Lesson 4

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.

• Groups of 2
• Display the image.

• “¿Qué observan? ¿Qué se preguntan?” // “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, “¿Qué grupos ven y de qué manera los ven?” // “What groups do you see and how do you see them?” (4 wholes, each whole has \(\frac{2}{3}\) shaded)
• “¿Cuántos tercios ven?” // “How many thirds do you see?” (8 thirds)
• “¿En qué se diferencian estos diagramas y los diagramas que hemos visto hasta ahora en esta unidad?” // “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.)
• “Hoy vamos a pensar en situaciones de grupos iguales, pero ahora cada grupo va a tener cantidades que son fracciones no unitarias” // “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 \(\frac{3}{4}\).

• “Trabajen en el problema individualmente. Expliquen o muestren su razonamiento para que los demás lo puedan entender. Después, compartan con su pareja cómo pensaron” // “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: “¿Cómo puedes representar (o imaginarte) 5 grupos de \(\frac{1}{4}\)?” // “How would you represent (or think about) 5 groups of \(\frac{1}{4}\)?” and “¿Cómo puedes basarte en esa representación (o estrategia) para encontrar cuánto hay en 5 grupos de \(\frac{3}{4}\)?” // “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.
• “¿Qué expresión de multiplicación puede representar la cantidad de mermelada que hay en los tarros? ¿Cómo lo saben?” // “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 \(\frac{3}{4}\).)
• “¿Dónde ven los 5 grupos en cada una de estas estrategias? ¿Dónde ven el \(\frac{3}{4}\)?” // “Where do you see the 5 groups in each strategy presented? Where do we see the \(\frac{3}{4}\)?”
• “¿En qué se parecen encontrar el valor de \(5 \times \frac{3}{4}\) y encontrar el valor de \(5 \times \frac{1}{4}\)?” // “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.)
• “¿En qué son diferentes?” // “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
• “Representemos más productos de un número entero por una fracción y encontremos sus valores” // “Let's represent some other products of a whole number and a fraction and find their values.”

• “En silencio, trabajen en la actividad durante unos minutos. Después, compartan sus respuestas con su compañero” // “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.
• “¿Cómo muestra su diagrama el valor de \(2 \times \frac{4}{6}\)?” // “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.
• “¿Por qué decidieron dibujar un diagrama para algunas expresiones y hacer algo diferente para otras?” // “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.
• “¿Cómo encontraron el valor de las expresiones sin haber dibujado diagramas?” // “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.