# Empty number chart

Empty number chart is a thinking mathematically context for practise, focussed on developing understanding of counting sequences, renaming and place value using a number chart maze as a tool.

Adapted from Teaching Mathematics by Siemon, Warren, Beswick, Faragher, Miller, Horne, Jazby, Breed, Clarke and Brady, 2020

## Syllabus

Syllabus outcomes and content descriptors from Mathematics K–10 Syllabus © NSW Education Standards Authority (NESA) for and on behalf of the Crown in right of the State of New South Wales, 2023

## Outcomes

- MAO-WM-01
- MA1-RWN-01
- MA1-RWN-02

## Watch

Watch Empty number chart video (6:47) to learn how to play.

[White text on a navy-blue background reads ‘Empty number chart’. Further white text reads ‘Siemon, Warren, Beswick, Faragher, Miller, Horne, Jazby, Breed, Clarke and Brady, 2020’. In the bottom right corner, the NSW Government red ‘waratah’ logo.]

[Black text on a white background reads ‘You will need…’ Three bullet points below (as read by speaker). On the right, a grid has various grey and yellow shaded squares on it. A number ‘147’ is written in the square in the bottom right corner.]

### Speaker 1

Alright, mathematicians, for this you will need an empty number chart. It might look like this one here on the screen, or you might have some other ones to puzzle with. You need a pencil or a pen, and if you'd like, someone to puzzle and talk with. So, let's puzzle.

[White text on a blue background reads ‘Let’s puzzle!’.]

[The grid square from earlier.]

### Speaker 1

Alright well, today we don’t have a game, but I thought we could play with a puzzle together.

### Speaker 2

OK.

### Speaker 1

So this comes from a book called 'Teaching Mathematics'. And this is an empty number chart challenge.

### Speaker 2

OK.

### Speaker 1

Yeah. So, well, first of all, what information can you see?

### Speaker 2

Well, I noticed right away that often number charts go to 100.

### Speaker 1

Yeah.

### Speaker 2

And this one has 147.

### Speaker 1

Oh, yeah.

### Speaker 2

So, it looks like it's a different range of numbers.

### Speaker 1

And also I can see some yellow, like they're shaded out. So, we have to figure out all the ones that aren't in yellow.

### Speaker 2

OK.

### Speaker 1

And so, usually what happens when we use representations of numbers like number lines and hundreds chart is the number to the left means it's one less than or the number before.

### Speaker 2

Yeah, you're right.

### Speaker 1

So, I think that would be 146 there.

### Speaker 2

I agree.

[A person writes ‘146’ on the grid in a grey square preceding the ‘147’ in the bottom right corner.]

### Speaker 1

OK, so, now we need to try to figure out maybe what the next number is. Because can you see how it looks like it goes in a bit of a path, actually.

[A finger traces the path of grey squares around the grid.]

### Speaker 2

Because we can get clues from...

### Speaker 1

Yeah.

### Speaker 2

Well, usually when you move up the number chart, the numbers get smaller, right? So, maybe this looks like 14 x 10 and 6 x 1. Maybe that would be 13 x 10 and 6 more?

### Speaker 1

So, it would definitely be 10 either more or less...

### Speaker 2

I think so.

### Speaker 1

Because there's 10 in each? Let's just check that. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. So, there's 10 in each row.

### Speaker 2

Yeah.

### Speaker 1

So that would mean that this square has to be 10 more or 10 less. And from your knowledge of a hundreds chart...

### Speaker 2

I've seen both, but often they...

### Speaker 1

Yeah, we use this one, that's OK, I'm happy to go with that.

### Speaker 2

Yeah.

### Speaker 1

OK, so you're saying that this would be 13 x 10 and 6, because it's one 10 less?

### Speaker 2

Yeah.

### Speaker 1

So, 13 x 10 and 6 more is 136.

[The number ‘136’ is written in a grey square immediately above the ‘146’.]

### Speaker 2

Yes.

### Speaker 1

Which would mean that this is then another 1 x 10 less, which would be 12 x 10, and 6 ones, which is 126.

### Speaker 2

OK, yeah, I agree.

### Speaker 1

Yeah.

[The number ‘126’ is written in a grey square immediately above the ‘136’. The remaining grey squares are worked out according to their relative positioning on the grid.]

### Speaker 2

And then that one's easy. That's just 127.

### Speaker 1

127, oh, yeah. And we can check that.

### Speaker 2

It matches, right, with...

### Speaker 1

The 7 matches down here in the one space.

### Speaker 2

So, we know our thinking is starting to make sense.

### Speaker 1

So, then this would be 110 less, which, if this is 127, we can rename that as 12 x 10 and 7. So, we could rename what's left as 11 x 10, and 7, which is 117. And then if we rename 117 as 11 x 10 and 7, 110 less is...?

### Speaker 2

That would be 10 x 10.

### Speaker 1

10 x 10 and 7 more, which is 107.

### Speaker 2

The renaming makes it a lot easier, doesn't it? Than actually then taking away 10 each time.

