identify the domain.Step 2: Look at the second number in each pair to identify the range.Solution: Th e domain is {1, 2, 4, 8}, and the range is {2, 3, 4, 5}

TRY THESE A

Determine the domain and range of the following sets.

a. {(2,4), (2,5), (2,6), (2,7), (2,8)}

b. {(3,12), (4,12), (12,12), (1,8)}

6. From Question 1, what would be the domain of the relation that associates Length of Index Finger to Height?

7. What would be the range of the relation that associates Length of Index Finger to Height?

A function is a special kind of relation. Like a relation, a function must match an input to an output, but functions have the additional restriction that each element in the input can match only one element in the output.

8. Is the relation that associated student numbers and their height a function? Explain your reasoning.

SUGGESTED LEARNING STRATEGIES: Question the Text, Marking the Text, Vocabulary Organizer, Note Taking, Think/Pair/Share, Summarize/Paraphrase/Retell, Quickwrite

Functions Who Am I?Who Am I?

A set is a collection of objects, like points, or a type of number. The symbols { } indicate a set.

MATH TERMS


ACADEMIC VOCABULARY

domainrangerelation


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My NotesMy Notes

EXAMPLE 2

Consider the relation {(1,2), (2,4), (4,5), (8,3)}.

Use mapping to determine if the relation is a function. Explain.

Step 1: Write all domain values in an oval.Step 2: Write all range values in another oval.Step 3: Connect the input values with their output values using

arrows.

1

2

4

8

2

3

4

5

Solution: From the mapping, we can see that every element in the input set is mapped to exactly one element in the output set. Th erefore, the relation is a function.

TRY THESE B

Use mapping to determine if the following are functions. Explain.

a. {(1,3), (1,4), (1,5)}

b. x + 2 for x = {0, 1, 2, 3, 4}

c. x 1 3 5 7y 2 5 3 2

9. Use mapping to determine if the relation that associates height to student number is a function. Explain your reasoning.

SUGGESTED LEARNING STRATEGIES: Marking the Text, Vocabulary Organizer, Note Taking, Create Representations, Quickwrite

Who Am I?Who Am I?Functions ACTIVITY 3.2

continued

One type of representation that helps to determine if a relation is a function is a mapping. Th e illustration to the right is a mapping. Th e particular relation that was mapped is a function. Every input (x-value) is mapped to exactly one output (y-value). Note that a y-value can be associated with more than one x-value. Each input (x) has exactly one output (y).

x y

ACADEMIC VOCABULARY

A function is a special kind of relation in which each element of the domain matches exactly one element of the range.


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My Notes

SUGGESTED LEARNING STRATEGIES: Create Representations, Look for a Pattern, Group Presentation

Another representation that helps to determine if a relation is a function is a table.

EXAMPLE 3

Determine if the following relations are functions.

a. {(1,2), (2,4), (4,2), (8,4)}. b. {(6,2), (6,3), (8,3)}

Step 1: Look at the number of output values for each input.Solution: Th e relation A is a function since each input has only one

output. Th e relation B is not a function because one input, 6, has two diff erent outputs.

TRY THESE C

Determine if the following relations are functions. Explain why they are or are not.

a. b. c. d.

10. Create a table of values that represents a function and a second table that does not represent a function. How would you identify any table that does not represent a function?

Another representation that helps to determine if a relation is a function is graphing. An example on how to use graphing this way is on the next page.

Functions Who Am I?Who Am I?


x y6 26 38 3

x y1 22 44 28 4

x y-3 2

5 52 83 -3

-5 -56 2

-2 58 8

x y2 34 56 78 93 114 135 156 17

x y-5 5-4 4-3 3-2 2-1 1

0 01 12 2

x y-8 9

4 -5-8 8

4 -4-8 7

4 -3-8 6

4 -2


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My Notes

EXAMPLE 4

Determine which of the following graphs represents a function.

Step 1: Plot the ordered pairs.Relation A Relation B

{(-2,-1), (-1,2), (0,0), {(-1,-1), (-1,2), (0,0), (1,-2), (2,2)} (1,2), (1,-2)}

Step 2: Look at the graph to determine if any of the x-values have more than one y-value.

Solution: Relation A is a function since each input has only one output. Relation B is not a function because at least one input has two diff erent outputs.

Relation C Relation Dy = |x| x = y 2

Solution: Relation C is a function since each input has only one output. Relation D is not a function because at least one input has two diff erent outputs.

TRY THESE D

Which of the following graphs represent functions? Explain your reasoning.a. b.



SUGGESTED LEARNING STRATEGIES: Look for a Pattern

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My Notes

SUGGESTED LEARNING STRATEGIES: Look for a Pattern

TRY THESE D (continued)

c. d.

11. When looking at a graph of a relation, how can you determine if it is a function?

Discrete data are data that can only have certain values such as the number of people in your class. On a graph there will be a space between every two possible values. Continuous data can take on any value within a certain range; for example, height. On a graph continuous data and continuous functions have no breaks, holes, or gaps. In the following example, Function A is discrete and Function B is continuous.

Function A Function B



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ACADEMIC VOCABULARY

Data are discrete if there are only a fi nite number of values possible or if there is a space on the number line or on a graph between each 2 possible values.

Data are continuous if there are no breaks in their domain or range or if the graph has no breaks, holes or gaps.


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My NotesFunctions can be represented by expressions.

12. If a function that is represented by the expression x + 5 has inputs labeled x and outputs labeled y, then the diagram below represents the mapping from the input x to the output y.

a. If x = 5 is used as an input in the diagram, what it the output?

b. If x = -3 is used as an input in the diagram, what it the output?

c. If x = 0.03 is used as an input in the diagram, what it the output?

d. If x = - 1 __ 2 is used as an input in the diagram,

what it the output?

e. Is there any limit to the number of input values that can be used with this expression? Explain your reasoning.

f. Is the function discrete or continuous? Explain.

SUGGESTED LEARNING STRATEGIES: Shared Reading, Think/Pair/Share

FunctionsACTIVITY 3.2continued Who Am I?Who Am I?

yx x + 5


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My Notes

SUGGESTED LEARNING STRATEGIES: Activate Prior Knowledge, Debrief

Mr. Walker collected the following data about shoe size and height from fi ve members in his class.

Shoe SizeApproximate Height

(in centimeters)6 1406.5 1447 1488 1569.5 168

13. Consider the relation that associates the shoe size of one of Mr. Walker’s student to his or her approximate height.

a. Use a mapping to determine if the relation is a function. Explain how you arrived at your answer.

b. Draw a graph and explain how it confi rms your answer to part a.

c. An expression that can be used to represent the relation is 8x + 92, where x represents the students’ shoe size. Is there any limit to the input values that can be used with this expression? Explain your reasoning.

d. Is the relation discrete or continuous? Explain your reasoning.




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Use notebook paper to write your answers. Show your work.

