Lesson 17: Squares and Cubes
About this lesson
In this lesson, students learn about perfect squares and perfect cubes. They see that these names come from the areas of squares and the volumes of cubes with wholenumber side lengths. Students find unknown side lengths of a square given the area or unknown edge lengths of a cube given the volume. To do this, they make use of the structure in expressions for area and volume (MP7).
Students also use exponents of 2 and 3 and see that in this geometric context, exponents help to efficiently express multiplication of the side lengths of squares and cubes. Students learn that expressions with exponents of 2 and 3 are called squares and cubes, and see the geometric motivation for this terminology. (The term “exponent” is deliberately not defined more generally at this time. Students will work with exponents in more depth in a later unit.)
In working with length, area, and volume throughout the lesson, students must attend to units. In order to write the formula for the volume of a cube, students look for and express regularity in repeated reasoning (MP8).
Note: Students will need to bring in a personal collection of 10–50 small objects ahead of time for the first lesson of the next unit. Examples include rocks, seashells, trading cards, or coins.
Lesson overview
 17.1 Warmup: Perfect Squares (5 minutes)

17.2 Optional Activity: Building with 32 Cubes (15 minutes)
 Includes "Are you Ready for More?" extension problem
 There is a digital applet in this activity.
 17.3 Activity: Perfect Cubes (10 minutes)

17.4 Activity: Introducing Exponents (15 minutes)
 Includes "Are you Ready for More?" extension problem
 Lesson Synthesis
 17.5 Cooldown: Exponent Expressions (5 minutes)
Learning goals:
 Generalize a process for finding the volume of a cube, and justify (orally) why this can be abstracted as \(s^3\).
 Include appropriate units (orally and in writing) when reporting lengths, areas, and volumes, e.g. cm, cm<sup>2</sup>, cm<sup>3</sup>.
 Interpret and write expressions with exponents <sup>2</sup> and <sup>3</sup> to represent the area of a square or the volume of a cube.
Learning goals (student facing):
 Let’s investigate perfect squares and perfect cubes.
Learning targets (student facing):
 I can write and explain the formula for the volume of a cube, including the meaning of the exponent.
 When I know the edge length of a cube, I can find the volume and express it using appropriate units.
Required materials:
 snap cubes
Required preparation:
 Prepare sets of 32 snap cubes for each group of 2 students.
Glossary:
 cubed  We use the word cubed to mean “to the third power.” This is because a cube with side length \(s\) has a volume of \(s \cdot s \cdot s\), or \(s^3\).
 exponent  In expressions like \(5^3\) and \(8^2\), the 3 and the 2 are called exponents. They tell you how many factors to multiply. For example, \(5^3\) = \(5 \cdot 5 \cdot 5\), and \(8^2 = 8 \cdot 8\).
 squared We use the word squared to mean “to the second power.” This is because a square with side length \(s\) has an area of \(s \cdot s\), or \(s^2\).
 Access the complete Grade 6 glossary.
Standards
 This lesson builds on the standards:CCSS.4.MD.A.3MS.4.MD.3MO.4.GM.C.8 CCSS.5.MD.C.5.aMS.5.MD.5a
 This lesson builds towards the standard:CCSS.6.EE.A.1MS.6.EE.1MO.6.EEI.A.2d
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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