Lesson 4

• Groups of 2
• Display the image.

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

• 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.”

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. 

• 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 \(\frac{3}{4}\).

• “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.

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 \(\frac{3}{4}\)?”

• 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.)  

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).

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

• “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.

• 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.