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Cubes


Before you read much further I think it will help if you get yourself three cubes that are the same size. Maybe if you are at home they could be the kind that very young toddlers have to play with, made from wood or plastic. If you are at school it should be no problem, but avoid those cubes that have bits for connecting together. Just three cubes of the same size.


pic 1

Place just one of these cubes down in front of you on the table. I'm going to ask a question that gets repeated each time we try a new idea. Here's the question:

"If you are allowed to walk around the table, bend down low, move your head around in any way that you like, get as near to the cubes as you like, etc. then how many square faces could you see altogether, without lifting up or touching the cubes in any way?''

Well I guess that with one cube it is quite easy, yes it's 5.

Now use two cubes and put them in any way that you like. Well, here are three arrangements I did earlier.

pic 2

The first one lets me see 8, the second one 9 and with the two on their own it's 10! So the answer to the question depends on how the cubes are arranged. I suppose that's pretty obvious. Well, what about three cubes? Here are some arrangements - just some.

pic 3

The first has 11; the second has 13; the 3 on their own has 15; the last one has 14 and two-thirds. Now that's a strange one - you can see why we need "non-connecting'' cubes. "How did I get 14 and two-thirds?'' you may be asking. It's the sort of question I ask the children I'm working with. I'll tell you. The red shows 5 whole faces, the green and the blue each show 4 whole faces. There is about 1/3 of the underneath of the red I could see if I creep down low and look up when I'm very close. There is 2/3 on top of both the blue and the green that are visible. That makes a total of 14 and 2/3.

So you take your 3 cubes and see what your answer to the question is each time you make a new arrangement. It's probably good to record your answers. This is very easy with some art packages on the computer. You might like to get an arrangement that produces a particular number of faces visible. See what your largest/smallest answer is. When you think you have got sufficient answers, probably something in the region of 10, then use a fourth cube and see how you get on now. Good Luck.


Why do this problem?

Use this activity not only as an investigation in itself but also to lay some foundations about (surface) area. It is also very useful in giving the opportunity for pupils to talk about fractions in a most thoughtful manner.

Possible approach

I tend to introduce this activity with a whole class of children sitting around in a circle and the cubes that we are looking at placed in the middle. (I've used 10 cm cubes.) I've asked, "Tell me what you can about this." [one cube placed on the floor] Someone always says that there are six faces or sometimes the word 'side' is used, particularly with younger children. I accept with interest all the answers that the children give, after all they are telling me what they know or can see. I focus in on the faces and ask the question set out in the challenge, emphasising that I can place my eyes almost anywhere. This sometimes requires me to lie on the floor!

When everyone is happy about the answer 5, I introduce a second cube and one by one go through the three examples that are set out. I emphasise that the three answers are different. From a teacher's point of view I ask the children to explain how they are counting. This brings out some very interesting points and you learn a lot about the children.

Moving on to the three cubes I usually do two or three examples and then ask for volunteers to do one for everyone to consider. The fraction examples usually come up and even with young children I find that they are very happy with adding up halves and giving good estimations.

It is probably a good idea not to be particular about the standard of recording in this activity. It is the activity itself in arranging and counting that is important and not artistic skills.

Having left the circle and begun to try their own three-cube arrangements, children get into heated discussions about counting because I ask them to check each other's counting. This difficulty has usually arisen whilst they are in the circle. They share ways of making sure that all are counted and that none are counted twice.

Key questions

Tell me about the parts that you can see.
Tell me about this bit here that is not a whole face.

Possible extension

You can give pupils the opportunity to categorise the shapes that they make and the sizes of areas they find. Of course you could move onto using more cubes.
 

For the exceptionally mathematically able

When working with a very able pupil I used the idea of a cube with a circular hole. There happened to be an old powder paint container around that gave me the idea. So, imagine (or use if you've got one) a cube that has a circular hole in the top. The hole is covered in a special paper - so that a "vertex down" cube can pierce the paper as it is lowered into the hole. The remaining paper stays intact!  Or you could think of it as the green cube being filled with sand, the red cube is pushed in and overspill-sand is wiped away.
What is the total area of the green and the red cubes' surfaces that are now visible?
This picture gives some idea of how the red cube is as it is about to be lowered into the cylindrical hole.

1st H CubeHoley Cube



Possible support

For a pupil who gets over anxious about the recording, use a digital camera to take photographs of his/her arrangements.