Let's talk 3

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 visualized dots moving from one ten frame to another, so they could use what they know to solve the problem. And the pony thought about eight and six as ten and four. She imagined two dots moving from the six so she could make one ten. Then she knew that one ten and four is renamed 14. Let's explore these ideas further.

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 eight and six. Mm-hmm. So I have eight, a collection of eight 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 eight and I have one dot, one counter for each dot, then that also is eight. And this is six. 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 five fingers, and then one more would be six. Uh-huh and I could also do the laying over strategy, so if I have one counter for each dot, we know this is six because that's the problem we were working on. I could use that to prove, yeah, I have six yellow counters.

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. And he said well I know eight and six is actually equivalent to, equivalent to seven and seven, which is double seven. And so what he did in his minds eye was, he went from this collection of eight. I'm gonna arrange it like that, and this collection of six, and he said, well if I take one of these here and move it across to here, I now have double seven. 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 seven is fourteen. Uh-huh.

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

Let's have a look at what the pony thought about. Because she visualized dots moving too but she did it differently. And she said well what I can see here is, I can see I would need two more to make a ten frame. Mm-hmm, to complete the ten frame. So she said if I slide these two 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 ten and four more, which we can call 14. 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.

So there's two 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 eight and six but think of it as seven and seven. And we can see eight and six and think of it as ten and four, but together it's still 14. 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.

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. 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.

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 eight and six is equivalent in value to seven and seven. Double seven. So let's put one peg on seven and, yes, this peg also needs to go on seven and let's check to see. Ahah, they are equivalent. Look, double seven is the same in value to eight and six. That's really cool.

And let's have a look at my little pony, there she is. Excuse me, pirate. And then she said eight and six is equivalent to ten and four. Let's have a look. So one 10, oh not equivalent yet, and one four. Let's see what happens. Aha, they are equivalent! Look the scale balances out. Mm-hmm, and you know what else mathematicians, here's our pirate, what would happen do you think if we kept the ponies ten and four and put the pirates seven and seven 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. So there's one seven, oh definitely, not in balance, not equivalent, and two sevens. Ah, so look at that, we could also draw one more conclusion from our experiment. And that is, that double seven is equivalent to 10 and four. 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. 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.