standard_description: Interpret multiplication as scaling (resizing), by:
- Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence a/b = (n×a)/(n×b) to the effect of multiplying a/b by 1.
- Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication.
problem_activity: Narrative
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.
Representation: Internalize Comprehension
. Synthesis: Invite students to identify which details were needed in order to solve the problem. Display the sentence frame: “The next time I need to find the volume of a figure composed of two rectangular prisms, I will . . .“
Supports accessibility for: Conceptual Processing, Organization, Memory
Required Materials
Materials to Copy
Isometric Dot Paper Standard
Launch
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.)
Activity
“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.
Student Facing
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Partner A: Find the volume of Figure 1.
Partner B: Find the volume of Figure 2.
Figure 1
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Figure 2
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Student Response
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Advancing Student Thinking
If students do not find the correct volume of the figure, ask, “Where do you see rectangular prisms in this figure?”
Activity Synthesis
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.)


aligns: no