## Little Boxes

I don't know whether you've had the same problem as me. I've found that I have loads of CDs and DVDs and they are not very tidy. I decided to collect them all together in their boxes and after putting them all the same way up I started putting them together into cuboid-type shapes. I found when I started that just two could be put together to give three different cuboids. Here's what they
looked like:-

I've used different colours to show each box, just to make it clearer to see, but we're going to imagine each box is identical.

I then tried it with an extra box making three altogether. I moved them around and found that again I could make three different cuboids. These are the ones that I found:-

But, perhaps like you, I've got lots of these boxes of CDs and DVDs. So I moved on to using four. Again I found three but realised that there were more that I could find. Here are the three:-

Well, you could try the same ideas. If you have not got CD/DVD boxes it does not matter. You could use old match boxes or books that are all the same size. In fact you can use any objects as long as they are cuboid shaped and you have plenty of them that are exactly the same. All you have to do is to find out how many different cuboids you can make when you use fourof them, then five of them,
then six *etc.*

But notice that when you've put them together it does not matter which way up they are, they count as being the same. For example these three would count as being the same:-

Now that you've tried some, how about getting other friends and people at home to have a go and see what they all come up with - maybe you could work together?

Well, Good Luck.

### Why do this problem?

### Possible approach

This activity is probably best done in groups of two or three in the classroom, for the discussion forms a very vital part of the work. The resource that you use does not matter too much but it is best to avoid cuboids that have three measurements that are related in some multiple/factor way. So a 2 by 4 by 10 or even a 3 by 5 by 9 would not be so good really.

When the children get to the four cuboids which can be arranged in more than three ways as I have shown, then the interest grows. If the children are encouraged to tabulate their results in some way then there are lots of fascinating things that crop up which are not too difficult to get to the bottom of, usually.

### Key questions

Do you know if you have found them all?

How do you know you have found them all?

What else could you find out?

### Possible extension

For those pupils who have confidently completed the activity with maybe up to 7 or 8 boxes then the time could be right for looking at the patterns of numbers generated and to see whether they can come up with any predictions.

### For the exceptionally mathematically able

These pupils could be challenged to consider the activity in which the dimensions of the boxes relate. For example suppose they were $2$x$4$x$8$ cm?What difference does it make? What sequence of numbers does it now generate as you increase the number of boxes? Why?

### Possible support

Some children might find it useful to have an adult to help organise the way they go about finding all the possibilities.