Monday, November 12, 2012

Nets of Prisms

I think that if you were to ask most people what a net of a cube looks like, they will say a cross shape. Very few though will know that there are in fact 11 different nets. I've challenged both students and adults to find all 11 and it never fails to get everyone engaged. But this is mainly because of the manipulative that we use: Polydrons. Here it is easy to quickly check if a net does or does not work. It is also great to have a good old think about what congruence means and how two 'different' net are in fact the same if one is a reflection or rotation of the other.
I was in a Grade 3/4 split class recently and we were hoping to do this nets of a cube investigation. However after a quick glance at the bucket of Polydrons, I realised that I would have to change things a bit: there were not enough 'square' Polydrons. We would have to find nets of triangular prisms instead. This was going to be a ton of fun for me too as, off the top of my head, I didn't know how many such nets existed.
I started the lesson by showing a net of a cube (the one shaped like a T) and asked them to predict what this 2D shape would fold into. Most predicted a cube (some said a box, some said a square) and everyone was quite happy when I showed them that indeed this was the case.
I then held up a triangular prism and gave them the challenge:
Find as many nets as you can for this shape.
There was one catch: they had to tell how many of what shapes they needed. Rather than giving them two triangles and three squares, I wanted them to think about what was required. They were all able to do this even though I didn't hold up the prism for very long. In fact, I could see a lot of students trying to visualise what was needed (you know that look, when they seem to stare at a spot on a wall and have their hands in front of them holding an invisible prism?)
In a short space of time we got the following:
This was pretty much the first one that most students did.
And each time I responded with "Great! Now make a different one." And they did!

When students came up with congruent nets we had great discussions that led us to agree that even though they looked a bit different, mathematicians say that congruent nets are in fact the same net.

The wonderful thing about Polydrons is that students can easily reason and prove if their nets do work as seen in this clip:

Planning the consolidation with other teachers and principals was fun. I was asked how many nets existed. "Well, I think there are six as that's how many I've found," I said. In a flash, my buddy Mansel had corrected me (with a little help from Google I think!): there are in fact eight. So we set as a challenge to the class: they had found four, now they had to find the other four.
The other thing we wanted to check is do they really know what a prism is? Their own explanations were on the right lines but far from exact. So I used an analogy which has helped many students (and teachers) in the past.
A prism is like a loaf of bread: you can cut it from front to back and each slice is always the same size and shape.
I have found that this draws their attention to two important attributes that in later grades will be very important for working out the volume: the slice of bread is the area of the cross section and the number of slices will represent the 'length' of the prism.
For this class though it was enough to say that the shape of the slice tells us the name of the prism. Holding up a cube, I asked them what this could also be called. After a quick discussion, most groups agreed that it could also be called a square prism.
Or squarular prism as one student said.
I later showed the adults some paper cups that you get in fast food joints for getting ketchup and asked them what is the net of this shape. Most people are surprised when you unwrap the cup. Try it!
I then showed this photo I took recently and asked "Is this a prism? And how might you work out its volume?"

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