Recently we covered the Pythagorean Theorem in class and I was looking for a proof to use to discover the theorem when my CT gave me one of the area based ones that she has used in the past. I decided to use that one and do a whole class discussion about using this for the proof. We started with this jumbled mess of pieces and then formed them into two squares with each having four of the red triangles.
After we got them in the squares I told students we were going to label each side with a letter, a for the shortest side, b for the middle side, and c for the longest side. I then asked students to tell me the area of each of the squares that we created. From this we obtained the following,
Students immediately saw that the area of the squares were equal. After this we talked about if there was anything special about the red triangles in each of our squares. Rather quickly one student explained how the triangles were congruent because all three sides were the same so we have eight congruent triangles so their areas are equal . After we had these two facts nailed down I started removing triangles.
I removed one triangle from each and asked if the areas of the shapes there were still equal and I got a resounding yes from my students. I then took away another triangle from each.
I asked again if these shapes still had the same area and got another yes from the students. After this step our discussion got rather interesting. I removed a third triangle from each to obtain this shape,
I asked again if these two shapes had the same area but now this time a vast majority of the class yelled NO! This struck me, as this wasn’t something that I would have thought would have been a common answer for the class, although I will admit my CT said this happens most years that she has done this activity(There were a few students who said yes but a majority of the class was not buying it). I asked for a few students to share why these shapes did not have the same area anymore and got very similar answers from all those that shared and it went a little like this. Students talked about how the pink square in the shape on the right was messing with the area because it was somehow different than it was before. The most I could get out of them on what they meant by this was that since it was not connected to a triangle anymore by a side it had to messing with the area. Just to make sure that we were all on the same page about the shape with two triangles I added the triangle back and asked if these had the same area and again they said yes. So I took away the triangle again and asked again if the shapes still had the same area and again I got a no. I asked the class if I had changed anything about the size of any of the shapes. “No”. Then we talked about what we knew about the triangles in each shape. One student said they were congruent and another added that congruence means that they are exactly the same. I thought finally we’re getting somewhere with this. I told students to think about what we knew about the previous shapes and what we just discussed about the triangles. As I waited hands started going up and various students shared that they shapes had to be the same area because the area was same before the third triangle was removed and since we removed congruent triangles the new shape also had to have the same area. I was really happy with the fact that they were able to figure this out! So we removed the last triangle and decided that their areas were also the same and after that triangle we were left with three squares.
Since we knew the area of each of the squares and that the area of the yellow square was the same as the green and pink squares combined we rearranged the squares and wrote an equation.
I asked students if they had ever seen this formula before and many stated that it was the Pythagorean Theorem! After we saw where the formula came from we spent some time using it.