Lesson 12
Volume of Right Prisms
12.1: Three Prisms with the Same Volume (5 minutes)
Warmup
The purpose of this warmup is to encourage students to think about possible heights of prisms with the same height and volume based on the area of a base. This is a review of previous work students have done with volume in which they found the volume of a rectangular prism by multiplying the area of a base and height. The ideas in this warmup are revisited later in this lesson, so it is important students can clearly explain how they ordered their prisms based on them having the same volume and how they found the height of the prism with base C.
Launch
Arrange students in groups of 2. Give students 1 minute of quiet work time followed by time to discuss their explanations with a partner. Follow with a wholeclass discussion.
Student Facing
Rectangles A, B, and C represent bases of three prisms.
 If each prism has the same height, which one will have the greatest volume, and which will have the least? Explain your reasoning.

If each prism has the same volume, which one will have the tallest height, and which will have the shortest? Explain your reasoning.
Student Response
For access, consult one of our IM Certified Partners.
Activity Synthesis
Select students to share the prism they found to have the greatest and least volume and the tallest and shortest height. Record and display their responses for all to see. Poll the class if they agree or disagree. If students all agree, ask a few students to share their reasoning. If they do not agree, ask students to share their reasoning until they reach an agreement.
If there is time, display this question for all to see: “If each prism has the same volume and the prism associated with base B has a height of 6 units, what is the height of the prism associated with base C?”
Have students share the volume of the prism with base C and their reasoning. Record and display the responses for all to see.
12.2: Finding Volume with Cubes (10 minutes)
Activity
In grades 5 and 6, students worked with the volume of rectangular prisms. In this activity, students extend their understanding to see that even when the base is not a rectangle, they can still calculate the volume of a prism by multiplying the area of the base times the height of the prism.
Students use snap cubes to build a prism with a base that matches the shape from the blackline master. Each group needs 30 snap cubes for the first question and a total of 60 snap cubes for the second question. If there are not enough snap cubes, two groups of 3 students may combine together after answering the first question to form one group of 6 students.
If using snap cubes that measure \(\frac34\) inch, make copies of the first page of the blackline master, with the slightly smaller shapes. If using snap cubes that measure 2 cm, make copies of the second page of the blackline master, with the slightly larger shapes.
Launch
Arrange students in groups of 3. Distribute copies of the blackline master, one halfpage to each group, and 30–60 snap cubes to each group. Give students 2–3 minutes of quiet work time followed by a wholeclass discussion.
For students using digital materials: depending on the needs of your class, either demonstrate how to build figures using the applet, or instruct students to read and follow the instructions for working the applet.
Student Facing
This applet has 64 snap cubes, all sitting in the same spot on the screen, like a hidden stack of blocks. You will always know where the stack is because it sits on a gray square. You can keep dragging blocks out of the pile by their red points until you have enough to build what you want.
Click on the red points to change from left/right movement to up/down movement.
There is also a shape on the grid. It marks the footprint of the shapes you will be building.
 Using the face of a snap cube as your area unit, what is the area of the shape? Explain or show your reasoning.
 Use snap cubes to build the shape from the paper. Add another layer of cubes on top of the shape you have built. Describe this threedimensional object.
 What is the volume of your object? Explain your reasoning.
 Right now, your object has a height of 2. What would the volume be

if it had a height of 5?

if it had a height of 8.5?

Student Response
For access, consult one of our IM Certified Partners.
Launch
Arrange students in groups of 3. Distribute copies of the blackline master, one halfpage to each group, and 30–60 snap cubes to each group. Give students 2–3 minutes of quiet work time followed by a wholeclass discussion.
For students using digital materials: depending on the needs of your class, either demonstrate how to build figures using the applet, or instruct students to read and follow the instructions for working the applet.
Student Facing
Your teacher will give you a paper with a shape on it and some snap cubes.
 Using the face of a snap cube as your area unit, what is the area of the shape? Explain or show your reasoning.
 Use snap cubes to build the shape from the paper. Add another layer of cubes on top of the shape you have built. Describe this threedimensional object.
 What is the volume of your object? Explain your reasoning.

Right now, your object has a height of 2. What would the volume be:

if it had a height of 5?

if it had a height of 8.5?

