Counting with understanding

Welcome back, little mathematicians. I hope you're having a really lovely day.

You know, some of you might have been with me recently, when we had this cup and it had some pasta shells in it, and we tipped them out and we had counted by ones to work out that there were 17. But it made me start thinking about something.

But before I share with you what made me curious, let's just double check that we still have 17 pieces of pasta. Will you count with me? Okay, great ready? I didn't hear you. Let's try again. Ready? 1, good now you keep going, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 and 17. 17 pieces of pasta. This is what this looks like.

But what I wondered is, when it was inside my cup, it didn't look like 17 was a very big number or very big quantity of pasta to have in my cup. And it looks like there's a lot of space left and what it made me start wondering about, I got really curious, is about how many pieces of pasta might fit in my cup all together. And so what I thought we could do is, find out. This is what mathematicians often do. They get curious about something and they notice something, and then they decide to test it out.

So this, I know there's 17 pieces in there, I know here there's quite a lot of space left, so I can use this information to help me estimate. So, can you estimate how many pieces of pasta you think would fit into my cup in total?

Okay. I'm gonna record our thinking. So, our estimates. Okay and what are you thinking now? Ohhh, you think there might be fifty? 50 pieces of pasta might fit into this cup? Okay, I can record fifty, like this, as one of our ways of thinking. Ohhh, some of you think 100 pieces of pasta. So I will record 100 like this. Huh, some of you think maybe, 30 pieces of pasta. Okay, that's how I would record the numeral for 30. Ahh, someone else is thinking maybe about 75, and that's how I would record 75. And I think, let's see if there's one more estimate that you may have. 200 pieces of pasta? Okay, let's write the number 200. 200.

Okay, so it's good that we've got our estimates but now we need to check. So I'll move these out of the way for a moment. And so I've got a big box of pasta and I'm gonna take a really big scoop. Oh that's right. I'll make sure that it's full, I think another one could fit there. Full but not overflowing. Do you think that's full? Oh, okay one more piece. Is that full now?

Okay, thank you. And now I'm gonna tip them out onto here. Oh my goodness. Actually seems like quite a lot of pasta. So would you change any of our estimates yet? Now that we have some more evidence of what it looks like over here? Ahh, I hear you too, I think now that I see this 30 is probably too small. Yeah, cuz when we saw what 17 looked like, 30 looks way too big, way too, way too, this looks like way too many pieces of pasta to be 30. So we might revise our estimate and say we don't think it's 30 anymore. But do you think the other quantities are still okay? Ahh, some of you think 200 might be too big now.

Okay, well as mathematicians we can revise our estimate so we can say, we think maybe it's 50, 100 or 75, or somewhere in that range of numbers.

All right, well, let's count to find out. And actually, we could count 1, 2, 3, 4, 5 but that's quite a lot of pasta, and so what I might do as a mathematician is use a structure to help me work out how many I have. And what's a structure that you think that we could use? Ahh, some of you are thinking we could use dice patterns, so I could arrange it so it looks like say for example 5 on a dice, or if I had one more 6 on a dice, or I could make 4 on dice.

Mm-hmm. Is there another structure that you think I could use? Ahh, some of you are thinking we could use a ten frame. I think that's a good idea too because it's a lot of pasta. So, I actually have some ten frames here and this is really helpful because when I see a ten frame like this, I know there's five at the top and there's five at the bottom, and there's ten boxes all together. And so if I put one piece of pasta in each of the boxes on my ten frame, I don't have to count by ones and I already know that I have 10.

So maybe I could do that again, I have a few other ten frames here. So while I fill this out I'd like you to think about how many will I have counted if both of these are full? I hear your thinking. I think, you're thinking that it would be 2 tens, 2 ten frames, 2 tens, and we call that 20. So 1 ten, 2 tens, same as saying 20. Ok, well, I have another ten frame so I can fill it out, and while I fill it out, can you think about how many pieces of pasta I will have counted once I filled out this ten frame as well?

Okay, it's another 10. That's right, it's 30. Because I have, now, 3 ten frames, and we call three tens 30. Okay, well, what about if I fill this one in? How many ten frames will I have filled with pasta? Mmm-hmmm. 4 and if I have 4 ten frames, that's the same as saying 4 tens and 4 tens is the same as saying 40. Yeah so, actually so far I don't have to count them, I can just use structure to help me. Okay I've got another ten frame so if I have 5 ten frames and they are fill, full of pasta, what number would that be, that I've counted so far? Well, worked out how many. Yeah, 50, because 5 tens is the same as saying 50.

I still have some more pasta so I can keep going but I've got a problem. I don't have any more ten frames left. But I've really liked this idea of filling them in. I wonder, I know, maybe we could draw our own ten frame? Yeah, can you help me draw it? What do you think I should do first? Okay, I should draw the rectangle around the outside, and then? Okay, I could do the line down the middle, cause look it has a line down the middle. Ohh, and then I need these lines, and how many are there? Let's check, one, two, three, four. Okay, I can use this one to help me too. One, two, three, four and so that should mean I have, one, two, three, four, five boxes at the top, and I'll have five down the bottom. So I have another ten frame.

