End-of-Unit Assessment

1.

Mai biked $6 \frac34$ miles today, and Noah biked $4 \frac12$ miles. How many times the length of Noah’s bike ride was Mai’s bike ride?

A.

$\frac23$ times as far

B.

$\frac32$ times as far

C.

$\frac94$ times as far

D.

$\frac{243}{8}$ times as far

Answer:

$\frac32$ times as far

Teaching Notes

Students who select choice A have reversed the order of division. Students who select choice C have answered the question, “How many miles longer was Mai’s ride?” Students who select choice D have multiplied the lengths of the two rides together.

2.

Priya opened a new bag of soil and used $2\frac{1}{2}$ pounds of soil, which is $\frac{5}{8}$ of the bag. How many pounds of soil were in the full bag?

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

Answer:

Sample response: $\frac{5}{8} \boldcdot {?} = 2\frac{1}{2}$ , or $2\frac{1}{2} \div \frac{5}{8} = {?}$

There were 4 pounds of soil in the bag, because $2\frac{1}{2} \boldcdot \frac{8}{5} = \frac{40}{10} = 4$ .

Teaching Notes

Students can reason about the size of 1 full bag of soil by dividing (

2\frac{1}{2} \div \frac{5}{8}={?}

) or by thinking about a unknown factor (

\frac{5}{8} \boldcdot {?} = 2\frac{1}{2}

). 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

2\frac{1}{2} \div 5

to find the weight of soil in

\frac{1}{8}

of a bag and then multiply the result (

\frac{1}{2}

) by 8 to find the weight in a full bag.

3.

Select all statements that show correct reasoning for finding $15 \div \frac29$ .

A.

Multiply 15 by 2, then divide by 9.

B.

Multiply 15 by 9, then divide by 2.

C.

Multiply 15 by $\frac19$ , then multiply by 2.

D.

Multiply 15 by 9, then multiply by $\frac12$ .

E.

Multiply

\frac{2}{9}

by 1, then multiply by

\frac{1}{15}

.

Answer: B, D

Teaching Notes

Students who select choice A or C have picked a statement that is equivalent to "Multiply by $\frac{2}{9}$ " instead of "Multiply by $\frac{9}{2}$ ". Students who select choice E have reversed the dividend and divisor, picking a way to find $\frac{2}{9} \div 15$ instead of $15 \div \frac{2}{9}$ .

4.

Divide.

$\frac34 \div \frac15$
$\frac92 \div \frac34$
$\frac{4}{9} \div \frac{8}{15}$
$5 \frac23 \div \frac32$

Answer:

$\frac{15}{4}$ or $3 \frac34$ (or equivalent)
6 (or equivalent)
$\frac{60}{72}$ or $\frac{5}{6}$ (or equivalent)
$\frac{34}{9}$ or $3 \frac79$ (or equivalent)

Teaching Notes

Students apply the algorithm for dividing fractions.

5.

Andre draws this tape diagram for $3 \div \frac{2}{3}$ :

Andre says that $3 \div \frac{2}{3} = 4 \frac{1}{3}$ because there are 4 groups of $\frac{2}{3}$ , with 1 group of $\frac{1}{3}$ left over. Do you agree with Andre? Explain or show your reasoning.

Answer:

Sample response: No, I disagree. There are 4 groups of $\frac{2}{3}$ in $2 \frac{2}{3}$ . Then there is $\frac{1}{3}$ left and this makes $\frac{1}{2}$ of another group. So, there are $4 \frac{1}{2}$ groups of $\frac{2}{3}$ in 3.

Minimal Tier 1 response:

Work is complete and correct.
Sample: No, because $3 \div \frac 2 3 = 4 \frac 1 2$ , not $4 \frac 1 3$ .

Tier 2 response:

Work shows general conceptual understanding and mastery, with some errors.
Sample errors: Disagreement with Andre but a minor error in logic or calculation leads to a different result other than $4 \frac 1 2$ ; agreement with Andre with a reasonable but flawed argument.

Tier 3 response:

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

Teaching Notes

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

6.

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

How many $\frac{1}{3}$ -inch cubes does it take to fill the box?

Answer:

560 (8 cubes fit along the width of the box, 10 cubes fit along the length, and 7 cubes fit vertically.)

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 $\frac13$ inch it will take to reach a length of $2\frac{2}{3}$ inches, etc.

7.

Lin has two small baking pans, each shaped like a rectangular prism. For each question, explain or show your reasoning.

The base of Lin’s first pan has an area of $11\frac{1}{4}$ square inches. The length of the pan is $4\frac{1}{2}$ inches. What is the width of the pan?
Lin’s second pan has a length of $\frac 8 3$ inches, a width of $\frac{15} 4$ inches, and a height of $\frac 3 2$ inches. What is the volume of the second pan?

Answer:

$2 \frac 1 2$ inches (or equivalent). The width is the solution to $4 \frac 1 2 \boldcdot w = 11 \frac 1 4$ . By writing each mixed number as a fraction, the problem is made simpler: $\frac 9 2 w = \frac {45} 4$ . Then $w = \frac 5 2$ .
15 cubic inches (or equivalent). The volume is the product of the pan’s length, width, and height: $\frac 3 2 \boldcdot \frac 8 3 \boldcdot \frac{15} 4 = 15$ .

Minimal Tier 1 response:

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

$\frac 5 2$ inches, because $11 \frac 1 4 \div 4 \frac 1 2 = \frac 5 2$ .
15 cubic inches, because $\frac 3 2 \boldcdot \frac 8 3 \boldcdot \frac {15} 4 = 15$ .

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 we expect students to understand the context of the problem, some may still have difficulty understanding without a diagram. The first problem is about area, even though the pan is described as a rectangular prism.