# Mathematics K–12

Stage 2 and 3 – ​A thinking mathematically context for practise focussed on developing flexible strategies and relationships to benchmarks

## 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

• MAO-WM-01
• MA2-AR-01
• MA2-AR-02
• MA2-RN-01
• MA2-RN-02

• MAO-WM-01
• MA3-AR-01
• MA3-RN-03

You will need:

## Watch

Watch the Dicey addition video (12:55).

Investigate creating equations using random numbers from 0-9.

(Duration: 12 minutes 55 seconds)

### Michelle

Welcome back mathematicians, and welcome back Barbara.

[Screen shows 3 pieces of paper. Two of the pieces of paper are A4, and one is a smaller piece of paper. The 2 A4 sheets of paper are side-by-side. On the left, there is a purple A4 piece of paper, and next to it there is a blue A4 piece of paper.

On the top of both the coloured pages, the number 1000 is written. Underneath there are 3 sets of dotted lines representing the place values of hundreds, tens, and ones, a plus symbol, another set of dotted lines representing hundreds, tens, and ones, a plus symbol, and a final set of dotted lines representing hundreds, tens, and ones followed by an equals symbol and an empty square box.

The small piece of paper underneath has a 0 to 9 spinner. Next to it, there is a paper clip and a pencil.]

Hi Michelle.

How are you?

I am excellent.

How are you?

I am very well.

### Michelle

You're looking interested.

### Barbara

Very interesting.

### Michelle

So, this is a game I got from Nrich math, and I thought we could play today.

Fantastic. I love it.

### Michelle

And I love competing with you.

### Barbara

I love beating you.

### Michelle

So, the way that we play is, you will need a zero to 9 spinner or dice.

Ok.

### Michelle

So, so you can go first.

So, if you put the pencil in the middle of the paper clip.

And then give the paper clip a flick.

Yes.

[Screen shows a paper clip each in the middle of the circle, Michelle puts the pencil in the end of the paper clip in the middle and spins the paper clip. The paper clip lands on 9.]

### Michelle

You've got 9.

Which is good right?

So, you can choose to put your 9 anywhere in this 3-digit number, anywhere in this 3-digit number and anywhere in this 3-digit number, but not here.

The goal is to be the person at the end to be the closest to a 1000.

[Screen shows Michelle circling the hundreds, tens, and ones spaces of each set of lines, indicating that Barbara can write 9 in any one of the spaces but not in the box.]

### Barbara

I'm going to think carefully about this, because if I put it in the hundreds in any of these, I'm going to get really close to a 1000, so I can't do that.

So, I might just put it in my tens so I've got 9 tens.

[Screen shows Barbara writing 9 in the tens space of the first 3-digit number.]

### Michelle

In the tens place

And then it's my turn to spin.

And I got an 8, so I I think I'm going to follow a similar idea, but I want to start with this number.

[Screen shows Michelle using the spinner and getting the number 8, which she writes in the tens space of the last 3 digit-number.]

### Barbara

Ok.

Just because I like my third number the most so far.

### Barbara

Well with addition, it doesn't matter what order you go in right?

### Michelle

That's exactly right.

### Barbara

2.

Ok, well I might um make this 200 because now I have 290 something which is not bad if I'm trying to get to 1000.

[Screen shows Barbara using the spinner and getting 2, which she writes in the hundreds space of the first 3-digit number.]

### Michelle

I think it's a good idea actually, because it's close to 300.

### Barbara

That's what I was thinking.

### Michelle

300, 300 and 300 is 900 which is pretty close to 1000 and so I think that's a pretty good spin, or decision.

A 3, well, I am going to follow my own advice and put 3 in the hundreds place of my middle 3-digit number.

[Screen shows Michelle’s spinner landing on the number 3, which she writes in the hundreds space of the second 3-digit number.]

### Barbara

Yeah, that makes sense because now it's, your similar to me.

You've got 300 and something and I've got nearly 300.

Ok.

2.

Ok.

I might put the 2 here in the tens place here.

[Screen shows Barbara’s spinner landing on the number 2, which she writes in the tens space of the last 3-digit number.]

