Unit 4: Roots, Pythagorean Theorem & Volume

This unit begins with solving simple quadratic & cubic equations visually using the meaning of root as the length of a single side of an actual square or cube.  This moves into attempting to solve x^2 = 2 through successive approximations before introducing using a double number line to approximate irrational numbers (based on linear regression).  Finding the radius from a cylinder or cone acts as a way to re-engage & apply square roots with rational and irrational values after learning the formulas conceptually first.  Likewise, finding the radius from a sphere acts as a way to re-engage & apply cube roots with rational and irrational values after learning the formula conceptually first.  The final application of the new learning is to the Pythagorean Theorem and its converse where rational and irrational side lengths are now both fair game.  Depth is explored by extending the Pythagorean Theorem to find the distance between any two points in the coordinate plane, converting & comparing fractions & decimals, and extending simple quadratic equations and simple cubic equations to binomials - time permitting.

This unit should be taught holistically & the three topics of Roots, Pythagorean Theorem & Volume should be interconnected & weaved together so that students are able to see how they relate to one another & connect to Unit 3.   Focusing some time on finding the side length from a given area or volume provides context and relevance for square roots and cube roots. Likewise, The Pythagorean Theorem & geometry are contexts that can necessiatate an understanding & application of irrational numbers making them relevant.