No 5 in a series of posts based on Thinking Mathematically (1985) by Mason, Burton, Stacey
I might use this task as an introduction before doing anything involving 3D visualisation. I’ve always felt that being able to conceptualise physical shapes is a skill that is very hard to teach in an instructive way. Tasks like this provide experiences that help to develop those skills. Here is the task:
I have 8 cubes all the same size. Two of them are painted red, two green, two blue and two yellow.
I wish to assemble them into one large cube with each colour appearing on each face.
In how many different ways can I assemble the cube?
The first challenge of this task is to interpret what it is asking and construct a mental image of that. The above image may help.
Then the question comes what do we mean by “different” large cubes, which is a conversation starter in itself. If we have the physical cubes in the form of multilink cubes we are likely to see that “different” means that one large cube cannot be simply rotated to form another one. The position of the small cubes actually have to be changed.
This 3D animation created on Geogebra (file here) may also help if real cubes are not available.
As John Mason points out in the book, whichever representation is used, it is important to record carefully the “different” cubes so they can be compared.
I came to the realisation that I need only look at the 4 small cubes on the front face because I can deduce where other 4 cubes on the rear face will be. Because no two small cubes of the same colour can share a face, the small cubes must be arranged so that they on diagonally opposite corners of the large cube.
So then it was a case of working out how many ways there are of arranging 4 things in a 2×2 arrangement where rotation is not allowed. I actually looked at it like this.
I convinced myself there are six ways of doing this. But I’m still not convinced that this means 6 is the answer to the question. I feel I would need to construct it to be sure I haven’t got any repeats within these 6.
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?
Origami is one of those things that I think I would love to spend more time exploring but rarely do. I have used the Origami Player with my students (it works really well as an App within Chrome), which gives excellent visual instructions on making things. The timings have been well thought out and it gives a little timer prompt so you know how long you have got to do that fold before looking up at the screen for the next one. It’s been an end of term, easy lesson. Nothing wrong with that.
The first session on the nRich Teacher Inspiration Day last week where we looked at some of the activities here got me thinking about how I could make it a slightly more meaningful learning experience.
Still not highly mathematical, but at least it gets students working together and struggling with something. To build resilience in our students they need see the struggle as a positive and not something to be avoided at all costs. It was a bit of a metaphor for all learning. A discussion that can be had with students when reflecting on this task might be along the lines of:
- Did you need help from someone at some point? (yes, good)
- Did you help someone else at some point? (yes, good)
- Did you struggle at some point? (yes, good)
- Did you give up? (hopefully no, good)
- Did you achieve something you didn’t think you could do before?
This type of discussion can be a powerful motivator and more useful than vague questions like “Did you enjoy it?” or “Did you have fun?”
There are lots of Origami ideas on this page of nRich’s new Wild Maths site. I really like the idea of modular Origami, i.e. each person makes a module and then they come together to create something beautiful. I have an end-of-term cross curricular session with Year 8 to plan. My Origami paper arrived this morning. You can use A4 paper and fold down the corner to make a square, but proper origami paper is really lovely and this pack was only £8 for 500 sheets. It looks great so time to start practising!
I will post an update on this as planning progresses and share some pictures of the final event. In the meantime, if anyone has an ideas for how to make Origami more mathematical (without spoiling the enjoyment of it!) then let me know.
The idea of these tasks is to get students to think more deeply about volume and to help visualise how different volumes “fit” together.
Questions 1-5 can be done by effectively counting cubes. Multilink cubes would be good to help with the visualisation at this point.
Question 6 is the one to get the discussion going. Is the answer 16 (if you picture them as solid cubes, rounding down) or 21.12 if you consider it as a flexible volume (e.g. a liquid)?
If you want to adapt this resource you can create your cuboids using this Geogebra resource.
Some questioning as a way to get to Platonic Solids:
- What is special about a cube? (compared to a cuboid)
- What is special about squares? (the sides on a square)
- What do we call 2D shapes where all the sides are the same?
- So, can we generalise, what is special about a cube?
- What other regular 2D shapes do you know. Draw them now.
- What other 3D shapes can you picture that can be constructed only with regular 2D shapes? These are called Platonic Solids.
Have a go. See if you can find some dusty old 2 shapes in a cupboard somewhere.
These are quite cheap or you could make your own from card using this template.
So, here they are in all their glory. The wikipedia page has some nice animations.
The Greeks discovered these 5 and mathematicians since have proved that there are definitely no more, whichever polygons we use.
Next you could complete this table.
Once you have done that, can you spot any patterns in a relationship between F, E and V? You might even discover Euler’s formula, which holds true for any convex polyhedra, not just the platonic solids.
V – E + F = 2
Or use (a lot!) of 2d shapes to make nets of these shapes.
This page gives all the answers in a nice table!