Lesson 9

The purpose of this Number Talk is for students to reason about place value relationships and the properties of multiplication. The elicited understandings and strategies will be helpful in later lessons and units when they multiply large numbers. In this unit, students produce and interpret multiplication expressions in terms of volume.

• Display one expression.
• “Give me a signal when you have an answer and can explain how you got it.”
• 1 minute: quiet think time

• Record answers and strategy.
• Keep expressions and work displayed.
• Repeat with each expression.

• “What patterns do you notice in the problems we solved?” (There is a 6 and 2 in each product. There are also factors of 10, and the 6 and 2 are sometimes multiplied by a factor of 10.)
• Consider asking:
  • “Who can restate _____’s reasoning in a different way?”
  • “Did anyone have the same strategy but would explain it differently?”
  • “Did anyone approach the problem in a different way?”

The purpose of this activity is for students to find the volume of figures composed of two non-overlapping right rectangular prisms by adding the volumes of the non-overlapping parts. There are different ways to decompose the figures. Monitor for students who break the figures apart differently and find the same total volume. To reinforce earlier work with cubic units of measure, ask students for the unit of measure in their response if they state the volume as only a number (MP2). If students finish early, give them isometric grid paper to draw a figure composed of two rectangular prisms for their partner to find the volume.

• Groups of 2
• Display the image from the student workbook with the missing side lengths.
• “How can we find the missing side lengths?”
• 1 minute: quiet think time
• 2 minutes: partner discussion
• Share and record responses on the image of the figure.  (We can subtract 5 from 7 in both cases.)

• “Now, each partner will find the volume of one of the figures and then switch papers. Your job is to then find the volume using a different decomposition strategy than your partner.”
• If there is time, the groups of 2 can make groups of 4 and share responses and strategies.
• 10 minutes: independent work time with partner discussion
• Give students access to isometric dot paper to draw the figures if they finish early.

If students do not find the correct volume of the figure, ask, “Where do you see rectangular prisms in this figure?”

• Display: Figure 2
• “How did you break up this shape to find the volume?” (I cut off the overhanging piece vertically to make two rectangular prisms. I cut off the bottom piece that the shape is resting on, horizontally.)
• “How did you find the side lengths of the rectangular prisms?” (Some of the lengths are provided, and I found the others by subtracting.)
• “Did you get the same volume when you broke up the figure differently? Why?” (Yes. The calculations were different but they both tell me how many cubic inches it takes to fill the shape.)

The goal of this activity is to represent expressions as decompositions of a figure made of two non-overlapping right rectangular prisms. This gives students an opportunity to interpret parentheses in expressions while also checking their understanding of different ways to represent the volume of rectangular prism; namely, length times width times height and area of a base times the corresponding height.
Students work abstractly and quantitatively in this problem (MP2) as they relate abstract expressions to decompositions of figures composed of two rectangular prisms.

• Groups of 2
• “Look at the first figure and think about how you might decompose it into 2 prisms.”
• 1 minute: quiet think time
• “Now you will consider how different expressions can represent the volume of figures.”

• 5 minutes: independent work time
• 5 minutes: partner discussion time
• Monitor for students who
  • use the numbers in the expressions to determine how to break up the figure
  • break up the figure and use this to identify the expressions.

If students do not describe how the expressions represent the volume of the figure, show them part of the expression, the numbers inside the parentheses for example, and ask them to explain which part of the figure is represented.

• Invite students to share how the expressions for the first prism represent its volume.
• “Why are there 3 factors in the expression \((2 \times 3) \times 4)\) but only 2 factors in the expression \((5 \times 6)\)?” (The 2, 3, and 4 are the length, width, and height of the prism in the back. The 5 is the height of the prism on bottom whose base area is 6.)
• “In the expression \((3 \times 3) \times 2\), what do the parentheses tell you?” (They tell me to first multiply 3 times 3, which gives me the area of the base, and then I multiply that by 2 to get the volume of the prism.)