Explaining the Pythagorean Theorem with Models & Diagrams

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  • 0:04 What Is the…
  • 0:56 Pythagorean Theorem Diagrams
  • 2:27 Pythagorean Theorem Models
  • 4:28 Lesson Summary
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Lesson Transcript
Instructor: Michael Quist

Michael has taught college-level mathematics and sociology; high school math, history, science, and speech/drama; and has a doctorate in education.

It's one of the most important math formulas they'll ever learn, yet many students can't effectively use the Pythagorean Theorem. This lesson will discuss and demonstrate how to use diagrams and models to explain the Pythagorean theorem.

What Is the Pythagorean Theorem?

Joe the Fireman has to carry Maruto, the 300-pound sumo wrestler, down a long ladder to the fire truck. If it's 40 feet to the ground and 60 feet from the building to the truck, how long is the ladder?

Triangle problems that include a right angle are possibly the most common geometric math challenge that we're likely to see in real life. The Pythagorean theorem tells us that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

But how do you get that across to your students? How do you get them to become so comfortable with the principles involved that they can automatically use them, like a familiar hammer or pair of scissors? Two good ways to do that involve diagrams, or pictures that illustrate the concept, and models, or real-life challenges that involve the concept under study.

Pythagorean Theorem Diagrams

Diagrams allow you to point out relationships within a concept. For example, using the Pythagorean Proof diagram, you can illustrate a visual proof for the theorem, giving the students a real reason to believe that the theorem actually works. Once they have a feel for how it works, they then will much more easily make it a part of their repertoire.

You can show the students a diagram of a graphed right triangle. You can then expand it out to something like the Triangle Squares diagram, which is on your screen now and which shows what squares you're talking about.

Triangle Squares
Squares of Sides in a Right Triangle

As you explain that the area inside the two smaller squares adds up to the larger square, you may find that some students don't think that's true. That's when you can pull out the Pythagorean Proof squares to illustrate, and have the students cut out triangles and fit them in to see it work.

  1. Show them that the two diagrams are the same size. They can measure, or they can cut them out and place one over the other.
  2. Have them cut out the triangles, and verify that they are also the same size. The triangles are supposed to be congruent, since they have the same variable letters, but have the students see for themselves that it's actually the case.
  3. Have the students notice what's left in the two boxes. The square that remains in the left diagram has to be the same area as the two squares in the right diagram, because that's all that is left of both diagrams, and you've taken the same things out of both. You've replaced matching amounts of space in both diagrams, and they started out equal, so what remains has to be the same area.

Pythagorean Theorem Models

Models can be a lot of fun for the students, and can create memorable experiences that cement the conceptual and procedural learning into their minds. When you're trying to teach them to use the Pythagorean theorem, you can give them a real-life problem to solve, one that will require the use of the theorem. Remember, the more real the model is, the more valuable it will be in reinforcing the learning.

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