61=6²+5²;60=2•6•5;11=6²-5². What is special about 61, 60, 11?

For geometry/algebra students.

Discovery Lesson: Consecutive Numbers and Pythagorean Triples

Grade Level: Middle School/HS/Geometry/Alg

[Assume they have seen the Pythagorean Theorem]

Goal: To guide students toward discovering a rule for generating a Pythagorean triple from any two consecutive positive integers using Socratic dialogue and algebra.

Warm-Up / Review (5 minutes)

Teacher: Why do we call 3,4,5 a Pythagorean triple or triplet? Now verify that 5,12,13 is also a triple.

Teacher: Good. Now let’s do something new with consecutive numbers…

Exploration (15–20 minutes)

Teacher: Let’s take two consecutive positive integers: 5 and 6.

Calculate:

• 5²= 25

• 6²= 36

Now, what do you get when you add the squares?

Student: 25 + 36 = 61

Teacher: Now subtract the squares. What is 6² -5²?

Student: 36 - 25 = 11

Teacher: Hmm… and finally, what if we multiply the product of six and five by two?

Student: That gives 60.

Teacher: Let’s look at those numbers: 11, 60, 61.

Do they satisfy the Pythagorean Theorem?

Let’s check:

11² + 60²= ?

121 + 3600 = 3721 = 61²

Student: It works!