1. grid fluff and fold charges $2.25 for each load of laundry. (lesson 1)
a. draw the graph of the proportional relationship between the two quantities, where x is the number of loads of laundry and y is the total cost.
graph titled laundry with x-axis number of loads (0 - 9) and y-axis total cost ($) (0 - 9)
b. describe how the unit rate is represented in the graph.
box with 33.75 written
2. open response daniella makes apple pies each fall. the cost at the local grocery store for x pounds of apples is shown in the table. what is the least amount of money daniella will spend for 15 pounds of apples? assume the relationship is proportional. (lesson 1)
| number of pounds, x | total cost ($), y |
| 2 | $4.50 |
| 3 | $6.75 |
empty box
3. multiple choice a turtle is crawling up a hill that rises 6 feet for every horizontal change of 36 feet. which of the following represents the slope of the hill, as a fraction in simplest form? (lesson 2)
a $\frac{6}{1}$ b $\frac{1}{6}$ c $\frac{36}{6}$ d $\frac{6}{36}$
4. open response the points in the table lie on a line. compute the slope of the line. (lesson 2)
| x | y |
| 6 | -3 |
| -2 | 1 |
| -4 | 2 |
box with $-\frac{1}{2}$ written
5. multiselect which statement is true about the graph? select all that apply. (lesson 3)
graph with a line and triangles
- the ratio of the rise to the run of each triangle is the same.
- the smaller triangle and the larger triangle shown are similar.
- the slope of the line is 2.
- the slope of the line is -2.
- the corresponding sides of the two triangles are not proportional.
module 4 • linear relationships and slope 249

Question

1. grid fluff and fold charges $2.25 for each load of laundry. (lesson 1)
a. draw the graph of the proportional relationship between the two quantities, where x is the number of loads of laundry and y is the total cost.
graph titled laundry with x-axis number of loads (0 - 9) and y-axis total cost ($) (0 - 9)
b. describe how the unit rate is represented in the graph.
box with 33.75 written
2. open response daniella makes apple pies each fall. the cost at the local grocery store for x pounds of apples is shown in the table. what is the least amount of money daniella will spend for 15 pounds of apples? assume the relationship is proportional. (lesson 1)
| number of pounds, x | total cost ($), y |
| 2 | $4.50 |
| 3 | $6.75 |
empty box
3. multiple choice a turtle is crawling up a hill that rises 6 feet for every horizontal change of 36 feet. which of the following represents the slope of the hill, as a fraction in simplest form? (lesson 2)
a $\frac{6}{1}$  b $\frac{1}{6}$  c $\frac{36}{6}$  d $\frac{6}{36}$
4. open response the points in the table lie on a line. compute the slope of the line. (lesson 2)
| x | y |
| 6 | -3 |
| -2 | 1 |
| -4 | 2 |
box with $-\frac{1}{2}$ written
5. multiselect which statement is true about the graph? select all that apply. (lesson 3)
graph with a line and triangles
- the ratio of the rise to the run of each triangle is the same.
- the smaller triangle and the larger triangle shown are similar.
- the slope of the line is 2.
- the slope of the line is -2.
- the corresponding sides of the two triangles are not proportional.
module 4 • linear relationships and slope 249

Accepted Answer

### Question 2 Solution (Open Response: Cost for 15 pounds of apples)
# Explanation:
## Step1: Find the unit rate (cost per pound)
The relationship is proportional, so we can find the unit rate using the given data. For $ x = 2 $ pounds, $ y = 4.50 $ dollars. The unit rate $ r $ is $ \frac{y}{x} $. So, $ r=\frac{4.50}{2}=2.25 $ dollars per pound. We can check with the other data: for $ x = 3 $ pounds, $ y = 6.75 $ dollars. $ \frac{6.75}{3}=2.25 $ dollars per pound, so the unit rate is consistent.

## Step2: Calculate cost for 15 pounds
Now, to find the cost for 15 pounds, we use the formula $ y = r\times x $, where $ r = 2.25 $ and $ x = 15 $. So, $ y = 2.25\times15 $. Calculating that: $ 2.25\times15 = 33.75 $ dollars.

# Answer:
$ \$33.75 $

### Question 3 Solution (Multiple Choice: Slope of the hill)
# Explanation:
## Step1: Recall the slope formula for a hill (rise over run)
Slope is defined as $ \text{slope}=\frac{\text{rise}}{\text{run}} $. The hill rises 6 feet (rise) for every horizontal change of 36 feet (run). So, the slope is $ \frac{6}{36} $.

