Check Your Readiness

1.

Select all the fractions that are greater than $\frac12$ .

A.

$\frac{50}{101}$

B.

$\frac{50}{100}$

C.

$\frac{12}{8}$

D.

$\frac{8}{18}$

E.

$\frac{8}{15}$

F.

$\frac{5}{8}$

Answer:

C, E, F

Teaching Notes

To make comparisons, students may use a fraction that is equivalent to $\frac{1}{2}$ , or they may simply look at whether the numerator is more or less than one-half of the denominator.

Students who struggle with this problem likely do not understand the part-whole interpretation of a fraction. They may benefit from spending more time working with visual representations of fractions, such as diagrams, fraction strips, and number lines.

2.

What is $2 \div 5$ ? Draw a diagram that explains how you know.

Answer:

$\frac{2}{5}$ . Sample reasoning:

$2 \div 5$ means 2 wholes that are each being divided into 5 equal pieces so they can be put into 5 groups. Taking 1 piece from each whole gives 5 groups 2 of the ⅕s, or $\frac{2}{5}$ , in each.
There are 10 fifths in 2 wholes. Dividing the 10 fifths into 5 groups means 2 fifths in each group.

Teaching Notes

In this problem, students demonstrate their understanding of integer division. From grade 5, students should have experience drawing diagrams to explain the operation of division. Their diagrams need not match the sample responses as long as they convey an understanding of division.

If most students struggle with this item, use the diagrams created in Lesson 2, Activity 2, to support student understanding of the two ways to think about division, and label the diagrams for clarity. This concept will continue throughout the next few lessons.

3.

Write $2\frac{1}{6}$ and $10\frac{2}{7}$ as fractions with only a numerator and a denominator.

Answer:

$\frac{13}{6}$ and $\frac{72}{7}$

Teaching Notes

Students will need to write equivalent fractions for mixed numbers when performing multiplication and division in this unit.

If most students struggle with this item, support students in understanding that a mixed number is a sum of a whole number and a fraction. Use fraction strips (such as provided in Lesson 5, Activity 1) or number line diagrams to revisit the idea that a whole number can be expressed as a fraction in the form of $\frac{a}{b}$ , which can then be added to the fractional part of a mixed number.

4.

Jada has 4 meters of ribbon. How many $\frac{1}{3}$ -meter pieces can she cut from the ribbon? Show your reasoning.

Answer:

12 pieces. Sample reasoning:

$12 \boldcdot \frac{1}{3} = 4$
Three $\frac13$ meter pieces can be cut from every 1 meter, so $4 \boldcdot 3$ , or 12, pieces can be cut from 4 meters.

Teaching Notes

To answer this question, students must in some way explain how to determine the number of times $\frac{1}{3}$ goes into 4. Their reasoning will provide insight into how students are thinking about situations that involve division. Students need not use the word “division” or perform the operation of division as part of their answer.

If most students struggle with this item, address any misconceptions during the discussion of Lesson 1, Activity 2.

5.

For each situation, write a multiplication expression or a division expression that represents a way to answer each question. Then, find the answer.

One weekend, Mai exercised for 60 minutes. That same weekend, Tyler exercised $\frac{3}{5}$ as long as Mai did. For how many minutes did Tyler exercise?
A rectangular park measures $\frac{1}{3}$ mile by $\frac{4}{5}$ mile. What is the area of the park in square miles?

Answer:

$\frac {3}{5} \boldcdot 60$ . Tyler exercised for 36 minutes.
$\frac{1}{3} \boldcdot \frac{4}{5}$ . The area is $\frac{4}{15}$ square mile.

Teaching Notes

Students interpret a situation that involves multiplicative comparison and decide on an expression that represents it.

If most students struggle with the first question, do the optional activity in Lesson 7 to support students in reasoning about multiplicative comparisons. If students struggle with the question about the area of a rectangle, before they find unknown side lengths by dividing, use one or more optional activities in Lesson 13 to solidify students’ understanding about finding unknown areas by multiplying.

6.

Find the value of each expression.

$\frac{3}{5} \boldcdot \frac{8}{12}$
$6 \div \frac{1}{3}$
$\frac{1}{8} \div 2$
$\frac{7}{8} + \frac{3}{5}$

Answer:

$\frac{24}{60}$ (or equivalent)
18
$\frac{1}{16}$
$\frac{59}{40}$ or $1 \frac{19}{40}$

Teaching Notes

This problem assesses students’ comfort with the operations on fractions they learned in grade 5, including division of a whole number by a unit fraction and division of a unit fraction by a whole number. In this unit, students will develop a more general algorithm for dividing fractions.

If most students struggle with the two division expressions, clarify any misconceptions during the Synthesis of Lesson 2, Activity 2. If they struggle to multiply two fractions, consider using diagrams to represent products of benchmark fractions such as $\frac{1}{2} \boldcdot \frac{2}{5}$ or $\frac{1}{3} \boldcdot \frac{1}{4}$ and supporting students in generalizing the process. If students struggle with addition of fractions with unlike denominators, find opportunities to write equivalent fractions before Lesson 16.

7.

A rectangular prism measures 6 cm in length, 7 cm in width, and 10 cm in height. What is its volume?

Answer:

420 cubic centimeters, or 420 cm³

Teaching Notes

In the last section of this unit, students apply their understanding of fraction division to find unknown area and volume measurements.

If most students struggle with this item, use this question as part of the Launch for Lesson 14, Activity 3. Ask students what the number 420 represents and what “cubic centimeters” mean.