TOPIC 1 From Proportions to Linear Relationships

TOPIC 1

From Proportions to Linear Relationships

Where might you see the sign shown? What can you say about the triangle on the sign? What do you think 8% represents?

Lesson 1

Post-Secondary Proportions Representations of Proportional Relationships . . . . . . . . . . . . . . . . . . . . . . . . . . . . M2-7

Lesson 2

Jack and Jill Went Up the Hill Using Similar Triangles to Describe the Steepness of a Line . . . . . . . . . . . . . . . . . M2-23

Lesson 3

Slippery Slopes Exploring Slopes Using Similar Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M2-43

Lesson 4

Up, Down, and All Around Transformations of Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M2-53

Post-

1

Secondary

Proportions

Representations of Proportional Relationships

WARM UP

Determine each equivalent

ratio.

1.

_7__ 16

5

_x__ 48

2.

__t _ 90

5

_5_ 9

3.

_1_0_ p

5

1

4.

250

5

_1_0_0_0_ q

LEARNING GOALS

? Represent proportional relationships with tables, lines, and linear equations.

? Compare graphs of proportional relationships. ? Compare two different proportional relationships

represented in multiple ways.

KEY TERMS

? proportional relationship ? constant of proportionality

You have studied proportional relationships in previous courses. How can you represent and compare proportional relationships using graphs, tables, and equations?

LESSON 1: Post-Secondary Proportions ? M2-7

Getting Started Ratio of Women to Men

Government agencies and civil rights groups monitor enrollment data at universities to ensure that different groups are fully represented. One study focused on the enrollment of women at a certain university. The study found that three out of every five students enrolled were women. Use the findings of the study to write each ratio. 1. the number of enrolled female students to the total number

of students

2. the number of enrolled male students to the total number of students

3. the number of enrolled female students to the number of enrolled male students

4. the number of enrolled male students to the number of enrolled female students

M2-8 ? TOPIC 1: From Proportions to Linear Relationships

ACTIVITY Representing Proportional

1.1 Relationships

Use the findings of the enrollment study to make predictions.

1. Determine the number of enrolled female students for each given total number of enrolled students. Explain your reasoning.

a. 15 total students

b. 250 total students

c. 4000 total students

2. Compare the total number of enrolled students to the number of enrolled male students.

a. Complete the table.

Total Students Enrolled in a University

0 250 6000

Male Students Enrolled in a University

6000

b. Explain how you calculated each value.

Does this represent a proportional relationship?

3. Determine the number of female students if 800 enrolled students are male. Show all work and explain your reasoning.

LESSON 1: Post-Secondary Proportions ? M2-9

NOTES

4. Choose the correct equation to match each description. Then compare the equations.

y 5 _25_x y 5 _25_x

y 5 _32_x

y 5 2x 1 3 y 5 2x 1 5

y 5 _53_x

y 5 _23_x y 5 3x 1 2

a. the number of female students enrolled, y, for x total number of students enrolled

b. the number of male students enrolled, y, for x total number of students enrolled

c. the number of female students enrolled, y, for x male students enrolled

d. the number of male students enrolled, y, for x female students enrolled

e. Describe the similarities and differences in each of the correct equations.

M2-10 ? TOPIC 1: From Proportions to Linear Relationships

5. Create graphs that display each ratio. Then compare the graphs.

a. the total number of female students enrolled, y, with respect to the total number of students enrolled, x

b. the total number of male students enrolled, y, with respect to the total number of students enrolled, x

c. Describe the similarities and differences of the two graphs.

In this lesson, you are studying relationships that are proportional. A proportional relationship is one in which the ratio of the inputs to the outputs is constant. For example, the ratio of women to men at a university is 3 : 2. Proportional relationships are always written in the form y 5 kx, where x represents an input value, y represents an output value, and k represents some constant that is not equal to 0. The constant k is called the constant of proportionality. 6. Identify the constant of proportionality for each relationship

in Question 4.

7. Identify the constant of proportionality, or rate of change, for each graph in Question 5. Then explain how to determine k from a graph.

You will sometimes hear a proportional relationship referred to as a direct variation.

LESSON 1: Post-Secondary Proportions ? M2-11

AC TIVIT Y

1.2

Comparing Ratios and Graphs

Can you determine proportionality or dependence?

Graphs provide a variety of information about relationships between quantities.

1. Examine the lines graphed on the coordinate plane. What can you determine about the relationships between the quantities by inspecting the graph?

y

Students

90

80

70

60

y1

50

40 y2

30

20

10

0 10 20 30 40 50 60 70 80 90

x

Total Number of Students at a University

The lines y1 and y2 each represent a proportional relationship. One line represents the proportional relationship between the number of females enrolled and the total number of students. The other line represents the proportional relationship between the number of males enrolled and the total number of students.

M2-12 ? TOPIC 1: From Proportions to Linear Relationships

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