Why the Pythagorean Theorem Needs a Visual Re-Visit

In many classrooms, the Pythagorean Theorem is taught as a calculation task: plug in the numbers, square them, and find the square root. However, the theorem isn't actually about the numbers; it’s about the areas of squares attached to the sides of a right triangle.

The most powerful visual for this concept is literal. If you have a right triangle, the "square" of side a (the a^2 part of the formula) is quite literally a square drawn on that side.

• The Concept: The area of the square on side a plus the area of the square on side b is exactly equal to the area of the square on the longest side, c (the hypotenuse).

• The Visual Proof: You can show students "proofs without words." Imagine the two smaller squares are containers filled with water. If you were to pour the water from both smaller squares into the large square on the hypotenuse, it would fill it perfectly.

Real-World "Visual" Applications

To make this stick, have students apply the visualization to scenarios where they can't just "see" the triangle immediately.

• The Ladder Problem: If a 10-foot ladder is leaning against a wall 6 feet away, how high does it reach? Visualizing the wall, the ground, and the ladder as a right triangle helps students see why we are solving for a "side" (b) rather than the "hypotenuse" (c).

• Screens and Ratios: Televisions are sold by their diagonal length. A "50-inch TV" is actually the hypotenuse of a right triangle. Visualizing the screen as two triangles joined at the hypotenuse helps students understand how the width and height relate to that 50-inch label.

When students see the squares on the sides, they stop asking, "Why am I squaring these numbers?" They realize that a2 is an area, and they are simply adding two smaller areas together to get a larger one. This geometric intuition makes the algebra $(a^2+b^2=c^2)$ feel like a natural consequence of the shape, rather than a rule they have to follow.

There are several misconceptions associated with the Pythagorean Theorem.  One is when students add the sides instead of the square so instead of a^2 + b^2 = c^2, they are thinking a + b = c.  When you create a square for each side, you can cut the squares loose and then move them to the hypothenuses so they can see they make a square there.

Another misconception is to solve for c^2 but forgetting to find the root.  When you show the largest square for c^2, they see it is the area of the square but we want to know the length of just one side. 

Thus providing visualization for the pythagorean theorem, students can relate that you're are adding areas together to find the area of the hypothenuse.  Or going the other way to show how to find a single side by taking away the area of the side you have.

Let me know what you think, I'd love to hear.  Have a great day.

 

Pythagorean Theorem. When Is The Best Time To Teach It?

  Today, I reviewed the distance formula in preparation for the semester final.  It got me to thinking when in the semester should I teach the Pythagorean theorem?

If I teach it at the beginning of the year when I do midpoint and distance, then it is quite applicable for distance. 

If I wait till later in the semester when I'm doing trig ratio's then I've missed the chance to relate the distance formula to it. 

I like to teach the Pythagorean theorem just after I teach classification of triangles, congruent and similar triangles because I have students use the  equal, less than or greater than to tell the type of triangle based only on measurements.

I love the theorem because it has so many possible applications from vectors to televisions to physics and it is good for them to know the basic formula.  Perhaps, this needs to be taught at different points throughout the geometry class with different applications so students see the formula as the course progresess.

Some real life applications include:
1. Road trips - finding the shortest route.
2. Painting buildings to help find the right sized ladder.
3. TV's and Computer Monitors
4. Navigation.
5. Surveying.

So if I consider that it is better to teach the theorem at several points throughout the course and I include the appropriate real life examples, then it might help students learn to use the theorem better and become familiar with it outside of the theoretical state.

If you have any suggestions, I'd love to hear from you.

Wow

  Today went very well and I think I found a balance for calculators in one class.  Tuesday,  I introduced/reviewed the Pythagorean Theorem in geometry. 

I had the students work some basic problems by hand.  I started by doing a few problems on the board that they copied down.  Next step was to have them try some themselves but after a couple minutes I worked it on the board in case they got stuck.  It worked quite well.

Today, I had them work more problems using a calculator so they could concentrate on when to add or subtract.  I gave them the if there is a hypotenuse length, they subtract to find the missing side and if they only have side lengths, they add to find the missing hypotenuse.  It was great because they spent a lot of time on learning when to add vs subtract and they didn't have to worry about their math.

At the end of the class period, a few students bemoaned the fact that class was over already.  That was so great.  In addition, the worksheet they used today included problems on using the theorem to determine if the triangle is acute, right or obtuse. 

Monday, we will stray to another worksheet long enough to learn to use the theorem to decide the type of triangle.  I will let them use the calculators again because it seems to take some stress off and allows them a chance to learn the material. 

From here, the students will be learning trig ratios and they need the Pythagorean Theorem for those.  I think I'll also look for some real life examples so they know there is a use for this material.