## Step2: Simplify the fraction
Simplify $ \frac{6}{36} $ by dividing both the numerator and the denominator by their greatest common divisor, which is 6. $ \frac{6\div6}{36\div6}=\frac{1}{6} $. Wait, no, wait: Wait, rise is vertical change (6 feet) and run is horizontal change (36 feet). Wait, actually, slope for a hill (a line) is rise over run, where rise is the vertical change (upward) and run is horizontal change (rightward). So, rise = 6, run = 36. So slope is $ \frac{6}{36}=\frac{1}{6} $? Wait, no, wait the options: Option B is $ \frac{1}{6} $, but let's check again. Wait, the problem says "rises 6 feet for every horizontal change of 36 feet". So rise is 6, run is 36. So slope is $ \frac{6}{36}=\frac{1}{6} $? Wait, but let's check the options. Wait, the options are A $ \frac{6}{1} $, B $ \frac{1}{6} $, C $ \frac{36}{6} $, D $ \frac{6}{36} $. But we need to simplify. $ \frac{6}{36}=\frac{1}{6} $, so the simplified form is $ \frac{1}{6} $, which is option B? Wait, no, wait: Wait, maybe I mixed up rise and run? Wait, no, slope is rise over run. Rise is vertical (6), run is horizontal (36). So slope is $ \frac{6}{36}=\frac{1}{6} $, which is option B. Wait, but let's check the options again. Wait, the problem says "as a fraction in simplest form". So $ \frac{6}{36} $ simplifies to $ \frac{1}{6} $, so the correct option is B.

# Answer:
B. $ \frac{1}{6} $

### Question 4 Solution (Open Response: Slope of the line)
# Explanation:
## Step1: Recall the slope formula between two points
The slope formula between two points $ (x_1,y_1) $ and $ (x_2,y_2) $ is $ m=\frac{y_2 - y_1}{x_2 - x_1} $. Let's take two points from the table. Let's take $ (6, -3) $ and $ (-2, 1) $.

## Step2: Apply the slope formula
Using $ (x_1,y_1)=(6, -3) $ and $ (x_2,y_2)=(-2, 1) $, we have $ m=\frac{1 - (-3)}{-2 - 6}=\frac{1 + 3}{-8}=\frac{4}{-8}=-\frac{1}{2} $. Let's check with another pair, say $ (-2, 1) $ and $ (-4, 2) $. Then $ m=\frac{2 - 1}{-4 - (-2)}=\frac{1}{-2}=-\frac{1}{2} $. So the slope is $ -\frac{1}{2} $.

# Answer:
$ -\frac{1}{2} $

### Question 5 Solution (Multiselect: True statements about the graph)
# Brief Explanations:
1. **"The ratio of the rise to the run of each triangle is the same."**: Since the triangles are formed by the line, the slope (rise over run) is constant for the line, so the ratio of rise to run for each triangle (similar triangles formed by the line and the axes) will be the same. This is true.
2. **"The smaller triangle and the larger triangle shown are similar."**: Similar triangles have proportional sides and equal angles. Since both triangles are formed by the same line (with the same slope), their angles are equal, and their sides are proportional (because the slope is the same, so rise/run is the same for both). So they are similar. This is true.
3. **"The slope of the line is 2."**: The line is decreasing (going from top - left to bottom - right), so the slope should be negative. So this is false.
4. **"The slope of the line is - 2."**: Let's check the slope. If we take two points on the line, say, from the graph, if we move 1 unit right, we move 2 units down (since it's a steep line). So slope is $ \frac{-2}{1}=-2 $. Wait, maybe my previous thought was wrong. Wait, let's see: if the line goes from (0, y - intercept) to (x, 0). Suppose the larger triangle has a run of, say, 1 and rise of - 2 (since it's decreasing). So slope is $ \frac{-2}{1}=-2 $. So this statement is true? Wait, no, let's re - evaluate. Wait, the first statement: "The ratio of the rise to the run of each triangle is the same." Since the slope is constant, the rise/run ratio (which is the slope) is the same for both triangles. So that's true. The second: similar triangles, yes, because they have the same angles (right angle and equal acute angles because slope is same). The fourth: slope is - 2. Let's confirm. If we take two points: suppose the line passes through (0, 4) and (2, 0). Then slope is $ \frac{0 - 4}{2 - 0}=\frac{-4}{2}=-2 $. So slope is - 2. So the statements that are true are: "The ratio of the rise to the run of each triangle is the same.", "The smaller triangle and the larger triangle shown are similar.", "The slope of the line is - 2."

Wait, let's re - check:

- "The ratio of the rise to the run of each triangle is the same." → True, because slope is constant (rise/run = slope).
- "The smaller triangle and the larger triangle shown are similar." → True, because they are right - angled and have the same acute angle (since slope is same, so the angle with the x - axis is same), so by AA similarity, they are similar.
- "The slope of the line is 2." → False, because the line is decreasing, slope is negative.
- "The slope of the line is - 2." → True, as calculated.
- "The corresponding sides of the two triangles are not proportional." → False, because similar triangles have proportional corresponding sides.

# Answer:
- The ratio of the rise to the run of each triangle is the same.
- The smaller triangle and the larger triangle shown are similar.
- The slope of the line is - 2.

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QUESTION IMAGE

Question

Explanation:

Question 2 Solution (Open Response: Cost for 15 pounds of apples)

Step1: Find the unit rate (cost per pound)

Step2: Calculate cost for 15 pounds

Step1: Recall the slope formula for a hill (rise over run)

Step2: Simplify the fraction

Step1: Recall the slope formula between two points

Step2: Apply the slope formula

Answer:

Question 3 Solution (Multiple Choice: Slope of the hill)