Find the domain and range for the data in questions 1 and 2.

1. {(-1,7), (-2,4), (-2,-3), (6,-3)} 2. x y

5 29 31 0

-2 9

Use mapping to determine if the information in Questions 3–5 represent functions.

3. {(-2,2), (-3,4), (-4,5), (-2,6), (-3,7)} 4. x + 5 for x = 3, 5, 7, 9, 11 5. x y

5 36 37 48 4

For Questions 6–8 determine if the relations represent a function. Explain your reasoning.

6. x y-2 5-5 7

8 822 17

-1 320 76

-12 017 22

7.

8.


How do the domain and range of a relation help to

determine if a relation is a function?



5

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Embedded Assessment 1 Linear Relationships and Functions

EDUCATION PAYS

Th e following data was taken from an article, “Education Pays,” by Sandy Baum and Jennifer Ma.

AgeMedian Annual Income with

a High School DiplomaMedian Annual Income

with a Bachelor’s Degree

22 $19,882 $26,547

23 19,882 26,547

24 19,882 26,547

25 27,713 41,593

26 27,713 41,593

27 27,713 41,593

28 27,713 41,593

29 27,713 41,593

30 27,713 41,593

1. Does the data relating age with income for those with a high school diploma represent a linear relationship? Explain your reasoning.

2. Is the median annual income for either the High School Diploma or the Bachelor’s Degree a function of age? Explain your reasoning.

3. Explain why the data in the table is considered discrete.

4. Th ree relations follow. For each relation, explain whether:

• Th e information represents a linear relationship.• Th e information is discrete or continuous.• Th e information represents a function.

a.

b. y = -2x + 3

Use after Activity 3.2.

x y

1 -3

2 -7

3 -11

4 -15

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Embedded Assessment 1 Use after Activity 3.2.

Linear Relationships and FunctionsEDUCATION PAYS

c.

Exemplary Profi cient Emerging

Math Knowledge#1, 2, 4

The student:• Correctly identifi es

the data as linear or nonlinear. (1)

• Correctly identifi es whether or not income is a function of age for both relations. (2)

• Correctly identifi es data as linear or nonlinear, discrete or continuous, and determines if the data represents a function. (4)

The student provides complete and correct identifi cation for two of the items.

The student provides at least two justifi cations, but only one is complete and correct.

Communication#1, 2, 3

The student:• Correctly explains

why the data is nonlinear or nonlinear.

• Correctly explains why income is or is not a function of age for both relations. (2)

• Correctly describes why the data is discrete. (3)

The student gives explanations for the three items, but only two are complete and correct.

The student gives at least two of the required explanations for questions 1, 2, and 3, but they are incomplete and incorrect.

4

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ACTIVITY

My Notes

ACTIVITY

3.3Exploring Slope High Ratio MountainSUGGESTED LEARNING STRATEGIES: Create Representations, Look for a Pattern, Activate Prior Knowledge, Discussion Group

Misty Flipp worked odd jobs all summer long and saved her money to buy passes to the ski lift at the High Ratio Mountain Ski Resort. In August, Misty researched the lift ticket prices and found several options. Since she worked so hard to earn this money, Misty carefully investigated each of her options.

1. Suppose Misty purchased a daily lift ticket each time she goes skiing. Complete the table below for the total cost of the lift tickets.

Number of Days 0 1 2 3 4 5 6

Total Cost of Lift Tickets

2. Use the table to complete the statement: When the number of days in the row increases by , Misty’s cost increases by .

3. Does the data in the table represent a linear relationship? Explain your reasoning.

4. Determine the following: a. Does the data represent a function?

b. Is the data discrete or continuous in this context?

High Ratio MountainSki Resort

Student Lift Ticket prices

Daily Lift Ticket

10-Day Package

$30

$80 upon purchase and$20 per day (up to 10 days)

Unlimited Season Pass $390


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My Notes

Exploring SlopeACTIVITY 3.3continued High Ratio MountainHigh Ratio Mountain

5. State the domain and the range of the data in the table.

6. Plot the data from the table on the grid below.

7. Label the left most point on the graph point A. Label the next 6 points, from left to right, points B, C, D, E, F, and G.

8. Use the graph to complete the statement: When the number of

days increases by , Misty’s cost increases by .

9. Describe how you move along the grid to get from one point to another.

From A to B: Go Up $ and Go Right Days

From B to C: Go Up $ and Go Right Days

From C to D: Go Up $ and Go Right Days

From D to E: Go Up $ and Go Right Days

From E to F: Go Up $ and Go Right Days

From F to G: Go Up $ and Go Right Days

SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge, Create Representations, Look for a Pattern

Days

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Exploring SlopeHigh Ratio MountainHigh Ratio Mountain

SUGGESTED LEARNING STRATEGIES: Marking the Text, Vocabulary Organizer, Think/Pair/Share, Look for a Pattern, Activating Prior Knowledge

10. Th e movements you traced in Question 9 can be written as a ratio, up _____ right . Write ratios in the form up _____ right that describe how to move from:

A to B: B to C:

C to D: D to E:

Another way to think of the movement “Go Up” is as the change in y. Similarly, the movement “Go Right” is the change in x. With this in mind, the ratio, up _____ right , can be rewritten as

change in y __________ change in x . Th e illustration to the right shows the change in

y and the change in x between two points on a line.

11. Find the change in y, the change in x, and write the ratio:

From A to C:

From B to E:

From A to E:

12. What do you notice about these ratios?

13. What are the units of the ratios you created?

14. Explain how the ratios relate to Misty’s situation.

15. Find the change in x, the change in y, and write a ratio:

From B to A:

From E to B:

16. How do these ratios compare to those you found in Question 10?

A ratio is an expression that compares quantities relative to each other.

The rate of change of a relation is a ratio.

MATH TERMS

Change in y

__________ Change in x = 3 __ 5

WRITING MATH

When writing a ratio, you can also represent the relationship by separating each quantity with a colon. For example, the ratio 1:4 is read “one to four.”

Chan

ge in

y

Change in x


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SUGGESTED LEARNING STRATEGIES: Quickwrite, Marking The Text, Vocabulary Organizer, Interactive Word Wall

Th e slope of a line is determined by the ratio change in y

__________ change in x between any two points that lie on the line. Th e slope is the constant rate of change of a line. All linear relationships have a constant rate of change. Th e slope of a line is what determines how steep or fl at it looks on a graph.

Th e y-intercept of a line is the y-coordinate when the x-coordinate is 0. It is the point at which the line crosses the y-axis, (0, y).

17. Let d represent the number of days Misty plans to ski and let C represent Misty’s total cost. Write an equation for C in terms of d.

TRY THESE A

Find the slope and y-intercept for the following.

a. b.

c. d.

e. John is longboarding at a constant rate down the road. If 2 min aft er he leaves his house he is 1000 ft away and at 5 minutes he is 2500 ft from his house what would his average rate of change be?

