Cubes

How many faces can you see when you arrange these three cubes in different ways?

Age

7 to 11

Challenge level

Exploring and noticing Working systematically Conjecturing and generalising Visualising and representing Reasoning, convincing and proving

Being curious Being resourceful Being resilient Being collaborative

Problem

In this activity we will be using cubes to make different arrangements. First, find three cubes that are all the same size and all have flat faces.

Start by looking at just one of these cubes.

Put it onto the table or another flat surface. How many square faces can you see? You are allowed to walk around the cube and bend down to see it, but you aren't allowed to pick the cube up. Click on the button below to check your answer:

There are five faces that we can see, because the sixth face is against the table.

Next, take two of the cubes. Put them down on the table, either separately or together. What different arrangements are there?

I've made these three arrangements with my cubes:

How many square faces can I see in each case? Have a go at using your imagination to visualise how many I can see, and try making these arrangements yourself to check your answers. Then click on the button below:

In the first example, I can see nine faces - there are four lots of two faces around the sides, then one face at the top.

In the second example, I can see eight faces - one at each end, then three lots of two faces.

In the third example, I can see ten faces - there are two lots of five faces.

These three arrangements each showed me a different number of faces. Are there any other ways in which I could have arranged the cubes? Would these show me a different number of faces?

Think about how we can tell whether or not two arrangements are the same. For example, I could turn the second arrangement so that the red cube is pointing to the left instead - would this count as a different arrangement? Why/why not?

Finally, take all three of the cubes. Arrange these in some different ways. It might help to write or draw each arrangement to keep track of the ways that you have already tried. Here are some of my arrangements:

How many square faces can I see in each of these three cases?

Now it's your turn! Use your three cubes every time, and see how many different ways there are to arrange them. How many faces can you see each time? Try to find all the possible answers.

Further challenge:

What happens if the faces don't line up completely? Look at the example below:

I can see $14\frac{2}{3}$ faces on this arrangement. See if you can work out how I counted them, then click on the button below to check your answer:

On the red cube, I can see five whole faces, and if I bend down I can also see $\frac{1}{3}$ of a face underneath the red cube. On the blue cube, I can see four whole faces around the sides and then $\frac{2}{3}$ of a face on top. The green cube is the same as the blue cube, so this also shows me $4\frac{2}{3}$ faces. Adding all of these together gives me:

$5\frac{1}{3}$ + $4\frac{2}{3}$ + $4\frac{2}{3}$ = $14\frac{2}{3}$

How many arrangements can you find where the faces don't line up completely? How many different ways of doing this do you think there are?

Teachers' Resources

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.

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.

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.

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.

If you want to see how I counted up that last one click here,

"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 the fourth cube and see how you get on now. Good Luck.

Why do this problem?

This activity is a good starting point for pupils to learn about surface area of shapes. The further challenge activity provides plenty of real-life opportunities for children to consolidate their fractions knowledge. Completing this task will involve pupils working systematically, using mathematical reasoning to explain how they know they have found all the possible solutions.

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 with 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.

If, as the adult, you need to see in advance what ideas may, and I emphasise may, come from the pupils then click here.

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. You could also move on to using more cubes.

For more extension work

When working with a very high-attaining 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.

Possible support

Some pupils will find it helpful to be able to keep their finished arrangements, instead of breaking them up. This will allow children to focus on building the arrangements and working out the totals rather than getting held up with drawing or recording their solutions. Taking photographs of the arrangements is another option for these pupils.

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Cubes

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Student Solutions

Teachers' Resources

Cubes

"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?''

Why do this problem?

Possible approach

Key questions

Possible extension

For more extension work

Possible support