## Song Book

A school song book contains 700 songs, numbered 1 to 700.

Each month at a special assembly, the children sing four different songs.

The numbers of the songs are displayed to the children by putting together single-digit cards like this:

The card for 6 may be turned upside down to serve as a 9.

What is the minimum number of digit cards that is needed to show any possible combination of four song numbers?

How many of each digit must there be?

### Why do this problem?

This problem encourages logical thinking and, to a lesser extent, an understanding of place value.

### Possible approach

This is an ideal activity for 'think, pair, share'. Give the children a chance to work alone and then in pairs, so that they can check out their thinking with each other. You could ask them to record their solutions (just the answers) on the board. Put pairs who have arrived at different solutions together to see if they can convince each other of who is correct - and find where their
reasoning differs.

Some children will be able to do this question with minimal recording and maximum mathematical reasoning. Others will need practical materials to get them started and will then move to abstract working. Others will need cards or other practical materials for the whole question. Try to move children on to working without materials as soon as possible so that they are beginning to generalise
(what's true for 1 is true for 2, 3, 4, 5 for example).

### Key questions

How could we start this problem?

What's the biggest number we're going to need? What's the smallest?

Are there any numbers that we'll need more of/fewer of?

How could we record this?

### Possible extension

What 'What if .. ?' questions could we ask?

Perhaps we could consider numbers up to 800 or 900 - can we now find a quicker way of doing this since we've done the numbers up to 700?

### Possible support

Children who need support might choose to use printed grids like

this for filling in possible numbers and lots of scrap paper cut up into squares for writing numbers on.

Or you could make the question simpler by asking them to work out what numbers would be needed for songs up to 100.