Parallelograms, you know, the weird quadrilaterals that look like a sheared rectangle. These:
I’ve never rally thought that deeply about them, to be honest. They have some uses in angle reasoning lessons, and we need to be able to find their area in the GCSE, but I’ve not thought too deeply about them recently at all.
When teaching how to find the area I normally do this:
It’s a fine method, and easy to show that it works by showing that you can cut the end off, stick.it in the other end and get a rectangle which is clearly of the same area.
But last week I marked a mock exam in which one of my year 11s had done this:
I love this method, it’s much, much nicer than the other. I couldn’t wait to question him. When I did he said that he “couldn’t remember” how to do it, but knee how to find the area of a non right angled triangle so split it into two of them which were congruent using SAS.
I asked him what would happen if you split the parallelogram across the other diagonal. He thought about it for a while, and eventually told me it would be fine because of “how the sine curve is” and because, “the angles add up to 180”.
I was impressed by his reasoning. He has clearly understood this method and generalised the area of a parallelogram in a way I’d never considered. I would have phrased is slightly differently though:
The area of a parallelogram is equal to the product of two adjecent sides multiple by the sine of one of the angles. (Either will so as Sin x = Sin (180 – x) )