8 grade math module 4 topic b

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8th Grade Math

Module 4

Topic B

Name_________________

Period________________

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Essential Question: What is a direct variation?

Direct Variation

A _________________ is the linear function modeled by the equation

y=kx, where k≠ 0. The coefficient k is the _____________________

You can find k using a graph.

Example 1) Finding the Constant of Variation.

Graph the direct variation. Find the constant of variation.

x Y

0 0

2 3

4 6

6 9

Use a point to find the

constant of variation.

The constant of variation is _________________.

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2) Write a direct variation equation that relates the two variables.

Then solve.

a) Suppose y varies directly as x, and y= 14 when x = 7. Find y when

x = 4.

b) Suppose a varies directly as b, and a= 9 when b=3. Find b when a=27.

3) The amount of money m you earn varies directly with h the number

of hours you work. You work for 6h and earn $43.50. Write an

equation to model the situation. How many hours do you need to work

to earn $246.50?

Summary: A direct variation is a ________________modeled by the

equation __________________, where k is the ________________.

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Lesson 10-A Critical Look at Proportional Relationships Essential Questions:

How is average speed found?

Discussion:

Consider the word problem

below.

Paul walks 2 miles in 25 minutes.

How many miles can Paul walk in

137.5 minutes?

How many miles would Paul be

able to walk in 50 minutes?

Explain.



Explain.



In 125 minutes?

How could we determine the

number of miles Paul could walk

in 137.5 minutes?

How many miles, y, can Paul walk

in x minutes?

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Example 1:

Let’s look at another problem where

only a table is provided.

We want to know how many miles, y,

can be traveled in any number of

hours x.

Using our previous work, what should

we do?

What does the equation mean in

terms of the problem presented?

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On your own:

Example 1:

Wesley walks at a constant speed from

his house to school 1.5 miles away. It

took him 25 minutes to get to school.

a. What fraction represents his

constant speed, C?

b. You want to know how many miles he

has walked after 15 minutes. Let y

represent the distance he traveled

after 15 minutes of walking at the

given constant speed. Write a

fraction that represents the constant

speed, C, in terms of y.

c. Write the fractions from parts a.

and b. as a proportion and solve to find

how many miles Wesley walked after

15 minutes.

d. Let y be the distance in miles that

Wesley traveled after x minutes.

Write a linear equation in two

variables that represents how many

miles Wesley walked after x minutes.

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Example 2:

Stefanie drove at a constant speed

from her apartment to her friend’s

house 20 miles away. It took her 45

minutes to reach her destination.

a. What fraction represents her

constant speed, C?

b. What fraction represents constant

speed, C, if it takes her x number of

minutes to get halfway to her friend’s

house?

c. Write a proportion using the

fractions from parts a. and b. to

determine how many minutes it takes

her to get to the halfway point.

d. Write a two-variable equation to

represent how many miles Stefanie can

drive over any time interval.

Summary:

Average speed is found by taking the total distance

traveled in a given time interval, ____________ by

the time interval.

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Lesson 11-Constant Rate Essential Questions:

What is constant rate?

Discussion:

Example 1:

Pauline mows a lawn at a constant rate. Suppose

she mows a 35 square foot lawn in 2.5 minutes.

What area, in square feet, can she mow in 10

minutes? 𝑡 minutes?

Write a two-variable linear equation that

represents the area of lawn, y, Pauline can mow

in ,x, minutes.

𝒕 (time in

minutes)

Linear Equation 𝒚 (area in

square feet)

0

1

2

3

4

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Vocabulary/Examples:

Average Rate:

Over a specific time interval

Constant Rate:

Over any time interval

The change in the value of a quantity divided by the elapsed

time

The constant rate of change with respect to the variable x

of a linear function y=f(x) is the slope of its graph.

What is the difference between

constant rate and average rate?

Average rate is the rate in which something can be done over

specific time interval

Constant rate assuming that the average rate is the same

over any time period.

Example 2:

Water flows at a constant rate

out of a faucet. Suppose the

volume of water that comes out

in three minutes is 10.5 gallons.

How many gallons of water

comes out of the faucet in 𝑡

minutes?

