## Secret Number

This is a game for two players and a simple calculator.

Annie and Ben are playing. Annie puts her secret number into the calculator without showing Ben.

Annie then asks Ben, "What do you want to add?"

Ben tells Annie the number he wants to add. "I want to add four."

Annie presses the 'add' button and then the four button. The calculator now shows '$4$'. Annie gives the calculator to Ben.

Ben presses the 'equals' button and the calculator gives the answer '$10$'.

What was Annie's secret number?

How do you know?

You could play this with a friend. If you work out your friend's secret number correctly, it is your turn to put in a secret number of your own. You could score a point for every one you get right.

A multiplication version of the game might go like this:

Charlie puts in a secret number and asks Dana, "What do you want to multiply it by?"

Dana replies, "Multiply it by $5$."

Charlie puts in 'times' and '$5$' and hands the calculator to Dana.

When Dana presses the 'equals' button the calculator shows '$35$'.

Dana now has to work out Charlie's secret number. What was it?

How do you know?

Try playing this version with your friend too!

### Why do this problem?

This game is a good one to play with young children once they are familiar with the basic number operations. They will like the idea of their number being "secret", and of course being able to work out someone else's "secret" number! Looking for a ''secret'' number is the basis for algebra and solving unknowns in
equations. So as well as enjoying what they are doing, your class will be engaging with some important mathematical ideas.

Possible approach

A good way to start might be for you either to enter your own secret number and invite the class to suggest what to add, or perhaps ask two children to come to the front to demonstrate. This activity will create a great opportunity for rich discussion amongst the class about how they can work out the secret number. You could ask the children to think for themselves first, then share their
ideas with a partner and finally with the whole group. (Think-pair-share.)

Playing this game is a good lead-in to talking about inverse operations. You could also introduce some element of recording, perhaps by asking the children to record what they do each time in their own way - this can help to reveal a lot about their thinking processes.

Key questions

What number added to $4$ makes $10$?

What number multiplied by $5$ makes $35$?

How could you check your answer?

Possible extension

The best extension for this is for children to play the game with a partner as they are invited to do in the original problem. For some, there need be no limit to the numbers or operations involved. However, it would be a good idea to get all "secret" numbers and working out recorded!

Possible support

Use a calculator openly with the children so that they can see exactly what is happening. When they understand the mechanism of the game then start using a "secret" number with very simple numbers. Learners could also use a number line or multiplication square to help.