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Speaker

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

Yes, that's great that you are. So I wanted to show you two things. This one to start with, it's a ten frame.

[Screen shows a blue piece of A3 paper in a landscape orientation folded down the centre vertically. Presenter places a ten-frame on the left-hand side of the paper showing 5 dots in the top row and one dot in the bottom row.]

Can you show me what that's representing? Can you show me using your fingers?

Ah, I can see some of you are holding your fingers like this. With five and one more, and I can see some of you are holding your fingers like this. With four and two more. Uh-huh. Which is the same as five and one more. Look, if I add one more here and get rid of one they're still the same isn't it? Mm-hmm.

[Presenter shows 6 fingers. The first time they hold up 5 fingers on one hand and one finger on the other hand. The next time they hold up 4 fingers on one hand and 2 on the other hand. The presenter then shows how if they put down one of the two fingers on one hand and put up a fifth finger on the other, that it is the same.]

And some of you are doing, oh I can see three and three. Uh-huh because it is six, yes, and you could represent it like five and one, or four and two, or three and three. Mm-hmm.

[Presenter shows 6 fingers again. This time, they hold up 3 fingers on one hand and 3 fingers on the other hand. They gesture toward the ten-frame showing 6 and then point to the 5 dots in the top row and one dot in the bottom row. The presenter continues to demonstrate the other combinations on their hands as before. They hold up 4 fingers on one hand and 2 on the other and 3 fingers on one hand and 3 on the other.]

Okay, how many dots can you see on this card? What's it representing?

[Presenter takes away the ten-frame showing 6 dots and places a ten-frame on the left-hand side of the paper showing 5 dots in the top row and 3 dots in the bottom row.]

Can you show me using your fingers again?

Oh, I see some 5 and 3s, to represent 8. Uh-huh, what else have you, ah, some 4 and 4s to represent 8.

[Presenter shows 8 fingers. The first time they hold up 5 fingers on one hand and 3 fingers on the other hand. The next time they hold up 4 fingers on one hand and 4 on the other hand.]

Uh-huh. Oh I see and someone is doing this, they're saying, look it's 10 and take away 2, to show that, I know, my finger I can't stop it moving! I know our hands do weird things sometimes. Yes, so, it's 8.

[Presenter shows 8 fingers. They hold up 10 fingers and then bend 2 down. When bending 2 fingers down the presenter's other fingers also move.]

So my question for you today is mathematicians, is how could we work out how many dots we have all together?

[Presenter places the ten-frame from earlier with 6 dots next to the ten-frame with 8 dots. They circle both ten-frames with the finger in the air.]

You know there's 8 here and there's 6 here, how could we work out how many there are all together?

[Presenter points to the ten-frame with 8 dots and then the ten-frame with 6 dots.]

Ah, yeah, so we're gonna do this a bit like a number talk. You might do this in your classroom, where we're thinking, we share our fist to say I'm thinking, and then when we have one strategy, one way of solving, we might go I have one way I could think about it. Mm-hmm. Yeah, when we've thought about one possible strategy, yeah, we're gonna look for a second one and maybe even a third one.

[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.]

Oh. I can see you're thinking hard! Okay, I think everybody's got one strategy now, uh-huh.

[Presenter makes a fist then lifts their thumb to show they have one strategy.]

Okay, well let's think about how some other people solved this problem. So we asked some students to, and they're going to be represented today by our pirate, ahharrr. Yes, I know, we really like pirates and a pony. Yes, because we love ponies too. I mean and what could you have, think of better friends, then for mathematics, than a pirate and a pony? Oh, I know Yoda. Aha, so Yoda also thought about how he would solve this problem.

[Presenter places a pirate, pony and Yoda figurine to the left of the page.]

So let's look at their three strategies and we might represent Yoda’s thinking in green, to match his body. Mm-hmm. So Yoda was thinking, or the team represented by Yoda, was thinking. First of all, he's lucky he's got a coat on because it's cold and wintry, and you know how lovely to have a snake as a, as a scarf, you know. Lucky it's not a real one!

[Presenter moves the Yoda figurine onto the page and holds up a green marker. They then pick up Yoda and fix its coat and play with its snake scarf.]

