The Upper Elementary guys have been working through the beginnings of proof making for a couple of weeks now. All of the layers upon tedious layers have come together this week in the creation of a formula for a ratio for a particular type of triangle.

We have discovered that an equilateral triangle may be divided several ways. One of these ways is to create four small equilateral triangles and another way is through three identical isosceles triangles. We then compared one of the four small triangles and the large triangle and figured that this is a 4:1 ration.

At that point, I introduced a new equilateral triangle – one that was slightly smaller than the larger one from before. Over several days, the children established that this new triangle was equivalent to three of the small triangles from before. It has a 3:1 ratio.

We then decided that the smaller triangle (1) plus the middle triangle (3) would equal the large triangle (4). I laid down the largest triangle and placed the other two triangles to form the other two legs of a right angled scalene triangle. I was able to do this because the children and I knew through a proof that the altitude largest triangle (T3) was equal to the side of the middle sized triangle (T2) and that the side of the smallest triangle (T1) is equal to half the side of T3. (Still confused? Refer to the photos of the Upper’s project proofs.)

Ahhhhh. Bells went off in LR’s mind. “Wait! Isn’t that a binomial equation.” I could have hugged him. Angles sang; birds danced; cows jumped over the moon.

Today we discussed the ratios of T1 plus T2 equaling T3 (1+3=4)

This is very basic when we were looking at only one triangle each. What would happen if we combined each triangle with another of its same size and formed a rhombus. Ahhhh. (2×1)+(2×3)=(2×4) – Two because there are two of each triangle in our new equation.

What happens if we add another triangle to each making a trapizoid. (3×1)+(3×3)=(3×4).

We did this pattern work all the way until we made a visual hexagon with each triangle and had this equation in our list (6×1)+(3×6)=(4×6).

Off they went to make their own versions. Making T2 is difficult; depending on age and understanding, each child found their own way to create these triangles.

There was some discussion of Mandelbrot sets. However that was quickly overshadowed by a statement that the three hexagons looked like gears. JV set out to explain that you can’t put three gears touching (sequence yes) which led us to the

Lego Antikythera (see below for a really cool video) which led us to Charles Babbage and his steam or hand powered computer in the mid-1800’s (and he had a female programer who was the child of Lord Byron), which led us to Lord Byron and the Ottoman Wars. Boy do we travel far in our random association moments.