Unit 2 Introducing Proportional Relationships — Unit Plan

Title

Assessment

Lesson 1

One of These Things Is Not Like the Others

Orangey-Pineapple Juice

Here are three different recipes for Orangey-Pineapple Juice. Two of these mixtures taste the same and one tastes different.

Recipe 1: Mix 4 cups of orange juice with 6 cups of pineapple juice.
Recipe 2: Mix 6 cups of orange juice with 9 cups of pineapple juice.
Recipe 3: Mix 9 cups of orange juice with 12 cups of pineapple juice.

Which two recipes will taste the same, and which one will taste different? Explain or show your reasoning.

Show Solution

Recipes 1 and 2 will taste the same. Sample reasoning: Recipe 3 is different because it requires $1\frac13$ cups of pineapple juice for every 1 cup of orange juice. Recipes 1 and 2 both require $1\frac12$ cups of pineapple juice for every 1 cup of orange juice.

recipe 1

orange juice (cups)	pineapple juice (cups)
4	6
2	3
1	$1 \frac12$

recipe 2

orange juice (cups)	pineapple juice (cups)
6	9
2	3
1	$1 \frac12$

recipe 3

orange juice (cups)	pineapple juice (cups)
9	12
3	4
1	$1\frac13$

Double number line diagrams can be used to compare the recipes, for instance, by noting that for Recipes 1 and 2, you use 2 cups of orange juice for every 3 cups of pineapple juice, whereas with Recipe 3, you use $2\frac14$ cups of orange juice for 3 cups of pineapple juice.

Section A Check

Section A Checkpoint

Problem 1

The table represents mixtures of black and white paint that produce the same shade of gray.

Complete the table as you answer the questions.

To make the same shade of gray, how many cups of white paint will they need to mix with 1 cup of black paint? Explain or show your reasoning.
How many cups of black paint will they need to mix with 16 cups of white paint? Explain or show your reasoning.
Make up a new pair of numbers that would make the same shade of gray.
What is the constant of proportionality?
What does the constant of proportionality mean in this situation?

black paint (cups)	white paint (cups)
$\frac12$	4
3	24
1
	16

Show Solution

8 cups. Sample reasoning: From the row showing $\frac12$ and 4, I multiplied both by 2 to get 1 and 8.
2 cups. Sample reasoning: I divided 16 by 8 to get 2.
Sample response: 10 cups of black paint with 80 cups of white paint. (Any pair of numbers that makes a ratio equivalent to $1:8$ is acceptable.)
8
For every 1 cup of black paint, mix in 8 cups of white paint.

black paint (cups)	white paint (cups)
$\frac12$	4
3	24
1	8
2	16
10	80

Lesson 4

Proportional Relationships and Equations

It’s Snowing in Syracuse

Snow is falling steadily in Syracuse, New York. After 2 hours, 4 inches of snow has fallen.

If it continues to snow at the same rate, how many inches of snow would you expect after 6.5 hours? If you get stuck, you can use the table to help.
Write an equation that gives the amount of snow that has fallen after $x$ hours at this rate.
How many inches of snow will fall in 24 hours if it continues to snow at this rate?

time (hours)	snow (inches)
	1
1
2	4
6.5
$x$

Show Solution

13 inches (Two inches fell in 1 hour, 6.5 is $1 \boldcdot (6.5)$ , and $2 \boldcdot (6.5) = 13$ .)
Sample response: $y=2x$ , where $x$ is the number of hours that have passed and $y$ is the inches of snow that has fallen.
48 inches ( $24 \boldcdot 2 = 48$ )

Lesson 5

Two Equations for Each Relationship

Flight of the Albatross

An albatross is a large bird that can fly 400 kilometers in 8 hours at a constant speed. Using $d$ for distance in kilometers and $t$ for number of hours, an equation that represents this situation is $d = 50t$ .

What are two constants of proportionality for the relationship between distance in kilometers and number of hours? What is the relationship between these two values?
Write another equation that relates $d$ and $t$ in this context.

Show Solution

50 and $\frac{1}{50}$ ; Sample response: They are reciprocals of each other.
$t = \frac{1}{50} d$

Lesson 6

Writing Equations to Represent Relationships

More Recycling

Glass bottles can be recycled. At one recycling center, 1 ton of clear glass is worth $25. (1 ton = 2,000 pounds)

How many pounds of clear glass is worth $10?
How much money is 40 pounds of clear glass worth?
Write an equation to represent the relationship between the weight of clear glass and the value of the glass.