𝒕 (time in

minutes)

Linear Equation V (in gallons)

0

1

2

3

4

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Example 2:

Emily paints at a constant rate. She can paint

32 square feet in 5 minutes. What area, A, in

square feet, can she paint in 𝑡 minutes?

a. Write the linear equation in two

variables that represents the number of

square feet Emily can paint in any given

time interval.

b. Complete the table below. Use a calculator and round your answer to the tenths place.

11

20

18

16

14

12

10

8

6

4

2

0 1 2 3 4 5 6 7 8 9 10

22

Are

a p

ain

ted

in s

qu

are

feet

, A

24

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c. Graph the data on a coordinate plane.

d. About how many square

feet can Emily paint in

2½ minutes? Explain.

Time in minutes, 𝑡

12

16

14

12

10

8

6

4

2

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140

Example 3:

Joseph walks at a constant

speed. He walked to a store that

is one-half mile away in 6

minutes. How many miles, 𝑚, can

he walk in 𝑡 minutes?

b. Complete the table below. Use a calculator and round your answer to the tenths place.

c. Graph the data on a coordinate plane.

Time in minutes, 𝑡

Dis

tan

ce in

mile

s, 𝑚

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d. Joseph’s friend lives 4

miles away from him.

About how long would it

take Joseph to walk to

his friend’s house?

Explain.

Summary:

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Lesson 12-Linear Equations in Two Variables Essential Questions:

What does graphing linear equations looks like?

Opening

Emily tells you that she scored 32

points in a basketball game with only

two- and three-point baskets (no

free throws). How many of each

type of basket did she score? Use

the table below to organize your

work.

Let 𝑥 be the number of two-

pointers and 𝑦 be the number of

three-pointers that Emily scored.

Write an equation to represent the

situation.

Vocabulary:

Linear equation in 2 variables

Standard Form

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Constants

Variables

What is a solution to a linear

equation in two variables?

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Exploratory Challenge

1. Find five solutions for the linear equation 𝑥 + 𝑦 = 3, and plot the solutions as points on a

coordinate plane.

x Linear Equation

y

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2. Find five solutions for the linear equation 2𝑥 − 𝑦 = 10, and plot the solutions as points on a

coordinate plane.

x Linear Equation

y

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Lesson 14-Graph of a Linear Equation-Horizontal and Vertical Lines

Essential Questions:

What are the equations for horizontal and vertical lines?

On Your Own

Exercises:

1. Find at least four solutions to

graph the linear equation 1𝑥 + 2𝑦

= 5.

What shape is the graph of

the linear equation taking?

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2. Find at least four solutions to

graph the linear equation 1𝑥 + 0𝑦 = 5.

3. What was different about the

equations in Exercises 1 and 2?

What effect did this change have

on the graph?

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Discussion: Case 1 𝑎𝑥 + 𝑏𝑦 = 𝑐

Where a, b, and c are constants and a = 1 and b=0

How have we found solutions to

this type of problem in previous

lessons?

What happens when we fix a

value for x?


value for y?

What happens when we pick

another value for y?

Pick 3 more values for y.

Graph the solutions on the

coordinate grid.

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Theorem:

On Your Own

4. Graph the linear equation

𝑥 = −2


𝑥 = 3

6. What will the graph of 𝑥 = 0

look like?

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7. Find at least four solutions

to graph the linear equation

2𝑥 + 1𝑦 = 2.

8. Find at least four solutions

to graph the linear equation

0𝑥 + 1𝑦 = 2

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9. What was different about

the equations in Exercises 7

and 8? What effect did this

change have on the graph?

Discussion: Case 2 𝑎𝑥 + 𝑏𝑦 = 𝑐

Where a, b, and c are constants and a = 0 and b=1

Let’s mimic what we did for

case 1,


value for y?


value for x?

Pick 4 more values for x

Graph the solutions on the

coordinate plane

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Theorem:

On Your Own


𝑦 = −2.


𝑦 = 3

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12. What will the graph of

𝑦 = 0 look like?

Summary Lesson 14:

A linear equation x=C will be a _______________ line passing through the point (C,0).

A linear equation y=C will be a _______________ line passing through the point (0,C)

8 grade math module 4 topic b

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