I can fill this in I think, and now if I have six ten frames, how manypieces are pasta will I know that I have? Yeah, 6 tens, which we rename as 60. Do you think I have enough for another ten as well? Okay, so should I draw another ten frame? Okay, so the rectangle around the outside you think? Ohh, you're right, I could also draw the four lines first, down the middle, and I'll use this one to help guide me, so I get my eye in. And then I could come back here and do the line down the middle that cuts each of these rectangles into smaller rectangles, that sort of look like squares.

Let's check, good thinking, one, two, three, four, five. Mm-hmm and if there's five on the top, that's right, there has to be five down the bottom, and that's a ten frame. So if I fill this in, how many ten frames full of pasta will I have now? Yeah. 7. Look, 1, 2, 3, 4, 5, 6, 7 ten frames.

I think I have enough space to draw, have another one, but what I'm wondering is do you think I need to draw it or could I use my mathematical imagination to imagine the ten frame there? Yeah, so imagine I'm drawing the rectangle, and the four lines in the middle and then the line across, that divides those into half, and I would have one, two, three, four, five on the top, and one, two, three, four, five on the bottom. So that would mean I have an 8 ten frame and I'm just imagining this in my mind's eye. Look.

Yeah, cuz I can use structures that I have, I can draw them, but I can also imagine that they are there, to help me work out how many. So let's just check this one because we're imagining a ten frame so we can count by twos. Two, four, six, eight, ten. Another ten frame. So now I have 1, 2, 3, 4, 5, 6, 7, 8 tens. Yes, and we call that 80, and 2 more. So that means that all together I had 82 pieces of pasta that could fit inside my cup.

So if I come back to what we were estimating, we thought, maybe 50, maybe 75 and maybe a hundred. And what we discovered, is that we actually have 82. And that these estimates are all actually pretty reasonable, aren't they? It's quite a lot of pasta that fits into a nice little pink cup and you know little mathematicians this is actually making me think of something else. What if I didn't have pasta, but what if I used something else?

What else would fit inside my cup? So if I get rid of these ones for a minute. I still have my 2 ten frames that I drew, but what if, I was thinking about how many teddy bears would fit in my cup? If I fill it, there's my cup full of teddy bears. And when we had pasta, we knew that we, we worked out that we could fit 82 pieces of pasta inside the same cup, oh you're right I could fit another one in there.

I wonder if we'll have more or less of the teddy's? What do you think? You think there'll be less? Mmm, I think so too because if I look at my piece of pasta and a teddy, I'll use the red one, so it's easier for you to see. They are about the same, well, the pasta is a little bit longer, so, but not much and the pasta is a bit skinnier than the teddy bear, and if I turn it this way the teddy bear's a bit fatter. But what happens with the teddy bears is they, sit like this with one another, where they don't nest inside. But with the pasta, look, they fit inside each other a little bit and so that might mean that more of them fit inside my cup.

Oh, ok, you would like another one in there. Now do you think that's full? Okay, how many teddy bears do you think would fit in my cup here? Do think less than 82? Okay, should we count to find out? Or, actually, okay so I'm going to tip them out here. How many do you think there are now? I can hear you, some of you are saying about 30, or maybe 50, or maybe 20.

Do you think we could use our ten frames to help us? Yeah, because if we have this one full, how many teddy bears would there be? Ten, and if this one is full? 20. Ok, so I can start by filling up our teddy bears on our ten frame for us. And that one's full, so I've counted already ten, or I've worked out that we have ten. And over here, we have another ten, so we know that one 10 and another 10 is called 20, and I could get another ten frame here. And we can try to fill this in and let's see what happens. And this is where structure is really helpful for me, because we can work out how many without actually having to count everything by ones.

Oh look. So how many 10 frames do we have? That's right, 3. 1, 2 and 3, and then 1 left over. So how many teddy bears fit into this cup? Mm-hmm, 31. So we had 82 pieces of pasta, and for teddy bears there were 31 teddy bears, that can fit inside this cup. I wonder if you've got a cup like this, or a different color, or one that you really love at home, that you could use to work out how many things it can hold. And can you find anything that's the same?

Over to you, little mathematicians. Okay mathematicians, what was some of the mathematics? So, using familiar structure helps us quantify a collection without having to count everything by one's. So in this case we used groups of 10 and we could use renaming. So for example, 8 tens and 2 ones, we could rename as 82. Yeah, and something else that we really realized today is that you can use structures that you have, like pre-made 10 frames, you can also draw your own structures, or you can imagine them. So it doesn't matter what equipment you have, you can still be, and think like, and imagine like a mathematician.

Okay, over to you mathematicians! Find a container or a cup at home and go find things that you can quantify! Over to you.