The purpose of this task is for students to understand the relationship between the volume and edge lengths of cubes with that of cube roots.

The purpose of this discussion is for students to make the connection that knowing the volume of a cube is sufficient information for stating the exact side length of the cube using cube roots. 

Here are some questions for discussion:

• “Which two consecutive integers is between?” ( is between 1 and 2 because and .)

• “Which two consecutive integers is between?” ( is between 2 and 3 because and .)

• “Name another volume for a cube with edge lengths between 2 and 3.” (Sample response: A cube with volume 26 has edge lengths between 2 and 3 since , meaning has a value slightly less than 3.)

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.)

Math Community
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 ."