# 2, 4, 6, 8

## Problem

Use these cards to make some number sentences:

For example:

In each number sentence you must only use a card once, but of course you can re-use them in your next number sentence if you want to.

Can you find a way to use all the cards in one number sentence?

Can you find all the possible ways to use the cards?

## Getting Started

How will you know you have found all the different number sentences?

Are there any other sums that have the $8$ in them, for example?

Are there any other addition sums?

Can you make any number sentences with all four numbers?

You could print out these cards.

## Student Solutions

Jane and Sanjev sent us the following answer:

First we tried to make a statement using the plus on one side of the equals sign and the minus on the other, but we couldn't find one.

Then we tried to use the plus and the minus on the same side, and we got 2 + 8 - 6 = 4.

Then we tried to see what statements we could make starting with the number 2 :

2 + 4 = 6

2 + 6 = 8

2 + 8 - 4 = 6

2 + 8 - 6 = 4

We could not find any more.

Then we tried to start with the number 4 :

4 + 2 = 6

4 + 6 - 8 = 2

4 + 6 - 2 = 8

and with the number 6 :

6 + 2 = 8

6 + 4 - 2 = 8

6 + 4 - 8 = 2

and with the number 8 :

8 + 2 - 4 = 6

8 + 2 - 6 = 4

8 - 2 = 6

8 - 6 = 2

And then we can write them all the other way round, so from 2 + 4 = 6 we get 6 = 2 + 4.

That way we made 28 statements.

We have then had a solution from Amelia Earhart Class Year 2 at Newport C of E Primary School, Isle of Wight

Thank you for your solution from the Isle of Wight.

## Teachers' Resources

**Why do this problem?**

This problem gives children the opportunity to create their own number sentences. The discussion it provokes will give them the opportunity to use vocabulary related to addition and subtraction, and to begin to develop ways of working systematically.

### Possible approach

Having the digits and operation signs on the interactive whiteboard will allow you flexibility to be led by the children as they tackle this task. This very simple interactivity could be useful. You could begin by dragging some of the numbers and operations to make an incorrect number sentence. Invite children to talk in pairs about what is on the board. Ask learners to share their thoughts with everyone which will no doubt mean that your error is pointed out. How could they make the sum correct? Invite a pair to do this alongside the incorrect number sentence so that the original version can still be seen. Ask if another pair can make it right in a different way. Collect some other suggestions and then set up the first task. Working with these number and operation cards, pairs can try out their ideas.

Gather the class together to talk about this first challenge and then introduce the second one so that this time they don't have to use all the cards, but must make as many true number statements as they can. As they work, encourage them to record what they have done, and look out of those that have a system of some description, for example finding all the sentences with $8$ to being with, then with $6$ etc.

### Key questions

### Possible extension

Children could be encouraged to consider a similar problem using the odd digits less than ten. Does this create difficulties? If so, why?

The set of numbers being considered can be extended in various ways. The children can be encouraged to think about extensions which generate interesting results and those that do not do so.

A similar idea could be introduced using multiplication and division as well as addition and subtraction.

### Possible support

Some children may need the support of concrete apparatus such as counters or Multilink cubes to make the calculations. Trying out some one step calculations using these digits would be a good introductory task.

You could break the task down a bit for some children, for example, by asking them to find all the number sentences which use the $2$ first, then the $4$ etc. Having the number and operation cards to manipulate will mean that children will find it easier to change their minds rather than feeling pressured to have a correct calculation
immediately.