So, so Yoda was thinking that he could count the dots. Uh-huh and he said, but hold on a second, I already know this is 8 and I already know this is 6, so I don't need to count all of them again, I just need to count some of them. Mm-hmm.

[Presenter points to the dots on the ten-frames then moves the Yoda figurine as if it is talking. Then the presenter uses their finger to circle the ten-frame with 8 dots and the ten-frame with 6 dots. They then point to the dots individually showing that they don’t need toy count them all.]

So he said this is 8 over here and so then I can say 8, 9, 10, 11, 12, 13, 14. 14 dots.

[Presenter uses their finger to circle the ten-frame with 8 dots again as they say the number word 8. They then pick up Yoda and touch him to each dot on the second ten-frame as they count on.]

So Yoda was thinking about, I know there's 8 and then he counted on 6 more. So he said, 8, 9, 10, 11, 12, 13, 14. Is that six number words? One, two, three, four, five, six. Yes, so he said 8 and 6 is equivalent to, equivalent to 14. Or 14 is 8 and 6. That's even more efficient way to write it. Mmm-hmm.

[Presenter writes the numeral ‘8’ in green marker on the right side of the paper. They then write the numerals ‘9, 10, 11, 12, 13, 14’ underneath. Then the presenter writes ‘8 and 6 is equivalent to 14’, before adding ‘14 is’ in front of the ‘8 and 6’ and crossing out ‘is equivalent to’.]

So that was Yoda's way of thinking, but we're also joined by the pony and the pirate. Who would you like to know next? Aha, the pony, okay, well the pony, who we can represent in pink, the pony was thinking about the dots.

[Presenter wiggles each figurine as they are mentioned. The presenter moves the pony figurine onto the page and holds up a pink marker. Presenter points to the dots on the ten-frames.]

Mm-hmm, and she said I want you to imagine something in your mind's eye, because if I moved 2 of these dots over here, I'd have one full ten-frame. Yes, and one not full ten-frame and that would help me, because if I move 2 dots over, how many dots would be here? Aha, 4. And how many dots would be here? 10! And one 10 and 4, we can just rename as 14.

[Presenter gets 2 white and 2 orange counters and places them on the paper. The presenter uses their hand to cover 2 of the 6 dots on the ten-frame on the right. Then they place 2 fingers in the empty boxes next to the 8 dots on the ten-frame on the left. The presenter indicates that the right ten-frame would then have 4 dots and the left ten-frame would be full with 10 dots.]

So in her mind what she did, was let's represent these 2 dots here, and she said I'm gonna move these over, so imagine them disappearing they're no longer fall over to here, and now I can see one 10 and 4 more and I know that's 14. Uh-huh.

[Presenter places the 2 orange counters over 2 of the 6 dots that they covered before with their hand on the ten-frame on the right. They then replace the 2 orange counters with 2 white counters and move the 2 orange counters into the empty boxes next to the 8 dots on the ten-frame on the left.]

So the pony said, well I know that 8 and 6 is equivalent to 10 and 4. Uh-huh and 14 is 10 and 4. Uh-huh, do you like that strategy by the pony? Yeah, she was a bit proud of herself.

[Presenter draws a vertical line with the pink marker to separate the green and pink ideas. They write ‘8 and 6’, then underneath write ‘10 and 4’, then underneath that write ‘14 is 10 and 4’. The presenter then moves the pony figurine under their written idea.]

Gee, so then the pirate came along, ahhh harrr, my hearties, and the pirate, I think we should represent him in red, hopefully you can discern the difference.

[Presenter draws a vertical line with the red marker to separate the pink and red ideas.]

Oh yeah. I know, Yoda had all the writing space and the ponies got a little bit less, and now the pirate got hardly any. But the pirate thought of a similar strategy to the pony, in the idea of moving dots, and he said what he could see is that inside of 8 there's actually 7, because look, if you remove one dot, you now have 7. Aha. And he said and if I put a dot over here, I now have 2 sevens. And he said I just actually know in my brain that 2 sevens is 14.

[Presenter places one white counter over one of the 8 dots in the ten-frame on the left. They then place one orange counter next to the 6 dots on the ten-frame on the right.]

