Inspired by this Monty Python sketch, I asked students to find the area of a triangle, if they know the lengths of the three sides. Gradually I chopped off a mathematical limb, removing ipads, calculators and compasses, before only a brain and a pencil remained.
Some great and unexpected learning:
- The compass was too small to draw an arc of 13cm. So we enlarged the shape by a factor of 0.5, and thought hard about how that would affect area
- We thought about how to accurately drop a perpendicular from a point to a line, when measuring the perpendicular height of a triangle
- We wanted to work out the square root of 21 x 8 x 7 x 6, but without using a calculator. Prime factors to the rescue!
I had given the class 5 minutes to try and find a way of calculating the area of the triangle, using only a pencil and a brain. Some wildly inventive methods, that involved inventing an adapted version of Pythagoras got students close to the correct area. Great creativity, but lacking in precision.
The start of understanding the proof:
- If a student knows every constituent skill then they can understand the proof. Whole is greater than sum of the parts – the multiple steps are difficult to keep in your working memory
- Students have well-trained instinct to expand brackets. Sometimes this isn’t helpful!
- One student spent a long time trying to prove that s = 1/2(a + b + c). The distinction between something you can define and something you have to prove.