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My 8th grades are really struggling with squares and square roots, and cubes and cube roots. I would love to have a task for them to compare square roots and cube roots.
I think that even kids who can say the square root of 64 is greater than the square root of 49 because 64 is greater than 49, could benefit so much from the unpacking of that. How do you know? How can you prove it? That leads to students accessing the task at their level– some will draw or build models, some may jump right in with numbers, an indirect measurement.
If they were comparing a square root to a cube root, they might have that same direct comparison, but it would be really interesting for them to build/discover/discuss that when we are finding the square root and the cube root, we are looking for the single attribute of one dimension of an array or 3D model and that is what’s being compared. I’m so curious– what would they do if asked to compare the square root of 64 to the cube root of 64?
Completing this task leads nicely into the standard of being able to order and compare irrational numbers.