Unit 3 Plan - Grade 8

Lesson 1

Understanding Proportional Relationships

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Turtle Race (1 problem)

This graph represents the positions of two turtles in a race.

On the same axes, draw a line for a third turtle that is going half as fast as the turtle described by line $g$ .
Explain how your line shows that the turtle is going half as fast.

graph. horizontal axis, distance traveled in centimeters, scale 0 to 18, by 2's. vertical axis, elapsed time in seconds, scale 0 to 6, by 1's. 2 lines graphed, labeled g and h.

Show Solution

A line through $(0,0)$ , $(1,1)$ , $(2,2)$ , etc.
Sample reasoning: After 2 seconds, the turtle described by line $g$ moved 4 cm, while the third turtle moved only 2 cm. This third turtle covers half the distance in the same amount of time.

Lesson 2

Graphs of Proportional Relationships

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Different Axes (1 problem)

Which one of these relationships is different from the other three? Explain how you know.

Graph A of 4 graphs labeled A, B, C, D. — A

Graph B of 4 graphs labeled A, B, C, D. — B

Graph C of 4 graphs labeled A, B, C, D. — C

Show Solution

Graph B is a representation of $y=5.5x$ or $\frac{y}{x}=\frac{55}{10}$ while Graphs A, C, and D are all representations of $y=5x$ or $\frac{y}{x}=5$ .

Lesson 3

Representing Proportional Relationships

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Graph the Relationship (1 problem)

Sketch a graph that shows the relationship between grams of honey and grams of salt needed for a bakery recipe. Show on the graph how much honey is needed for 70 grams of salt.

salt (grams)	honey (grams)
10	14
25	35

Show Solution

Possible graph: Axes labeled from 0 to 140, with grams of salt on the horizontal axis and grams of honey on the vertical. Coordinate points may include $(0,0)$ , $(10,14)$ , and $(70, 98)$ .

Lesson 4

Comparing Proportional Relationships

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Different Salt Mixtures (1 problem)

Here are recipes for two mixtures of salt and water that taste different.

Information about Mixture A is shown in the table.

Mixture B can be described by the equation $y=2.5x$ , where $x$ is the number of teaspoons of salt, and $y$ is the number of cups of water.

salt (teaspoons)	water (cups)
4	5
7	$8\frac34$
9	$11\frac14$

If you used 10 cups of water, which mixture would use more salt? How much more? Explain or show your reasoning.
Which mixture tastes saltier? Explain your reasoning.

Show Solution

Mixture A uses 4 more teaspoons of salt than Mixture B. Sample reasoning: Mixture A would use 8 teaspoons of salt because I can double the row with 4 and 5 to get 8 and 10. Mixture B would use 4 teaspoons of salt because $10=2.5(4)$ .
Mixture A tastes saltier because it uses more salt for the same amount of water. Sample reasoning: Mixture A uses 8 teaspoons of salt for 10 cups of water and Mixture B only uses 4 teaspoons of salt for the same amount of water.

Section A Check

Section A Checkpoint

Problem 1

Jada and Noah count the number of steps they take to walk a set distance. To walk the same distance, Jada takes 8 steps while Noah takes 10 steps. Then they find that when Noah takes 15 steps, Jada takes 12 steps.

Write an equation that represents this situation. Use $n$ to represent the number of steps Noah takes and $j$ to represent the number of steps Jada takes.
Create a graph that represents this situation and can be used to determine how many steps Noah will take if Jada takes 100 steps.

Show Solution

$n=\frac54j$ or $j=\frac45n$ (or equivalent)
Sample graph. If Jada takes 100 steps, Noah will take 125.

Problem 2

Diego and Priya are filling buckets of the same size with water from two different hoses.

Diego can fill 20 buckets in 5 minutes.

The equation $y=3x$ describes how Priya can fill buckets, where $x$ represents the time in minutes, and $y$ represents the total number of buckets she has filled.

Who is filling buckets faster? Explain your reasoning.

Show Solution

Diego is filling buckets faster. Sample reasoning: Diego can fill 20 buckets in 5 minutes but Priya can only fill 15 buckets in 5 minutes.

Lesson 5

Introduction to Linear Relationships

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Stacking More Cups (1 problem)

A different style of cup is stacked. The graph shows the height of the stack in centimeters for different numbers of cups. How much does each cup after the first add to the height of the stack? Explain your reasoning.

Graph of line. Points plotted on line include 3 comma 5 and 5 tenths and 8 comma 8.

Show Solution

Each cup after the first adds 0.5 centimeters (or equivalent). Since 5 cups add 2.5 centimeters to the height of the stack, each cup adds 0.5 centimeters.

Lesson 6

More Linear Relationships

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Savings (1 problem)

The graph shows the savings in Andre’s bank account.

Calculate the slope and explain what it represents in this situation.
Determine the vertical intercept and explain what it represents in this situation.

Graph, horizontal axis, time in weeks, scale 0 to 10, by 1's. vertical axis, savings in dollars, scale 0 to 80, by 20's.