Ok.

### Barbara

'Cause now I'm scared if I put it in the hundreds, it would be too low.

### Michelle

I think that's a 9.

### Barbara

I think that was a 9.

### Michelle

Yeah, I think that's a 9.

I'm going to put my 9 here so I have 9 ones now with 8 tens which is 89.

[Screen shows Michelle’s spinner landing on the number 9, which she writes in the ones space of the final 3-digit number.]

### Barbara

So, you're close to 100 there.

Yeah.

### Barbara

Is that a 9 or a zero?

### Michelle

Re-spin that one.

It was right on the line.

Ok.

### Barbara

Zero.

Ok.

So, zero.

Oh, that's interesting, Ok.

I can put it in the ones or in the tens.

You know what I'm going to put it here because I want it to be close to 300 and

I'm pretty close to 300 at that spot.

[Screen shows Barbara spinning and landing on zero, which she writes in the ones space in the first 3-digit number.]

### Michelle

I got a seven.

So, I'm gonna put that, well, I definitely don't want to put it in my hundreds place.

[Screen shows Michelle spinning and getting 7, which she writes in the tens space of the second 3-digit number.]

### Barbara

That's how I felt about the 9.

### Michelle

'Cause then already it would be, if that was 789, then another 300 would be a 1008, it would already be over a thousand.

And even though the game is closest, I feel like that's already too much, so I might put it in the tens place of this number now, so I already know I've got 37 tens, so it'll be 1000 and seventy something.

### Barbara

Ok.

2.

Ok.

I might put this in the hundreds and then when I, if I get a larger number this can, I could, I could then put that in the other hundreds.

[Screen shows Barbara using the spinner and landing on 2, which she writes in the hundreds space of the first 3-digit number.]

Ok.

5 and a half.

### Michelle

2.

I'm going to put that here.

Oh look, my number 289 is only one number away from 290.

[Screen shows Michelle’s spinner landing on the number 2, which she writes in the hundreds space of the last 3-digit number.]

### Barbara

Oh, and you've got here these 2.

That's nearly 300 and that's pretty close to 400.

### Michelle

Oh yes, I'd better be careful.

### Barbara

Ok, see now I want a big number, something larger than 6, I think.

Seven is, is 7 perfect?

Ooh it might be a bit too big,'cause I've got about 500

Oh Ok.

Ok, so I'm going to put my 7 in my tens in the second number.

[Screen shows Barbara’s spinner landing on the number 7, which she writes in the tens space of the second 3-digit number.]

### Michelle

And a one.

I'm happy with that.

But now I've got to make a hard choice.

So about 700 and if I put that there, here.

Oh yeah.

[Screen shows Michelle’s spinner landing on the number one, which she writes in the tens space of the second 3-digit number.]

### Michelle

I could get some big numbers to get about 200.

You know if I got like 190 something.

Yeah.

### Michelle

But it could be too soon to get a one like, ideally it would be a 2 but my chances of getting a 2 is one in 10.

### Barbara

Of course, ok.

So, what are you thinking?

### Michelle

I think, usually I risk it.

### Barbara

You can risk it.

I'm happy for you to risk it.

### Michelle

I know you're happy.

### Barbara

If I beat you that works with my plan.

### Michelle

I am not going to be conservative, but I will put the one here so that the number isn't too big.

I think.

### Barbara

I should have put it in the hundreds Michelle.

Then I could have definitely beat you.

### Michelle

Well, this was my fear.

### Barbara

Oh, a 2.

I would like a bigger number.

So, oh, I'm running out of chances though.

Now it does feel like I'm taking a lot of chances here 'cause I want like, I want 5.

I want like 500 I think, so definitely not 2.

So, I might.

[Screen shows Barbara’s spinner landing on the number 2. Barbara then writes 2 in the ones space of the last 3-digit number.]

### Michelle

Oh, you're taking risks too.

I am.

A 4.

A 4 is good.

### Michelle

Four is good and it's going here.

'Cause you can go over 1000.

It's the closest 1000.

So, I like this 'cause look, 407, 400.

