End-of-Unit Assessment

1.

Priya ran $1\frac12$ miles today, and Tyler ran $3\frac34$ miles. How many times the length of Priya’s run was Tyler’s run?

A.

$\frac{45}{8}$ times as far

B.

$\frac{9}{4}$ times as far

C.

$\frac{5}{2}$ times as far

D.

$\frac{2}{5}$ times as far

Answer:

C

Teaching Notes

Students who select choice A have multiplied the lengths of the two runs together. Students who select choice B have answered the question, “How many miles longer was Tyler’s run?” Students who select choice D have reversed the order of division.

2.

Diego has run $3\frac{1}{3}$ kilometers, which is $\frac{5}{6}$ of the distance in a race. How long is the whole race?

Write a multiplication equation or a division equation to represent the question, and then find the answer.

Answer:

Sample response: $\frac{5}{6} \boldcdot {?} = 3\frac{1}{3}$ , or $3\frac{1}{3} \div \frac{5}{6} = {?}$

The whole race is 4 kilometers, because $\frac{10}{3} \div \frac{5}{6} = \frac {10}{3} \boldcdot \frac{6}{5} = \frac{60}{15} = 4$ .

Teaching Notes

Students can reason about the length of the entire race by dividing ( $3\frac{1}{3} \div \frac{5}{6} = {?}$ ) or by thinking about a unknown factor ( $\frac{5}{6} \boldcdot {?}= 3\frac{1}{3}$ ). Some students may find the answer by reasoning indirectly (and independently of the equation they write to represent the question). For example, they may calculate $3\frac{1}{3} \div 5$ to find $\frac{1}{6}$ of the distance in the race and then multiply the result ( $\frac{2}{3}$ ) by 6 to find the distance of the whole race.

3.

Select all statements that show correct reasoning for finding

12 \div \frac{3}{8}

?

A.

Multiply 12 by 8, then divide by 3.

B.

Multiply 12 by 3, then divide by 8.

C.

Multiply

\frac{3}{8}

by 1, then multiply by

\frac{1}{12}

.

D.

Multiply 12 by

\frac{1}{8}

, then multiply by 3.

E.

Multiply 12 by

\frac{1}{3}

, then multiply by 8.

Answer: A, E

Teaching Notes

Students who select choice B or D have picked a statement that is equivalent to "Multiply by

\frac{3}{8}

" instead of "Multiply by

\frac{8}{3}

," Students who select choice C have reversed the dividend and divisor, picking a way to find

\frac{3}{8} \div 12

instead of

12 \div \frac{3}{8}

.

4.

Divide.

$\frac{2}{3} \div \frac{1}{4}$
$\frac{10}{3} \div \frac{2}{6}$
$\frac{5}{6} \div \frac{7}{12}$
$2\frac{3}{4} \div \frac{4}{3}$

Answer:

$\frac{8}{3}$ (or equivalent)
$\frac{60}{6}$ (or equivalent)
$\frac{60}{42}$ (or equivalent)
$\frac{33}{16}$ (or equivalent)

Teaching Notes

Students apply the algorithm for dividing fractions.

5.

Lin draws this tape diagram for $4 \div \frac34$ :

Lin says that $4 \div \frac34 = 5\frac14$ because there are 5 groups of $\frac34$ with $\frac14$ left over. Do you agree with Lin? Explain or show your reasoning.

Answer:

Sample response: No, I disagree. There are 5 groups of $\frac34$ in $3\frac34$ . Then there is $\frac14$ left, and this makes $\frac13$ of another group. There are $5\frac13$ groups of $\frac34$ in 4.

Minimal Tier 1 response:

Work is complete and correct.
Sample: No, because $4 \div \frac34 = 5\frac13$ , not $5\frac14$ .

Tier 2 response:

Work shows general conceptual understanding and mastery, with some errors.
Sample errors: Disagreement with Lin, but a minor error in logic or calculation leads to a different result other than $5\frac13$ ; agreement with Lin, with a reasonable but flawed argument.

Tier 3 response:

Significant errors in work demonstrate lack of conceptual understanding or mastery.
Sample errors: Disagreement with Lin based on a major flaw in logic or calculation; agreement with Lin with a badly flawed argument; agreement or disagreement without any stated justification.

Teaching Notes

Students who agree with Lin’s statement may have a fundamental misunderstanding of division that may come from working with remainders of whole number division.

6.

A box has a width of $2\frac{1}{4}$ inches, a length of $2\frac{1}{2}$ inches, and a height of $1\frac{3}{4}$ inches.

How many $\frac{1}{4}$ -inch cubes does it take to fill the box?

Answer:

630 cubes (9 cubes fit along the width of the box, 10 cubes fit along the length, and 7 cubes fit vertically.) $9 \boldcdot 10 \boldcdot 7 = 630$

Teaching Notes

Students will need to use fraction division to calculate how many cubes will fit along each side of the box. Rather than using the standard algorithm, some students may visualize or otherwise reason conceptually about how many cubes with side length $\frac14$ inch it will take to reach a length of $2\frac14$ inches, etc.

7.

Elena has two aquariums, each shaped like a rectangular prism. For each question, explain or show your reasoning.

One aquarium has a length of $\frac72$ feet, a width of $\frac43$ feet, and a height of $\frac32$ feet. What is the volume of the aquarium?
Elena paints the entire back side of the second aquarium, which has a height of $1\frac{3}{4}$ feet. The painted area is $5\frac{5}{6}$ square feet. What is its length?

Answer:

7 cubic feet (or equivalent). The volume is the product of the aquarium's length, width, and height: $\frac72 \boldcdot \frac43 \boldcdot \frac32 = 7$ .
$3\frac13$ feet (or equivalent). The length is the solution to $\ell \boldcdot 1\frac34 = 5\frac56$ . By writing each mixed number as a fraction, the problem is made simpler: $\frac74 \ell = \frac{35}{6}$ . Then $\ell = \frac{10}{3}$ .

Minimal Tier 1 response:

Work is complete and correct, with complete explanation or justification.
Sample:
7 cubic feet, because $\frac72 \boldcdot \frac43 \boldcdot \frac32 = 7$ .
$\frac{10}{3}$ feet, because $5\frac56 \div 1\frac34 = \frac{10}{3}$ .

Tier 2 response:

Work shows good conceptual understanding and mastery, with either minor errors or correct work with insufficient explanation or justification.
Sample errors: Correct answers without justification; one or two errors in calculation, such as incorrect rewriting of mixed numbers, but correct equations or representations used.

Tier 3 response:

Work shows a developing but incomplete conceptual understanding, with significant errors.
Sample errors: One incorrect answer with invalid work or no work shown; any incorrect choice of multiplication or division; invalid method used to multiply or divide fractions or mixed numbers; more than two errors in calculation.

Tier 4 response:

Work includes major errors or omissions that demonstrate a lack of conceptual understanding and mastery.
Sample errors: Two incorrect answers with invalid work or no work shown; consistently incorrect choices of multiplication or division; repeated use of invalid methods to multiply or divide.

Teaching Notes

While most students should understand the context of the problem, some may still have difficulty understanding without a diagram. The second problem is about area, even though the aquarium is described as a rectangular prism.