End-of-Unit Assessment

1.

Which table represents a proportional relationship?

A.

0	0
2	4
3	9
4	16

B.

0	0
2	6
3	9
4	12

C.

0	12
2	6
3	3
4	0

D.

0	6
2	12
3	15
4	18

Answer:

B

Teaching Notes

Students identify a proportional relationship from a table. Students can correctly exclude C 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 C or D may know the graph of a proportional relationship is linear, but may not recall that line must contain $(0,0)$ . Students who select A 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 peanuts, a proportional relationship.

Select all the true statements.

A.

2.5 pounds of peanuts costs $1.00.

B.

1 pound of peanuts costs $2.50.

C.

5 pounds of peanuts cost $12.50.

D.

9 pounds of peanuts cost $19.50.

E.

The point (4,10) is on the graph of the proportional relationship.

Answer:

B, C, E

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 select A are using the unit rate in reverse, and may have the variables confused. Students who do not select B may not have determined the correct unit rate of $2.50. Students who do not select C may have made a calculation error. Students who select D have a misunderstanding about the nature of proportional relationships, since they added 2 to each quantity. Students who do not select E might be incorrectly eyeballing the graph or estimating: since the cost of 4 pounds is $10, the point $(4,10)$ should be on the graph.

3.

Kiran walked at a constant speed. He walked 1 mile in 15 minutes.

Which of these equations represents the distance $d$ (in miles) that Kiran walks in $t$ minutes?

A.

d= t + 14

B.

d=t - 14

C.

d=15t

D.

$d=\frac{1}{15}t$

Answer:

$d=\frac{1}{15}t$

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.

Students who select C have probably reversed the variables in context; they may have written $t = 15 d$ and ended there. They may also have misread the description as 15 miles in 1 minute. Students who select B have made an error involving the proportional relationship: even though 1 mile in 15 minutes makes this equation true, it is not a generally correct equation. Students who select A have likely made two errors: the error from B, along with the error in reversing the variables’ meanings.

4.

The two lines represent the distance, over time, that two cars are traveling. Which car is traveling faster? Explain how you know.

<p>A graph. Distance. Miles. Time. Hours.</p>

Answer:

Car B is traveling faster. In each graph, the constant of proportionality represents the rate at which the cars are traveling. The car that is traveling faster is the one for which the graph is steeper: Car B is traveling faster than Car A. Alternatively, choose a time and see how many miles the two cars have traveled at that time. This graph shows the distance traveled at the given time $t$ .

Because $b>a$ at time $t$ , Car B is moving faster than Car A.

Minimal Tier 1 response:

Work is complete and correct.
Sample: Car B, because the graph for Car B is steeper.

Tier 2 response:

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

Tier 3 response:

Significant errors in work demonstrate lack of conceptual understanding or mastery.
Sample errors: Picking Car A with an incomplete or omitted explanation; claiming both cars are traveling with equal speed.

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 bananas at a grocery store.

number of bananas	weight in pounds
3
5	1.6
12

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

Answer:

number of bananas	weight in pounds
3	0.96
5	1.6
12	3.84

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 0.32 pounds per banana, then multiply to determine the cost for 3 and for 12 bananas.

6.

The equation $S=50+45w$ represents the savings, $S$ , after $w$ weeks working and depositing money into a savings account at the bank.

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

Answer:

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.

number of weeks savings in dollars

row 1 0 50

row 2 10 500

row 3 20 950

In the second row, $500=10\boldcdot50$ , but in the third row, $950=20\boldcdot(47.5)$ . Because 50 is different from 47.5, this is not a proportional relationship.

The first row of the table also shows that the relationship cannot be proportional because 50 is not a multiple of 0.

	number of weeks	savings in dollars
row 1	0	50
row 2	10	500
row 3	20	950

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, reversal of quantities $S=45+50w$ or 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 a smoothie calls for 5 cups of strawberries for every 2 cups of bananas. The line represents the relationship between the amount of strawberries and the amount of bananas needed to make a smoothie according to this recipe. The point $(1, 2.5)$ is on the line.

Label the axes of the graph.
Write an equation that represents the proportional relationship between strawberries and bananas. Explain the meaning of each variable.
Explain the meaning of the point $(1, 2.5)$ in terms of the situation.

Answer:

The vertical axis should be labeled “strawberries (cups),” and the horizontal axis should be labeled “bananas (cups).”
Answers vary. Sample response: If $y$ is the number of cups of strawberries, and $x$ is the number of cups of bananas, then $y=2.5x$ .
The point $(1,2.5)$ indicates that the recipe works with 1 cup of bananas and 2.5 cups of strawberries. 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 “strawberries (cups)” on the vertical and bananas (cups)” on the horizontal.
$s=2.5b$ , where $s$ is cups of strawberries and $b$ is cups of bananas.
You could make the recipe with 1 cup of bananas and 2.5 of strawberries

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.