The purpose of this activity is for students to find quotients of a whole number by a unit fraction and observe patterns in how the size of the numerator and denominator influence the size of the quotient. In a previous lesson, students were provided a tape diagram. In this lesson, students draw a diagram but they may also reason about the size of the quotients in other ways. When students notice a pattern or repetitive action in computation, they are looking for and expressing regularity in repeated reasoning (MP8).

This activity uses MLR7 Compare and Connect. Advances: Representing, Conversing.

• “Work with your partner to create a visual display that shows your thinking about how the problem sets are the same and different. You may want to include details such as notes, diagrams, drawings, etc., to help others understand your thinking.”
• 2–5 minutes: independent or group work
• 5–7 minutes: gallery walk
• “What is the same and what is different between the two sets of problems?”
• 30 seconds: quiet think time
• 1 minute: partner discussion
• Additional connections could include:
  • “What is changing in each problem set?” (The size of the number being divided is changing in problem set A and the size of the pieces the number is being divided into is changing in problem set B.)
  • “If the patterns continued, what would be the next equation in each set?”
• “Why is the quotient getting larger in both sets of problems?” (In problem set A, it is getting larger because you have 4 more size pieces in each additional whole. In problem set B, it is getting larger because the size of the pieces are getting smaller, so you have more of them.)

The purpose of this activity is for students to match situations to expressions and then find the value of the expressions (MP2). Students see expressions that show both quotients of a whole number by a fraction and quotients of a fraction by a whole number. Students need to think carefully about the situations to make sure the expression they choose matches the situation (MP2).

• “How did you decide to match each situation to an expression?”
• Refer to each situation and ask, “How does the situation represent division?”

• Ask previously selected students to share their solutions.
• “How did you decide which expression matched which situation?” (I thought about the number of things being divided and whether it was a fraction or a whole number. I thought about the size of the pieces.)
• Display: 
• “Describe how this equation represents the situation in the first problem.” (There are 3 cups of kernels in the bowl and a serving is a cup, so there are 12 servings in the bowl.)
• Display:
• “How do you know that this equation does not match the situation?” (The answer is a fraction. That doesn’t make sense because there is more than 1 serving in 3 cups.)