That's 800, 300 and we're at 1100 which is 1100.

It's pretty good so far.

### Michelle

There's only 2 spins left

You could get a 9 and a 9, and then I win.

### Barbara

Yeah, it looks like I think I was showing off a little bit too much.

This sometimes happens when we play.

I show off a lot and then I lose.

### Michelle

Oh yes, I'm so happy.

### Barbara

Ok, well the 8 has to go here.

And then I just have to really hope for.

[Screen shows Barbara’s spinner landing on the number 8, which she writes in the ones space of the second 3-digit number.]

Yeah.

### Barbara

For certain numbers.

### Michelle

You've got like 300 and 200, right?

### Barbara

So, 500.

So, I'd be happy with 500.

I would go over but I'd be happy.

I'd be happy with 400.

I'd be really happy with 4, I think.

### Michelle

Six and above I win.

### Barbara

Look at my spin, so many 2’s.

I need to change my spin strategy.

### Michelle

A, 3 well, I mean this isn't going to make much difference for me, right?

'Cause they're both going to be 3 in the ones place.

[Screen shows Michelle’s spinner landing on the number 3, which she writes in the ones space of the second 3-digit number.]

Alright.

### Michelle

So, it doesn't make any difference to my total.

### Barbara

I've got one spin left.

Come on 9.

### Barbara

Ok mathematicians wish me luck.

Come on 9.

### Barbara

Michelle says she, I always beat her, but I don't, ok, cheer for me.

I'm the underdog.

Yes.

### Barbara

Oh no!

Oh, so far away even, look at that number, that's already nearly 1000. Wow.

[Screen shows Barbara’s spinner landing on the number 9, which she writes in the hundreds space of the second 3-digit number.]

### Michelle

Look, I don't even need to spin 'cause I've already won but I will do it anyway.

8.

[Screen shows Michelle’s spinner landing on the number 8, which she writes in the ones space of the first 3-digit number.]

Ok, so, so now the challenge is we need to work out exactly what our numbers are but, but we can do that, or in fact what we can do is get the mathematicians out there to help us.

Ok, great.

### Michelle

So, here's a piece of paper for you.

You make a game board and I'll make one.

Ok.

### Michelle

And then the mathematicians at home can play it.

### Barbara

Ok, I've got an idea.

### Barbara

I liked 1000 as the target number, but I thought we could use subtraction.

[Screen shows Barbara creating her own game board. On the top of the page, the number 1000 is written. Underneath there are 2 sets of dotted lines. There are dotted lines representing the place values of thousands, hundreds, tens, and ones, a minus symbol, and a final set of dotted lines representing thousands, hundreds, tens, and ones, followed by an equals symbol, and an empty box.]

### Michelle

Really nice idea.

### Barbara

And then 4-digit numbers.

### Michelle

Ok, well I wanted to show that you could have any number as a target number.

### Barbara

Ok, great, oh yeah.

### Michelle

But you could also do multiplication.

[Screen shows Michelle creating her own game board. On the top of the page the number 420 is written. Underneath there are 2 sets of dotted lines. There are dotted lines representing the place values of tens and ones, a multiplication symbol, a dotted line representing a single-digit number followed by an equals symbol, and an empty box.]

### Barbara

Wow, that's really.

### Michelle

So, what could you multiply a 2-digit by one digit to get as close as possible to 420.

### Barbara

And I like your, your game's a quick game.

### Michelle

Yeah, only 3 digits.

### Barbara

Only 3 rolls.

So, you really need to, where we had a lot of chances before, we got to think about.

So, that makes it really, whereas mine has got 8 rolls, So that's a different way of

playing, isn't it?

### Michelle

Maybe makes it too tricky.

You could do something like you get 5 spins and you have to choose 3 of them.

### Barbara

Oh, that's a good idea.

### Michelle

Yeah, alright.

Over to you mathematicians to have, have a play.

You can use the version that Barbara and I used to start with and then start modifying and making your own game boards.

### Barbara

Good luck.

I hope you can beat Michelle.

### Michelle

Happy mathsing.

