In grades 5 and 6, students calculated the volume of rectangular prisms. In this lesson, students learn that they can calculate the volume of any right prism by multiplying the area of the base times the height of the prism. Students make sense of this formula by picturing the prism decomposed into identical layers 1 unit tall. These layers are composed of a number of cubic units equal to the number of square units in the area of the base. The height of the prism tells how many of these layers there are. Therefore, multiplying the number of cubic units in one layer times the number of layers gives the total number of cubic units in the prism, regardless of the shape of the base.
Given some threedimensional figures that are prisms and some that are not, students decide whether they can apply the formula \(V=Bh\) to calculate the volume. If so, they identify the base and measure the height, before calculating the volume. Students also apply the formula \(V=Bh\) to find the height of a prism given its volume and the area of its base.
Lesson overview
 12.1 Warmup: Three Prisms with the Same Volume (5 minutes)

12.2 Activity: Finding Volume with Cubes (10 minutes)
 There is a digital applet in this activity.

12.3 Activity: Can You Find the Volume? (15 minutes)
 Includes "Are you Ready for More?" extension problem
 There is a digital applet in this activity.
 12.4 Optional Activity: What’s the Prism’s Height? (10 minutes)
 Lesson Synthesis
 12.5 Cooldown: Octagonal Box (5 minutes)
Learning goals:
 Determine the volume of a right prism by counting how many unit cubes it takes to build one layer and then multiplying by the number of layers.
 Generalize (orally) the relationship between the volume of a prism, the area of its base, and its height.
 Identify whether a given figure is a prism, and if so, identify its base and height.
Learning goals (student facing):
 Let’s look at volumes of prisms.
Learning targets (student facing):
 I can explain why the volume of a prism can be found by multiplying the area of the base and the height of the prism.
Required materials:
 snap cubes
 copies of blackline master
 copies of blackline master
 preassembled polyhedra
 rulers marked with centimeters
Required preparation:
 You will need the Finding Volume with Cubes blackline master for this lesson.
 You will only use one of the two pages.
 If your snap cubes measure \(\frac34\) inch, print the first page of the blackline master, with the slightly smaller shapes.
 If your snap cubes measure 2 cm, print the second page of the blackline master, with the slightly larger shapes.
 Make sure to print the blackline master at 100% scale so the dimensions are accurate.
 Prepare 1 copy for every 6 students, and cut the pages in half so that each group of 3 students has one halfpage.
 Print, cut, and assemble the nets from the Can You Find the Volume? blackline master.
 Card stock paper is recommended.
 Make sure to print the blackline master at 100% scale so the dimensions are accurate.
 Prepare 1 polyhedron for every 2 students (1 copy of the entire file for every 1218 students).
 Make sure students have access to snap cubes and rulers marked in centimeters.
Glossary:
 volume  Volume is the number of cubic units that fill a threedimensional region, without any gaps or overlaps. For example, the volume of this rectangular prism is 60 units\(^3\), because it is composed of 3 layers that are each 20 units\(^3\).
 Access the complete Grade 7 glossary.
IM 6–8 Math was originally developed by Open Up Resources and authored by Illustrative Mathematics, and is copyright 20172019 by Open Up Resources. It is licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0). OUR's 6–8 Math Curriculum is available at https://openupresources.org/mathcurriculum/.
Adaptations and updates to IM 6–8 Math are copyright 2019 by Illustrative Mathematics, and are licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).
Adaptations to add additional English language learner supports are copyright 2019 by Open Up Resources, and are licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).
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