Course Guide - USAID Teacher Education Project

Unit 3 GeometryWeek 5, Session 3: Introduction to the Pythagorean Theorem1. What are the important concepts?a) Right triangles have a base, a height, and a hypotenuse.b) A right triangle's hypotenuse can be calculated by using the squares of its base andheight. This is known as the Pythagorean Theorem.c) There are certain right triangles whose base, height, and hypotenuse are wholenumbers (or multiples of whole numbers).2. How do children think about these concepts?a) Most adults remember "A squared + B squared = C squared" from high schoolgeometry. However few adults know why this equation makes sense. Thus, whenyoungsters are given this formula without having opportunities to explore proofs ofthe Theorem, they have little understanding of it.b) Even if youngsters have seen a visual proof of the Pythagorean Theorem (such assquares being drawn on each side of a triangle), they are unaware that there areliterally hundreds of proofs for the Theorem. It is mathematically important thatstudents realize a theorem can be proved in multiple ways.c) Youngsters may overgeneralize and assume that if the Pythagorean Theoremapplies to all right triangles, then it must apply to all triangles. Having youngstersexplore this will help them understand the power of a counterexample.3. What is essential to know or do in class?a) Have students use what they learned about building upright and tilted squares andthe distance between points on a grid to build squares on the sides of right triangles.b) Have students notice the additive relationship among the squares they drew on thesides of right triangles.c) Have students devise the formula for the Pythagorean Theorem.4. Class Activitiesa) Remind students of how they were able to draw a "tilted square" with an area of 2,5, 8, and 10 units in the prior session.b) Distribute dot paper and ask students to draw a right triangle with a base of 1 and aheight of 1. Then direct them to draw a square on each side of the triangle. What dothey notice about the three squares that they drew? What is each square's area? Isthere a relationship between the three numbers? How does this relate to one of thetilted squares they drew in the prior session?