# Let's talk 2 – number talk (5 x 9) Stage 2

Stage 2 – A thinking mathematically targeted teaching opportunity focussed on reasoning with, comparing and communicating strategies to solve 5 x 9

## 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-MR-01

You will need:

• pencils

## Watch

Watch the Let's talk 2: number talk (5 x 9) Stage 2 video (6:11).

Investigate strategies to solve 5 nines

### Speaker

Hello there, mathematicians, I hope you're having a really lovely day today.

[Screen shows a blank piece of paper which fills the screen. There is a red card at the top of the screen which reads, ‘5 nines’ and 5 times 9 in brackets, underneath.]

To get our brains thinking mathematically we have this question here for you today. And that is how many strategies could you use to solve 5 nines?

Yes, so it's a different question to what is 5 times 9? What we're thinking about is what are the different strategies you could use to solve it. Mhm.

[Screen shows the presenter’s hand. They hold out their fist.]

So if you've done number talks in your class, it's like doing a number talk, so we're going to use this symbol to say I'm still thinking.

[Screen shows the presenter give a thumbs up signal.]

And this would mean, yeah, I have one strategy.

[Screen shows the presenter put 2 fingers out, the thumb and forefinger.]

And this could mean I have two strategies, for example, and you could keep thinking.

OK, you've got one. Great.

[Screen shows the presenter’s hand grab hold of three mini figures and place them side by side on the middle of the paper.]

So, we asked some students this question too and they can't be here so we're going to use these minifigs to help us represent their thinking.

[Screen moves one of the figures to the left-hand side of the page.]

So, the first one goes over here. They look like construction workers I think.

[Screen shows the presenter point to the red card at the top of the page that reads, ‘5 nines’.]

So, the construction workers’ teams were thinking about this idea of 5 and what they know about 5 is that it's half way to 10.

[Underneath the construction worker figure the presenter writes the number sentence, ‘5 is half of 10’.]

So, they said that they know that 5 is half of 10.

[Underneath the number sentence the presenter draws two horizontal lines and two vertical lines to create a rectangle. The top horizontal line slants to the left. The presenter draws a horizontal line in the middle of the rectangle and labels the bottom half ‘10’. They draw a rectangle underneath in half the length and label it ‘5’.]

So, if I have, oh that's a bit wonky, but if I had 10 of something, then this portion is 5.

[Presenter writes the number sentence, ‘10 nines equals 9 tens’.]

And they said that they could use them that to help them solve this idea 'cause they said they know that 10 nines is the same as saying 9 tens because of the commutative property.

[Presenter adds to the number sentence so that now it reads, ’10 nines equals 9 tens equals 90’. They point to the ‘9 tens’, then to ‘90’ at the end of the sentence and back to ‘9 tens’.]

And so they know that they could just use place value then to rename 9 tens as 90.

[Presenter points to ‘90’, then points to the number ‘10’ at the beginning of the number sentence and up to ‘5’ at the start of the first number sentence.]

And then what they have to do is halve 90 'cause they doubled 5 to get 10. So that now they have to halve 90.

[Presenter writes the number sentence, ‘90 divided by 2 equals 45’.]

So, 90 divided by 2 which is the same as halving and they said they knew that as 45 and that's one strategy that you could use to solve 5 nines.

[Presenter draws a green line beside the working out, from the top to the page to the bottom. They move a figure to the right of that line.]

So they used their knowledge of tens to solve fives. Yeah, that's an interesting idea. Was that like yours? No, you had a different way.

Well, that's good cause this guy also had a different way. I think he looks like a foreman on a building project. So let's call him the foreman.

[Presenter points to the red card which reads, ‘5 nines’. Presenter writes the number sentence, ‘5 nines equals 5 tens minus 5 ones.’]

And the foreman was thinking about 5 nines and what these guys were thinking about is that 9 is pretty close to 10. So we could think about 5 nines as being 5 tens, minus 5 ones.

[Presenter writes the number sentence, ‘5 times 10 equals 50’.]

Yeah, and they said they wanted to use this because they know something about 5 tens. They said 5 times 10 or 5 tens. They know that is 50 because of place value.

[Presenter writes the number sentence, ‘5 tens equals 50’ and draws brackets around this number line.]

You're right because 5 tens we renamed as 50, uhm.

[Presenter points to where they wrote ‘5 ones’ on the first number line. Underneath their working out so far, they write, ‘5 one equals five’. The presenter points to the ‘5’ and then the ‘50’. They write, ‘50 minus 5 equals 45’.]

And then they said they know 5 ones, 5 ones is 5. And so they just needed to now subtract 5 from 50. So 50 subtract 5 is equivalent to 45.

[Presenter takes their marker to the top of the working out and writes the number sentence, ‘5 times 9 equals 5 times 10 minus 5 times 1’ using mathematical symbols.]

Yeah, I'll write this in symbols for you. OK, so this would be 5 times 9 is equivalent in value to 5 times 10 minus 5 times ones. Yeah, that's how we would symbolically represent their idea.

[Presenter draws a green line beside the working out, from the top to the page to the bottom. They move the final figure to the right of that line, which is now the last third of the page. Presenter writes the number sentence, ‘9 equals 8 plus one’.]

And over here the blue team. They look like scientists I think. The scientist teams they were thinking about this number of 9 as well, but they were thinking about this. That 9 is composed of 8 and one more.

[Presenter writes the number sentence, ‘5 nines equals 5 eights plus 5 ones’.]

So they were thinking that 5 nines is equivalent to 5 eights plus 5 ones.

[Presenter writes, ‘5 times 8’.]

Yes, and then what they know about multiplying by 8 is that you can double, repeatedly double and you could also use the doubling and halving strategy.

[Presenter adds to the number sentence which now reads, ‘5 times 8 equals 10 times 4 equals 20 times 2 equals 40’. Underneath this they write, ‘5 ones equals 5’.]

So what they did here was say, well 5 eights is equivalent to 10 times 4. And if they keep going, that's 20 times 2, which is 40, uhm and then 5 ones is 5.

[Presenter writes, ’40 plus 5 equals 45’.]

And then they just need to join these quantities to have 40 and 5 more equivalent to 45.

[Presenter shows 3 fingers above the working out and then shows two fingers.]

Aha so there you go mathematicians, there are three different strategies that we use and because you know, we like to embrace our inner George Polya as we work as mathematicians. And he said that's really good to solve one problem in five different ways.

So here's three strategies, our challenge to you is can you think of another two?

[Presenter places a red card on the bottom of the page, out of the way of the working out. The red card reads, ‘6 nines’ and ‘6 times 9’ in brackets, underneath.]

And how could you use any of these strategies to think about 6 nines (6 x 9).

[Screen reads, ‘Over to you!’ on a blue background.]

Over to you mathematicians.

[Screen shows text that represents what the presenter is saying.]

So as you know, we always love to ask what was the mathematics? So a couple of things here. Two important ones. One was that as a mathematician, you can think flexibly about numbers and situations. So when you see 5 nines you can think about it as 5 tens minus 5 ones, or 5 eights and 5 ones more.

We also found 3 different strategies we could use to solve the problem and you were asked to find 2 and this is really important because as Cathy Fosnot says, this is what it means to be a mathematician when we can look to the context of the problem and make decisions about what strategies we used to solve them?

So back to you.

[End of transcript]

## Instructions

• We found 3 different strategies we could use to solve the same problem.
• Can you think of another 2 strategies to solve the problem 5 nines (5×9)?