Alright mathematicians, we thought we’d come together to talk about some strategies we might use to join these 3 collections because it might be something that you would use a written strategy for, but there's some nice things happening with yours, Barbara, that you could use just mental strategies.

So, what is it that you noticed?

### Barbara

I noticed that um these numbers here, so this is 222 and this is 978.

But if I was able to move this 22 across to this 978 and that becomes 1000.

And this becomes 200, and they're both really easy to add with.

[Screen shows Barbara’s game board from the first game. At the top of the page, the number 100 is written. Underneath it has 290 plus 978 plus 222, an equals symbol and an empty box. Barbara points to the number 222 and the number 978 and traces the numbers 22 and 78 to signify that they can be added together easily.]

### Michelle

Two more would make that 80, 980 and then the other 20 more would make that 1000.

So, why don't you rewrite your number sentence down below 'cause you can do that as a mathematician.

You can adjust the numbers to work for you.

### Barbara

So, I've still got 290.

[Screen shows Barbara’s game board from the first game. It has 290 plus 978 plus 222, an equals symbol and an empty box. Underneath, Barbara rewrites her number sentence. She writes 290 plus 1000 plus 200. Barbara points to the number 1000 in her new number sentence, and then points to the number 200 to show her mental addition strategies. She then points to the number 290 to show how she adds the numbers together. Barbara draws the equals symbol and writes the number 1490 next to it. The rewritten number sentence now reads 290 plus 1000 plus 200 equals 1490.]

### Michelle

290.

But this, now I'm adding 1000.

Now, I have to remember that now that's not 222, now that's 200.

### Michelle

Yeah, 'cause you just took the 22 from here and put it onto this.

Right?

### Michelle

Yeah, and so now that looks so much friendlier to work with.

So how would you work with it from here?

### Barbara

Ok, so I've got um 1000 and I've got 200, so I could say that I got 1200 and 1400 and then 1490.

How would you solve it?

### Michelle

I would do it a little bit differently.

I would take, I'd probably take a chunk out of the 200.

Ok.

So just for my.

### Barbara

Would you still keep that number sentence, that first part the same or would you change it?

### Michelle

No, I would change it actually.

So, I would write this, rewrite it as 300 plus 1000 plus 190.

Because what I know then is it that's 490 and then with 1000 it's just 1490.

So, the same sort of idea just, just re thinking.

I would just do another step of rethinking.

[Screen shows Michelle writing 300 plus 1000 plus 200. Underneath she draws lines from the numbers 300 and 190 to show her mental addition. Where the lines meet, she writes the number 490. She then draws lines from the number 490 and the number 1000 to display her mental addition and writes the number 1490 where the lines meet. Michelle then moves to her rewritten number sentence and writes the equals sign followed by the number 1490.]

Ok.

### Michelle

And so, what looks like a really scary thing to start with if you just use that idea of adjusting numbers, and you know, thinking about landmarks, you can add 3, 3-digit numbers in your head and you don't need a written strategy.

Well done us.

### Michelle

Well done mathematics.

Over to you mathematicians.

[Screen reads: Over to you! What’s (some of) the mathematics?]

Alright, so what's some of the mathematics in this game?

So, one of the things we really saw today was that adjusting numbers by using your knowledge of landmark numbers and bridging to tens and other number facts that you know can really help you work more confidently with problems.

So, when we rethought 290 plus 978 plus 222 in a different way, it made it look much more friendly for us to work with.

[Screen reads: For example: 290 plus 978 plus 222 equals 290 plus 1000 plus 200.]

We also realise that this game um provides a really fun context to use and enhance your knowledge of place value and the operations.

As well as like with most of the games that we play in mathematics, a really nice space to develop some serious mathematical reasoning that always needs a little bit of luck.

Have fun playing mathematicians.

[End of transcript]

## Instructions

• Find a partner and a 0-9 dice or spinner.
• Draw your gameboard so you each have the same one.
• Each player takes a turn to spin the spinner and decide where to play that digit in your number sentence (equation).
• Spin the spinner 9 times each.
• The person whose sum is closest to 1000 is the winner!