18. Draw a line that contains the points you plotted in Question 6. Using the graph, fi nd the slope and y-intercept of the line.

x y0 01 2.52 54 10

x y-1 4

0 21 03 -4

CONNECT TO PHYSICAL PHYSICAL ACTIVITYACTIVITY

Longboards are larger than the more trick-oriented skateboards. Longboards are heavier and sturdier than skateboards. Some people even use them instead of bicycles.

READING MATH

The slope of a line,

change in y

__________ change in x , is also expressed

symbolically as ∆y

___ ∆ x .

∆ is the Greek letter, delta. 4

3

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–4 –2 –1–3 21 3 4–1

–2

–3

–4

y

x

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–1–2–3 1 2 3–1

–2

y

x

ACADEMIC VOCABULARY

Slope is the ratio of vertical change to horizontal change or

change in y

__________ change in x · y-intercept


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SUGGESTED LEARNING STRATEGIES: Create Representations, Look for a Pattern, Shared Reading, Interactive Word Wall

19. Suppose Misty purchased the 10-Day Ticket Package that costs $80 plus $20 per day.

a. Complete the table below for the total cost of the lift tickets in the 10-day package for 0 through 6 days. Be sure to include the initial cost of $80.

Number of Days 0 1 2 3 4 5 6

Total Cost of Lift Tickets

b. Explain how you know the data in the table above is linear.

20. Plot the data from the table on the given axes.

21. Draw a line that contains the points you plotted in Item 20.

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SUGGESTED LEARNING STRATEGIES: Quickwrite, Group Presentation, Create Representations, Summarize/Paraphrase/Retell

22. Find the slope and the y-intercept of the line that contains the points in the graph for Question 20, and explain how they relate to Misty’s situation.

23. Compare and contrast the lines associated with the data for the Daily Lift Tickets in Question 6, and the data for the 10-Day Package.

24. Let d represent the number of days Misty plans to ski and let K represent Misty’s cost. Write an equation for K in terms d for Misty’s cost.

25. Although it seemed like a lot of money, Misty thought about the unlimited season pass for $390.

a. First, she compared the season pass to the daily lift tickets at $30 each. How many times would Misty have to go skiing before she would save money with the $390 season pass? Show your work.

b. Next, Misty compared the price of an unlimited season pass to two 10-Day packages that she would use for 20 days of skiing. Which package would be the best buy? Explain your reasoning.


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SUGGESTED LEARNING STRATEGIES: Create Representations, Identify a Subtask, Discussion Group, RAFT

26. If Misty skis the following number of days, which of the three packages should she purchase? Explain why.

a. 6 days

b. 8 days

c. 13 days

d. 16 days

27. Write a persuasive letter to Misty based on your analysis that makes a recommendation of which package she should purchase. Include multiple representations (graphs, tables, and/or equations) to support your reasoning.


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Misty determined that she gets 64 miles on 2 gallons of gas from her car as she drives from her house to go skiing.

1. Create a ratio of Misty’s miles per gallon. 2. Using the ratio you found in Question 1,

determine how far Misty can go on 1 gallon of gas.

3. How many miles could Misty travel on a full tank of 12 gallons of gas?

4. What is the slope of the line shown?a. -4

b. - 3 __ 2

c. - 2 __ 3

d. 3 __ 2 e. 4

5. Find the slope and y-intercept of the following:a. x y

0 13 76 13

b. x y-3 10

0 44 -4

c.

d.

6. If a line has a slope of 3 __ 4 , and contains the point (3, 1), then it must also contain which of the following points? a. (-2, -2)b. (-1, -2)c. (0, -3)d. (2, 2)e. (7, 3)


How does the steepness of a line aff ect the slope

of the line?

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ACTIVITY

3.4Slope Intercept FormThe Leaky Bottle

Owen’s water bottle leaked in his bookbag. He did the following experiment to fi nd how quickly water drains from a small hole placed in a water bottle.

1. Follow the steps below and fi ll in the table.• Get a water bottle and a container to catch the water.• Poke a small hole in the bottom of the water bottle • Ensure the hole is facing down, and open the bottle cap.• Draw a line on the bottle every 5 seconds to mark the

water level.• Aft er the water is drained from the bottle, measure the

heights at each of the times that you marked.

SUGGESTED LEARNING STRATEGIES: Use Manipulatives, Create Representations, Discussion Group, Think/Pair/Share, Activating Prior Knowledge

The y-intercept of a line is the y-value when x = 0. It is the place where the line hits the y-axis.

MATH TERMS

Time in Seconds 0 5 10 15 20 25 30 35 40Height of Water

(cm)

2. Make a scatter plot of the data on the grid below.

3. Does the relationship between time and the height of the water appear to be linear? Explain your reasoning.

4. Is the data you collected continuous or discrete? Explain your reasoning.

5. Draw a line through the points on the scatterplot you created.

a. Find the slope of the line you drew.

b. Find the y-intercept of the line you drew.

Time

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2010 30

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Slope Intercept FormACTIVITY 3.4continued The Leaky BottleThe Leaky Bottle

6. Write an equation that gives the height of the water H given the time t.

7. How does the coeffi cient t in your equation relate to the experiment? Be certain to include appropriate units in your answer.

8. How does the constant term in the equation relate to the experiment? Be certain to include appropriate units in your answer.

9. For each linear equation below:• Make a table of values.• Graph using a diff erent color for each line.• Determine the slope.

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

x y = x-3-2-1

0123

x y = 2x-3-2-1

0123

x y = 4x-2-1.5-1

0122.5

a. y = x b. y = 2x c. y = 4x

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Slope Intercept FormThe Leaky BottleThe Leaky Bottle

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

10. How does the slope you found for each linear equation relate to the coeffi cients of x in the equations for Question 9?


x y = -x-3-2-1

0123

x y = -2x-3-2-1

0123

x y = -4x-2-1.5-1

012

12. How does the slope you found relate to the coeffi cients of x in the equations for Question 11?

13. Write an equation of a line that is:

a. Steeper (increasing) than the ones you graphed in Question 9.

b. Steeper (decreasing) than the ones you graphed in Question 11.

a. y = -x b. y = -2x c. y = -4x

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


15. Compare and contrast the slopes you found in Questions 9, 11, and 14. Refer to the representations you’ve created in your comparisons. What conclusions can you draw about the slope of lines?

16. Write the equation of a line that is steeper than 1 __ 2 but less than one.

a. y = 1 __ 2 x b. y = 1 __ 4 x c. y = 1 __ 5 x

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

17. For each linear equation below:• Make a table of values.• Graph using a diff erent color for each line.• Determine the y-intercept.• Determine the slope.