Yes, and so he said, well eight and, oh that's pink, sorry pirate, 8 and 6 is the same as saying 7 and 7. And that 14 is 7 and 7, and that was the pirate's strategy.

[Presenter starts to write in pink marker, then changes to red and crosses out pink writing. They write ‘8 and 6’, then underneath write ‘7 and 7’, then underneath that write ‘14 is 7 and 7’. The presenter then moves the pirate figurine under their written idea.]

So isn't that really cool, little mathematicians, to think about the idea that you can visualize the dots moving. So even though you see 8, you could change the 8 into 14 if you wanted by visualizing these 2 dots here moving over here.

[Presenter shows the pony strategy again by placing the 2 orange counters over 2 of the 6 dots on the ten-frame on the right. They then replace the 2 orange counters with 2 white counters and move the 2 orange counters into the empty boxes next to the 8 dots on the ten-frame on the left.]

Aha, or if you know double 7, then you could imagine this dot, moving over here to make double 7.

[Presenter shows the pirate strategy again by placing one orange counter over one of the 8 dots in the ten-frame on the left. They then replace the orange counter with a white counter and move the orange counter next to the 6 dots on the ten-frame on the right.]

So what's some of the mathematics here? Yeah, we saw that you can solve the same problem in different ways. So here are three different ways that we could think about combining 6 with 8.

We also saw that you can use numbers flexibly and today we saw this when the pony and the pirate visualized dots moving from one ten-frame to another, so they could use what they know to solve the problem.

The pony thought about 8 and 6 as 10 and 4, and she imagined 2 dots moving from the 6, so she could make one 10. She then what knew, used what she knew about one 10 and 4 and she renamed that as 14.

Have a great day mathematicians!

[End of transcript]

Speaker

So let's investigate this idea a little further. We said that you can use numbers flexibly, and we said that we saw this when the pony and the pirate visualised dots moving from one ten-frame to another, so they could use what they know to solve the problem.

And the pony thought about 8 and 6 as 10 and 4. She imagined 2 dots moving from the 6 so she could make one 10. Then she knew that one 10 and 4 is renamed 14. Let's explore these ideas further.

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

Hello there, mathematicians! We hope you're having a really nice day today.

We thought we'd come back and investigate a little bit more on the strategies that were used by our pony and our mathematical pirate when we were thinking about how we could solve 8 and 6.

[Screen shows a blue piece of A3 paper in a landscape orientation folded down the centre vertically. There are two ten-frames at the top of the screen with counters underneath. The ten-frame on the left shows 8 black dots and has 8 red counters spread out under it. The ten-frame on the right shows 6 black dots and has 6 yellow counters spread out under it. The presenter points to the pony and pirate figurines.]

Mm-hmm. So I have 8, a collection of 8 here. I can prove that it's 8 actually, because I could lay one over the top of each dot even though they're a little bit bigger, since I know that's 8 and I have one dot, one counter for each dot, then that also is 8.

[DECSRIPTION: Presenter uses their finger to circle the 8 red counters. They then lay a red counter over each dot on the ten-frame with 8 dots. Once proved, they take the ten-frame away and leave the counters in a pile.]

And this is 6. Hmm I could prove that a different way actually, I could think about what I know about my fingers. So I know on one hand, I have 5 fingers, and then one more would be 6. Uh-huh and I could also do the laying over strategy, so if I have one counter for each dot, we know this is 6 because that's the problem we were working on. I could use that to prove, yeah, I have 6 yellow counters.

[Presenter uses their finger to circle the 6 yellow counters. They show there are 6 counters by placing 5 fingers on one hand and a thumb on the other hand on each of the counters. They then lay a yellow counter over each dot on the ten-frame with 6 dots. Once proved, they take the ten-frame away and leave the counters in a pile.]

Okay, so let's think about what the pirate did. The pirate said, ahh harrrr my hearties. That's right, that's how he does maths! You could try using your best pirate voices for maths too.

[Presenter picks up the pirate figurine and makes a pirate voice.]

And he said well I know 8 and 6 is actually equivalent to, equivalent to 7 and 7, which is double 7.

