Lesson 10

The purpose of this warm-up is for students to notice that figures composed of two right rectangular prisms can be decomposed in different ways which will be useful when students find the volume of figures composed of two right rectangular prisms in a later activity. While students may notice and wonder many things about these images, comparing the side lengths of the two figures is the important discussion point.

• Groups of 2
• Display the image.
• “What do you notice? What do you wonder?”

• 1 minute: quiet think time
• 1 minute: partner discussion
• Share and record responses.

• “Do you think the pictures show the same figure? Why or why not?” (Yes, they look the same. Yes, I can use the given side lengths to calculate that they are the same.)

The purpose of this activity is for students to find the volume of a figure in different ways. The given figure can be decomposed in two ways into rectangular prisms by making different cuts. However, it can also be found using a single, larger rectangular prism by removing a smaller rectangular prism. This provides an opportunity to express its volume as a difference of volumes of rectangular prisms. Students may notice this feature, and it is highlighted in the activity synthesis.

When students decide whether or not they have the same expressions, they need to reason carefully about what “the same” means. They consider if the order of the factors is different, is it the same expression and if the order of the addends is different, is it the same expression. Students use what they know about volume, geometric figures, and the properties of operations to justify the equivalence of the expressions and critique their peers' reasoning (MP2, MP3, MP7).

• Groups of 2
• “You are going to look for different expressions to calculate the volume of a figure.”

• 5 minutes: partner work time
• Monitor for students who draw a vertical line to show where they decompose the figure to share during the synthesis.

• Invite students to share different expressions for the volume of the figure.
• Display expression: \((3 \times 5 \times 5) + (2 \times 2 \times 5)\)
• “How does the expression represent the volume of the figure?” (\(3 \times 5 \times 5\) represents the prism that is 3 by 5 by 5 cubes tall and the \(2 \times 2 \times 5\) represents the prism that is 2 by 2 by 5 cubes tall.)
• If no student wrote the expression \((5 \times 5 \times 5) – (2 \times 3 \times 5)\), display this expression.
• “How does the expression show the volume of the figure in cubic units?” (There is a 5 by 5 by 5 cube and a piece has been taken away. The piece that is taken away measures 2 cubes by 3 cubes by 5 cubes.)

The purpose of this activity is for students to write equivalent expressions in order to find the volume of a figure composed of two right rectangular prisms. Students decompose the figure in two different ways, and write matching expressions to find the volume. For extra support, provide students with colored pencils to shade the two parts of the prism before finding the side lengths they need to calculate the volume.
Monitor and select a student with each of the following strategies to share in the synthesis:  

• decomposed the figure into a prism with the side lengths 4 ft by 4 ft by 3 ft and a prism with the side lengths 10 ft by 4 ft by 3 ft and wrote this expression (or one written in a different order) to represent the volume: \((4 \times 4 \times 3) + (10 \times 4 \times 3)\)
• decomposed the figure into a prism with the side lengths 4 ft by 8 ft by 3 ft and a prism with the side lengths 6 ft by 4 ft by 3 ft and wrote this expression (or one written in a different order) to represent the volume: \((4 \times 8 \times 3) + (6 \times 4 \times 3)\)

• Groups of 2

• 10 minutes: individual work time
• 5 minutes: partner discussion
• As students work, consider asking, “Why did you choose to decompose the prism that way?”

• Ask the two selected students to display their work side by side for all to see.
• “How are the diagrams the same? How are they different?”
• “How do the expressions relate to the diagrams?”
• Display: \((10 \times 8 \times 3) - (6 \times 4 \times 3)\)
• “How does this expression represent the volume of the prism?” (The larger rectangular prism has the side lengths \(10 \times 8 \times 3\) cubic feet. We can subtract a rectangular prism with the side lengths \(6 \times 4 \times 3\) cubic feet.)
• “What is the value of \((10 \times 8 \times 3) - (6 \times 4 \times 3)\)?” (168)