Lesson 4

Equal Groups of Non-Unit Fractions

Warm-up: Notice and Wonder: Thirds (10 minutes)

Narrative

This warm-up prompts students to examine a diagram representing equal groups of non-unit fractions. The understandings elicited here allow students to discuss the relationship between the product of a whole number and a unit fraction and that of a whole number and a non-unit fraction with the same denominator.

Launch

• Groups of 2
• Display the image.

Activity

• “What do you notice? What do you wonder?”
• 1 minute: quiet think time
• 1 minute: partner discussion
• Share and record responses.

Student Facing

What do you notice? What do you wonder?

Activity Synthesis

• If no students notice or wonder about equal groups, ask, “What groups do you see and how do you see them?” (4 wholes, each whole has $$\frac{2}{3}$$ shaded)
• “How many thirds do you see?” (8 thirds)
• “How are these diagrams different than those we've seen so far in this unit?” (Previously, each whole has only one shaded part. These have two shaded parts each.)
• “Today, we will think about situations that involve equal groups but now each group has non-unit fractions.”

Activity 1: Jars of Jam (15 minutes)

Narrative

In this 5 Practices activity, students reason about a situation that involves finding the product of a whole number and a non-unit fraction. They may rely on what they previously learned about multiplying a whole number and a unit fraction, but can reason in any way that makes sense to them. The goal is to elicit different strategies and help students see the connections between strategies and with their earlier work. Students reason abstractly and quantitively as they solve the problem (MP2) and construct arguments (MP3) as they share their reasoning during the synthesis.

Monitor for the students who:

• draw a drawing or a diagram to show 5 groups with three $$\frac{1}{4}$$s in each group and count the total number of fourths
• reason additively, by finding the value of $$\frac{3}{4} + \frac{3}{4} + \frac{3}{4} + \frac{3}{4} + \frac{3}{4}$$, or by adding smaller groups of $$\frac{3}{4}$$ at a time, for instance, 2 groups of $$\frac{3}{4}$$, another 2 groups, and 1 more group
• reason multiplicatively, for instance, by thinking of $$\frac{3}{4}$$ as $$3 \times \frac{1}{4}$$ and then finding $$5 \times 3 \times \frac{1}{4}$$, or by reasoning about $$5 \times \frac{3}{4}$$

Students who see the situation as $$5 \times \frac{3}{4}$$ may, based on their earlier work, generalize that the value is $$\frac{5 \times 3}{4}$$. Encourage them to clarify how they know this is the case.

During the synthesis, sequence student presentations in the order listed.

Reading: MLR6 Three Reads. “We are going to read this 3 times.” After the 1st Read: “Tell your partner what this situation is about.” After the 2nd Read: “List the quantities. What can be counted or measured?” (number of jars, number of friends, number of cups of jam). After the 3rd Read: “What strategies can we use to solve this problem?”
Representation: Internalize Comprehension. Synthesis: Invite students to identify details they want to remember. Display the sentence frame: “The next time I need to represent the product of a whole number and a fraction, I will . . . .”
Supports accessibility for: Conceptual Processing, Organization, Memory

Launch

• Groups of 2
• Read the first problem as a class.
• Invite students to share what they know about homemade jams or any experience in making them.
• If needed, remind students that measuring cups come in different fractional amounts, such as $$\frac{1}{4}$$, $$\frac{1}{2}$$, and $$\frac{3}{4}$$.

Activity

• “Work independently on the problem. Explain or show your reasoning so that it can be followed by others. Afterwards, share your thinking with your partner.”
• 5 minutes: independent work time
• 2–3 minutes: partner discussion
• Monitor for the strategies listed in the activity narrative.

Student Facing

Elena fills 5 small jars with homemade jams to share with her friends. Each jar can fit $$\frac{3}{4}$$ cup of jam. How many cups of jam are in the jars? Explain or show your reasoning.

If you have time: Elena still has some jam left. She takes 2 large jars and puts $$\frac{5}{4}$$ cups of jam in each jar. How many cups of jam are in the jars?

Student Response

If students are not sure how to represent or reason about 5 groups of $$\frac{3}{4}$$, consider asking them: “How would you represent (or think about) 5 groups of $$\frac{1}{4}$$?” and “How can you build on that representation (or strategy) to find how much is in 5 groups of $$\frac{3}{4}$$?”