Student Response
For access, consult one of our IM Certified Partners.
Activity Synthesis
Select students to share their reasoning.
Consider asking some of the following questions:
 “How do you know this figure is a prism?” (Cross sections parallel to the base are identical copies.)
 “What is the area of the base of this figure?” (It is the number of cubes in one layer of the prism.)
 “How do you calculate the total number of cubes to make the prism?” (Multiply the number of cubes in one layer by the number of layers.)
 “What is the volume of this prism?” (The volume is the same as calculating the number of cubes to make the prism.)
 “If you find the area of the base, how do you use that information to calculate the volume of the prism?” (Multiply the area of the base by the height of the prism.)
 “How would the volume of the prism change if we changed the shape of the base but still used 27 cubes to build it?” (The volume would not change.)
If not mentioned by students, explain that calculating the total number of cubes to make the prism is the same as calculating the volume of the prism. We can find the area of the base of the prism and multiply that by the number of layers in the prism which is the same as the height of the prism. The height of the prism is measured in units, the area of the base is measured in units^{2} and the volume of the prism is measured in units^{3}.
Design Principle(s): Optimize output (for explanation); Maximize metaawareness
12.3: Can You Find the Volume? (15 minutes)
Activity
The purpose of this activity is for students get handson experience with polyhedra, recognizing whether a figure is a prism and if so, determining which face is the base of the prism. Once students determine which face is the base, they use a ruler marked in inches to measure the height of the prism. The area of each face is labeled on the shape so that students do not get bogged down with calculating the base area and can focus on using the area to find the volume.
Instead of creating enough sets of polyhedra for every group to have one of every shape at the same time, consider having the students pass the shapes from one group to the next or rotate around to different stations so that fewer sets of shapes have to be constructed.
Launch
Arrange students in groups of 3–4. Distribute the threedimensional figures that were already assembled from the nets in the blackline master and rulers marked in centimeters. Give students 1–2 minutes of quiet work time with the polyhedra given to their group, have groups exchange objects so that each group gets to examine each figure. Follow with a wholeclass discussion.
Students using the digital version have an applet with the six polyhedra in 3D. Students can rotate the view using the tool marked by two intersecting, curved arrows. Note that each polyhedron has only one label per unique face. If no other measurements are shown, the faces are congruent. Students can use the distance tool, marked with the "cm," to find the height or length of any segment. Troubleshooting tip: the cursor must be on the 3D Graphics window for the full toolbar to appear.
Student Facing
The applet has a set of threedimensional figures.
 For each figure, determine whether the shape is a prism.
 For each prism:
 Find the area of the base of the prism.
 Find the height of the prism.
 Calculate the volume of the prism.
Is it a prism?  area of prism base (cm^{2})  height (cm)  volume (cm^{3}) 

 Begin by grabbing the gray bar on the left and dragging it to the right until you see the slider.
 Choose a figure using the slider.
 Rotate the view using the Rotate 3D Graphics tool marked by two intersecting, curved arrows.
 Note that each polyhedron has only one label per unique face. Where no measurements are shown, the faces are identical copies.
 Use the distance tool, marked with the "cm," to click on any segment and find the height or length.
 Troubleshooting tip: the cursor must be on the 3D Graphics window for the full toolbar to appear.
Student Response
For access, consult one of our IM Certified Partners.
Launch
Arrange students in groups of 3–4. Distribute the threedimensional figures that were already assembled from the nets in the blackline master and rulers marked in centimeters. Give students 1–2 minutes of quiet work time with the polyhedra given to their group, have groups exchange objects so that each group gets to examine each figure. Follow with a wholeclass discussion.
Students using the digital version have an applet with the five polyhedra in 3D. Students can rotate the view using the tool marked by two intersecting, curved arrows. Note that each polyhedron has only one label per unique face. If no other measurements are shown, the faces are congruent. Students can use the distance tool, marked with the "cm," to find the height or length of any segment. Troubleshooting tip: the cursor must be on the 3D Graphics window for the full toolbar to appear.
Student Facing
Your teacher will give you a set of threedimensional figures.
 For each figure, determine whether the shape is a prism.
 For each prism:
 Find the area of the base of the prism.
 Find the height of the prism.
 Calculate the volume of the prism.
Is it a prism?  area of prism base (cm^{2})  height (cm)  volume (cm^{3})  

