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 volumes 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.
Yellow rectangular prism with blue rectangular prism on top. Yellow prism: 10 feet by unknown width by 3 feet. Blue prism: 5 feet by 3 feet by 5 feet. Height of both prisms combined is 8 feet.
Blue rectangular prism joined on the side with a yellow rectangular prism. Blue prism: 5 feet by 3 feet by 8 feet. Yellow prism: 5 feet by 3 feet by 3 feet.
Student Response
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Advancing Student Thinking
Activity Synthesis
“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.)
Activity 1
Standards Alignment
Building On
Addressing
5.MD.C.5.c
Recognize volume as additive. Find volumes of solid figures composed of two non-overlapping right rectangular prisms by adding the volumes of the non-overlapping parts, applying this technique to solve real world problems.
Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. For example, express the calculation “add and , then multiply by ” as . Recognize that is three times as large as , without having to calculate the indicated sum or product.
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. The volume also can be found by removing a smaller rectangular prism from a single, larger rectangular prism. This provides an opportunity to express the volume of the figure as a difference of the 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 the reasoning of their peers (MP2, MP3, MP7).
MLR8 Discussion Supports. Display sentence frames to support partner discussion: “First, I _____ because . . . .” “_____ and _____ are the same because . . . .” “_____ and _____ are different because . . . .” Advances: Conversing, Representing
Launch
Groups of 2
“You are going to look for different expressions to calculate the volume of a figure.”
Activity
5 minutes: partner work time
Monitor for students, who draw a vertical line to show where they decompose the figure, to share during the Activity Synthesis.
Write an expression to represent the volume of the figure in unit cubes.
Figure composed of two attached prisms. First prism: 3 cubes by 5 cubes by 5 cubes. Second prism: 2 cubes by 2 cubes by 5 cubes.
Compare expressions with your partner.
How are they alike?
How are they different?
If they are the same, try to find another way to represent the volume.
Student Response
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Advancing Student Thinking
Activity Synthesis
Invite students to share different expressions for the volume of the figure.
Display the expression:
“How does the expression represent the volume of the figure?” ( represents the prism that is 3-units-by-5-units-by-5-units tall and the represents the prism that is 2-units-by-2-units-by5-units tall.)
If no student wrote the expression , display this expression.
“How does the expression show the volume of the figure in cubic units?” (There is a 5-unit-by-5-unit-by-5-unit cube, and a piece has been taken away. The piece taken away measures 2 units by 3 units by 5 units.)
Activity 2
Standards Alignment
Building On
Addressing
5.MD.C.5.c
Recognize volume as additive. Find volumes of solid figures composed of two non-overlapping right rectangular prisms by adding the volumes of the non-overlapping parts, applying this technique to solve real world problems.
Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. For example, express the calculation “add and , then multiply by ” as . Recognize that is three times as large as , without having to calculate the indicated sum or product.
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 they find the side lengths needed to calculate the volume.
Monitor and select a student, with each of the following approaches, to share in the Activity Synthesis:
Decompose the figure into a prism, with the side lengths 4 feet by 4 feet by 3 feet and a prism with the side lengths 10 feet by 4 feet by 3 feet, and write this expression (or the equivalent) to represent the volume: .
Decompose the figure into a prism with the side lengths 4 feet by 8 feet by 3 feet and a prism with the side lengths 6 feet by 4 feet by 3 feet, and write this expression (or the equivalent) to represent the volume: .
Enclose the figure to create a prism with side lengths 10 feet by 8 feet by 3 feet, and subtract the part of the prism with side lengths 6 feet by 4 feet by 3 feet, using a subtraction equation like Mai’s in problem 3 to represent the volume.
The approaches will be displayed later, side by side with Mai’s approach, to help students connect how they decompose each figure and the expression they use to find the volume. If a student uses an approach like Mai’s in the first problem, then invite them to display their original thinking in the Activity Synthesis discussion. Aim to elicit both key mathematical ideas and a variety of student voices, especially students who haven’t shared recently.
Action and Expression: Internalize Executive Functions. Invite students to plan a strategy, including the tools they will use, for finding the volume of the figure. If time allows, invite students to share their plan with a partner before they begin. Supports accessibility for: Organization, Memory, Attention
Launch
Groups of 2
Activity
10 minutes: individual work time
5 minutes: partner discussion
As you monitor for the approaches listed in the Activity Narrative, consider asking:
“Why did you choose to decompose the prism that way?”
“How does your expression connect to the way you decomposed the figure?”
Find the volume by decomposing the figure in as many ways as you can. Show your thinking. Organize your work so it can be followed by others.
6-sided rectangular prism. Straight sides. All side lengths meet at right angles. Side lengths. Bottom, 10 feet. Right side rises 4 feet, then goes left 6 feet, goes up an unlabeled distance, then goes left an unlabeled distance, then goes down 8 feet. Width shown as 3 feet.
6-sided rectangular prism. Straight sides. All side lengths meet at right angles. Side lengths. Bottom, 10 feet. Right side rises 4 feet, then goes left 6 feet, goes up an unlabeled distance, then goes left an unlabeled distance, then goes down 8 feet. Width shown as 3 feet.
Write expressions to represent each way that you decompose the figure.
Mai used this expression to find the volume of the figure:
.
Use the diagram to interpret Mai's expression. Show your thinking. Organize your work so it can be followed by others.
6-sided rectangular prism. Straight sides. All side lengths meet at right angles. Side lengths. Bottom, 10 feet. Right side rises 4 feet, then goes left 6 feet, goes up an unlabeled length, then goes left an unlabeled length, then goes down 8 feet. Width shown as 3 feet.
Activity Synthesis
Invite previously selected students to display their work side by side for all to see, without sharing their thinking. Also display Mai’s approach if no one in the class uses this subtraction strategy.
“Take a minute to look at the diagrams and expressions.”
Connect students’ approaches by asking:
“How are the diagrams the same?” (Each way to think about the prism has 2 other prisms.)
“How are they different?” (In each of two diagrams, the figure is decomposed into 2 rectangular prisms to find the volume of each, and then the volumes are added. Mai’s diagram has a larger rectangular prism, with the side lengths 10 feet by 8 feet by 3 feet. It shows subtracting a rectangular prism with the side lengths 6 feet by 4 feet by 3 feet.)
Connect students’ approaches to the learning goal by asking:
“How do the expressions with addition relate to the diagrams?” (The prism is decomposed into 2 smaller prisms. Multiply the side lengths of each prism, and then add them together.)
“How does Mai’s expression represent the volume of the prism?” (Multiply the side lengths of the larger and smaller rectangular prisms, and then subtract the smaller volume from the larger volume to find the volume of the figure.)
“What is the value of ?” (168, I calculated it, but I also know this because it represents the volume. The value is the same as the other expressions for volume for this figure.)
Lesson Synthesis
“Today we decomposed the same figure in different ways and wrote expressions to represent volume.”
“Which decomposition strategy did you prefer to use? Why?” (It depends on the numbers. I decompose the figure in the way that gives me the friendliest numbers.)
“Do you get the same expressions, using either decomposition? Why?” (No, because the figure is broken into rectangular prisms with different side lengths.)
“The expressions are different, depending on how we decompose the shape, but the volume is the same. Why is that?” (The volume doesn’t change. We just decompose the figure in different ways. The expressions are equal.)
Standards Alignment
Building On
Addressing
Building Toward
5.MD.C.5.c
Recognize volume as additive. Find volumes of solid figures composed of two non-overlapping right rectangular prisms by adding the volumes of the non-overlapping parts, applying this technique to solve real world problems.
