In this 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 approaches and help students make connections with their earlier work. Students reason abstractly and quantitatively as they solve the problem (MP2), and construct arguments (MP3) as they share their reasoning during the Activity Synthesis.

Monitor for and select to share in the Activity Synthesis students with the following approaches:

• Reason about equal groups and unit fractions, by creating a drawing or a diagram to show 5 groups, with three s in each group, and counting the total number of fourths.
• Reason additively, by finding the value of , or by adding smaller groups of at a time, for instance, 2 groups of , another 2 groups, and 1 more group.
• Reason multiplicatively, for instance, by thinking of as and then finding , or by reasoning about .

The approaches are sequenced from more concrete to more abstract to elicit the different ways students may be thinking about equal groups and multiplying a whole number and a non-unit fraction. Students who see the situation as may, based on their earlier work, generalize that the value is . Encourage them to clarify how they know this is the case. Aim to elicit both key mathematical ideas and a variety of student voices, especially students who haven't shared recently.

• Invite previously selected students to share in the given order. Record or display their work for all to see.
• Connect students’ approaches by asking:
  • “¿Dónde ven los 5 grupos en cada una de estas estrategias?” // “Where do you see the 5 groups in each strategy presented?” (5 groups in a diagram, adding 5 times or multiplying by 5)
  • “¿Dónde ven el ?” // Where do we see the ?” (3 one-fourths and that’s or in each group, or added 5 times)
• Connect students’ approaches to the learning goal by asking:
  • “¿Qué expresión de multiplicación puede representar la cantidad de slime que hay en los tarros? ¿Cómo lo saben?” // “What multiplication expression can represent the amount of slime in the jars? How do you know?” ( or , because there are 5 equal groups of .)
  • “¿En qué se parecen encontrar el valor de y encontrar el valor de ?” // “How is finding the value of like finding the value of ?” (Both are about finding the total amount in equal groups. 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, building on their work with diagrams that represent products of a whole number and a unit fraction. They begin to generalize that the number of shaded parts in a diagram that represents is , and explain that generalization (MP8).

• Discuss the four multiplication expressions in the third problem.
• Select 1–2 students who might have drawn diagrams for all four expressions.
• “¿Cómo muestra su diagrama el valor de ?” // “How does your diagram show the value of ?” (There are 2 groups of , so there are 8 sixths shaded, which is .)
• Select 1–2 students who drew a diagram for some expressions and reasoned 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 four 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.