Lesson Plan – Pythagorean Triples

Name: Andrew Janick

Date: 11/26/02

Course: Geometry

Number of Students: 17

I.  Goal(s)

·  To develop a geometric understanding of the Pythagorean Theorem and its application into finding Pythagorean triple combinations

II.  Objective(s)

·  Given three squares of a different size, students will be able to determine if the squares form a Pythagorean triple

·  Students will form their own squares to attempt to satisfy the Pythagorean theorem

·  Students will form a general conjecture about the process of finding Pythagorean triples

III.  Materials

·  Graph paper – two sheets per student

·  Scissors – one pair per group of two students

·  Scrap paper – to work out triples numerically

IV.  Motivation

1.  Explain that the square of a number is actually based in early Greek mathematics for finding area of a square, Since the area of a square is one side length multiplied by another, and a square’s side lengths are equal, therefore the number is ‘squared.’ (Ask if there are any comments)

2.  Ask students if they can create a geometric representation of the Pythagorean theorem.

3.  Explain that we will use squares to find whole number representations of the Pythagorean theorem, called Pythagorean triples.

Transition: Ask students to pair up while I hand out materials.

V.  Lesson Procedure

4.  Have groups cut out a 3X3, 4X4, and 5X5 square and form a right triangle with the side lengths.

5.  Explain that this is the most basic Pythagorean triple, but there are many more combinations.

6.  Have the students cut the smaller squares in some way to fit them perfectly inside the largest square. Have students make guesses at possible triples on scrap paper and test them geometrically.

7.  Allow students to work in groups for small period of time, walking around observing students for their thought processes.

8.  Ask student to tell me some of their successful combinations, documenting them on the chalkboard. Ask if the other students agree with all these combinations.

9.  Ask students come up with a possible rule for finding triples.

10.  If this seems to stump them, give this hint: There is an algebraic formula for each a, b, and c in a Pythagorean triple. Given two positive integers, u and v, where u>v, then a=2uv, b=u^2-v^2, c=u^2+v^2.

11.  Do not give equations; just explain that there are possible equations. This is a difficult so I might possibility give them the equations, depending on how much they seem lost.

12.  If students are able to come up with the equations on their own, have them present their ideas to the class.

VI.  Closure

13.  Explain that by using the three equations all possible Pythagorean triples can be found, and all these combinations can be described geometrically.

VII.  Extension

If time permits, have students use equations to find a Pythagorean triple and test those geometrically.

VIII.  Reflections (after lesson)