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 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.
MLR7 Compare and Connect.
Invite students to think about details they can include that will help others understand their thinking. For example, specific language, using different colors, shading, arrows, labels, notes, diagrams or drawings. Give students time to investigate each others’ work. During the whole-class discussion, ask, “What kinds of details or language helped you understand your partner’s work?”
Advances: Representing, Conversing
Launch
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.”
Activity
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.
Student Facing
Explain how each expression represents the volume of the figure. Show your thinking. Organize it so it can be followed by others.
\(((2 \times 3) \times4) + ((3 \times 3) \times 2)\)
\((5 \times 6) + (3 \times 4)\)
[IMG]
How does each expression represent the volume of the prism? Explain or show your thinking. Organize it so it can be followed by others.
\((5\times8\times6) + (5\times4\times9)\)
cubic inches
\((5\times4\times3) + (5\times12\times6)\)
cubic inches
[IMG]
Student Response
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Advancing Student Thinking
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.
Activity Synthesis
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.)


aligns: no