[Presenter gets a red marker and writes ‘8 and 6 is equivalent to 7 and 7’ on the right side of the paper. They then write ‘Double 7’ underneath.’]

And so what he did in his minds eye was, he went from this collection of 8. I'm gonna arrange it like that, and this collection of 6, and he said, well if I take one of these here and move it across to here, I now have double 7.

[Presenter arranges the counters into 2 rows on the left and right side. They put the 8 red counters on the left in 2 rows of 4 and the 6 yellow counters on the right in 2 rows of 3. The presenter moves one red counter across to join the yellow counters and flip it over, making it yellow. There are now 7 counters on each side – 4 in the top rows and 3 in the bottom rows.]

Aha, so even though he moved a counter across, this side decreased by one and this side increased by one, we still have the same number all together. Yeah. So he said double 7 is 14. Uh-huh.

[Presenter moves the counter back to its original position, making it red again. They then demonstrate the strategy again and move one red counter across to join the yellow counters and flip it over, making it yellow. The presenter points to the left side that had 8 counters and now has 7, showing it decreased by one. They then point to the right side that had 6 counters and now has 7, showing it increased by one. The presenter circles both groups of counters together with their finger, highlighting that there is still the same number of counters. They write ‘14’ underneath where they had written ‘Double 7.]

Do you wanna see that one more time? Look, he said here's 8 and here's 6, and I know inside 8 I can see 7. Look, there it is and so if I take this one counter and put it to here I now have 2 sevens and I know 2 sevens is the same as double 7, which is 14. Ahh, and so he visualised that happening in his mind. Mm-hmm.

[Presenter moves the counter back to its original position, making it red again. They point to the 8 red counters on the left and then the 6 yellow counters on the right. They move one red counter away from the row of 8 counters to show 7. They presenter moves the red counter across to join the yellow counters and flips it over, making it yellow. The presenter circles both groups of counters with their finger and then points to where they have written ‘Double 7’ explaining it is the same a 2 sevens. They then point underneath to where they have written ‘14’. Presenter moves the counter back to its original position, making it red again.]

Let's have a look at what the pony thought about. Because she visualised dots moving too but she did it differently. And she said well what I can see here is, I can see I would need 2 more to make a ten-frame. Mm-hmm, to complete the ten-frame.

[Presenter swaps the position of the pony and the pirate figurines. They then point to the empty spaces next to each row of the 8 red counters on the left.]

So she said if I slide these 2 over here, I don't have any more new counters all together. That's right, I haven't added any or taken any away, I've just moved some across, and so now, what I have is one 10 and 4 more, which we can call 14.

[Presenter slides 2 of the 6 yellow counters from the right side over to the left side to join the 8 red counters. They then use their finger to circle around the groups of counters showing they don't have any new counters. The presenter moves the 2 counters back to the original position and then slides them across again, this time flipping them over, making them red. They use their finger to circle the groups as they say one 10 and 4 more. Presenter moves the counters back to the original position, making them yellow again.]

Mm-hmm. So let's have a think about how we could write that down. So she said I know eight and six is equivalent to ten and four. Uh-huh, because she slides this across in her mind's eye and then it became a ten and four more. Mm-hmm. And she said she knows one ten and four is renamed 14.

[Presenter gets a pink marker and writes ‘8 and 6 is equivalent to 10 and 4’ on the left side of the paper. They then write ‘1 ten and 4 is renamed 14.’]

So there's 2 different strategies but both of them, yet, used the mathematical imagination didn't they, where we moved quantities around. And so that's really interesting to us, isn't it, that we can see 8 and 6 but think of it as 7 and 7. And we can see 8 and 6 and think of it as 10 and 4, but together it's still 14.

[Presenter points to the 8 red counters on the left and then the 6 yellow counters on the right. They move one red counter across to join the yellow counters and flip it over, making it yellow. Presenter moves the counters back to the original position, making it red again. They point to the 8 red counters on the left and then the 6 yellow counters on the right again. Presenter slides 2 of the 6 yellow counters from the right side over to the left side to join the 8 red counters, and flips them over, making them red.]

Cause look here, that's 14, that's 14 and that's still 14. Because I haven't added any new counters to my whole collection and I haven't taken any counters away from my collection.