Activity Synthesis

• Select previously identified students to share their responses. Display or record their work for all to see.
• “What multiplication expression can represent the amount of jam in the jars? How do you know?” ($$5 \times \frac{3}{4}$$ or $$5 \times 3 \times \frac{1}{4}$$, because there are 5 equal groups of $$\frac{3}{4}$$.)
• “Where do you see the 5 groups in each strategy presented? Where do we see the $$\frac{3}{4}$$?”
• “How is finding the value of $$5 \times \frac{3}{4}$$ like finding the value of $$5 \times \frac{1}{4}$$?” (They're both about finding the total amount in equal groups. They both involve a whole number of groups and a fraction in each group.)
• “How is it different?” (The amount in each group is a non-unit fraction instead of a unit fraction.)

Activity 2: How Do We Multiply? (20 minutes)

Narrative

The purpose of this activity is for students to use diagrams to reason about products of a whole number and a non-unit fraction with diagrams, building on their work with diagrams that represent products of a whole number an a unit fraction. They begin to generalize that the number of shaded parts in a diagram that represents $$n \times \frac{a}{b}$$ is $$n \times a$$ and to explain that generalization (MP8).

Launch

• Groups of 2
• “Let's represent some other products of a whole number and a fraction and find their values.”

Activity

• “Take a few quiet minutes to work on the activity. Afterwards, share your responses with your partner.”
• 5–7 minutes: independent work
• 2–3 minutes: partner discussion
• Monitor for the strategies students use to reason about the last two problems.
• Identify students who reason visually (using diagrams), additively, and multiplicatively to share in the synthesis.

Student Facing

1. This diagram represents $$\frac{2}{5}$$.

1. Show how you would use or adjust the diagram to represent $$4 \times \frac{2}{5}$$.
2. What is the value of the shaded parts in your diagram?
2. This diagram represents $$\frac{5}{8}$$.

1. Show how you would use or adjust the diagram to represent $$3 \times \frac{5}{8}$$.
2. What is the value of the shaded parts in your diagram?
3. Find the value of each expression. Draw a diagram if you find it helpful. Be prepared to explain your reasoning.

1. $$2 \times \frac {1}{6}$$
2. $$2 \times \frac{4}{6}$$
3. $$2 \times \frac {5}{6}$$
4. $$4 \times \frac{5}{6}$$
4. Mai said that to multiply any fraction by a whole number, she would multiply the whole number and the numerator of the fraction and keep the same denominator. Do you agree with Mai? Explain your reasoning.

Activity Synthesis

• Discuss the four multiplication expressions in the third problem.
• Select 1–2 students who might have drawn diagrams for all expressions.
• “How does your diagram show the value of $$2 \times \frac{4}{6}$$?” (There are 2 groups of $$\frac{4}{6}$$ , so there are 8 sixths shaded, which is $$\frac{8}{6}$$.)
• Select 1–2 students who drew a diagram for some expressions and reason numerically for others.
• “Why did you choose to draw a diagram for some expressions and to do something else for others?” (After drawing the first two diagrams, I realized that I'd have to draw a lot of groups or parts, so I thought about the numbers instead.)
• Select 1–2 students who reasoned about all expressions numerically.
• “How did you find the value of the expressions without drawing diagrams at all?” (I saw a pattern in earlier problems, that we can multiply the whole number and the numerator of the fraction and keep the denominator.)
• Discuss the last problem in the lesson synthesis.

Lesson Synthesis

Lesson Synthesis

“Mai said she can multiply any fraction by a whole number by multiplying the whole number by the numerator and keeping the denominator.”

Invite students to share whether they agree or disagree with Mai's statement and to explain their reasoning.

“Let’s discuss Mai’s reasoning using the expression $$4 \times \frac{2}{3}$$ and the diagram from today’s warm-up.”

“Why can we multiply $$4 \times 2$$ to get the numerator of the product?” (We can think in terms of thirds. The diagram shows 4 groups of 2 thirds, or 8 thirds total.)

“Why is the denominator of the product the same as the fraction in the expression?” (The denominator represents the size of the equal parts in each group. The size of the part doesn’t change when the number of groups increases.)