Mid-Unit Assessment

1.

Which expression has a value that is less than 1?

A.

$4 \div \frac23$

B.

$\frac34 \div \frac23$

C.

$\frac{14}{8} \div \frac{15}{8}$

D.

$2 \frac48 \div 1 \frac58$

Answer:

C

Teaching Notes

Students who select choice A may be reversing the order of the division. Students who select choice B may think that since both the dividend and divisor are less than 1, the quotient must be less than 1. Students who select choice D may have made an arithmetic calculation error, but should have been able to determine the answer without calculation.

2.

Andre used $4\frac12$ cups of vinegar to make multiple batches of a cleaning liquid. The recipe says to use $\frac{3}{4}$ cup of vinegar for each batch. How many batches did he make?

A.

6 batches

B.

$4\frac12$ batches

C.

$3\frac38$ batches

D.

$\frac16$ of a batch

Answer:

A

Teaching Notes

Students divide a fraction by a fraction to answer a “How many groups?” question. Students who select choice B may have assumed that each batch used 1 cup of vinegar. Students who select choice C multiplied the numbers in the problem, perhaps misunderstanding what information was given and what they needed to find. Students who select choice D reversed the numbers when dividing.

3.

Kiran has used $\frac45$ of the pieces in his jigsaw puzzle. He has used 120 pieces. How many pieces are in the whole puzzle?

A.

96

B.

125

C.

150

D.

144

Answer:

C

Teaching Notes

Students who select choice A might have misinterpreted the question and find $\frac{4}{5} \boldcdot 120$ , thinking that 120 is the total number of pieces in the puzzle. Students who select choice B may be taking a ballpark guess or may have miscalculated. Students who select choice D may have thought that $\frac{1}{5}$ of 120 needed to be added to the 120 pieces that were used.

4.

Priya used $1\frac{1}{2}$ gallons of paint to paint $\frac{3}{5}$ of a shed.

How many gallons of paint will it take to paint the whole shed?

Answer:

$2\frac{1}{2}$ gallons.

Teaching Notes

Students divide a fraction by a fraction to answer a “How much in one group?” question.

5.

Write a multiplication equation that represents the question “How many $\frac23$ s are in $\frac98$ ?”
Write a division equation that represents the question “How many $\frac23$ s are in $\frac98$ ?”

Answer:

$? \boldcdot \frac23 = \frac98$ (or equivalent)
$\frac98 \div \frac23 = {?}$ (or equivalent)

Teaching Notes

Students write multiplication and division equations for fraction division problems.

6.

What is $\frac{5}{6} \div \frac{1}{12}$ ? Draw a diagram to show your reasoning.

Answer:

10. (Students may draw a number line, a tape diagram, or a different representation.) Sample reasoning:

Minimal Tier 1 response:

Work is complete and correct.
Sample: $\frac{5}{6}$ is equivalent to $\frac{10}{12}$ , and there are 10 groups of $\frac{1}{12}$ in $\frac{10}{12}$ .

Tier 2 response:

Work shows general conceptual understanding and mastery, with some errors.
Acceptable errors: Minor errors in representation cause an error in determining the answer.
Sample errors: Representation includes minor errors such as a mislabeled number line or tape diagram; correct answer with no diagram; correct answer based on an equation only.

Tier 3 response:

Significant errors in work demonstrate lack of conceptual understanding or mastery.
Sample errors: Incorrect answer with no diagram; incorrect answer with work based on an equation only; major error in representation that shows a lack of understanding of fractions, such as counting on a number line by denominators.

Teaching Notes

This problem asks students to demonstrate a visual understanding of the division of fractions, rather than to just supply the result of a calculation.

7.

Noah is painting boxes that are all the same size. He used 2 ounces of paint to cover $1\frac{2}{3}$ boxes. How many ounces of paint are needed for 1 box?

Write a multiplication or division equation to represent the question, then find the answer. Show your reasoning.

Answer:

Equations: $2 \div \frac{5}{3} = {?}$ , or $\frac{5}{3} \boldcdot {?} = 2$ .

$\frac{6}{5}$ or $1\frac{1}{5}$ ounces of paint are needed. Sample reasoning:

$1\frac{2}{3}$ is $\frac{5}{3}$ . If 5 thirds boxes require 2 ounces of paint, then each 1 third requires $\frac25$ ounces. There are 3 thirds in 1, so $3 \boldcdot \frac{2}{5}$ , or $\frac{6}{5}$ , ounces are needed.

Minimal Tier 1 response:

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

See diagram. $\frac65$ (or equivalent) ounces of paint

Tier 2 response:

Work shows good conceptual understanding and mastery, with either minor errors or correct work with insufficient explanation or justification.
Sample errors: Miscounting the number of segments in a tape diagram or including one too many or too few segments, mistakes converting mixed numbers to improper fractions; equation is correct or shows a minor error.

Tier 3 response:

Work shows a developing but incomplete conceptual understanding, with significant errors.
Sample errors: Arithmetic errors that show a difficulty interpreting fractions; errors that involve inverting the wrong fraction or reversing the order of division; a part of the diagram or explanation is off the mark.

Tier 4 response:

Work includes major errors or omissions that demonstrate a lack of conceptual understanding and mastery.
Sample errors: Diagram, equation, and explanation are significantly off; work does not show evidence of framework for understanding division.

Teaching Notes

This problem asks students to write a multiplication or division equation to describe a situation, and then to solve the problem using a method of their choice and showing their reasoning.