# That number square

## Problem

When you arrive in the classroom one morning you discover all the numbers have fallen off the class number square and they are in a heap on the floor. All that is left on the wall is a blank grid!

There are five minutes to go before the first lesson starts.

**Your challenge is to find a quick way of putting the numbers back in their right places on the grid.**

Where will you start?

You might find it helpful to print out a blank grid and number tiles to cut out.

Once you have had chance to think about how you will put the numbers in place, talk to some of your friends about what they would do.

Or, you could click below to see how some other children got started on this challenge.

Omar and Isla said:

We started with the tens column, counting in tens.

Julia and Chris said:

We started along the top row. We filled in 1 to 10, then you can fill in each column by adding 10 more each time.

Michaela and Nathan said:

We counted in ones because we can do this super-fast!

You could try each of these starting points.

What would you do next? Would this be a quick way for you to put the hundred square back together?

Is there a way of making the hundred square that you think is a particularly 'good' way?

Why? What makes it a 'good' way?

## Getting Started

Does your class number square start with 0 or 1? Or with a different number?

How will you start? Why?

## Student Solutions

Thank you to everybody who sent us their strategies for putting the hundred square back together.

Benji and Elizabeth from Trefonen Primary School in England sent in this explanation of their strategy:

We placed the numbers 1-10 first. This helped us to place other numbers below each of these in their columns with the same digits in the ones column. We also placed the numbers which were multiples of ten in the last column as they were easy to replace.

Well done for using this pattern in the hundred square to replace the numbers quickly!

The children in Puffins Class at Trefonen Primary School had a competition to replace the numbers in the hundred square as quickly as possible:

We used multiples of 10 to fill in the end column; then we filled in the rest of the top row; then we could do it randomly really quickly. We worked in small groups and timed which group was the fastest and we thought that this was probably the best strategy because it was the quickest, they were called 'The Number Bonds FC'.

One group had a really good idea to cover the whole 100 square in glue, at the start.

These are some good ideas - I wonder what the winning strategy was?

We also received some solutions from children at Maria L Baldwin School in the USA. Lillian said:

I filled in all the ones then the twos etc.

Thank you all for sharing your strategies with us. I wonder what Lillian means by 'ones' and 'twos'?

D from the UK shared a pattern they found:

I saw a pattern on the 100 square. My dad and I talked about this and he helped me write down my thoughts.

In each column, the 10s of the numbers change by 10 and the 1s stay the same.

E.g. The 8th column from the left side have these numbers in order from 8 to 98 from top to bottom. The tens increase with each number - 8, 18, 28, 38, 48, 58, 68, 78, 88, and 98.

When I pick a number up from the pile, I put it in the right place by looking at the 10s of the number and I know which row it goes in by counting down to the row of this digit plus 1. And I know which column it goes in, by using the 1s of the number, by counting from the left.

E.g. If I pick 78, it goes in the 8th column and the 8th row.

Then my Daddy told me this doesn't work for 30. He asked me to look at this and this is what I saw.

The 10th column is different because all these numbers have a zero on the end. The row that they go into is done by counting down from the top row.

E.g. With 50, it goes in the 10th column and the 5th row because it has 5 10s and a zero on the end.

This is faster because if I tried to find the number 1 first then I might have to check 100 numbers to find the right one.

When I use my method, I only do 2 checks for each number.

Very clearly explained, D. You've thought really carefully about how to find the right place for each number on the grid.

Do you agree with D that the quickest method is to pick up a number from the pile and work out where to place this number on the grid, or have you found another strategy that you think is quicker? We'd love to hear your thoughts - please email us if you'd like to share your ideas.

## Teachers' Resources

**Why do this problem?**

This activity also offers an opportunity to discuss the strategies that children come up with - what is a good strategy for putting the number tiles back in the correct places as quickly as possible? What makes one strategy 'better' than another?

Possible approach

Print and cut out the number tiles and blank grid, leaving out either the 0 tile or the 100 tile. Alternatively, you may have your own sets of blocks, tiles or cards that the children could use. Explain the task and, without saying anything more, invite learners to try getting started in pairs. As they work, walk around and observe how they set about the task.

After a little while, discuss different approaches together, drawing on what you have observed. For example:

- Did you try and place the first tile in the random pile on the floor on the number grid first? If so, how did you decide where to put it?
- Or did you sift through the number tiles to find one that you know exactly where to place on the number grid?
- Which one did you choose? Why that one?
- Were there any key number tiles to get in place on the number grid that help you place the rest? What is special about those 'key' tiles?

You may also choose to discuss the starting points offered in the problem itself.

Ask children to devise some good strategies between them, and then encourage them to experiment with different approaches to see which ones help them put the number tiles back as fast as possible. You may find that different children find different strategies useful. Challenge them to articulate *how they know* where to put a particular number tile. Encourage explanations that
focus on pattern, place value and multiples.

### Key questions

*(Not to be asked only when a slip-up has occurred as then children often learn that such questions indicate that something is wrong! Asking it when they are right too helps you to explore their reasoning.)*

### Possible extension

What happens if it is a 50 - 149 number square, for example?

Can you use the same strategies?

What happens if the number square is not 10 by 10 but only 6 squares wide, for example?

How do you know where to place the number tiles now?

Which widths of number square are harder/easier than the 10 by 10 grid?

### Possible support

You may like to give some children a number grid where some numbers remain. Some children could work with the number grid up to 50 or up to 30. Working with a partner gives children the opportunity to discuss where to place a number, and why.