Unit 3 Linear Relationships — Unit Plan

Title

Assessment

Lesson 1

Understanding Proportional Relationships

Turtle Race

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

Different Axes

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

Graph the Relationship

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

Different Salt Mixtures

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

Stacking More Cups

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

Savings

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

Filling a Tank

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 $y=mx+b$

Similarities and Differences in Two Lines

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

The Slopes of Graphs

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

Different Slopes

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

Another Way

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

Five Lines

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

Identify the Points

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

More Solutions to Linear Equations

Intercepted

Does the graph of the line for $3x-y=\text-6$ pass through the points $(\text-2,0)$ and $(0,\text-6)$ ? Explain your reasoning.

Show Solution

The graph passes through the point

(\text-2,0)

but not through the point

(0,\text-6)

. Sample reasoning: Since

3(\text-2)-0=\text-6

, the point

(\text-2,0)

is a solution to the equation and will lie on the line. Since

3(0)-(\text-6)=6

, and not

\text-6

, the point

(0,\text-6)

is not a solution and will not lie on the line.

Section D Check

Section D Checkpoint

Problem 1

This graph shows the line represented by the equation $y=8-3x$ .

Select all points that lie on this line.

(0,8)

(6,0)

(3,\text-1)

(20,\text-52)

(\frac{\text-4}{3}, 12)

(\text-5, \text-7)

Show Solution

A, C, D, E

Lesson 15

Using Linear Relations to Solve Problems

No cool-down

Unit 3 Assessment

End-of-Unit Assessment

Problem 1

Select all the points that are on the graph of the line $2x + 4y = 20$ .

$(0,5)$

$(0,10)$

$(1,2)$

$(4,2)$

$(5,0)$

$(10,0)$

Show Solution

A, F

Problem 2

For two weeks, the highest temperature each day was recorded in four different cities, represented by the lines $\ell$ , $m$ , $n$ , and $p$ . Which statement is true?

graph. horizontal axis, time passed in days. vertical axis, temperature in degrees. 4 lines labeled l, m, n, p.

The high temperature in the city represented by line $\ell$ increased as time passed.

The high temperature in the city represented by line $m$ decreased steadily.

Initially, the high temperature was warmer in the city represented by line $p$ than in the city represented by line $m$ .

The high temperature in the city represented by line $p$ increased faster than the high temperature in the city represented by line $n$ .

Show Solution

The high temperature in the city represented by line $m$ decreased steadily.

Problem 3

Jada earns twice as much money per hour as Diego. Which graph best represents this scenario?

graph with no grid. Horizontal axis money earned in dollars. Vertical axis time worked in hours. 2 lines starting at origin. — A

graph with no grid. Horizontal axis time worked in hours. Vertical axis money earned in dollars. 2 lines starting at origin. — B

A.Graph A

B.Graph B

C.Graph C

D.Graph D

Show Solution

Graph B

Problem 4

Write an equation for each line.

Show Solution

line $\ell$ : $y = 4$ (or equivalent), line $m$ : $y = 4 - 2x$ (or equivalent), line $n$ : $y = x - 1$ (or equivalent), line $p$ : $x = \text -4$

Problem 5

Three runners are training for a race. One day, they all run a lap around a track, each at their own constant speed.

The graph shows the distance in meters that Runner #1 runs with respect to the time in seconds.

graph. horizontal axis, time in seconds, scale = 0 to 70 by 10's. vertical axis, distance in meters, scale 0 to 400, by 50's. line passing through origin and 50 comma 200.

Runner #3’s information is in the table:

time (seconds)	distance (meters)
9	45
11	55
25	125
45	225
60	300

The equation that relates Runner #2’s distance (in meters) with time (in seconds) is $d=6.5t$ .

Which of the three runners runs the fastest? Explain your reasoning.

Show Solution

Runner #2 runs the fastest. Sample reasoning: Using the points $(0,0)$ and $(50,200)$ from Runner #1’s graph, the slope is 4, showing that they run 4 meters every second. Runner #2’s equation shows that they run 6.5 meters every second. Within the table, the unit rate for Runner #3 is 5 meters per second because $45\div9=5$ .

Since Runner #2 travels farther every second, Runner #2 is the fastest.

Minimal Tier 1 response:

Work is complete and correct.
Sample: Runner #1 goes at 4 meters per second, Runner #2 goes at 6.5 meters per second, and Runner #3 goes at 5 meters per second. Runner #2 is the fastest because they travel the greatest distance in 1 second.