18. How is the y-intercept related to the constant term in the equations?

19. Identify the slope and y-intercept in each of the following equations.

a. y = 2 __ 3 x + 5 b. y = -x + 1 c. y = 4x - 3

a. y = 2 __ 3 x + 3 b. y = 2 __ 3 x + 6 c. y =

2 __ 3 x - 3

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SUGGESTED LEARNING STRATEGIES: Shared Reading, Interactive Word Wall, Discussion Group, Create Representations

20. Identify and plot the y-intercept of the equation y = 1 __ 2 x + 3 on the coordinate grid and use the slope to fi nd two more points on the line.

21. Sketch a line through the two points.

Equations of the form y = mx + b are written in slope-intercept form, where m is the slope of the line, and b is the y-intercept of the line.

22. Use the y-intercept and the slope to graph the following equations of lines.

a. y = 1 __ 3 x - 2 b. y = -2x + 1 c. y = -3x + 4

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SUGGESTED LEARNING STRATEGIES: Work Backwards, Quickwrite, Group Presentation

23. Owen found that the equation y = -3x + 24 represented the water leaking from his bottle.

a. What is the y-intercept, and what would it represent in this context?

b. What is the slope, and what would it represent in this context?

c. Explain to Owen what would have to happen to the bottle for the slope to change to -4.

24. Explain how to graph the equation y = 2x - 3 without using a table of values.


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1. Find the slope.

x y0 103 8.56 7

Graph the linear equations.

2. y = 3x 3. y = -5x 4. y = -

1 __ 2 x 5. Write an equation of a line that has a slope

that is greater than 1 but less than 2. 6. Write the equation of the line graphed

below.

Graph the linear equations.

7. y = 2x + 4 8. y = -3x + 2 9. y = 2 __ 3 x - 5 10. Write an equation for the line graphed

below.


Explain two ways to graph a linear equation

of the form y = mx + b, where m and b represent any real number.

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ACTIVITY

My Notes


SUGGESTED LEARNING STRATEGIES: Shared Reading, Markingthe Text, Create Representations, Interactiving Word Wall, Vocabulary Organizer, Activating Prior Knowledge

3.5Intercepts, Horizontal and Vertical Lines Drive Time

Matt is driving from Tucson to Flagstaff , Arizona. Aft er driving 20 miles on two-lane roads, he gets on the interstate highway where he will drive 65 mph.

1. Write a linear equation that gives Matt’s distance from Tucson given the number of hours since Matt has been driving on the interstate.

2. What are the slope and y-intercept of the line in Question 1?

Th e x-intercept of a line is the point where the line crosses the x-axis. Its coordinates will be in the form (c, 0) where c is a real number.

EXAMPLE 1

Find the x-intercept on a graph.

Step 1: Find the intersection of the line with the x-axis.

Solution: Th e x-intercept is 2, or the point (2, 0).TRY THESE A

Find the x- and y-intercepts of the graphs below.

a. b.

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ACADEMIC VOCABULARY

x-intercept


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154 SpringBoard® Mathematics with Meaning™ Level 3

Intercepts, Horizontal and Vertical Lines ACTIVITY 3.5continued Drive TimeDrive Time

To fi nd the x-intercept of a line algebraically, use the fact that the intercept lies at the point (c, 0).

EXAMPLE 2

A. Find the x-intercept of the line y = 4x - 24 algebraically.Step 1: Substitute 0 for y. 0 = 4x - 24Step 2: Solve for x. 0 = 4x - 24

+24 +2424 ___ 6 = 4x ___ 4

6 = xSolution: Th e x-intercept is 6. Th e coordinates are (6, 0).

To fi nd the y-intercept of a line algebraically, use the fact that the intercept lies at the point (0, d).

B. Find the y-intercept of the line y = 4x - 24 algebraically.Step 1: Substitute 0 for x. y = 4(0) - 24Step 1: Solve for y. y = -24Solution: Th e y-intercept is -24. Th e coordinates are (0, -24).

TRY THESE B

Find the x- and y-intercepts of the following equations.

a. y = -5x - 10 b. y = 1 __ 4 x + 5 c. y = 0.5x + 2

d. y = 7 + 2x e. 2x + 3y = 9

3. Find the x- and y-intercepts of the equation you found in Question 1 algebraically.

SUGGESTED LEARNING STRATEGIES: Discussion Group, Think/Pair/Share, Quickwrite


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Intercepts, Horizontal and Vertical Lines Drive TimeDrive Time

4. Graph each of the following equations.

a. y = 1 __ 2 x b. y = 1 __ 5 x c. y = 1 ___ 10 x

5. What happens to the graph of the equation of a line as the slope gets closer to zero?

6. Predict what a line with a slope that is equal to zero would look like.

SUGGESTED LEARNING STRATEGIES: Create Representations, Look for a Pattern

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SUGGESTED LEARNING STRATEGIES: Create Representations

7. Fill in the table values for the following equations.

a. y = 0x + 3 b. y = 0x + 6 c. y = 0x - 3

8. Graph the equations from Question 7.

9. Simplify and rewrite the equations in Question 7. What patterns do you notice about equations of lines that have a slope of zero?

x y-3-2-1 0 1 2 3

x y-3-2-1 0 1 2 3

x y-3-2-1 0 1 2 3

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10. Graph each of the following equations.

a. y = 5x b. y = 7x c. y = 10x

11. What happens to the graph of the line as the slope gets larger?

12. On the coordinate grid above, draw a vertical line through the point (5, 0).

13. Using the line you drew in Question 12:

a. Plot and label the coordinates of 4 additional points on the line.

b. Express the slope of the line in the form ∆y

___ ∆ x .

As the slope of a line increases, the line becomes closer to a vertical line. When the denominator of a slope is zero, the slope is said to be undefi ned. Th e slope of a vertical line is undefi ned.

14. Look at the line you drew in drew in Question 12.

a. What do you notice about the y values?

b. What do you notice about the x values?

c. Why do you think the equation of the line is x = 5?

SUGGESTED LEARNING STRATEGIES: Create Representations, Look for a Pattern, Interactive Word Wall, Vocabulary Organizer

A line is horizontal if it is fl at, level, or parallel with the horizon.

A line that is vertical goes straight up. It is perpendicular to the horizon.

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15. Graph the following horizontal and vertical lines.

a. x = -3

b. y = -2

c. x = 6

d. y = 4

16. Write the equations of the following horizontal and vertical lines.

a. b.

SUGGESTED LEARNING STRATEGIES: Look for a Pattern, Create Representations

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EXAMPLE 3

Graph y = 2x + 6 using x- and y-intercepts.

Step 1: Find the x- and y-intercepts algebraically.Find the x-intercept Find the y-intercept

0 = 2x + 6 y = 2(0) + 6

-6 = 2x y = 6

-3 = x

Step 2: Plot the coordinates of the x- and y-intercepts.Step 3: Connect the intercepts with a line.Solution:

TRY THESE C

Graph the equations of the following lines using x and y-intercepts.

a. y = x + 5

b. y = -x - 4

c. y = -3x + 6

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Find the x- and y-intercepts of the following graphs.

