This activity serves two purposes: to explicitly bridge “How many of this in that?” questions to division expressions, and to explore division situations in which the quotients are not whole numbers.

Once again, students move from reasoning concretely and visually to reasoning symbolically. They start by thinking about “How many rhombuses are in a trapezoid?” and then express that question as multiplication (, or ) and division (). Students think about how to deal with a remainder in such problems.

As students discuss in groups, listen for their explanations for the question “How many rhombuses are in a trapezoid?” Select two students to share later—one person to elaborate on Diego’s argument, and another to support Jada’s argument.

This activity works best when each student has access to pattern blocks. If pattern blocks are not available, consider using the digital version of the activity. In the digital version, students use an applet to investigate fractional relationships with pattern blocks. The applet allows students to place and compare pattern blocks with or without a grid.

This activity gives students another opportunity to use diagrams and equations to make sense of division involving fractions. Reasoning repeatedly about “How many of these are in that?” questions in different situations allows students to notice regularity in how the quantities can be related by multiplication and division (MP8).

For each situation, many kinds of visual representations are possible, but creating a meaningful representation may be challenging nonetheless. Urge students to use the contexts to generate ideas for useful diagrams, and to start with a draft and modify it as needed. Students may also use the fraction strips in the Warm-up as a starting point for drawing diagrams.

Monitor for diagrams that:

• Clearly show the given quantities in the situation (including the size of 1 group and a total amount).
• Effectively communicate the answer to the question (the number of groups).
• Can be generalized to different situations. Tape diagrams, and number lines are some possible examples.

Use Compare and Connect to help students compare, contrast, and connect the different diagrams and equations. Invite groups to prepare a display that shows the representations they created to answer their assigned question. Encourage students to include details that will help others interpret their thinking. For example, specific language, use of different colors, shading, arrows, labels, and notes. Give students time to investigate each other’s work. Urge them to pay attention to two things: how the number of groups, the size of each group, and a total amount can be seen in each diagram, and whether the equations make sense.

During the whole-class discussion, ask questions such as:

• “How are the diagrams (or equations) for different situations alike? How are they different?”
• “How do the size of each group, the number of groups, and a total amount show up in each diagram (or equation)?”
• “Are there any benefits or drawbacks to one type of diagram compared to another?”

As time permits, invite previously selected students to share features of their diagrams and how the diagrams support their reasoning.