Copyright © University of Cambridge. All rights reserved.

## How Many?

We present this activity as a way for adults to engage with just a few pupils at a time. It works well to encourage the youngsters to talk about their thoughts and how they have reached their conclusion/s.

There are a lot of variations but the open questions allow the children to speak freely. Please watch this very short video as a stimulus for what you could do with your own pupils. However, it would be worth reading the

Notes first.

**If you can see this message Flash may not be working in your browser**
**Please see http://nrich.maths.org/techhelp/#flash to enable it.**
**Why do this problem?**

This month's NRICH site has been inspired by the way teachers at Kingsfield School in Bristol work with their pupils. Following an introduction to a potentially rich starting point, a considerable proportion of the lesson time at Kingsfield is dedicated to working on questions and ideas generated by children.

Working on this project can encourage learners to work together, discuss ideas, test things out, and explore further. This is how it is to be a mathematician, working alongside other mathematicians.

This activity can do well to replace questions set out like; $3 + ? = 8$, $? + 4 = 8$ etc. Those written examples can be very confusing to decipher, whereas the early thinking of many children enables them to manage very confidently the more practical approach outlined in the video. This project will help to develop concepts of algebraic thinking.

**Possible approach**

With about four pupils sitting down with you and some counters or cubes and a cloth available, as in the video, the activity can begin. Often, we might give children a number of cubes to count. We then give them some more cubes and ask them how many there are in total. In contrast, this is transformed into a richer activity by covering up a number of cubes with a cloth. You can then invite a
child to take some of the cubes out from under the cloth and ask the group how many there are left. This encourages algebraic thinking. You could go further by saying "I'm covering up $15$ cubes. How many would I have to take out for there to be $8$ left under the cloth?". Then on another occasion, you could try giving children some cubes from under the cover, telling them how many are hidden
altogether, and asking, "How many did I have to start with?".

Encourage pupils to think about what other children have said. When a child makes a suggestion, it can be useful to ask why s/he thinks that but repeating what s/he said word for word, without feeling the need to paraphrase. For example, a child might say "there should be $8$ there" and the teacher might respond: "why do you say 'there should be $8$
there'?".

**Key questions**

Do you agree with what Sara said?

Why do you think there are four hidden?

Tell me about that (thought).

Do you all agree then that ...?

How can we find out?

What questions would you like to ask?

### Possible extension

### Possible support

We should like to thank the OU and the BBC for the programme which inspired this project.