# Let's talk 4 – number talk (8+6+8)

A thinking mathematically targeted teaching opportunity focussed on investigating strategies to add 8, 6 and 8 and exploring other equivalent quantities

## 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
- MA1-CSQ-01

## Watch

Watch Let's talk 4 Part 1 video (12:01).

### Speaker

Hello there, mathematicians.

We hope you are having a really lovely day today no matter where you might be as you're watching this.

So, I have some cards here and I'd like you to tell me how many dots you can see? Or the quantities that's represented by them. So, what can you see here?

[Presenter shows a ten-frame with 8 black dots – 5 in the top row and 3 in the bottom row.]

Mm-hmm, can you show me using your fingers?

Uh-huh. Oh yes, I can see some people are showing me 5 fingers and 3 more, and that's 8.

[Presenter holds up 5 fingers on one hand and 3 fingers on the other.]

Uh-huh, yeah and some of you actually showed me 4 and 4 which is also 8.

[Presenter holds up 4 fingers on one hand and 4 fingers on the other.]

Mm-hmm and, oh, I was gonna say 6 and 2 but that's a bit. Oh maybe, look, I could do 6 like this and 2 more. Maybe that's 6 and 2.

[Presenter holds up 5 fingers on one hand and one pinkie finger on the other hand to make 6. They then hold up their thumb and pointer fingers to show 2 more.]

Uh-huh, all right, so this is 8 and we could also see there's 2 less than 10. How could you show that on your fingers? 2 less than 10?

[Presenter points to the 2 empty boxes on the ten-frame.]

Oh yeah, that's a good idea! Look, you could, if you had cubes, you could say well that's 10 and those 2 don't count.

[Presenter holds up 5 fingers on one hand and 5 fingers on the other, showing 10. Two of their fingers have red cubes on the end.]

Uh-huh, okay so we know this is 8. I'll just move it over to here. How, what about on this card? What can you see? How many dots are there? Can you show me using your fingers?

[Presenter moves the ten-frame showing 8 dots to the top left corner. They show another ten-frame with 6 black dots – 5 on the top row and one on the bottom row.]

Oh okay, some of you have got this, 5 and 1, and some have 3 and 3.

[Presenter holds up 5 fingers on one hand and one finger on the other. They then hold up 3 fingers on one hand and 3 fingers on the other.]

Uh-huh and some of you have, oh yeah, 4 and 2 and they're all 6. Uh-huh, and that's right, it's 4 less than 10.

[Presenter holds up 4 fingers on one hand and 2 finger on the other. They point to 6 dots on the ten-frame then point to the 4 empty spaces on the ten-frame indicating it is 4 less than 10. The presenter moves the ten-frame showing 6 dots to the left side under the other ten-frame.]

Mm-hmm, and what about this quantity? How many dots can you see here? Show me on your fingers.

[Presenter shows another ten-frame with 8 black dots – 4 on the top row and 4 on the bottom row.]

Uh-huh, it is 5 and 3. It's 8. Yeah, like this one isn't it, just arranged differently, and I know they're the same quantity, because there's 2 dots missing and 2 dots missing. Mm-hmm, or 4 and 4.

[Presenter holds up 5 fingers on one hand and 3 fingers on the other. They point to the other ten-frame on the left which also has 8 dots. The presenter uses 2 fingers to point to the 2 empty boxes on both ten-frames with 8 dots. They then hold up 4 fingers on one hand and 4 fingers on the other.]

Aha, and what's another way to show 2 dots missing? Oh yeah, you could do something like this, where you're like trying to show, look, it's 10 and 2 missing. 10 and 2 missing. And I know that finger just moves! It has a mind of its own.

[Presenter holds up 5 fingers on each hand then bends down 2 fingers on one hand. Then the presenter moves the ten-frame showing 8 dots to the left side under the other ten-frames.]

Okay, so now our, my question for you, mathematicians, is if I wanted to join all of these quantities together, how many dots would there be in total? And what are some different strategies that we could think of to solve this problem?

[Presenter circles the 3 ten-frames with their finger in the air then moves them side by side along the top of the screen.]

