# Engaging students: Deriving the Pythagorean theorem

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Haley Higginbotham. Her topic, from Geometry: deriving the Pythagorean theorem.

How could you as a teacher create an activity or project that involves your topic?

An interesting hands-on activity would be to do a visual proof of Pythagorean Theorem by using just paper, scissors, a ruler, and a pencil. Starting with a square piece of paper, the students will make a square with a length in the bottom left corner and a square of b length in the upper right hand corner, similar to the picture below on the right-hand side. Then the students would cut out the squares, and end up with two squares and two rectangles. The students would then be instructed to cut the both rectangles along their diagonals. Then the challenge is to make a square that contains a square inside by only using the triangles they have cut out. The level of difficult of the challenge will depend on the grade level and on the caliber of students, but it’s still more interesting than writing out a formal proof. Then after everyone has made something similar to the picture below on the left-hand side, I would ask them if they know why this proves the Pythagorean Theorem. If a student has a good explanation, I would ask them to demonstrate their explanation to the rest of the class. If no one figures it out, I would suggest they label the different lengths and see if they figure it out then.

How has this topic appeared in high culture?

The Pythagoras tree is a fractal constructed using squares that are arranged to form right triangles. Fractals are very popular for use in art since the repetitive pattern is very aesthetically pleasing and fairly easy to replicate, especially using technology. The following picture is an example of a Pythagoras tree sculpture extended into 3 dimensions. There is also the Pythagorean snail, which is constructed by making isosceles right triangles in a circular pattern, keeping the smallest leg of each triangle the same size. With this basic design, you can create a variety of designs, an example is pictured below. Even though the base is a bunch of triangles in a spiral, the design overlaid on top of it takes it from purely mathematical to a piece of art. Of course, one could argue that mathematics itself is an art, but the general population would agree that the design really makes it a work of art.

How can technology be used?

I think I would use Desmos to extend the activity described in A2, since they have seen it works for their particular choice of a and b, but they might not see how it works for all choices of a and b (as long as the triangles they have are right triangles). By using Desmos, I can use an activity that allows students to drag the different sides to see that the relationship holds no matter how a and b change. I think something similar to this activity would work: https://teacher.desmos.com/activitybuilder/custom/5adc7bfced2ada678516940e, except I would modify it so it was closer to the activity that we did with paper during class. I would also show them the other explanation of the squares aligned with the lines of the triangles. It’s great because Desmos has activities that you can use but you can also customize the activities however you want to fit your specific ideas. You could also ‘code’ from scratch your own activity.

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