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### Number and algebra

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### Working mathematically

### For younger learners

### Advanced mathematics

# Largest Even

## Largest Even

**Why do this problem?**

### Possible approach

### Key questions

### Possible support

### Possible extension

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Age 5 to 11

Challenge Level

- Problem
- Getting Started
- Student Solutions
- Teachers' Resources

How do you know whether a number is even?

To start with, the interactivity below gives you one random digit.

Your task is to find the **largest possible two-digit even number** which uses the computer's digit, and one of your own.

Have a go several times.

**How are you deciding which digit to choose?**

**Can you decribe a strategy that means your first 'guess' is always correct?**

Clicking on the purple cog allows you to change the settings.

For a follow-on challenge, choose an *odd* two-digit target number (select 'Even False' in the settings).

**How are you deciding which digit to choose now?**

**Can you decribe your new strategy so your first 'guess' is always correct?**

You can explore the settings further to change:

- the number of digits in your target number
- the number of digits provided by the computer.

Let us know how your settings change your strategy.

This problem encourages children to work together to develop a method for finding a solution which will *always* work. It will also help to reinforce understanding of odd and even numbers.

You could introduce the challenge using two sets of 0-9 digit cards. Hold up a 3 and ask the class which digit they could choose to put with the 3 to make a two-digit even number. Take a few suggestions and encourage learners to find all the possibilities (30, 32, 34, 36, 38), listing them on the board as you go. How do they know that each of these is even? What do they notice about these numbers? Encourage the class to explain why the 3 must go in the tens column if we are making an even number.

Repeat this, perhaps by holding up an even digit, for example 6, this time. Now they are many more possibilities as the 6 can go in the ones column or the tens column. Again, you can list them on the board as they are suggested by the class.

Then, introduce the interactivity, explaining that now we are being given a digit by the computer and our challenge is to find the *largest* possible even number with another digit of our choice. Children could write the answer to the computer-generated problem on individual whiteboards, or they could discuss with a partner before offering their suggestion. If the computer does not
agree with the class' suggestion, take some time to discuss why. You will find that the interactivity does not allow a zero to be used as the tens digit of a two-digit number. If this comes as a surprise to the class, encourage them to consider why.

Ideally at this point, each pair of learners would have access to the interactivity on a computer or tablet so they can have a go at some more examples. Invite them to develop a strategy for *always* finding the highest even number on their first attempt. If you do not have access to multiple devices, you could continue working as a whole class using the interactivity
displayed, or you could ask pairs to use digit cards to model what the interactivity does.

Bring everyone together after a suitable length of time to share ways of working. Encourage a few pairs to articulate how they make sure their first 'guess' is always the highest possible two-digit even number. You could compare and contrast the different approaches.

As a final challenge, give everyone time to consider how their strategy would change if they were asked to create the highest possible two-digit *odd* number. You can change the settings of the interactivity by clicking on the purple cog in the top right corner, and selecting 'Even False'. The observant amongst your class may notice that the interactivity now offers an extra button
which says 'I don't think there is such a number'. When will this be useful? Why?

What do you know about even numbers?

What do you know about the ones digit of even numbers?

Where could the digit the computer has given us go - in the ones or tens? Why?

How can we make a large number?

How do we know that is the largest even number we can make?

Some children might benefit from having equipment to help them check whether a number is even, for example multilink cubes or counters that can be put in pairs.

Change the settings on the interactivity (click on the purple cog) to challenge children to extend their strategy for three-digit and four-digit numbers. They can choose to be given one, two or three digits by the computer (but the number of digits provided must be fewer than the target).

The problem Dozens includes a similar interactivity but with a more challenging starting point and the scope to explore multiples of any number, not just odd/even numbers. You could take a look at the Teachers' Resources of Dozens to see how that interactivity might be used in the classroom.

Vincent and Tara are making triangles with the class construction set. They have a pile of strips of different lengths. How many different triangles can they make?