Uh-huh, so it's exactly like a number talk if you've done one of those before. So, this means we're thinking hard about what could be one possible strategy. Uh-huh, and when we thought of a strategy we could use, we can show our thumb to say, I've got one way of thinking about this. Yeah, and our second finger would show, actually I've got a second strategy I could use, and you might even have a third strategy.

[The presenter makes a fist to show they are thinking. Then they keep their hand in a fist and lift one thumb to show they have one strategy. They lift a second and third finger to show they have 2 and 3 strategies.]

Okay, because remember we're thinking about what are the strategies you could use, that's what we really want to focus on. Ah okay, shall we talk together? Great!

Okay, so to help us with our thinking today, I can't, I don't have any students here with me today, but we did ask them and they're gonna be represented today by the pirate, aharrr harrr. You should be doing your pirate sound too. You should give me your best ahaarrr me hearties. Oh yeah, that was pretty good from some of you, some of you need some practice. We also have a pony, you know cause, this is what you need in mathematics. Pirates and ponies. Uh-huh, and you know to offer us some great wisdom, we also have Yoda.

[Screen shows a pirate, pony and Yoda figurine.]

Yes, so let's actually start off with Yoda's strategy. So, Yoda was working with some students, and they said well we know we could do some stuff with counting, actually, because we, we know stuff about numbers, and we could count them. And they said they know this is 8, and they know this is 6, and they know this is 8 so they don't actually have to count all of them again. But they might count some of them.

[Presenter moves Yoda figurine to the right side of the screen and takes the pirate and pony away. They point to the right ten-frame showing 8 dots, then the middle ten-frame showing 6 dots and then the left ten-frame showing 8 dots.]

Mm so Yoda was thinking well if we know this is 8, we could then count the rest. So, let's have 8 and then let's count together. Ready? 8, Yoda is gonna try to help, he's got little arms, but they're not very good at pointing. 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22.

[Presenter points Yoda towards the right ten-frame and counts 8. They then touch Yoda to each dot as they count on by ones in the middle and left ten-frames.]

Uh-huh, so we're going to represent Yoda's thinking in green, and I guess a number line might be a really good way to show their thinking. They started with 8, and then they counted on.

[Presenter uses a green marker to draw a number line consisting of a horizontal line and a vertical dash towards the start of the line with the number ‘8’ underneath.]

So how many did they count on here? 6? Okay, so I need to do 6 little jumps. 9, 10, 11, 12, 13, 14 and that's my 6, and then, mm-hmm, another 8. So, 14, 15, 16, 17, 18, 19, 20, 21, 22. 22.

[Presenter draws 6 little semicircles on top of the number line starting from the dash where the 8 is labelled. They write ‘14’ at the end of the sixth semicircle underneath the number line. The presenter draws another 8 little semicircles on top of the number line continuing from the number 14. The presenter using their other hand to keep track as they count on by raising their fingers until they get to 5 then lowering 3. They write ‘22’ at the end of the eighth semicircle underneath the number line.]

And that's a jump of 6 ones, and that's a jump of 8 ones, to work out that 8 combined with 8 combined with 6 is equivalent in value to 22.

[Presenter draws a curved line above the semicircles from the 14 to the 8 and writes ‘plus 6 ones’ above it. They then draw another curved line above the semicircles from the 22 to the 14 and write ‘plus 8 ones’ above it. Underneath the number line they write ‘8 plus 8 plus 6 equals 22.’]

Okay so that was Yoda’s way of thinking but what about, who would you like to hear from next? The pirate or the pony?

Oh, the pirate. Nice choice. So, we're going to represent the pirates thinking in red. That's a good idea, isn't it? Yes, so the pirate actually thought quite differently. He said, well I know stuff about counting but I also know stuff about numbers.

[Presnter shows the pirate figurine and a red marker.]

Mm-hmm, and he said well actually what I might think about first is double 8, because I know that double 8 is 16. That's a number fact I have in my head and that, then what I'm going to think about the 16 is that 16 and 4 more would make 20. So, inside my 6 I can see 4 and 2 more. So, the 16 and the 4 combines to make 20 and then 2 more is 22.

