In this transitional lesson, students conclude their work with area and begin to explore volume of rectangular prisms. First, they extend their work on area to include triangles, using division to find the length of a base or a height in a triangle when the area is known. Second, they undertake a key activity for extending their understanding of how to find the volume of a prism.
In previous grades, students learned that the volume of a prism with wholenumber edge lengths is the product of the edge lengths. Now they consider the volume of a prism with dimensions \(1\frac12\) inch by 2 inches by \(2\frac12\) inches. They picture it as being packed with cubes whose edge length is \(\frac12\) inch, making it a prism that is 3 cubes by 4 cubes by 5 cubes, for a total of 60 cubes, because \(3 \cdot 4 \cdot 5 = 60\). At the same time, they see that each of these \(\frac12\)inch cubes has a volume of \(\frac18\) cubic inches, because we can fit 8 of them into a unit cube. They conclude that the volume of the prism is \(60\cdot\)\(\frac18\)=\(7\frac12\) cubic inches.
In the next lesson, by repeating this reasoning and generalizing (MP8), students see that the volume of a rectangular prism with fractional edge lengths can also be found by multiplying its edge lengths directly (e.g., (\(1\frac12\))⋅2⋅(\(2\frac12\))=\(7\frac12\)).
Lesson overview
 14.1 Warmup: Area of Triangle (5 minutes)
 14.2 Activity: Bases and Heights of Triangles (10 minutes)

14.3 Activity: Volumes of Cubes and Prisms (20 minutes)
 Includes "Are you Ready for More?" extension problem
 There is a digital applet in this activity.
 Lesson Synthesis
 14.4 Cooldown: Triangles and Cubes (5 minutes)
Learning goals:
 Apply dividing by fractions to calculate the base or height of a triangle, given its area and the other measurement.
 Determine the volume of a rectangular prism by counting how many 12inch or 13inch cubes it takes to build, and explain (orally and in writing) the solution method.
 Generalize that the volume of a rectangular prism with fractional edge lengths can be found by multiplying the edge lengths.
Learning goals (student facing):
 Let’s explore area and volume when fractions are involved.
Learning targets (student facing):
 I can use division and multiplication to solve problems involving areas of triangles with fractional bases and heights.
 I can explain how to find the volume of a rectangular prism using cubes that have a unit fraction as their edge length.
 I know how to find the volume of a rectangular prism even when the edge lengths are not whole numbers.
Required materials:
 \(\frac12\)inch cubes
 geometry toolkits
Required preparation:
 For the Volumes of Cubes and Prisms activity, prepare 20 halfinch cubes for every group of 3–4 students.
 Wooden ones are available inexpensively at craft stores.
 If you have access to centimeter cubes, you could use those instead.
 Tell students that we will consider them halfinch cubes for the purposes of that activity.
Glossary:
 Access the complete Grade 6 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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