In this lesson, students reason abstractly and quantitatively from the meaning of cube and cube root to solve equations of the form \(x^3 = a\) and \(\sqrt[3]{x} = a\), where can be positive or negative (MP2). Students use the graph of \(y=x^3\) to find that all numbers have exactly one cube root.
Note that there are claims like, “All numbers have exactly one cube root,” which would be more precisely stated as “All real numbers have exactly one real cube root.” However, students don’t know about any numbers other than real numbers, so it does not make sense to make this distinction at this time. In upcoming lessons, students will expand their concept of number to include imaginary and complex numbers.
Some of the activities in this lesson work best when each student has access to devices that can run the Desmos applets because students can attend to the level of precision in making estimates from a graph (MP6).
Lesson overview
 8.1 Warmup: Put Your Arm(s) Up (5 minutes)

8.2 Activity: Finding Cube Roots with a Graph (10 minutes)
 There is a digital applet in this activity.

8.3 Optional Activity: Cube Root Equations (10 minutes)
 There is a digital applet in this activity.

8.4 Activity: Solve These Equations With Cube Roots in Them (20 minutes)
 Includes "Are you Ready for More?" extension problem
 Lesson Synthesis
 8.5 Cooldown: Cube It (5 minutes)
Learning goals:
 Calculate solutions to equations involving cubes and cube roots and explain the solution method used.
 Compare and contrast the processes of solving equations with square roots and with cube roots.
 Justify using a graph that every number, positive or negative, has exactly one cube root.
Learning goals (student facing):
 Let’s compare equations with cubes and cube roots.
Learning targets (student facing):
 I can solve equations by cubing or finding cube roots.
Standards:
 This lesson builds on the standards:CCSS.8.EE.A.2MS.8.EE.2MO.8.EEI.A.2aMO.8.EEI.A.2b
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