[Presnter points to 2 ten-frames showing 8 dots and rearranges them so they are side by side and the ten-frame with 6 dots is at the end. They then place 4 yellow cubes over 4 of the 6 dots on the last ten-frame. The presenter then places 2 red cubes over the 2 dots remaining on the ten-frame. They then move the four yellow cubes over on top of the 2 ten-frames with 8 dots showing a total of 20. Then they highlight the 2 red cubes left which makes 22.]

Yeah, so let's think about how we could record that. So, what he said was I'm gonna rethink 8 plus 6 plus 8 as 8 plus 8 plus 6.

[Presenter writes ‘8 plus 6 plus 8 equals 8 plus 8 plus 6’.]

Now I'm gonna use the commutative property, he said, good little pirate, knows his mathematical vocabulary, mm-hmm and then he said double 8 is 16.

[Presenter writes ‘Double 8 equals 16’ underneath the other recordings.]

Mm-hm and then he said I know 6 is 4 and 2 so then 16 plus 4 combines to make 20.

[Presenter writes ‘6 equals 4 and 2’ underneath and then ‘16 plus 4 equals 20’ underneath that.]

And 20 and 2 combines to make 22.

[Presenter writes ‘20 plus 2 equals 22’ underneath again.]

Uh-huh, yeah, and I could draw that as a tree diagram for you. So, 8 plus 6 plus 8. So, he said double 8 is 16 and the 4, I, the 6 I can partition into 4 and 2 more. 16 and 4 is 20. And 20 and 2 more is 22. Uh-huh, that just fits on the page, I think. Oops, not quite, I'll do it sideways. 16 and 4 is 20 and then 2 more makes 22.

[Presenter writes ‘8 plus 6 plus 8’ underneath and then uses lines to make a tree diagram. First, they draw lines underneath the 2 eights, joining them together and write ‘16’. Next, they draw 2 lines splitting off underneath the 6 and write ‘4’ and ‘2’. Then, they draw 2 lines from the 16 and the 4 joining them together and write ‘20’. Finally, they draw 2 lines from the 20 and the 2, joining them together, and write ‘22’.]

Uh-huh, so that was another way from the pirate, and he was thinking about using what he knew and so then along came the little pony. And we'll represent the pony in pink. It's a good idea and we'll move these blocks and we'll put our question back. We'll re-commute it.

[Presenter shows the pony figurine and finds a pink marker. They rearrange the ten-frames at the top so that the ten-frame with 6 dots is in the middle of the 2 ten-frames with 8 dots.]

And the pony was thinking, well, hold on a second, I can imagine something happening to the numbers. And she said, imagine here on my 6 if I moved some of these around. So actually, here in my 6 I can see 4 and 2 more, or 2 and 2 and 2.

[Presenter places orange counters over 4 of the 6 dots on the middle ten-frame.]

And if these dots here moved across over here, like this, that would now be blank. Mm-hmm, so now I have a 10, a 4 and an 8. Aha, can you imagine what's happening next?

[Presenter moves 2 orange counters from the middle ten-frame with 6 dots to the left ten-frame with 8 dots. They put white counters over the 2 dots they have moved. The presenter then points out they have now got a ten-frame with 10 dots, a ten-frame with 4 dots and a ten-frame with 8 dots.]

Yes! Then she said, well I can move these 2 over to here. I really need some other white counters, I'll use blocks, and said well now I've got a full ten-frame and a full ten-frame and 2 more.

[Presenter moves 2 more orange counters from the middle ten-frame with 4 dots remaining to the right ten-frame with 8 dots. They put white blocks over the 2 dots they have moved. The presenter then points out they have now got 2 full ten-frames, and a ten-frame with only 2 remaining.]

And actually, I'm going to join those together first, a bit like the pirate did, and say well one 10 and 2 tens, we just call that 20. And 2 more is called 22.

[Presenter rearranges the ten-frames so that the 2 full ten-frames are next to each other. Then the ten-frame with 2 left is at the end.]

Aha, so she actually thought about the problem of 8 plus 6 combined with 8 is equivalent in value to 10 plus 10 plus 2. Uh-huh, and then she said I know 2 tens is renamed as 20. And 20 and 2 more is just called 22.