[Presenter circles both groups of counters together with their finger, highlighting there are 14. They then move back the 2 counters they had moved across to the red side, and make them yellow again, showing it is still 14. They then move one red counter over to the yellow side, and flip it over to make it yellow, showing it still makes 14.]

Do wanna have a look at that on the balance scale? Cool!

Okay there mathematicians, let's have a look at this together. So this is my balance scale and what it shows me is, things that are equivalent. So for example, if I put a 5 on this side and a 5 on this side, it will balance, to say to me 5 is equivalent in value to 5.

[Screen shows the blue paper with written strategies from before at the top of the screen. It also shows a balance scale 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 a blue peg on the number 5 hook on each side and the scale balances.]

Mm-hmm. Or I could know something else about 5, like what are two numbers that combine to be 5? Ah, 4 and 1. Mm-hmm so if I move my peg to four now, oh, that means they're not equivalent. But if I put another one here, what happens? Uh-huh, that's telling me that this side is equivalent in value to this side.

[Presenter moves the peg on the right-hand side from the number 5 hook to the number 4 hook and the scale tips up. They then add a blue peg to the number 1 hook on the right-hand side and it balances. Presenter points to the right side, then the left side highlighting they are equivalent in value.]

Okay, so let's have a look at 8 and 6, which is our problem. So let's put a peg on 8 and a peg on 6. And here's our little pirate friend, let's think about his strategy. He said 8 and 6 is equivalent in value to 7 and 7. Double 7. So let's put one peg on 7and, yes, this peg also needs to go on 7and let's check to see. Ahah, they are equivalent. Look, double 7is the same in value to 8 and 6. That's really cool.

[Presenter places a blue peg on the number 8 and number 6 hook on the left-hand side. They then show the pirate figurine and place it down on the right side of the table. The presenter refers to the pirate’s strategy written on the paper from before and points to what is written. They place one blue peg and then a second blue peg on the number 7 hook on the right-hand side and the scale balances.]

And let's have a look at my little pony, there she is. Excuse me, pirate.

[Presenter swaps the pirate figurine for the pony and sits it on the right side of the table.]

And then she said 8 and 6 is equivalent to 10 and 4. Let's have a look. So one 10, oh not equivalent yet, and one 4. Let's see what happens. Aha, they are equivalent! Look the scale balances out.

[The presenter refers to the pony's strategy written on the paper from before and points to what is written. The blue pegs are still in position on the number 8 and number 6 hook on the left-hand side. They then move the pegs on the right-hand side so that there is a blue peg on the number 10 and number 4 hooks, and the scale balances. The presenter lines their arm up straight in-line with the balance scale, showing it is balanced.]

Mm-hmm, and you know what else mathematicians, here's our pirate, what would happen do you think if we kept the ponies 10 and 4 and put the pirates 7 and 7 over on this side? Do you think it would be balanced or do you think one side will be bigger, taller than the other? Shall we look together? Let's see.

[Presenter gets the pirate figurine and sits it on the left side of the table. They point to the pony and the blue hooks hanging from the number 10 and 4 on the right side of the balance scale. The presenter then points to the pirate and refers to the ‘7 and 7’ written on the paper from before.]

So there's one seven, oh definitely, not in balance, not equivalent, and 2 sevens. Ah, so look at that, we could also draw one more conclusion from our experiment. And that is, that double 7 is equivalent to 10 and 4.

[Presenter moves the pegs on the left-hand side so that there is one blue peg on the number 7 hook and the balance scale tips. Then they add a second blue peg to the number 7 hook and the scale balances. The presenter writes in black marker along the bottom of the paper ‘Double 7 is equivalent to 10 and 4’.]

Nice work mathematicians!

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.

[Screen shows a screenshot of the balance scale with blue pegs on the number 8 and 6 hooks on the left-hand side and blue pegs on the number 10 and 4 hooks on the right-hand side. The scale is balanced.]

We can imagine objects moving to help us use numbers flexibly too. This means that if we can imagine things we have the power to move them.

[Screen shows screenshots of the different combinations of ways to make 14 using counters. The first image shows 8 red counters and 6 yellow counters. The second image shows 10 red counters and 4 yellow counters.]

[End of transcript]