1.

2.

For 3–6, fi nd the x- and y-intercepts of the equations.

3. y = 8x + 24 4. y = 3x + 4 5. y = -2x + 5 6. 5x + y = 8

Graph the lines that have the following intercepts.

7. x-intercept: 6y-intercept: -2

8. x-intercept: -3y-intercept: 7

Graph the following lines

9. y = 6 10. x = -4 11. Write the equation of the lines graphed

below.

a.

b.


When would it be easier to graph a line using its

slope and y-intercept than to graph it using its x- and y-intercepts? Explain your reasoning.

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Slopes and InterceptsLINEAR KINDNESS

Ben’s Bells, a community service organization, started hanging ceramic wind chimes randomly in trees, on bike paths, and in parks around the country in 2003 with a written message to simply take one home and pass on the kindness. Th e linear equation y = 2000x + 1000 represents the total number of bells, y, that have been hung by the project given the years, x, since 2003.

1. What is the slope of the line and what does it represent?

2. What is the y-intercept of the line?

3. Graph the equation on the grid below.

4. Write the equation of the line represented by:

a. the data in the table. x y

-2 5

-1 7

0 9

1 11

2 13

4 8521 6 93 7

100020003000400050006000700080009000104

11 •10412 •10413 •10414 •10415 •10416 •10417 •10418 •10419 •10420 •104

10

y

x

161-162_SB_MS3_3-EA2_SE.indd 161161-162_SB_MS3_3-EA2_SE.indd 161 4/4/09 12:19:28 AM4/4/09 12:19:28 AM

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Slopes and InterceptsLINEAR KINDNESS

4. b. the data in the graph.


Math Knowledge#1, 2, 3, 4

The student:• Correctly determines

the slope of the line and what it represents in the problem situation. (1)

• Correctly identifi es the y-intercept of the line. (2)

• Correctly graphs a line given its equation. (3)

• Correctly determines the equation of a line from a table of values. (4a)

• Correctly determines the equation of a line given its graph. (4b)

The student provides complete or correct answers for three or four of the items.

The student provides at least two answers for the fi ve items, but they may be incorrect or incomplete

Representation #3

The student correctly represents the equation as a graph.

The student is unable to produce a graph of the equation.

Communication#1

The student correctly explains what the slope of the line represents in the context of this problem situation. (1)

The student is unable to explain what the slope represents in this context.

8

10

6

4

2

–4–5 –3 –2 –1 1 2 3 4 5–2

–4

–6

–8

–10

y

x


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ACTIVITY

3.6Analyzing Bivariate DataSue SwandiveSUGGESTED LEARNING STRATEGIES: Shared Reading, Role Play

Th e famous bungee jumper, Sue Swandive, is coming to visit your community to promote her new doll line. Th ere will be a bungee competition with the new doll. Th e winning group will get a special prize. Rumor has it that they may get to go bungee jumping with Sue herself.

Th e competition rules are as follows:a. Attach a rock to the back of a Sue Swandive doll.b. Make a bungee cord by connecting rubber bands and attach

it to the doll.c. Drop the doll, with bungee cord attached, from a height

specifi ed by your teacher. Height:_______. d. Th e winning group’s Sue doll will come as close to the

ground as possible without hitting her head.

To help your group predict how long to make the bungee cord for the competition, you will collect data in your classroom fi rst. You will use this data to make a prediction for the number of rubber bands it will take to win the competition. When it is time for your doll to bungee from the height your teacher specifi ed, you will use the prediction your group made.

Begin the classroom part of your experiments as follows:

• With one rubber band attached to the Sue doll, have a student hold the end of the rubber band and the doll’s feet at the 0 position on the tape measure.

• Let go of the doll’s feet but not the bungee cord. • Have your group watch carefully to record the height of

the doll’s head at its lowest position. (It may be helpful to tie the doll’s hair back.)

• Be prepared to repeat each jump a few times to get an accurate measurement.

• Record your fi ndings in the table on the next page.• Add rubber bands and continue to take readings until just

before Sue’s head touches the fl oor.

andunderneathitself.

over thegray,

Pull theblack bandthrough,

How to Tiea Slipknot

3.

2.

1.


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Analyzing Bivariate DataACTIVITY 3.6continued Sue SwandiveSue Swandive


1. Number of Rubber Bands Attached to

the Sue DollLength of Bungee

Jump

1

2

3

4

5

6

7

8

9

10

Th e data you have recorded is an example of bivariate data. Bivariate data is data with two variables.

2. Create a scatter plot of the data on the grid below.

3. Does the data represent a linear relationship? Explain your answers using both the scatterplot and the table.

Number of Rubber Bands

Leng

ht o

f Bun

gee

Jum

p

4 8521 6 93 7

10

20

30

40

50

10

ACADEMIC VOCABULARY

Bivariate data can be written as ordered pairs where each numerical quantity represents measurement information recorded about a particular subject.


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Analyzing Bivariate DataSue SwandiveSue Swandive

SUGGESTED LEARNING STRATEGIES: Quickwrite, Interactive Word Wall, Create Representations, Think/Pair/Share

4. Describe how the length of the bungee jump changes as the number of rubber bands increases.

5. What type of association does the data represent?

A trend line is a line that indicates the general course or tendency of data.

6. Use a tool like spaghetti or a ruler, and place it on the scatter plot in a position that has about the same number of points above and below the line. On the coordinate grid, mark two points that the line passes through. Th ey do not have to be data points.

7. Draw the line that passes through the two points.

8. Write an equation for your trend line in slope intercept form.

9. Explain what the variables in the equation of your line trend line represent.

10. How does the slope relate to the Sue Doll situation?

A collection of data points is said to have a positive association if it has the property that y tends to increase as x increases. It is said to have a negative association if y tends to decrease as x increases. Finally, if the data has no clear relationship, it is said to have no association.

MATH TERMS

10

10Positive Association

10

10Negative Association

10

10No Association

ACADEMIC VOCABULARY

trend line


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SUGGESTED LEARNING STRATEGIES: Quickwrite, Think/Pair/Share, Work Backwards

11. Could you use the equations you wrote to predict the length of the bungee jump with 3.5 rubber bands?

12. Use your equation to predict how many rubber bands it will take to give Sue a maximum bungee jump without touching the ground in the contest.

Th e following data was collected on a group of students. Th ere are many possible ways to pair the data: TV to homework, homework to TV, TV to test scores, test scores to TV, homework to test scores, test scores to homework.

