Here is a simple set of slides created using Geogebra. I would normally just do this directly on Geogebra but as I needed to prepare some slides for our department planning, I thought I would share them. I have added some suggestions for how to run these in the notes section
I want students to make the connection that if the y-value doesn’t change for a bunch of coordinates (in this base the y-value is always 4) then the line that those coordinates all sit on is y=4. I’ve even made a little GIF where the point deliberately slides off the the right for a little while. Gotta love a GIF!
I built this on Geogebra. It’s pretty simple but might be a good way in to 3D coordinates and more generally explaining the concept of dimensions.
You can download and open the original Geogebra file here which gives more control than just playing the gif.
- Start with all sliders at zero.
- As you increase slider a, talk about the first dimension. Any point in the 1D world can be described by a single number which shows how far along the line you are. Every object in a 1D world is just a line. Long or short. It can be described by a single number which we can call length.
- Once a has reached 1, talk about the second dimension. This is now like a floor, or the surface of the earth. We call this a plane.
- Increase the slider b.
- The world in 2D contains two dimensions, which we can call length and width. There are other words: e.g. breadth, depth.
- Every point in a 2D world can be described by 2 coordinates. These are the x and y coordinates. It’s important to notice that the x-direction (i.e. the x-axis) and the y-directions (the y-axis) are a right-angles to each other or orthogonal. Why is this?
- Once b has reached 1, what shape to we have? How many vertices does this shape have? How many edges? What are the coordinates of its vertices? Do we need 3 figure or 2 figure coordinates for a 2D shape?
- Now we can bring things into the real world in which we live, the 3D world where shapes also have height.
- Increase the slider c to grow the cube upwards.
- When c has reached 1, what shape do we have? How many edges, vertices, faces does it have? What are the coordinates of the vertices of this shape?
From there, you could always say:
Why stop at 3 Dimensions?
A friend of mine (who is not a maths teacher) recently sent me this article that Marcus Du Sautoy wrote in 2013 when he was president of the MA. He makes the case for a “Mathematical Literature” GCSE to sit alongside the utilitarian curriculum a bit like English Literature sits alongside English Language GCSE. It should aim to develop a student’s love of mathematics through doing maths for enjoyment to foster an appreciation of our rich and varied mathematical heritage.
At the end of the article Du Sautoy he says he is a mathematician not an educationalist. But I think the example he gives in here is a great one. There is definitely a lesson in it, and not just for higher attaining classes. Once the basics of plotting coordinates in a Cartesian system are secure, I think this would be accessible to any KS2/3 class upwards. But this comes with a health warning: I haven’t tried it yet, so (as with anything you find on the Internet) proceed at your own risk!
The lesson plan
- Pupils need to know how to plot coordinates in 2D (1st quadrant only)
- They need to know what a square is and be able to reason why the basic square has coordinates (0,0), (0,1), (1,0), (1,1). You could maybe start with something like this from Don Steward, although this is already maybe more than is required.
- They need a systematic way of finding arrangements of things. e.g. how many ways are there of arranging the letters ABC? ABCD? etc.
And that is about it.
So next, we can introduce the 3D coordinate system. Visualising things in 3d can be hard; often it is not closely correlated to mathematical attainment. But there are a wealth of options now to show and “spin” a 3D object on a screen. Here is one I created on Geogebra:
I would spend some time on this, looking at systematic ways to make sure we have the coordinates of all 8 vertices. You might want to take the approach I wrote about here. I would try to convince students that they could have done this systematically with having had to see the cube in 3D space. Because what they are going to do next is find the coordinates of an object that they definitely can’t see – a 4D cube.
Although it’s impossible to see in our 3D world, we can use maths to work out the coordinates of each vertex of a 4D cube. Again, if this is done systematically students will hopefully soon see that the number of vertices doubles each time we add another dimension. Here is 2D, 3D and 4D:
The next part of the article is where it really gets interesting. Because as well as this being an important piece of maths in its own right, it also has an application in computing. It is used in error correction when sending digital signals. The rules are reasonably straightforward and the article provides an example of a piece of code that contains an error. Can your students apply the rule to find the incorrect bit?