standard_description: Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. For example, calculate mortgage payments.
problem_activity: Narrative
The purpose of this activity is for students to find the volume of a figure composed of two rectangular prisms and recognize that volume is additive. In the previous activity, students physically put two rectangular prisms together and then decomposed a figure into two rectangular prisms. In this activity, they are given an image of a figure composed of two rectangular prisms and asked to find the volume. They can do this in many ways. During the synthesis, connect the different strategies students used to the decomposition of the figure into two rectangular prisms.
As students experiment with different ways to group the cubes to efficiently count them, applying what they already know about the volume of rectangular prisms, they are looking for and making use of the structure of geometric objects (MP7). As students begin to generalize the idea that volume is additive, they are looking for and expressing regularity in repeated reasoning (MP8).
MLR7 Compare and Connect.
Synthesis: Invite students to prepare a visual display that shows the strategy they used to find the volume of each figure. Encourage students to include details that will help others interpret their thinking; for example, using specific language, or using different colors, shading, or arrows. Give students time to investigate each other’s work.
Advances: Representing
Launch
Groups of 2
Activity
5 minutes: individual work time
5 minutes: partner discussion
As students work, monitor for students who:
find two (or more) rectangular prisms that can be put together to make the figure and adding their volumes.
find the number of cubes in a base layer and multiplying by the number of layers, even though the base layer is not a rectangle (figure d).
Student Facing
Find the volume of each figure. Explain or show your reasoning.
a.
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b.
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c.
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d.
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Student Response
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Advancing Student Thinking
If students did not decompose the figures into rectangular prisms, encourage them to build the figures with connecting cubes and ask, “how can you decompose the figure into two rectangular prisms?”
Activity Synthesis
Display Figures c and d.
Ask selected students to share their way of splitting each figure.
“Who broke the figure up the same way? Who broke it up differently?”
“Can you think of other ways you could break up these figures?“ (I can cut them into several layers—3 horizontal layers for c, and 5 vertical layers for d. Each layer is a rectangular prism.)


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