Readiness Check

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.

If most students struggle with this item, support them in their understanding of fractions by visually representing the fractions. Students will benefit from placing fractions on a number line and thinking about each fraction by comparing it to 0, $\frac{1}{2}$, and 1.

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.

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.

To answer this question, students must in some way explain how to determine the number of times $\frac{1}{2}$ goes into 5. 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.

Students interpret situations that involve multiplicative comparison and the area of a rectangle, both of which can be reasoned by multiplication. Students write expressions to reason about the situations and answer the questions.

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.

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. Students will develop a more general algorithm for dividing fractions in this unit.

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.

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 720 represents and what “cubic centimeters” mean.