13. Which pairs of data seem to have a positive association? Explain your reasoning.

14. Which pairs of data seem to have a negative association? Explain your reasoning.

15. Which pairs of data seem to have no association? Explain your reasoning.

Hours of TV per Week

32 13 28 19 11 21 15 11 15 12 17 20

Percent of Homework Completed

58 82 65 87 98 78 75 92 75 91 90 81

Test Score 66 85 75 85 100 88 85 90 90 95 85 85


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Analyzing Bivariate DataSue SwandiveSue Swandive

SUGGESTED LEARNING STRATEGIES: Create Representations, Activating Prior Knowledge, Discussion Group

16. For each pair of variables listed below, create a scatter plot with the fi rst variable shown on the x-axis and the second variable on the y-axis. Find a trend line that represents the data.

a. Hours of TV per week versus the percent of homework completed

b. Hours of TV per week versus Test Score

c. Percent of homework done versus Test Score

a.

b.

c.

17. One student came in late to take the test. He had watched 30 hours of TV during the week, but he scored 100 on the test. How would adding this student’s data change the trend line?

2010 30

20

40

60

80

100

2010 30

20

40

60

80

100

20 40 60 80 100

20

40

60

80

100


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SUGGESTED LEARNING STRATEGIES: Discussion Group, Think/Pair/Share

18. Does the data tell you that watching TV causes you to score lower on tests? Explain your reasoning.



Determine if the following graphs have a positive, negative, or no association.

1.

2.

3.

4.

5. Find the equations of the trend lines for any of the questions, 1–4, that had a positive or negative association.


What does the association of a set of

bivariate data indicate about the slope of the trend line?

4 8521 6 93 7

2

4

6

8

10

5 10

24

6

8

12

10

14

15

Graph 1

10 3020

2040

60

80

120

100

140

40

5 2010 15

10

20

30

40

25


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ACTIVITY

3.7Systems of Linear Equations Systems of TreesSUGGESTED LEARNING STRATEGIES: Marking the Text, Summarize/Paraphrase/Retell, Work Backwards, Create Representations, Group Presentation, Quickwrite, Activating Prior Knowledge

Bob decided to plant some trees in his yard. He bought a 10-gallon mesquite tree and a 50-gallon desert willow and planted them in his yard. Aft er one year he was shocked at the growth of both trees, so he measured their heights. Th e mesquite was 5 ft tall, and the desert willow was 8 ft tall. Th e next year he measured again and found the mesquite was 6 ft 6 in. tall, and the desert willow was 8 ft 8 in. tall.

1. List all the numerical information associated with each tree.

2. What information in the paragraph is not needed to fi nd an equation that will predict the height of the trees in a given year?

3. If the trees grew at a constant rate the fi rst two years, how tall were they when Bob planted them?

4. Let M be the height of the mesquite tree in inches. Find a linear equation that represents the height of the tree in a given year, t.

5. Find a linear equation that represents the height, W, in inches of the desert willow in a given year.

6. Could you use the equations you came up with in Questions 4 and 5 to predict the height at 1.5 years?

7. Is the domain continuous or discrete? Explain your reasoning.

8. What is the domain of the functions M and W?

MesquiteLeaves

Desert WillowLeaves


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Systems of Linear Equations Systems of TreesSystems of Trees

9. Use the table below to help explain how the height of the mesquite tree compares to the height of the willow over time.

Year M (inches) W (inches)

10. Graph each of the equations on the following grid and use the graph to determine in what year the mesquite reaches the same height as the desert willow.

11. When the mesquite tree and the desert willow are the same height, what is true about the values of W and M?

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

4 82 6

50

100

150

200

250

10


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12. Write and solve an equation to fi nd the value of t when the mesquite tree and the desert willow are the same height.

13. What is the meaning of your solution in Question 12?

14. How does the solution you found in Question 12 relate to the table and the graph?

One way to categorize equations M and W is as a system of linear equations. Th e solution to a system of linear equations will always be the point where the two lines intersect. Th e value you determined in Question 12 was the solution to this particular system of linear equations.

Systems of linear equations can be solved in many diff erent ways. One way is numerically.

15. Determine which ordered pair in the set {(2,2), (2,3), (2,4),(3,3)} is the solution to the system of linear equations.

{ y = -x + 5 y = x + 1

16. Create a table of values to fi nd the solution to the following system of equations.

{ y 1 = -x - 2

y 2 = 2 __ 3 x + 3

SUGGESTED LEARNING STRATEGIES: Create Representations, Quickwrite, Shared Reading, Interactive Word Wall, Think/Pair/Share



x y 1 y 2

-6

-5

-4

-3

-2

-1

0

1

ACADEMIC VOCABULARY

A system of linear equations is a collection of equations which are all considered simultaneously.

The word linear indicates that there will only be equations of lines in this collection.

A point, or set of points, is the solution to a system of equations in two variables, when it makes both equations true.

WRITING MATH

When working with two or more sets of data in a system of equations, the output variables can be differentiated by writing them with subscripts. For instance, y1 and y2 are used in problem 16.


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My NotesMy Notes



17. Create a table of values to fi nd the solution to the following system of equations (use the My Notes space):

{ y 1 = 5x + 4

y 2 = 2x + 1

18. What problems came up while solving the systems of equations numerically?

Another way to solve systems of linear equations is by graphing.

19. Graph the following system of equations and write out the solution.

{ y = 2x - 4 y = -

1 __ 2 x + 1


{ y = 3x - 4 y = 3x + 2


6

4

2

–10 –2–4–6–8 2 4 6 8 10–2

–4

–6

y

x

6

4

2

–10 –2–4–6–8 42 6 8 10–2

–4

–6

y

x


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SUGGESTED LEARNING STRATEGIES: Think/Pair/Share, Quickwrite, Interactive Word Wall, Note Taking




{ y = 1 __ 3 x + 2

y = -x - 3

22. What problems came up while solving the systems of equations graphically?

You can also solve a system of linear equations algebraically by using the transitive property of equality.

EXAMPLE 1

Solve the following system of equations algebraically. { y = 4x - 1 y = -x + 4

Step 1: Set the equations equal to each other. 4x - 1 = -x + 44x - 1 = -x + 4+x +x

Step2: Solve for x. 5x - 1 = 4 +1 +1 5x = 5 x = 1

Step 3: Substitute x into one of the original equations, and solve for y.

y = 4(1) - 1 y = 3

Step 4: Check your solution using the other equation.

3 � -1 + 4 3 = 3

Solution: Write the solution as an ordered pair. Th e lines intersect at the point (1,3).

The transitive property of equality states:

If a = b and b = c, then a = c.

MATH TERMS

6

4

2

–10 –2–4–6–8 2 4 6 8 10–2

–4

–6

y

x


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TRY THESE A

Solve the following systems of linear equations algebraically.

a. { y = -x + 1 y = -5x - 17 b. { y = 1 __ 2 x + 4

y = - 3 __ 2 x - 4

c. { 2x - 3y = -1

y = x - 1 d. { 4x + y = 6

-5x - y = 21





Determine what information is needed to solve the following problem. Do not solve the problem.

