End-of-Unit Assessment

1.

Priya has a recipe for banana bread. She uses $7 \frac 1 2$ cups of flour to make 3 loaves of banana bread.

Andre will follow the same recipe. He will make $b$ loaves of banana bread using $f$ cups of flour.

Which of these equations represents the relationship between $b$ and $f$ ?

A.

$b = \frac 2 9 f$

B.

$b = \frac 2 5 f$

C.

$b = \frac 5 2 f$

D.

$b = \frac 9 2 f$

Answer:

$b = \frac 2 5 f$

Teaching Notes

Students who select choice D may have subtracted $7 \frac 1 2 - 3 = \frac 9 2$ . Students who select choice C reversed the relationship between the variables. Students who select choice A may have reversed the variables while also making the mistake of subtracting to determine $\frac 9 2$ .

2.

Diego measured the length of a pen to be 22 cm. The actual length of the pen is 23 cm.

Which of these is closest to the percent error for Diego’s measurement?

A.

4.3%

B.

4.5%

C.

95.7%

D.

104.5%

Answer:

4.3%

Teaching Notes

Students who select choice B may be calculating the percent error relative to the measurement and not the actual value. Students who select choice C may be finding what percentage 22 is of 23. Students who select choce D may be finding what percentage 23 is of 22.

3.

A car is 180 inches long. A truck is 75% longer than the car.

How long is the truck?

A.

135 inches

B.

240 inches

C.

255 inches

D.

315 inches

Answer:

315 inches

Teaching Notes

Students who select choice A computed 75% of the car’s length, but did not add 180 to it. Students who select choice B solved the problem “180 is 75% of what number?” instead of the actual problem. Students who select choice C may have added $180 + 75$ to get 255.

4.

A circular running track is $\frac 1 4$ mile long. Elena runs on this track, completing each lap in $\frac 1 {20}$ of an hour.

What is Elena’s running speed? Include the unit of measure.

Answer:

5 miles per hour. (We can divide the number of miles by the number of hours to find the number of miles per hour: $\frac 1 4 \div \frac 1 {20}$ .)

Teaching Notes

The most likely error is students answering $\frac 1 5$ mile per hour.

Some students may answer $\frac 1 5$ hour per mile, which is correct, but it is a pace and not a running speed; his is for the teacher to decide whether to mark this answer as correct, but it is not a correct answer to the question asked. Similarly, it is for the teacher to decide what to do if a student answers “5” instead of “5 miles per hour”; this is not a correct answer to the question asked.

5.

Today, everything at a store is on sale. The store offers a 20% discount off the regular price.

The regular price of a T-shirt is $18. What is the discounted price?
If the regular price of an item is $x$ dollars, what is the discounted price in dollars?
The discounted price of a hat is $18. What is the regular price?

Answer:

$14.40 (80% of $18)
$0.8x$ or $(x - 0.2x)$ or $\frac 4 5 x$ (or equivalent)
$22.50 ( $0.8x = 18$ , so $x = 22.5$ .)

Teaching Notes

The last part of this item can be answered in several different ways, including a double number line, a table, or solving an equation.

6.

Lin’s father is paying for a $20 meal. He has a 15%-off coupon for the meal. After the discount, a 7% sales tax is applied.

What does Lin’s father pay for the meal? Explain or show your reasoning.

Answer:

$18.19. Sample reasoning: The coupon takes away 15% of $20, which is $3. The cost after the coupon is $17. The added tax is $1.19 (7% of $17). The total is $18.19, the sum of $17 and $1.19.

Minimal Tier 1 response:

Work is complete and correct.
Sample: $20 - 3 = 17$ , and $17 \boldcdot (1.07) = 18.19$ .

Tier 2 response:

Work shows general conceptual understanding and mastery, with some errors.
Sample errors: Calculating the tax to be $1.40, using 20 instead of 17; answering $1.19, using the amount of the tax as the final answer; answering $18.19 without showing or explaining work; errors in computation but not in concept.

Tier 3 response:

Significant errors in work demonstrate lack of conceptual understanding or mastery.
Sample errors: Ignoring the 15% discount or the 7% tax altogether; adding 15% to the total; using 15% of $20 as the amount before the tax; errors because of major misunderstandings about operations, such as adding $7 as the tax instead of 7%.

Teaching Notes

Use this problem as a check on students’ ability to add and multiply with decimals, and as a check on their understanding of discounts and tax.

7.

Tyler’s brother works in a shoe store.

Tyler’s brother earns a commission that is 2.5% of the amount he sells. Last week, he sold $900 worth of shoes. How much was his commission?
The store buys a pair of shoes for $50 and sells it for $80. What percentage is the markup?
Tyler’s brother earns $12 per hour. The store offers him a raise—a 5% increase. After the raise, how much will Tyler’s brother make per hour?

Answer:

$22.50. The commission was $22.50, because $900 \boldcdot (0.025) = 22.5$ .
60%. The markup is $30, and $30 is 60% of $50.
$12.60. Because 5% of $12 is $0.60, Tyler’s brother now makes $12 + 0.60$ dollars per hour.

Minimal Tier 1 response:

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

$900 \boldcdot (0.025) = 22.5$
60%, a $30 increase.
$12.60. The raise is $0.60.

Tier 2 response:

Work shows good conceptual understanding and mastery, with either minor errors or correct work with insufficient explanation or justification.
Sample errors: One or two calculation errors in executing the parts of the problem; stating 60 instead of 60% for the markup without a very clear description that the 60 refers to 60%; answering 0.60, the value of the raise, instead of the total after the raise.

Tier 3 response:

Work shows a developing but incomplete conceptual understanding, with significant errors.
Sample errors: One major conceptual error, perhaps due to a misinterpretation of the meaning of one of the terms in the problem; omission of one part of the problem; repeated calculation errors in execution.

Tier 4 response:

Work includes major errors or omissions that demonstrate a lack of conceptual understanding and mastery.
Sample errors: Two or more parts of the problem are omitted or with major conceptual errors.

Teaching Notes

While this problem does not require students to know the term "commission" from this unit, students who remember the term may be more successful here. The term "markup" is required for the second part.