Show Solution

800 pounds, because $2,000 \div 25 = 80$ and $80 \boldcdot 10 = 800$
$0.50, because $40 = 80 \boldcdot 0.50$
Sample response: If $v$ represents the value, in dollars, of $p$ pounds of clear glass, then the equation could be either $p = 80v$ or $v = 0.0125p$ .

Section B Check

Section B Checkpoint

Problem 1

Elena is riding her bike around the park at a constant pace. She completes 5 laps in 20 minutes.

Write an equation that shows the time in minutes it takes Elena to complete x laps at the same pace. (If you get stuck, consider completing the table.)

number of laps	time in minutes
5	20
10
11
1
$x$

Show Solution

y=4x

(where y represents time in minutes)

Problem 2

Hawaiians have a unique system for measuring lengths. Two of their units are called “muku” and “anana.”

The equation $a = 0.75m$ gives the relationship between a length measured in muku, $m$ , and the same length measured in anana, $a$ .

How many anana are in 24 muku?
How many muku are in 24 anana?

Show Solution

18 anana
32 muku

Lesson 7

Comparing Relationships with Tables

Apples and Pizza

Based on the information in the table, is the cost of the apples proportional to the weight of apples?

pounds of apples cost of apples

2 $3.76

3 $5.64

4 $7.52

5 $9.40
Based on the information in the table, is the cost of the pizza proportional to the number of toppings?

number of toppings cost of pizza

2 $11.99

3 $13.49

4 $14.99

5 $16.49
Write an equation for the proportional relationship.

Show Solution

Yes, because the cost per pound of apples is the same in each row, 1.88 dollars per pound.
No, because the cost per topping is not the same in each row. (An equation is $C = 1.50T + 8.99$ but students do not need to provide an equation.)
$c = 1.88p$ , where $c$ represents the cost of the apples and $p$ represents the pounds of apples.

Lesson 8

Comparing Relationships with Equations

Tables and Chairs

Andre is setting up rectangular tables for a party. He can fit 6 chairs around a single table. Andre lines up 10 tables end-to-end and tries to fit 60 chairs around them, but he is surprised when he cannot fit them all.

Write an equation for the relationship between the number of chairs $c$ and the number of tables $t$ when:
- the tables are apart from each other:
- the tables are placed end-to-end:
A rectangular figure with arcs that represent a rectangular table and chairs. The figure shows 3 rectangles placed end-to-end. There are 6 arcs on each horizontal side and 1 arc on each vertical side.
Is the first relationship proportional? Explain how you know.
Is the second relationship proportional? Explain how you know.

Show Solution

When the tables are apart: $c = 6t$ (or $t = \frac16 c$ ).
When the tables are together: $c = 4t + 2$ (or $t = \frac14 c - \frac12$ ).
This relationship is proportional. Sample reasonings:
- It can be represented with an equation of the form $c = kt$ (or $t = kc$ ).
- There are 6 chairs per table no matter how many tables.
This relationship is not proportional. Sample reasonings:
- The number of chairs per table changes depending on how many tables there are.
- The quotient of chairs and tables is not constant.
- The relationship cannot be expressed with an equation of the form $c=kt$ .

As shown in this table, the number of chairs per table is the same when the tables are apart, but it is not the same if the tables are pushed together.

With tables apart:

tables	chairs	$\frac{\text{chairs}}{\text{tables}}$
1	6	6
2	12	6
3	18	6
4	24	6
10	60	6
$t$	$6t$	6

With tables end-to-end:

tables	chairs	$\frac{\text{chairs}}{\text{tables}}$
1	6	6
2	10	5
3	14	4.667
4	18	4.5
10	42	4.2
$t$	$4t+2$

Lesson 9

Solving Problems about Proportional Relationships

Folding Programs

Lin is folding programs for the school music concert. She wants to know how long it will take her to finish folding all the programs. What information would you need to know to write an equation that represents this relationship?

Show Solution

Sample responses:

Is Lin folding the programs at a constant rate?
How long does it take her to fold 1 program?
How many programs can she fold in 1 minute?
How many programs are there total?

Section C Check

Section C Checkpoint

Problem 1

A fashion designer orders leather by the square meter. The table shows their last four orders. They are wondering if they get a better deal when they place a larger order.

Do the values in the table show evidence of a proportional relationship? Explain or show your reasoning.

square meters of leather	price in dollars
40	740.00
20	370.00
35	647.50
52	962.00

Show Solution

Yes. Dividing the price by the square meters of leather gets 18.5 for each row.

Problem 2

The equation $y = 16 \div x$ represents the relationship between the length, $x$ , and width, $y$ , of a rectangle whose area is 16 square units.

