Mid-Unit Assessment

1.

Select all the expressions that have a value greater than 1.

A.

$5 \div \frac{9}{10}$

B.

$\frac{3}{5} \div 6$

C.

$\frac{4}{5} \div \frac{3}{7}$

D.

$2\frac{3}{5} \div 1\frac{3}{5}$

E.

$\frac{12}{5} \div \frac{13}{5}$

Answer: A, C, D

Teaching Notes

Students who select choice B or E, or who fail to select choice A, may be reversing the order of the division. This is notable in A and B, where whole numbers are used. Students who fail to select choice C may assume that, because the dividend is less than one, the quotient is automatically less than one. Students who fail to select choice D may have made an arithmetic calculation error, but should have been able to determine the answer without calculation.

2.

A recipe for homemade glue calls for $\frac{2}{3}$ of a cup of water per batch. Elena used $5\frac{1}{3}$ cups of water to make multiple batches of glue. How many batches did she make?

Answer:

8 batches

Teaching Notes

Students divide a fraction by a fraction to answer a “How many groups?” question.

3.

Jada has read $\frac{3}{5}$ of a book. She has read 75 pages so far. How many pages are in the whole book?

A.

45 pages

B.

105 pages

C.

120 pages

D.

125 pages

Answer:

125 pages

Teaching Notes

Students might select choice A if they misinterpreted the question and found $\frac{3}{5} \boldcdot 75$ , thinking that 75 is the total number of pages in the book. Students who select choice B may have thought that $\frac{2}{5}$ of 75 needed to be added to the 75 pages that were read. Students who select choice C may be taking a ballpark guess or may have miscalculated.

4.

Han used $1\frac{1}{4}$ gallons of paint to paint $\frac{5}{7}$ of a fence. How many gallons will it take to paint the whole fence?

A.

$\frac{1}{2}$ gallon

B.

$\frac{4}{7}$ gallon

C.

$\frac{7}{4}$ gallons

D.

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

Answer:

$\frac{7}{4}$ gallons

Teaching Notes

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

Students who select choice A may be thinking about the amount of additional paint needed to complete the painting. Students who select choice B may have reversed the quantities or interpreted the relationship as $1 \frac{1}{4} \boldcdot ? = \frac{5}{7}$ . Students who select choice D may have estimated the amount of paint needed as being a little more than $\frac{1}{4}$ .

5.

Write a multiplication equation that represents the question: How many $\frac 3 8$ s are in $\frac 5 4$ ?
Write a division equation that represents the question: How many $\frac 3 8$ s are in $\frac 5 4$ ?

Answer:

$?\, \boldcdot \frac38 = \frac54$ (or equivalent)
$\frac54\div \frac38 = \, ?$ (or equivalent)

Teaching Notes

Students write multiplication and division equations for fraction division problems.

6.

What is $\frac{3}{4} \div \frac{1}{8}$ ? Draw a diagram to show your reasoning.

Answer:

6. (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: $\frac18$ fits in $\frac14$ twice, and $\frac34$ is three $\frac14$ s. $2 \boldcdot 3 = 6$ .

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.

7.

Clare is painting some doors that are all the same size. She used 2 liters of paint to cover $1\frac{3}{5}$ doors. How many liters of paint are needed for 1 door?

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

Answer:

Equations: $2 \div 1\frac{3}{5} = {?}$ , or $1\frac{3}{5} \boldcdot {?} = 2$ .

$1\frac{1}{4}$ or $\frac{5}{4}$ liters are needed to paint 1 door. Sample reasoning:

$1\frac{3}{5}$ is $\frac{8}{5}$ . If 8 fifths doors require 2 liters of paint, then each 1 fifth requires $\frac{2}{8}$ or $\frac14$ liter of paint. There are 5 fifths in 1, so $5 \boldcdot \frac{1}{4}$ , or $\frac{5}{4}$ , liters are needed.

Minimal Tier 1 response:

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

See diagram. $1\frac14$ or $\frac54$ liters.

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 solve the problem using a method of their choice and showing their reasoning.