End-of-Unit Assessment

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

A.

A

B.

B

C.

C

D.

D

Answer:

C

Teaching Notes

Students identify a proportional relationship from a graph. Students can correctly identify C and exclude A, B, and D without a complete understanding of proportional relationships by using the general rule that the graph of a proportional relationship lies on a ray in the first quadrant starting at $(0,0)$ .

Students who select A or B may know the graph of a proportional relationship is linear, but may not recall that line must contain $(0,0)$ . Students who select D may know the graph contains $(0,0)$ but may not know that the graph must be linear.

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>

A.

1 pound of blueberries costs $2.75.

B.

2.75 pounds of blueberries cost $1.

C.

5 pounds of blueberries cost $15.50.

D.

12 pounds of blueberries cost $33.

E.

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

Answer: A, D

Teaching Notes

Students use the graph of a proportional relationship in order to find the unit rate, then to apply the unit rate to other questions about the situation. They are given a single point on the graph and work with these numbers to calculate the unit rate.

Students who do not select A may not have determined the correct unit rate of $2.75. Students who select B have used the unit rate in reverse, and may have the variables confused. Students who select C have a misunderstanding about the nature of proportional relationships, since they subtracted 1 from each quantity. Students who do not select to select D may have made a calculation error, or failed to use the proportionality of doubling the values given in the table. Students who select E might be eyeballing the graph, or estimating. The actual cost of 3 pounds is $8.25, so $(3,9)$ is not on the graph.

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?

A.

$t = 5d$

B.

$t = \frac 1 5 d$

C.

$t = d + 4$

D.

$t = d - 4$

Answer:

$t = 5d$

Teaching Notes

Students need to recognize that when traveling at constant speed, the amount of time traveled is proportional to the distance traveled. No graphing or explanation is required and the context is a familiar one, although the information is given in terms of pace in minutes per mile rather than speed in miles per minute.

Students who select B have probably reversed the variables in context. They may have written $d = \frac 1 5 t$ and ended there. They may also have misread the description as 5 miles in 1 minute, or may just be writing $\frac 1 5$ using their knowledge that division is a typical operation in rate contexts. Students who select C have made an error involving the proportional relationship: even though 1 mile in 5 minutes makes this equation true, it is not a generally correct equation. Students who select D have likely made two errors: the error from C, along with the error in reversing the variables’ meanings.

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>

Answer:

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.

Teaching Notes

This task requires students to interpret graphs of proportional relationships without numerical values. The rates at which the containers fill correspond to the constants of proportionality for the two relationships: the greater rate corresponds to the steeper graph.

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.

Answer:

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

Teaching Notes

This item assesses students’ ability to work with a proportional relationship defined by a table. While there is more than one possible way to complete the problem, the most likely method is to determine the unit rate of $0.12 per apple, then multiply to determine the cost for 2 and for 12 apples.

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.

Answer:

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.

temperature (degrees C)	temperature (degrees F)
0	32
10	50
20	68

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.

Teaching Notes

This item requires an explanation. Students’ explanations involving graphs and tables should more clearly indicate their overall understanding of the concept of a proportional relationship. Students have done similar work in this unit, with a more directed table-building exercise, so this version is more open-ended.

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.

Answer:

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.

Teaching Notes

This item is more challenging than earlier items. The axes are not labeled, and the coordinates of the point on the graph tell the amount of one quantity given one unit of the other quantity, rather than the values in the task statement. There are several ways students might reason about this problem, but however they do it, they need to explain their reasoning, requiring an understanding of the relationship between the constant of proportionality and the graph of the corresponding proportional relationship.