### Speaker 1

Especially over the decade.

### Speaker 2

Yeah.

### Speaker 1

Yeah, but we'll see when we go around. So, this number then would be 106. Because it's one less than 107. OK, now try renaming here.

### Speaker 2

OK, so, at the moment we have 10 x 10 and 6.

### Speaker 1

Yeah.

### Speaker 2

Now we need 9 x 10 and 6.

### Speaker 1

OK, which is 96.

### Speaker 2

And that was easy.

### Speaker 1

Yeah.

### Speaker 2

OK, then 8 x 10 and 6.

### Speaker 1

8 x 10 and 6, which is 86.

### Speaker 2

And now we just can count backwards. So, 85, 84, 83.

### Speaker 1

Ooh.

### Speaker 2

OK. And well, now we've got a couple of things to help us. We can think about it as 9 x 10 and 3.

### Speaker 1

Yes.

### Speaker 2

And we can see that we're correct because along this row we've got 9 x 10 here with the 96.

### Speaker 1

So, actually, we can now start to use other information.

### Speaker 2

Yeah.

### Speaker 1

Alright, so, I think what you're saying is that because this is 9 x 10, this would also have to be 9 x 10 and 3 more.

### Speaker 2

Yeah.

### Speaker 1

And the 3 remains the same because it will hold in that...

### Speaker 2

That's right.

### Speaker 1

..column, and the 9 would stay in that row.

### Speaker 2

Yeah. But I don't think all the way through, because... I think that's what makes it...

### Speaker 1

It will get interesting what happens here.

### Speaker 2

Yeah.

### Speaker 1

So, I think that we could actually fill in these ones really confidently and go, all of these will end with a 4 because of that. And that will be 10 x 10 and 4, 11 x 10 and 4, 12 x 10 and 4, 13 x 10 and 4.

### Speaker 2

Yeah, definitely.

### Speaker 1

So, those 10 x 10 and 4, which is 104, 11 x 10 and 4, which is...

### Speaker 2

114

### Speaker 1

12 x 10… oops… 12 x 10 and 4, which is...

### Speaker 2

124

### Speaker 1

When we rename, and 13 x 10 and 4 can be renamed as...

### Speaker 2

134.

[White text on blue reads ‘Over to you!’.]

[White text on blue reads ‘What’s (some of) the mathematics?’]

### Speaker 1

What's some of the mathematics?

[Black text on a white background reads ‘What’s (some of) the mathematics?’ Further text below read by speaker. At the bottom, a colour image of the number grid from earlier. The second column from the right is highlighted in red.]

### Speaker 1

In this puzzle, there are 10 boxes in each row. That means that moving up or down in one column, the numbers in each box will increase or decrease by one 10 each time we move up or down one row. Look, here I can see 86 is one 10 less than the number below it, 96. I can see that 96 is one 10 less than the number below it, 106. And I can see that 106 is two 10s less than the number two rows below it, 126.

[On the left, the red outlined column on its own. Text on the right (as read by speaker).]

### Speaker 1

We can also see that we can work out one 10 more or one 10 less in different ways. For example, we can use renaming to help us work out the number that is one 10 more or one 10 less.

[Below the text, blocks of yellow squares and columns move about.]

### Speaker 1

For example, because I know 106 can be renamed as 10 x 10 and 6 more, I can remove one 10 and I'll have 9 x 10 and 6 x 1 left. Then that can just be renamed as 96. I can also use knowledge of counting sequences to work out the number that is one 10 more or one 10 less. For example, I know that counting by 10 means the number in the ones place doesn't change. So, counting backwards by 10 from 146 means I would say: 136, 126, 116, 106, 96 and 86.

This puzzle helps us see something that great mathematicians do. Think about what do they know and how could they use it here. So, mathematicians, back to puzzling for you and us too.

[The NSW Government waratah logo turns briefly in the middle of various circles coloured blue, red, white and black. A copyright symbol and small blue text below it reads ‘State of New South Wales (Department of Education), 2021.’]

[End of transcript]

## How to play

- You can complete these mazes on your own or work with someone else.
- Your challenge is to determine the number sequence through the maze.
- Use what you know to decide what numbers are missing and to justify the number you placed.
- You do not need to follow the maze in sequential steps if you know the value of a place on the maze. For example, on a standard hundreds-chart, I know the number 2 boxes directly above 147 is 127.

## Other ways to play

- You can complete these mazes on your own or work with someone else.
- Your challenge is to determine the number sequence through the maze.
- Use what you know to decide what numbers are missing and to justify the number you placed.
- You do not need to follow the maze in sequential steps if you know the value of a place on the maze. For example, on a standard hundreds-chart, I know the number 2 boxes directly above 147 is 127.

## Reflection

What helped you to determine the numbers on the maze?

How would your numbers change if it was a bottom-up hundreds chart?

Show me how you used your knowledge of the hundreds-chart to complete the maze.

How did you use your knowledge of patterns on the number charts to know where to place numbers?