## the tetrahedron

This is a reflection on a lesson I did when asked to cover for another teacher. I was asked to teach

I gave groups of students a piece of coloured A4 paper, then asked them to work out the area of the piece of paper. All the students measured the sides, and came up with various (different) answers

I then gave them the information that a piece of A0 paper has area 1 sq metre, and mentioned the fact that paper sizes work by halving the area each time. After a while, one group used a calculator to come up with the answer 0.0625 sq m. We then discussed that this was also 625 sq cm, which reinforced earlier work they had done on units of area.

At this point I mentioned that this was a lesson on area and volume, and that we were going to use origami. I asked them to fold their sheet of coloured paper like this:

**surface area and volume of similar shapes**.I gave groups of students a piece of coloured A4 paper, then asked them to work out the area of the piece of paper. All the students measured the sides, and came up with various (different) answers

*around*the correct answer of 625 sq cm, demonstrating effectively the inaccuracy of measurement.I then gave them the information that a piece of A0 paper has area 1 sq metre, and mentioned the fact that paper sizes work by halving the area each time. After a while, one group used a calculator to come up with the answer 0.0625 sq m. We then discussed that this was also 625 sq cm, which reinforced earlier work they had done on units of area.

At this point I mentioned that this was a lesson on area and volume, and that we were going to use origami. I asked them to fold their sheet of coloured paper like this:

I asked them what angle(s) this fold creates. They had a few guesses, including the correct answer for the bottom left corner of 30 degrees. I then asked how could they prove it? A few said they could measure it, but I reminded them that measuring was not accurate. Then a couple of groups came up with the interesting idea of using trigonometry, and one group measured the sides of the triangle and correctly used tan to find an angle of 30.1 degrees. A great effort, but again, not accurate!

I then showed them a proof using similar triangles (can you see how?).

After this, we then made tetrahedra using a pattern from Mathematical Origami by David Mitchell that requires two pieces of paper. This naturally suggested that students work in pairs; modular origami is excellent for getting students to work together.

After we made the tetrahedron, we discussed how to use Pythagoras find the area of one face of the tetrahedron (supposing the length of the edges is 1) and then multiplied by 4 to get the total surface area (which is root 3).

I then asked them what would be the surface area for a similar tetrahedron made out of A5, or A6 would be? They were not sure and were thinking that it depended on how we had folded the paper, how many sheets of paper we had used, etc. I then showed them an A6 tetrahedron I had made earlier; they could see clearly that the lengths of it's sides were halved and the surface area was one quarter (you could fit four of them on a face).

Then I asked how many of the small A6 tetrahedra you could fit

I think they might have learnt something; there was lively discussion and I know they had fun doing it! We touched on quite a lot of important ideas, including measurement, proof, trigonometry, angles in triangles, surds, and, of course, area and volume.

I would like to extend this lesson idea further and investigate various angles (for example, the dihedral and tetrahedral angles). A nice way of doing this might be by dipping tetrahedron frames in soapy water and looking at the bubbles!

Or perhaps try this origami dissection puzzle. And I would like to have shown them this interesting tetrahedron video by Burkard Polster.

I then showed them a proof using similar triangles (can you see how?).

After this, we then made tetrahedra using a pattern from Mathematical Origami by David Mitchell that requires two pieces of paper. This naturally suggested that students work in pairs; modular origami is excellent for getting students to work together.

After we made the tetrahedron, we discussed how to use Pythagoras find the area of one face of the tetrahedron (supposing the length of the edges is 1) and then multiplied by 4 to get the total surface area (which is root 3).

I then asked them what would be the surface area for a similar tetrahedron made out of A5, or A6 would be? They were not sure and were thinking that it depended on how we had folded the paper, how many sheets of paper we had used, etc. I then showed them an A6 tetrahedron I had made earlier; they could see clearly that the lengths of it's sides were halved and the surface area was one quarter (you could fit four of them on a face).

Then I asked how many of the small A6 tetrahedra you could fit

**inside**the larger A4 one and they were pretty confident it was 8. (This is true by volume although you can't actually fit 8 of them in!).I think they might have learnt something; there was lively discussion and I know they had fun doing it! We touched on quite a lot of important ideas, including measurement, proof, trigonometry, angles in triangles, surds, and, of course, area and volume.

I would like to extend this lesson idea further and investigate various angles (for example, the dihedral and tetrahedral angles). A nice way of doing this might be by dipping tetrahedron frames in soapy water and looking at the bubbles!

Or perhaps try this origami dissection puzzle. And I would like to have shown them this interesting tetrahedron video by Burkard Polster.

If all this has got you excited about 3D shapes, you should have a go at making polyhedra.