[Presenter writes ‘8 plus 6 plus 8 equals 10 plus 10 plus 2’. Then underneath they write, ‘2 tens equal 20’. Then underneath that they write, ‘20 plus 2 equals 22.’]

Ahhh so she used her mathematical imagination to visualize the quantities moving. Yeah, would you like to see that one more time? To see what she did?

[Presenter removes counters and blocks and rearranges the ten-frames at the top so that the ten-frame with 6 dots is in the middle of the 2 ten-frames with 8 dots.]

Mm-hmm so she said, well I know something about these numbers. I can be flexible with them, so I can use them. I don't have to use them exactly as they are and, and what I can do is move these 2 dots here, to come here. So, let's move those 2 dots from 6, because 6 is made up of 4 and 2, and move them across which now means I have a 10, a 4 and an 8.

[Presenter moves 2 orange counters from the middle ten-frame with 6 dots to the left ten-frame with 8 dots. They put white blocks over the 2 dots they have moved. The presenter then points out they have now got a ten-frame with 10 dots, a ten-frame with 4 dots and a ten-frame with 8 dots.]

And then she said well I could do this again, look, I could do this again. Inside my 4 is 2 and 2 so, I can slide those across to here, and now I have 2 full ten frames and 2 more. Look, it's 2 full ten frames. 20, 2.

[Presenter moves 2 more orange counters from the middle ten-frame with 4 dots remaining to the right ten-frame with 8 dots. They put white counters over the 2 dots they have moved. The presenter then points out they have now got 2 full ten-frames, and a ten-frame with only 2 remaining. They then move the 2 full ten-frames together and the ten-frame with 2 left is at the end.]

What's some of the mathematics here?

Yeah, you can solve the same problem in different ways. And today we saw 3 different ways that we could think of combining 8 and 6 and 8.

[Screen shows a screenshot of the 3 ten-frames and recordings of the strategies used by the pony, pirate and Yoda from earlier in the video.]

Yes, and you can use numbers flexibly and today we saw this when the pony and the pirate visualised dots moving from 1 ten frame to another so they could use what they know to solve the problem. The pony thought about 6 as 2 and 2 and 2, so she imagined 2 dots moving to form 8 into one 10. She imagined another 2 dots moving to form the other 8 into another 10 and then she knew that 2 tens and 2 more can be renamed as 22.

[Screen shows a screenshot of 3 ten-frames side by side from earlier in the video. The ten-frame on the left and in the middle have 8 black dots and 2 orange counters, making 10 each. The ten-frame on the right has 6 black dots, but 4 dots are covered by white counters or blocks, so only 2 are showing.]

Have a great day mathematicians.

[End of transcript]

## Instructions

Watch and follow along with this number talk to think about, hear and see how 8 + 6 + 8 may be solved.

## Watch

Watch Let's talk 4 Part 2 video (4:44) to investigate some ideas further.

### Speaker

Let's investigate this idea a little further. Recently we said that you can use numbers flexibly. And we saw this when the pony and the pirate visualised the dots moving from one ten-frame to another, so they could use what they know to solve problems. And the pony thought about 6 as 2 and 2 and 2. And she imagined 2 dots moving to form 8 into one 10 then another 2 dots moving to make 8 into another 10. And then she just had 2 tens and 2, which she renamed as 22.

So, let's have a look at how these strategies work by exploring them on an equal arm balance.

[Screen shows a screenshot from the ‘Let’s talk 4 – part 1’ video. There are 3 ten-frames side by side. The ten-frame on the left and in the middle have 8 black dots and 2 orange counters, making 10 each. The ten-frame on the right has 6 black dots, but 4 dots are covered by white counters or blocks, so only 2 are showing.]

Hello there, mathematicians!

We thought we'd have a further look at these strategies, to think about how we could prove, mm-hmm, that you can think about 8 and 8 and 6 as 10 and 10 and 2 for example. So that was the pony's strategy, remember here's our little pony. Good morning, or good afternoon, depending on what time you're watching!