Complete the $y$ column of the table with values that make the equation $y = 16 \div x$ true.
Is there a proportional relationship between $x$ and $y$ ? Explain how you know. (It may help to complete the $\frac{y}{x}$ column.)

$x$	$y$	$\frac{y}{x}$
1
2
4
8

Show Solution

16, 8, 4, 2
No. Sample reasoning: $\frac{y}{x}$ is not the same for every pair of values that make the equation true.

Lesson 10

Introducing Graphs of Proportional Relationships

Which Are Not Proportional

Which graphs cannot represent a proportional relationship? Select all that apply. Explain how you know.

A coordinate graph, all on an xy planes, origin O. — A

Show Solution

B and C. Sample reasoning: Since graph B does not go through the origin, it cannot be a proportional relationship. Since the points in graph C cannot be connected by a single, straight line, it cannot be a proportional relationship.

Lesson 12

Using Graphs to Compare Relationships

Revisiting the Amusement Park

Noah and Diego left the amusement park’s ticket booth at the same time. Each moved at a constant speed toward his favorite ride. After 8 seconds, Noah was 17 meters from the ticket booth, and Diego was 43 meters away from the ticket booth.

Which line represents the distance traveled by Noah, and which line represents the distance traveled by Diego? Label each line with one name.
The graph of two lines in the coordinate plane with the horizontal axis labeled "elapsed time in seconds" and the vertical axis labeled "distance traveled, in meters." One line begins at the origin and moves steeply upwards and to the right. The other line also begins at the origin and moves steadily upwards and to the right.
Explain how you decided which line represents which person’s travel.

Show Solution

The steeper line represents the distance traveled by Diego.
Sample reasoning: Diego had gone farther after 8 seconds. If you pick a time and look at which line represents a person who has gone farther, that is the steeper graph. So that must be Diego’s line.

Section D Check

Section D Checkpoint

Problem 1

Decide whether each graph represents a proportional relationship.

Show Solution

Problem 2

A cyclist is training for a race. The graph represents the relationship between her distance traveled and elapsed time during a training ride.

Explain what the labeled point $(4, 1\frac13)$ represents in this situation.
Use the graph to find the constant of proportionality for this relationship.

Show Solution

It takes her 4 minutes to travel $1\frac13$ miles.
The constant of proportionality is $\frac13$ . This is seen on the graph at the point $(1, \frac13)$ .

Lesson 14

Four Representations

Explain Their Work

Choose a relationship that another group found and explain why it is a proportional relationship. Make sure to include the quantities they used and any important constants of proportionality.

Show Solution

Sample response: In a 100-yard, three-legged race, distance in yards and time in minutes are proportional since each value of distance could be multiplied by $\frac{1}{40}$ to get the time. The constant of proportionality they used was $\frac{1}{40}$ .

Lesson 15

Using Water Efficiently

No cool-down

Unit 2 Assessment

End-of-Unit Assessment

Problem 1

Which graph represents a proportional relationship?

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

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

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

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

Show Solution

Problem 2

The graph shows the cost $C$ in dollars of $w$ pounds of blueberries, a proportional relationship.

Select all the true statements.

Graph of a linear function and a point. Horizontal axis w. Vertical axis C. Function starts at (0 comma 0) and rises to point (6 comma 16 point 5 0), then continues rising.<br>

1 pound of blueberries costs $2.75.

2.75 pounds of blueberries cost $1.

5 pounds of blueberries cost $15.50.

12 pounds of blueberries cost $33.

The point $(3,9)$ is on the graph of the proportional relationship.

Show Solution

A, D

Problem 3

Andre rode his bike at a constant speed. He rode 1 mile in 5 minutes.

Which of these equations represents the amount of time $t$ (in minutes) that it takes him to ride a distance of $d$ miles?

$t = 5d$

$t = \frac 1 5 d$

$t = d + 4$

$t = d - 4$

Show Solution

$t = 5d$

Problem 4

The two lines represent the amount of water, over time, in two tanks that are the same size. Which container is filling more quickly? Explain how you know.

Two linear functions on coordinate plane, A and B.<br>

Show Solution

Container A is filling more quickly. In each graph, the constant of proportionality represents the rate at which the water is flowing into the containers. The container that is filling more quickly is the one for which the graph is steeper: Container A is filling more quickly than Container B. Alternatively, choose a time and see how much water is in the two containers at that time. This graph shows the amounts of water in the containers at a given time $t$ .

<p>Graph. Time in minutes. Water in gallons.</p>

Because $a > b$ at time $t$ , more water has flowed into Container A than into Container B.

