Real Numbers and the Pythagorean Theorem
Lessons in this chapter
About Real Numbers and the Pythagorean Theorem
Real Numbers and the Pythagorean Theorem covers two connected ideas: the real number system in full, and the one geometric formula that most often needs square roots to solve.
Finding Square Roots and Finding Cube Roots cover both the calculation and the inverse relationship to squaring and cubing, before Rational Numbers and Irrational Numbers classify every real number as one or the other: rational if it can be written as a fraction, irrational if its decimal expansion never repeats or terminates (like most square roots).
The Pythagorean Theorem introduces a² + b² = c² for right triangles, using square roots from earlier in the chapter to solve for a missing side, while The Converse of the Pythagorean Theorem reverses the logic, using the same formula to test whether a triangle with three given side lengths is a right triangle at all.
The six lessons here are 8th grade math worksheets worked in the page: Finding Square Roots, Irrational Numbers and The Pythagorean Theorem each hand your child a set of math problems to answer directly rather than a page to read. Answers are marked one at a time, so using the Pythagorean Theorem with the hypotenuse in the wrong place is caught on that question. All six are free to try, with no login needed for your first 3.
Frequently asked questions
Finding Square Roots and Finding Cube Roots come first, undoing squaring and cubing and introducing perfect squares and cubes. Rational Numbers and Irrational Numbers then sort every number your child has met into those that can be written as a fraction and those that cannot, such as the square root of 2 and pi. The Pythagorean Theorem and The Converse of the Pythagorean Theorem close the chapter, finding a missing side of a right triangle and then testing whether a triangle is right-angled at all. This chapter spans three Common Core domains: The Number System (8.NS) for rational and irrational numbers, Expressions and Equations (8.EE) for square and cube roots, and Geometry (8.G) for the Pythagorean Theorem.
A rational number can be written as a fraction of two integers (including whole numbers, terminating decimals, and repeating decimals); an irrational number cannot: its decimal expansion goes on forever without repeating, like most square roots.
Estimate between the two nearest perfect squares: the square root of 50 is between 7 (49) and 8 (64), closer to 7, since 50 is closer to 49, Finding Square Roots has kids refine this estimate further using that reasoning.
For any right triangle, a² + b² = c², where a and b are the two shorter sides (legs) and c is the longest side (the hypotenuse, opposite the right angle).
Plug the two known sides into a² + b² = c² (making sure c is used only for the hypotenuse), solve for the missing value, then take the square root to find the actual side length.
The converse reverses the logic: given three side lengths, if a² + b² = c² is true (with c the longest side), then the triangle is guaranteed to be a right triangle, which is how The Converse of the Pythagorean Theorem tests a triangle’s shape from its measurements alone.