[Screen shows the paper with written strategies from the ‘Let’s talk 4 – part 1’ video at the top of the screen. It also shows an equal arm balance with numbers on both arms of the scale. The numbers start at zero in the centre and go up to 10 on each side, with hooks under each number. Presenter places the pony figurine on the table and points to the pony’s strategy written on the paper from the previous video.]

So, let's have a look at this idea over here. So, let's make our original problem which was 8 and 8 and 6. And we can tell it's now not equivalent, that's right, because the balance scale isn't equal, isn't even.

[Presenter places 2 blue pegs on the number 8 hook and one blue peg on the number 6 hook on the right side of the balance. The equal arm balance is tipped down towards the right.]

Mm-hmm and what the pony was saying was, inside of 6 I know that I can break it up into 2 and 2 and 2. Let's have a look at that, to start with. Look here's 6. And what they're saying is 6 is equivalent to 6, which we know that balances. But she's also thinking that inside of 6 is 2 and 2 and 2. So 6 is equivalent to 3 twos or 3 twos is 6.

[Presenter takes the 2 pegs off the number 8 hook and leaves the peg on the number 6 hook on the right side. They then place a blue peg on the number 6 on the left side to show they are equivalent and balance. The presenter then removes the peg from the number 6 and places 3 blue pegs on the number 2 hook on the left side and they balance.]

And so, what the pony did was use this knowledge. Because when she says 6 and 8 and 8, she said what I know is that 8 and 2 makes 10 so I can take 2 from the 6 and add it to the 8 and make a 10 and I can do that again and then I'll have 2 left. Aha, and it balances. And so, what we can see here is that 8 and 6 and 8 is equivalent to 10 and 10 and 2. Yeah, and then she just renamed that. 2 tens is 20 and 2 more is 22.

[Presenter removes the pegs from the equal arm balance. They place one blue peg on the number 6 hook and 2 blue pegs on the number 8 hook on the right side. The presenter then places 2 blue pegs on the number 10 hook and one on the number 2 hook on the left side. The two sides of the scale balance. The presenter refers to the pony's strategy written on the paper from the last video and points to the mathematics recorded as they touch each peg on the balance.]

Let's have a look at the pirates thinking. I'm going to leave those pegs over there, and here's our pirate, ahh haarr. And what the pirate was thinking about first, was that he knew double 8 was 16, so he, he kept the double 8.

[They swap the position of the pony and the pirate figurines, placing the pirate on the left side of the table and removing the pony. The presenter places 2 blue pegs on the number 8 hook on the left side.]

Mm-hmm and then he said I know 6 is composed of 4 and 2. So let's have a look at that together. There's our 6 and we know this is equivalent to 3 twos, cause we checked that. Look, mm-hmm, that works. The pirate said 6 is also 4 and 2. So let's see what happens.

[Presenter removes all pegs from the equal arm balance except for the peg on the number 6 hook on the right side. They place 3 blue pegs on the number 2 hook on the left side and the scale balances. They then move one blue peg to the number 4 hook and one to the number 2 hook and it balances too.]

Aha and so here you use this knowledge. So, he said over here there's double 8 and double 8 is 16, I know this. And I know inside of 6 is 4 and 2 and 16 and 4 more is 20 and then I still have the 2 and that will make 22.

[Presenter removes all pegs from the equal arm balance and places one blue peg on the number 6 hook and 2 blue pegs on the number 8 hook on the right side. They then place 2 blue pegs on the number 8 hook, one blue peg on the number 4 hook and one blue peg on the number 2 hook on the left side. The two sides of the scale balance.]

And so that's how we can prove here, this idea, that 8 and 6 and 8 is equivalent to double 8 plus 4 plus 2. Nice work, mathematicians.

[The presenter refers to the pirate's strategy written on the paper from the last video and points to the mathematics recorded.]

So, what's some of the mathematics?

We can use an equal arm balance to investigate equivalence. This helps us see that we can think flexibly about numbers when solving problems. Allowing us to use what we know to work out what we don't know yet.

So young mathematicians, what other quantities can you find that are equivalent in value to 8 plus 8 plus 6?

Over to you mathematicians!

[End of transcript]

## Instructions

What other quantities can you find that are equivalent in value to 8+8+6?