figure A  
figure B  
figure C  
figure D  
figure E  
figure F 
Student Response
For access, consult one of our IM Certified Partners.
Student Facing
Are you ready for more?
Imagine a large, solid cube made out of 64 white snap cubes. Someone spray paints all 6 faces of the large cube blue. After the paint dries, they disassemble the large cube into a pile of 64 snap cubes.
 How many of those 64 snap cubes have exactly 2 faces that are blue?
 What are the other possible numbers of blue faces the cubes can have? How many of each are there?
 Try this problem again with some largersized cubes that use more than 64 snap cubes to build. What patterns do you notice?
Student Response
For access, consult one of our IM Certified Partners.
Anticipated Misconceptions
Some students may say that Figure A is not a prism because it is a cube. Ask them whether the cross sections would be identical if you made various cuts parallel to one side. Explain that a cube is a special type of square prism where the height of the prism matches the side lengths of the base. Consider making a comparison to the fact that a square is a special type of rectangle.
Activity Synthesis
Poll the class for answers to the first column of the table. Make sure the class agrees about answers before proceeding. Select students to provide their answers to each part of the table and an explanation. Display answers on the table for all to see. Ask students:
 “What is different about the structure of nonprisms in comparison to prisms?" (The prisms have multiple layers of the same base, where the nonprism does not have that.)
 “Why can’t you use ‘area of the base times the height’ to calculate the volume of the figures that were not prisms?" (Because the nonprism isn't made up of multiple layers of the same base.)
Design Principle(s): Optimize output (for explanation); Maximize metaawareness
12.4: What’s the Prism’s Height? (10 minutes)
Optional activity
The purpose of this activity is for students to work backwards from the volume to the height of a prism. Students see that for two prisms to have the same volume, the one with the smaller base has the taller height and the one with the larger base has the shorter height. The grid helps students find the area of the base so they can focus their attention on what it means to have a prism made out of stacks of layers of the same base.
Launch
Arrange students in groups of 2. Give students 5 minutes of quiet work time followed by time to discuss their thinking with a partner. Follow with a wholeclass discussion.
Supports accessibility for: Conceptual processing
Student Facing
There are 4 different prisms that all have the same volume. Here is what the base of each prism looks like.

Order the prisms from shortest to tallest. Explain your reasoning.

If the volume of each prism is 60 units^{3}, what would be the height of each prism?
 For a volume other than 60 units^{3}, what could be the height of each prism?

Discuss your thinking with your partner. If you disagree, work to reach an agreement.
Student Response
For access, consult one of our IM Certified Partners.
Activity Synthesis
Select students to share their responses and reasoning. If not brought up in student's explanation, explain that for the last problem, there is more than one possible correct answer. The smallest possible volume that involves all whole number side lengths is \(120 \text{ units}^3\), but there is nothing in the problem that requires all the heights to be whole numbers.
To highlight connections to calculating volume, ask:
 “How do you calculate the volume of a prism?” (Display the equation for all to see: \(V = B \boldcdot h\).)
 “Since \(V = B \boldcdot h\), how could we find the area of the base if we knew the volume and height of the prism?” (Display the equation for all to see: \(B = V \div h\).)
 “If we keep the volume the same, what happens to the height when we increase the area of the base?” (It decreases.)
 “If we keep the height the same, what happens to the volume when we increase the area of the base?” (It increases.)
Design Principle(s): Support sensemaking
Lesson Synthesis
Lesson Synthesis
 “What information do we need to calculate the volume of a prism?” (Area of the base and the height)
 “Explain how you could use layers to find the volume of a prism.” (If you look at the first layer of a prism, you can find how many cubes are in that layer by finding the area of the base. Once you find the number of cubes on the first layer, you multiply that by the number of layers it takes to stack up to the height of the prism.)
 “Two prisms have the same base area and height, but different base shapes. Which prism has a greater volume? Explain.” (The two prisms have the same volume. The shape of the base does not matter if it is a prism, only the base area matters.)
 “Two clay prisms use the same amount of clay to make them, but the first has a larger height than the second. Which prism has a larger base area?” (The second prism will have a larger base area since a shorter height means a larger base area if the volume is held constant. Imagine squashing the first one down in a nice way to make a shorter, fatter version.)
12.5: Cooldown  Octagonal Box (5 minutes)
CoolDown
For access, consult one of our IM Certified Partners.
Student Lesson Summary
Student Facing
Any cross section of a prism that is parallel to the base will be identical to the base. This means we can slice prisms up to help find their volume. For example, if we have a rectangular prism that is 3 units tall and has a base that is 4 units by 5 units, we can think of this as 3 layers, where each layer has \(4\boldcdot 5\) cubic units.
That means the volume of the original rectangular prism is \(3(4\boldcdot 5)\) cubic units.
This works with any prism! If we have a prism with height 3 cm that has a base of area 20 cm^{2}, then the volume is \(3\boldcdot 20\) cm^{3} regardless of the shape of the base. In general, the volume of a prism with height \(h\) and area \(B\) is
\(\displaystyle V = B \boldcdot h\)
For example, these two prisms both have a volume of 100 cm^{3}.