Introducing rekenreks

Hello there mathematicians. I hope you're having a gorgeous day today. Let's explore the rekenrek.

So this is what my rekenrek looks like at home, and here's another one I drew on the computer; and we're going to use the one I drew on the computer to do some mathematical investigating today. So before we get started like great mathematicians, we often ask this question, what do you notice? So before we begin imagining and playing with and thinking about the rekenrek what do you notice about it? What are some of its features?

Ok, so see if you've got one idea. Can you come up with another idea? What else did you notice? Ah yes, and sometimes we can notice things that aren't mathematical. See if you can notice some mathematical things.

Ok, shall we share some of those ideas together 'cause I can hear some of you saying that you notice that there are different coloured beads? Yeah, did you notice that too? There's some red beads and there's some light orange beads. Yes, and that each colour represents a collection of five. Look, there's a chunk of red, and there's five red beads inside that blue circle. And each chunk of colour represents another five.

Yes there's five in there, and that chunk is also five. Yeah, ah and that led some of you to notice this, that there are two fives on the top row. Look one, five, two, fives and two, fives on the bottom row. one, five, two, fives. Yes, and some of you said hey, that's the same as saying there are ten on the top row and ten on the bottom row. And yes, and that helped us to realise that there are twenty beads in total.

All right, then I was thinking if we're introducing rekenreks, well, how can we use rekenreks? How can we use them? So let's have a look. Yes, and what we do is we move beads across to represent quantities. So at the moment over here there's there are no beads, so my rekenrek is representing zero. But if I move one across over here now, I'm representing the quantity of, one. Yes, what about if I do this and move another bead across?

Over here I'm now representing two, Yes. And yes, I can represent two in a different way. In this case, now I'm representing two but I have one at the top and one down the bottom. Ok, mathematicians, I think you're ready. It's time to start imagining. Here's my rekenrek, and I'd like to move some beads across and I'd like you to imagine that I'm moving five beads across. So close your eyes and use your mathematical imagination to imagine you're moving those five beads across my rekenrek, and how are you moving them.

Ok, let's have a look. This is how I imagined it. Yes, and some of you imagined it like that, but some of you imagined it in a different way. Yeah, which is also awesome because we can think about things differently. But over here I have five. Ok, let's get our mathematical imaginations again. Are you ready? And this time I'd like to imagine four.

So close your eyes. And in your minds eye, imagine you're touching the rekenrek and yes, I can see it. Put your arms up. Yes, and move across four beads. Ok and let's see if you imagined it in the same way I did. Here's how I imagined it.

Ah yes, and I have four. Ah ha, and some of you like me, thought about this one left behind strategy because we know that chunk of red is a five and because four is one less than five, if I just leave one behind, I can slide across four. Yes, the one left behind strategy. Ok, and that helps me to see that four is made up of four and zero.

Alright. Let's imagine again, so over here on this time. Let's imagine four but in a different way. Ok, close your eyes. Imagine the rekenrek and you're sliding across four beads. How are you doing it? Oh. Ok, let's have a look if we imagined it in the same way this time. Here we go. Oh yes, I had four as double two and some of you thought about it like that way too. Yes, and I took away their highlighting 'cause I think we know what area to attend to now we're Ok.

Alright, let's slide those beads back. Ok, and why not? Let's keep playing with four. So close your eyes mathematicians and imagine four moving across but in a different way again. Oh, Ok, and some of you are moving some beads on the top and some of you are moving some beads down the bottom. I can see you're imagining it. Ok, let's have a look together. Did we imagine it in the same way? Ready? Ah ha, some of you did. We had four as three and one.

Ok, let's try again ready? Um, yeah, let's think about four in another new way. So imagine it in your mind's eye. You're moving beads across. Ah ha. Ok, let's have a look together. Is that what you imagined? Oh, some yes and some still no. That's Ok, mathematicians. It's still four. Yes, and in this case we have one and three. Yeah, and actually this is making me realise something which is, I think, pretty cool. Rekenreks can help us see important things about numbers. So for example, we can see the smaller numbers that hide inside the bigger numbers. Look.

Here's where we have, we were thinking about four and we can see four is four and zero. We can see four is three and one. We can see four is double two, 2 and 2. And we could also see four as one and three. And yeah, can you see that pattern too? Look? Look at the, look over at the left over here. There's four at the top and zero at the bottom. Then there's three at the top and one at the bottom.

And yes, two at the top and two at the bottom and one at the top and three at the bottom.

Yes, and if you just look at the top numbers. Look. four, three, two, one. Yes, like the pattern in the backward count.

Ok mathematicians, I think there's another way that we could make four still. So let's imagine it. Let's close our eyes and imagine what four could look like on a rekenrek that we haven't done yet. Are you imagining it? That idea of the pattern and the backward count on the top row might have helped you.

Ok, pick up your pencil mathematicians. And draw it on a piece of paper. What do you think another way of representing four on the rekenrek might have looked like? There's a new way we haven't done yet.

You're thinking hard I can see. That's right, 'cause we had one where we had four at the top and zero at the bottom. Then three at the top and one at the bottom. Two at the top and two at the bottom. One at the top and three at the bottom. Yes, and this time the pattern if we kept going with it, there would be zero at the top and four at the bottom. Yes it would look a little bit like this.

Uh-huh is that what you imagined and drew? Yes, and you're right. We could use that one left behind strategy again, 'cause we know there's a chunk of five and four is one less than five. So we can just leave one behind and slide the others over. And voila, we know that's four. I know it's a little bit like mathematical magic.

Ok, so mathematicians here's your challenge. I'd like you to think about this problem with me. So, mum gave us some baby carrots for a snack. There were eight in total. Some carrots were on my brothers plate and some carrots were on my plate. So here are the baby carrots or a picture of the baby carrots. And here's a picture of my brother's plate. And here's a picture of my plate.

And we can also use the rekenrek to help us imagine solving this problem too. Yeah, 'cause we know we have eight carrots and if I slide eight carrots across, I could say those eight carrots represent what's on my plate and the ones down the bottom represent what's on my brothers plate, which is zero.

And then I can draw a picture to record my thinking like a mathematician. So mathematicians, it's now over to you. How many carrots were on my plate? How many carrots were on my brother's plate? And show as many solutions as you can.

OK mathematicians over to you.

So, what's some of the mathematics here? So what the rekenreks can help us see is that bigger numbers are made up of smaller numbers, so we can see that inside a four there's four and zero. But we can also see sitting inside of four is 3 and one. Four is also two and two. Inside of four is one and three. And we also started to play around with this idea today that when each chunk of colour represents five. We can make four by thinking about one left behind. And that's a really interesting strategy for us to explore.

OK, mathematicians, have a lovely day. See you next time.