1. A boat on a river traveled 16 miles in 20 minutes going downstream. Th e boat can hold 15 gallons of gas. It takes 30 minutes for the boat to travel back upstream to where it started. Find the speed of the current.

2. Determine which of the following points {(1,-2), (-1,2), (1,2), (-1,-2)} are solutions to the system of equations.

{ 3x - y = 5

x + 4y = -7

3. Create a table of values to fi nd the solution to the system of equations.

{ y = 5x - 3

y = 2x - 6

Solve the systems of equations graphically.

4. { y = x + 2

y = 2x + 3

5. { 2x - 3y = 0

x + 3y = 9

Solve the following systems algebraically.

6. { y = -x + 5

y = x + 3

7. { x + y = 8

y = -x + 4

8. { x - 6y = -3

2x + 3y = 9


Which method(s) you have learned for solving

systems of equations do you prefer? Explain why.


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Bivariate Data and SystemsIS IT HOT IN HERE OR IS IT ME?

Th e weather at places around the world changes daily, sometimes hourly. Th e average temperature over a period of several years is used to study weather trends. Th e average temperatures for two cities, one in the northern hemisphere and one in the southern, are shown below.

Guaymas, MX

Month 1 2 3 4 5 6

Temp ˚F 64 66 68 75 79 87

Johannesburg, SA

Month 1 2 3 4 5 6

Temp ˚F 69 68 66 61 57 51

1. Plot the data from both cities on the grid below. Use dots for Guaymas and boxes for Johannesburg.

Temperature

10

20

30

40

50

60

70

80

90

Month1 2 3 4 5 6

2. Describe the associations for each city.

3. Draw a trend line for each set of data.

4. Find the equations of the trend lines for both cities.

5. Explain what the y-intercept means for each line in this context.

6. Explain what the slope of each line represents in this context.

7. Determine the month in which the temperatures of both cities are the same.

8. Solve the following system of equations algebraically.

{ y = 3x - 2 y = 5x - 8


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Bivariate Data and SystemsIS IT HOT IN HERE OR IS IT ME?

9. Solve the following system of equations graphically.

y = -3 x = 5


Math Knowledge#2, 4, 7, 8, 9

• Correctly identifi es associations of data on the graph (2)

• Correctly determines both equations of trend lines (4)

• Correctly identifi es the correct month when temperatures are the same (7)

• Correctly solves the system of equations (8)

• Correctly solves the system of equations graphically (9)

• Can only identify one of the two associations on the graph

• Determines the correct equation for one of the trend lines

• Identifi es the common temperature but not the month

• Identifi es only one coordinate of the solution to the system

• Graphs the equations but does not provide the correct solution

• Is unable to identify the associations present in the graph

• Is unable to determine the equation of either line

• Does not identify the common temperature or the month

• Is unable to provide a solution to the system

• Does not graph the equations

Problem Solving#2, 8

• Correctly interprets data on a graph to describe both associations (2)

• Correctly uses an appropriate method to solve the system of equations (8)

Solves one of the two items correctly and completely

Is unable to solve either of the two problems correctly

Representation#1, 3, 9

• Create representation of data (1)

• Correctly represents associations with trend lines (3)

• Correctly graphs the system of linear equations and determines the correct solution (9)

Provides appropriate representations for two of the three problems

Provides one of the required representations

Communication#2, 5, 6

• Correctly describes association of data plotted on the graph (2)

• Correctly explains the meaning of the y-intercept for both trend lines (5)

• Correctly explains the meaning of the slope for both of the trend lines (6)

Clearly communicates an explanation for two of the three items

Clearly communicates an explanation for only one of the items


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UNIT 3Practice


ACTIVITY 3.1

1. Looking at the graph, what do you notice about the relationship between x and y?

6

4

2

–10 –2–4–6–8 2 4 6 8 10–2

–4

–6

y

x

2. Looking at the graph, what do you notice about the relationship between x and y?

6

4

2

–10 –2–4–6–8 2 4 6 8 10–2

–4

–6

y

x

Graph the following data sets and identify each as linear or non-linear.

3. {(2,-3), (4,-2), (-2,-5), (0,4)} 4. {(3,0), (2,4), (-1,-4), (-2,1)} 5. {(0,5), (4,-3), (3,-1), (2,1)} 6. Determine which of the following

expressions displays a linear relationship. Use multiple representations to explain your reasoning.

a. √

__ x

b. 1 __ 2 x c. 3 + 0.5x d. x + 7 7. Explain how you can determine if an

expression represents a linear pattern.

ACTIVITY 3.2

Find the domain and range for the data in Questions 8 and 9

8. {(11,2), (2,-14), (-5,13), (58,33)} 9.

Use mapping to determine if the information in Questions 10–12 represent functions.

10. {(-3,4), (-6,1), (6,0), (-1,5), (-6,4)} 11. x - 9 for x = -1, -3, -5, -7, -9 12.

For Questions 13–15 determine if the relations represent functions. Explain your reasoning.

13.

x y

1 -83 -65 -47 -3

x y

1 -83 -65 -47 -3

x y

-1 9-5 0

9 0-1 4

x y

-1 9-5 0

9 0-1 4

x y

0 72 5

-7 06 -50 125 2

-1 41 8

x y

0 72 5

-7 06 -50 125 2

-1 41 8

177-182_SB_MS3_U3PP_SE.indd 177177-182_SB_MS3_U3PP_SE.indd 177 4/4/09 12:21:55 AM4/4/09 12:21:55 AM

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UNIT 3 Practice


14. 4

2

–2–4–6 2 4 6–2

–4

–6

–8

–10

y

x

15. 4

2

2 4 6 8–2 10 12–2

–4

–6

–8

y

x

ACTIVITY 3.3

Veronika rides her bike 24 miles in 2 hours.

16. Create a ratio of Veronika’s miles per hour.

17. Using the ratio you found in Question 16, determine how far Veronika can ride in 5 hours.

18. If Veronika rode her bike for 42 miles at the rate you found, how long was she riding?

19. Find the slope and y intercept of the following:

a.

b.

c.

6

4

2

–1–2–3 1 2 3–2

–4

–6

y

x

d. 6

4

2

–1–2–3 1 2 3–2

–4

–6

y

x

20. If a line with a slope of - 1 __ 2 contains the point (2, 3), then it must also contain which of the following points?

a. (-2, 6) b. (0, 5) c. (1, 2) d. (4, 2) e. (8, 0)

x y

0 -32 -14 1

x y

0 -32 -14 1

x y

-1 50 94 25

x y

-1 50 94 25


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UNIT 3Practice


ACTIVITY 3.4

21. Find the slope.

x y

0 112 7.54 4

Graph the following linear equations.

22. y = 5x 23. y = -4x 24. y = 1 __ 5 x 25. A line with a slope of -2 goes through the

point (3, 5). It also goes through the point (-2, p). What is the value of p?

