Eighth Grade · Math · week 23

Square and cube roots

Evaluate square roots and cube roots of perfect squares and cubes, and solve equations of the form x^2 = p and x^3 = p.

A square root undoes squaring: sqrt(64) asks "what number times itself gives 64?" (8, since 8 x 8 = 64). A cube root undoes cubing: the cube root of 27 asks "what number times itself three times gives 27?" (3, since 3 x 3 x 3 = 27).

x^2 = p has TWO solutions (a positive and a negative one), since both a positive number and its negative give the same result when squared. x^3 = p has only ONE solution, since a negative number cubed stays negative and can never accidentally also solve a positive-cube equation.

📋 For the teacher
Teach it (10–15 min):
  • Introduce sqrt as a symbol that literally means "undo the square" -- pair every root fact with its squaring fact (9x9=81, so sqrt(81)=9) so it never feels like a separate list to memorize.
  • Cube roots undo cubing the same way -- picture (or build) a small stack of unit cubes for 2x2x2=8 to make "the cube root of 8 is 2" concrete before it becomes a memorized fact.
  • The plus-or-minus rule matters: x^2 = p has TWO answers, but x^3 = p has only ONE, because a negative number cubed is still negative.
  • Keep this week's numbers to perfect squares and perfect cubes they can verify by hand (4, 9, 16, 25, 36, 49, 64, 81, 100 and 1, 8, 27, 64, 125) -- messier roots are next week's estimation skill.
Talk about it:
  • Why does x^2 = 49 have two solutions, but x^3 = 8 has only one?
  • How would you check that the cube root of 27 really is 3, without a calculator?
  • What is the connection between knowing your multiplication facts and knowing your square roots?

Mastery looks like: They evaluate square and cube roots of perfect squares/cubes and correctly solve x^2 = p (both signs) and x^3 = p (one sign).

Common stumbles: Giving only the positive solution for x^2 = p and missing the negative one; giving two solutions for a cube-root equation when only one exists.

Homework for this lessonType it, write it with a stylus, or print it for pencil & paper.open →
Records attendance + 15 min of Math for this learner.
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