Show Solution

The slope is 5 and means that Andre saves 5 dollars every week.
The vertical intercept is 40 and means that Andre initially had 40 dollars in his bank account.

Lesson 7

Representations of Linear Relationships

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Filling a Tank (1 problem)

The graph shows the relationship between the gallons of water in a tank and time as it is filling.

What is the slope and what does it mean in this situation?
What is the vertical intercept and what does it mean in this situation?

Show Solution

The slope is 6 and means that 6 gallons of water are added to the tank each minute.
The vertical intercept is 20 and means that the tank already had 20 gallons in it before it started filling.

Lesson 8

Translating to $y=mx+b$

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Similarities and Differences in Two Lines (1 problem)

Describe how the graph of $y=2x$ is the same and different from the graph of $y=2x-7$ .

Show Solution

Sample responses:

Both lines have a slope of 2, but one line has a $y$ -intercept of 0 while the other has a $y$ -intercept at -7.
Both lines have the same slope but different vertical intercepts.
The lines are parallel to each other, with one line being a translation of the other line.
Both lines have the same rate of change, but cross the $y$ -axis (or $x$ -axis) at different points.

Section B Check

Section B Checkpoint

Problem 1

A new park is planted with grass seed. Line $\ell$ shows the height of the grass every week:

A field nearby already has grass that is currently 3 inches tall. This grass grows 4 inches taller every week. Graph the height of this grass on the same set of axes as the grass just planted in the new park and label it line $k$ .
Write an equation that represents the grass growing in the field where $w$ is the number of weeks and $h$ is the height of the grass in inches.
Which set of grass is growing faster? Explain how you know.

Show Solution

See graph
$h=4w+3$ (or equivalent)
Both sets of grass are growing at the same rate. Sample reasoning: The slope of line $\ell$ is 4, which means it is growing 4 inches every week. This is the same for the grass in the nearby field. The lines are parallel, so they have the same slope. This also means they have the same rate of change.

Lesson 9

Slopes Don't Have to Be Positive

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The Slopes of Graphs (1 problem)

Match each graph with the situation that could describe the line.

A tank is set up to collect rainwater. During a storm, 3 gallons of rainwater is collected each minute.
After the storm, no water is used and no additional water is collected.
Several days later, rainwater from the tank is used to irrigate a garden at a rate of 8 gallons of water per minute.

Show Solution

Graph A: Situation 3

Graph B: Situation 2

Graph C: Situation 1

Lesson 10

Calculating Slope

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Different Slopes (1 problem)

Find the slope of the line that passes through each pair of points.

$(0,5)$ and $(8,2)$
$(2,\text-1)$ and $(6,1)$
$(\text-3, \text-2)$ and $(\text-1,\text-5)$

Show Solution

$\text-\frac38$ (or equivalent)
$\frac12$ (or equivalent)
$\text-\frac32$ (or equivalent)

Lesson 11

Line Designs

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Another Way (1 problem)

Draw the line that has a slope of $\text-\frac13$ and passes through the point $(5,4)$ .
Describe this same line in a different way.

coordinate grid, horizontal axis 0 to 11, by 1's. vertical axis -0 to 11, by 1's

Show Solution

Sample responses:
- a line with slope $\text-\frac13$ and passing through the point $(8,3)$
- a line with slope $\text-\frac13$ and vertical intercept at $(5\frac23,0)$
- the line of $y=5\frac23-\frac13x$
- the line that goes through the points $(2,5)$ and $(5,4)$

Lesson 12

Equations of All Kinds of Lines

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Five Lines (1 problem)

Here are 5 lines in the coordinate plane:

Write equations for lines $a$ , $b$ , $c$ , $d$ , and $e$ .

Show Solution

line $a$ : $x=\text-4$ , line $b$ : $x=4$ , line $c$ : $y=4$ , line $d$ : $y=\text-2$ , line $e$ : $y=\frac {\text{-}3}{4} x +1$ (or equivalent)

Section C Check

Section C Checkpoint

Problem 1

The graph shows the altitude of a helicopter.

Write an equation that represents the situation where $x$ is the time in minutes and $y$ is the altitude of the helicopter in feet.
What does the slope of the line represent in this situation?
A different helicopter is flying at a constant altitude of 7,000 feet. Draw a line on the same coordinate plane showing the altitude of this helicopter after $x$ minutes.
What is the slope of this line and why does it make sense in this situation?

Show Solution

$y=10,000-800x$ (or equivalent)

Sample response: The slope of the line represents how many feet the helicopter is descending, or going down, every minute.

The slope of this line is 0. This makes sense because the helicopter is not going up or down in altitude.

Lesson 13

Solutions to Linear Equations

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Identify the Points (1 problem)

Select all the coordinates that represent a point on the graph of the line $x-9y=12$ .

$(12,0)$
$(0,12)$
$(3,\text-1)$
$\left(0,\text-\frac43\right)$
$(\text-3,1)$

Show Solution

A, C, D

Lesson 14

Unit 3 Linear Relationships — Unit Plan