Tier 2 response:

Work shows general conceptual understanding and mastery, with some errors.
Sample errors: Work contains correct unit rates for all three runners but concludes that runner #1 or #3 is the fastest or does not name a fastest runner; one unit rate is incorrect (possibly with an incorrect fastest runner identified as a consequence); insufficient explanation of work.

Tier 3 response:

Significant errors in work demonstrate lack of conceptual understanding or mastery.
Sample errors: Two or more incorrect unit rates; the correct runner is identified but with no justification; response to the question is not based on unit rates or on similar methods, such as calculating which runner has gone the farthest after 10 miles.

Problem 6

A store is selling notebooks for $1.00 and pencils for $0.25. Jada has $10.00 to spend on school supplies.

Complete the table showing three ways Jada can spend all $10.00 on notebooks and pencils.

number of notebooks ( $n$ ) number of pencils ( $p$ )

6

4

2
Write an equation describing the number of notebooks $n$ , and pencils $p$ that Jada can buy for $10.00.
Draw a graph of the solutions to your equation.
Notebooks still cost $1.00 and pencils still cost $0.25, but now Jada has $15 to spend on supplies. How would a graph representing this new situation be the same and different from the graph representing when Jada had $10 to spend?

Show Solution

number of notebooks ( $n$ ) number of pencils ( $p$ )

6 16

9 4

2 32
$n+0.25p=10$ (or equivalent)
Sample graph:
Sample responses: The two graphs would have the same slope but different vertical intercepts. The graph of the new line would be parallel to the graph of the original line.

Minimal Tier 1 response:

Work is complete and correct.
The graph may be a continuous line, or it may consist only of points representing whole numbers of notebooks and pencils.
The graph may have the axes and labels reversed, where the number of pencils is on the horizontal axis, and the number of notebooks is on the vertical axis.
Sample: See solution on graph.

Tier 2 response:

Work shows general conceptual understanding and mastery, with some errors.
Sample errors: Graph contains only the three points from the table; scale chosen does not allow for all relevant data to be displayed; reasonable work in part 1 but results in an incorrect equation.
Acceptable errors: Equation and graph are correct based on an incorrect proportional relationship in the table; graph is correct based on an incorrect equation.

Tier 3 response:

Significant errors in work demonstrate lack of conceptual understanding or mastery.
Sample errors: Work does not factor in the $10 constraint; equation is not linear or does not make sense in this situation; incorrect answers in 2 or more representations.

Tier 4 response:

Work includes major errors or omissions that demonstrate a lack of conceptual understanding and mastery.
Sample errors: Does not come up with an equation or graph; omission of or incorrect work in three or more problem parts.

Problem 7

Han has a music playlist and each day he adds more songs to his list. The equation $y=4x+20$ describes Han’s playlist, where $x$ is the number of days, and $y$ is the total number of songs.

Tyler also has a music playlist. A graph representing the number of songs on Tyler’s playlist each day has a vertical intercept at $(0,12)$ and is parallel to the graph describing Han’s music playlist.

Tyler says that in 3 days he will have more songs on his playlist than Han will have on his. Do you agree or disagree with Tyler? Explain your reasoning.

Show Solution

I disagree with Tyler. Sample reasoning: The equation that represents Han’s music playlist has a slope of 4 and a $y$ -intercept of 20, which means that Han starts out with 20 songs on his playlist and adds 4 songs each day. The graph representing Tyler’s playlist has a $y$ -intercept of 12 and is parallel to a graph of Han’s playlist, which means that Tyler starts out with 12 songs on his playlist and also adds 4 songs each day. Both Han and Tyler add the same amount of songs each day, so the rate of change for both situations is the same. However, since Han started out with more songs, Tyler will never catch up or have more songs on his playlist than Han.

Minimal Tier 1 response:

Work is complete and correct, with complete explanation or justification.
An explanation that compares the slopes and vertical intercepts without providing numbers, or an explanation that includes a graph or equation is also acceptable.
Sample: See solution above.

Tier 2 response:

Work shows general conceptual understanding and mastery, with some errors.
Sample errors: Work does not explicitly state agreement or disagreement with Han but can be inferred from the reasoning; work in the form of a table, graph, or completed calculations that shows Han is incorrect, but the explanation is incomplete.

Tier 3 response:

Significant errors in work demonstrate lack of conceptual understanding or mastery.
Sample errors: Work does not include any explanation; work shows that Han’s statement is correct.