Students sort square or cube root values, equations of the form or , and number lines into matching sets of three during this activity. A sorting task gives students opportunities to analyze representations, statements, and structures closely and make connections (MP2, MP7).

Once all groups have completed the Card Sort, select groups to share one of their sets of three cards and how they matched the value, number line, and equation. Discuss the following: 

• “Which matches were tricky? Explain why.” (Two of the number lines have values between 4 and 5, and it took some extra reasoning to figure out that the one with the value closer to 4 was  while the one with the point closer to 5 was .) 

• “Did you need to make adjustments in your matches? What might have caused an error? What adjustments were made?” (I initially matched with the number line close to 7, but then I noticed it was a cube root and not a square root, so I adjusted my match.)

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Conclude the discussion by inviting 2–3 students to share a norm they identified in action. Provide this sentence frame to help students organize their thoughts in a clear, precise way:

• "I noticed our norm '' in action today and it really helped me/my group because ."

The purpose of this activity is for students to think about cube roots of numbers in relation to the two integer values they are closest to. Students are encouraged to use the fact that is a solution to the equation . Students can draw a number line if that helps them reason about the magnitude of the given cube roots, but this is not required. 

Monitor for students that multiply non-integers by hand to try and approximate. While this isn’t the preferred strategy, their work could be used to think about which integer the cube root is closest to during the Activity Synthesis.

The purpose of this discussion is for students to share the strategies they used to approximate cube roots by looking at perfect cubes. For each question, ask 1–2 students:

• “What strategy did you use to figure out the two whole numbers?” (I made a list of perfect cubes and then found which two the number was between.)

During the discussion, make a class display of perfect cubes up to at least , or add them to the existing display of perfect squares. This display should be posted in the classroom for the remaining lessons within this unit.

In this activity, students use rational approximations of irrational numbers to place both rational and irrational numbers on a number line. This reinforces the definition of a cube root as a solution to the equation of the form . This is also the first time that students are asked to think about negative cube roots. 

The purpose of this discussion is to make sure students understand how they can use rational numbers to approximate irrational numbers. Display the number line from the Task Statement for all to see. Select 1–2 groups to share their reasoning for each value placement and record their values on the displayed number line. After each placement, ask if any students used different reasoning and invite them to share with the class.