Minimal Tier 1 response:

Work is complete and correct.
Sample: Container A, because the graph for Container A is steeper.

Tier 2 response:

Work shows general conceptual understanding and mastery, with some errors.
Sample errors: Picking Container A with an incomplete or omitted explanation; picking Container B with a complete explanation of how a correct rate (minutes per gallon, for example) is larger.

Tier 3 response:

Significant errors in work demonstrate lack of conceptual understanding or mastery.
Sample errors: Picking Container B with an incomplete or omitted explanation; claiming both containers are filling equally quickly.

Problem 5

The table shows the weights of apples at a grocery store.

number of apples	weight in kilograms
2
5	0.60
12

Complete the table so that there is a proportional relationship between the number of apples and their weight.

Show Solution

number of apples	weight in kilograms
2	0.24
5	0.60
12	1.44

Problem 6

The equation $F = \frac{9}{5}C + 32$ relates temperature measured in degrees Celsius, $C$ , to degrees Fahrenheit, $F$ .

Determine whether there is a proportional relationship between $C$ and $F$ . Explain or show your reasoning.

Show Solution

The relationship between degrees Celsius and degrees Fahrenheit is not proportional. Explanations vary. Sample explanations:

Using a graph: Drawing the graph of the relationship shows it is not a line through the origin.
Using a table: These three rows are generated from the equation.

temperature (degrees C) temperature (degrees F)

0 32

10 50

20 68

In the second row, $50 = 10 \boldcdot 5$ , but in the third row, $68 = 20 \boldcdot (3.4)$ . Because 5 is different from 3.4, this is not a proportional relationship.

The first row of the table also shows that the relationship cannot be proportional because 32 is not a multiple of 0.

Minimal Tier 1 response:

Work is complete and correct.
Sample: No, because this graph is not a line through $(0,0)$ . (Response includes a correct graph.)
Acceptable errors: Axes of graph or headers of table are unlabeled.

Tier 2 response:

Work shows general conceptual understanding and mastery, with some errors.
Sample errors: Calculation errors while generating table or graph, including ones that lead to an incorrect conclusion based on the data (for example, accidentally using $F = \frac 9 5 C \boldcdot 32$ ); incorrectly drawn graph.

Tier 3 response:

Significant errors in work demonstrate lack of conceptual understanding or mastery.
Sample errors: Statement of a proportional relationship, unless based on incorrectly calculated data; answering either yes or no without explanation; answering based only on the equation, without building a table or graph.

Problem 7

A recipe for salad dressing calls for 3 tablespoons of oil for every 2 tablespoons of vinegar. The line represents the relationship between the amount of oil and the amount of vinegar needed to make salad dressing according to this recipe. The point $(1,1.5)$ is on the line.

A line is graphed in a coordinate plane. The line begins at the origin. The line moves steadily upward and to the right passing through the point with coordinates 1 comma 1 point 5.

Label the axes of the graph.
Write an equation that represents the proportional relationship between oil and vinegar. Explain the meaning of each variable.
Explain the meaning of the point $(1,1.5)$ in terms of the situation.

Show Solution

The vertical axis should be labeled “oil (tablespoons),” and the horizontal axis should be labeled “vinegar (tablespoons).”
Answers vary. Sample response: If $y$ is the number of tablespoons of oil, and $x$ is the number of tablespoons of vinegar, then $y = 1.5x$ .
The point $(1,1.5)$ indicates that the recipe works with 1 tablespoon of vinegar and 1.5 tablespoons of oil. This point gives a unit rate.

Minimal Tier 1 response:

Work is complete and correct, with complete explanation or justification.
Sample:

The graph is labeled “oil (tablespoons)” on the vertical and “vinegar (tablespoons)” on the horizontal.
$o = 1.5v$ , where $o$ is oil and $v$ is vinegar.
You could make the recipe with 1 tablespoon of vinegar and 1.5 tablespoons of oil.

Tier 2 response:

Work shows good conceptual understanding and mastery, with either minor errors or correct work with insufficient explanation or justification.
Sample errors: Lack of units in either graph or description; one reversal of the quantities in the three parts.

Tier 3 response:

Work shows a developing but incomplete conceptual understanding, with significant errors.
Sample errors: 2 or 3 reversals of the quantities (in total for the three parts); one part omitted or containing a more significant error than quantity reversal; two error types from Tier 2 response.

Tier 4 response:

Work includes major errors or omissions that demonstrate a lack of conceptual understanding and mastery.
Sample errors: More than one part omitted or containing a more significant error than quantity reversal; statements suggesting a nonproportional relationship.