26. Write the equation of the line graphed below.

6

4

2

–10 –5 5 10–2

–4

–6

y

x

Graph the following linear equations.

27. y = 5x - 2 28. y = 2x + 10 29. y = -25x + 100 30. Write an equation for the line graphed

below.

6

4

2

–10 –5 5 10–2

–4

–6

y

x

ACTIVITY 3.5

Find the x- and y-intercepts of the following graphs.

31. 4

2

–2–4–6 2 4 6 8–2

–4

–6

y

x

32.

6

4

2

–1–2–3–4–5 1 2–2

–4

–6

y

x

Find x- and y-intercepts of the following equations.

33. y = 6x + 30 34. y = 3x + 12 35. y = -7x - 21 36. 9x + y = 72For Questions 37 and 38, graph the line with the given intercepts.

37. x-intercept: 7 y-intercept: 4 38. x-intercept: -5 y-intercept: 3Graph the following equations of lines.

39. x = -7 40. y = 2


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UNIT 3 Practice


For 41–42, write the equation of the line in the graph.

41. 10

5

–10 –5 5 10

–5

–10

y

x

42. 10

5

–10 –5 5 10

–5

–10

y

x

ACTIVITY 3.6

Determine if the graphs for Questions 41 through 44 have a positive, a negative, or no association.

43.

8

10

6

4

2

–10 –5 5 10–2

–4

–6

–8

–10

y

x

44.

8

10

6

4

2

–10 –5 5 10–2

–4

–6

–8

–10

y

x

45.

8

10

6

4

2

–10 –5 5 10–2

–4

–6

–8

–10

y

x

46.

8

10

6

4

2

–10 –5 5 10–2

–4

–6

–8

–10

y

x

47. Find the equations of the trend lines for any of the graphs in 41–44 that had a positive or negative association.


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UNIT 3Practice


ACTIVITY 3.7

Determine what information is relevant to solve the following problem. Do not solve the problem.

48. A monkey weighs 10 pounds. He eats 2 pounds of bananas in a day. How many pounds of bananas will he eat in 1 week?

49. Determine which of the following points {(0,3), (-1,4), (3,0), (4,-1)} are solutions to the system of equations.

{ 3x + 2y = 5 x + 2y = 7

50. Determine which of the following points {(-3,5), (3,-5), (3,5), (-3,-5)} are solutions to the system of equations.

{ 3x - y = -4 2x - 5y = 19

Solve the following systems of equations by graphing.

51. { y = 3x + 2 y = -2x - 8

52. { x + y = -1

2x + 2y = 4

Solve the following systems of equations algebraically.

53. { y = 4 - x

y = x - 2

54. { y = -3x + 6

3x + y = 5

55. { 4x - y = 1

6x + y = -6


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UNIT 3 Reflection


An important aspect of growing as a learner is to take the time to refl ect on your learning. It is important to think about where you started, what you have accomplished, what helped you learn, and how you will apply your new knowledge in the future. Use notebook paper to record your thinking on the following topics and to identify evidence of your learning.

Essential Questions

1. Review the mathematical concepts and your work in this unit before you write thoughtful responses to the questions below. Support your responses with specifi c examples from concepts and activities in the unit.

Why is it important to consider slope, domain, and range in problem situations?

How can graphs be used to interpret solutions of real-world problems?

Academic Vocabulary

2. Look at the following academic vocabulary words: bivariate data range system of linear continuous data rate of change equations discrete data relation trend line domain slope x-intercept function solution of a system of y-intercept linear data linear equations Choose three words and explain your understanding of each word and why

each is important in your study of math.

Self-Evaluation

3. Look through the activities and Embedded Assessments in this unit. Use a table similar to the one below to list three major concepts in this unit and to rate your understanding of each.

Unit Concepts

Is Your Understanding Strong (S) or Weak (W)?

Concept 1

Concept 2

Concept 3

a. What will you do to address each weakness?

b. What strategies or class activities were particularly helpful in learning the concepts you identifi ed as strengths? Give examples to explain.

4. How do the concepts you learned in this unit relate to other math concepts and to the use of mathematics in the real world?

177-182_SB_MS3_U3PP_SE.indd 182177-182_SB_MS3_U3PP_SE.indd 182 4/7/09 1:56:39 PM4/7/09 1:56:39 PM

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Unit 3

Math Standards Review

1. Which situation, when graphed, would be non-linear?A. the amount of water in a tub as it drains

B. the height of a wedding cake as 5-inch layers are added

C. the speed of each car passing through an intersection

D. the weight of a sandbag as shovelfuls of dirt are added

2. What is the slope of the graph y = -2x + 6?

3. Jimmy joined Rhapsody internet music service at a cost of $12.99 per month. He received an MP3 player for a gift and wanted to start downloading songs. Rhapsody charges $0.99 per downloaded song.

Part A: Complete the table for the cost of downloading 2, 3, 4, or 5 songs in a month

# of songs Cost

12345

Part B: List the domain and range of the function from the table. Write an equation that Jimmy can use to deter-mine the cost C of any number of downloads d.

Answer and Explain

Domain:

Range:

1. Ⓐ Ⓑ Ⓒ Ⓓ

2.

○‒ ⊘⊘⊘⊘○• ○• ○• ○• ○• ○•⓪⓪⓪⓪⓪⓪①①①①①①②②②②②②③③③③③③④④④④④④⑤⑤⑤⑤⑤⑤⑥⑥⑥⑥⑥⑥⑦⑦⑦⑦⑦⑦⑧⑧⑧⑧⑧⑧⑨⑨⑨⑨⑨⑨Read

Explain

Solve

183-184_SB_MS3_U3_MSR_SE.indd 183183-184_SB_MS3_U3_MSR_SE.indd 183 4/10/09 11:17:14 AM4/10/09 11:17:14 AM

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Math Standards ReviewUnit 3 (continued)

4. Itmar was 63 inches tall in August at the start of 8th grade. His best friend Megan was 65 inches tall at that time. Itmar grew an average of one-half of an inch each month through May. Megan grew one-fourth of an inch each month through May.

Part A: Write two equations, one to show Itmar’s height at any time during the school year and one to show Megan’s. Use h for height and m for number of months since August.

Answer and Explain

Part B: Graph each student’s height from August to May on this graph.

Month

A S O N D J F M A M

Hei

ght i

n In

ches

64

65

66

68

70

Part C: Will Itmar be taller than Megan by the end of 8th grade in May? If so, describe the point where their heights are the same.

Solve and Explain

Read

Explain

Solve

183-184_SB_MS3_U3_MSR_SE.indd 184183-184_SB_MS3_U3_MSR_SE.indd 184 4/15/09 12:45:38 AM4/15/09 12:45:38 AM

equations and the 3 coordinate plane - mater …...2012/05/23 · rate of change relation slope...

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