Counting with understanding up to 100

A thinking mathematically targeted teaching opportunity focussed on quantifying collectioning, renaming numbers and number word sequences

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

• MAO-WM-01
• MA1-RWN-01
• MA1-RWN-02
• MA1-CSQ-01

Collect resources

You will need:

• pencils or markers
• a collection of objects like dried pasta

Watch

Watch Counting with understanding up to 100 video (16:31).

Explore ways to quantify collections by creating, using or imagining structures.

Michelle

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

Do 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.

[Screen shows a cup with 17 pasta shells in it.]

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, get 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.

[Michelle tips the pasta shells out of the cup and counts them out moving them, one at a time, counting 1 through to 17.]

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.

[Michelle gathers the pasta shells and places them back into the 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.

[Michelle picks up the cup, indicating the amount of space still left in it.]

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?

[Michelle places a writing pad down and writes the heading: ‘Our estimates’.]

Oh, you think there might be 50. 50 pieces of pasta might fit into this cup? Okay, I can record 50, like this, as one of our ways of thinking.

[Michelle points to the cup and writes 50 under the heading.]

Oh, some of you think 100 pieces of pasta. So, I will record 100 like this.

[Michelle writes 100 on the writing pad.]

Huh, some of you think maybe, 30 pieces of pasta.

[Michelle writes 30 on the writing pad.]

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.

[Michelle writes 75 on the writing pad.]

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.

[Michelle writes 200 on the writing pad.]

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.

[Michelle moves the writing pad to the side and places down a large box of pasta shells.]

And so, I've got a big box of pasta and I'm gonna take a really big scoop.

[Michelle takes the cup and scoops out pasta shells until the cup is full.]

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?

[Michelle adds one more pasta shell to the cup.]

Okay, thank you. And now I'm gonna tip them out onto here. Oh, my goodness. Actually, seems like quite a lot of pasta.

[Michelle tips the pasta shells out and moves the cup to the side.]

So, would you change any of our estimates yet? Now that we have some more evidence of what it looks like over here?

[Michelle places the writing pad down next to the pasta shells.]

Ahh, I hear you too, I think now that I see this, 30 is probably too small.

[Michelle points to the pasta shells and the number 30, on the writing pad and takes a red marker and strikes through 30 and circles the other quantities.]

Yeah, cause 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.

[Michelle pushes the pasta shells around, indicating a larger number.]

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.

[Michelle takes the red marker and strikes through 200.]

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.

Alright, 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.

[Michelle counts out 5 shells and moves them to the side.]

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.

[Michelle moves the 5 shells into a dice pattern, moving the pasta shells around. She adds one more to form a 6 dice pattern, and then she removes 2 to make a 4 dice pattern.]

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 5 at the top and there's 5 at the bottom, and there's 10 boxes all together.

[Michelle introduces a ten-frame on a piece of paper. She points to the 5 boxes on top and the 5 boxes underneath and she circles the 10 boxes.]

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.

[Michelle places a pasta shell in each box of her ten-frame.]

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?

[Michelle introduces another ten-frame and places a shell into each box.]

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, one 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?

[Michelle introduces another ten-frame and places a shell into each of boxes.]

Okay, it's another 10. That's right, it's 30. Because I have, now, 3 ten-frames, and we call 3 tens 30.

Okay, well, what about if I fill this one in? How many ten-frames will I have filled with pasta?

[Michelle introduces another ten-frame and places a shell into each of boxes.]

Mm-hmmm. 4 and if I have 4 ten-frames, that's the same as saying, 4 ten’s and 4 tens is the same as saying 40.

So, actually, so far, I don't have to count them, I can just use structure to help me.

Okay, I have 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.

[Michelle introduces another ten-frame and places a shell into each of boxes.]

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, because look it has a line down the middle. Oh, and then I need these lines, and how many are there? Let's check, 1, 2, 3, 4.

Okay, I can use this one to help me too. One, 2, 3, 4 and so that should mean I have, 1, 2, 3, 4, 5 boxes at the top, and I'll have 5 down the bottom. So, I have another ten-frame.

[Michelle draws a ten-frame, by drawing a rectangle, a horizontal line in the middle and 4 vertical lines, evenly spaced across.]

I can fill this in I think, and now if I have 6 ten-frames, how many pieces are pasta will I know that I have? Yeah, 6 tens, which we rename as 60.

[Michelle now adds a shell to each of the boxes in the ten-frame she just drew.]

Do you think I have enough for another 10 as well? Okay, so should I draw another ten-frame? Okay, so the rectangle around the outside, you think?

Oh, you're right, I could also draw the 4 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.

[Michelle now draws another ten-frame. She points and counts out the 10 boxes. She now adds a pasta shell to each box.]

Let's check, good thinking, 1, 2, 3, 4, 5. Mm-hmm and if there's 5 on the top, that's right, there has to be 5 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 4 lines in the middle and then the line across, that divides those into half, and I would have 1, 2, 3, 4, 5 on the top, and 1, 2, 3, 4, 5 on the bottom.

[Michelle draws an imaginary ten-frame with her finger on the paper under the other ten-frames.]

So that would mean I have an 8 ten-frame and I'm just imagining this in my mind's eye.

[Michelle now places a shell in each of the imaginary boxes.]

Look.

Yeah, cause 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, 4, 6, 8, 10.

[Michelle circles 2 shells with her finger in the imaginary ten-frame, counting by twos to 10.]

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.

[Michelle points to each of the 8 ten-frames and counts them out, and then points to the 2 shells left.]

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.

[Michelle places the writing pad down and writes 82 in red marker to the right of the number’s list.]

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.

[Michelle removes the writing pad and places the empty pink cup down.]

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.

[Michelle pushes the pasta and paper ten-frames out of the screen, leaving the 2 ten-frames she drew.]

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.

[Michelle introduces a container of teddy bears and fills the cup. On the writing pad she places a pasta shell above the 82 shells of pasta. She then points and indicates that 82 pasta shells fit into the cup.]

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 bears? What do you think?

You think, there'll be less? Mm, 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 bears 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.

[Michelle places a pasta shell and a teddy bear alongside each other and compares the size of the 2 items, moving them around. She then picks up another pasta shell to indicate how they fit into each other.]

Oh, ok, you would like another one in there.

[Michelle places another teddy bear in the cup and pats down the top to ensure it’s full, but not overflowing.]

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?

[Michelle points to the cup of teddy bears and then points to 82 on the writing pad.]

Okay, should we count to find out? Or, actually, okay so I'm going to tip them out here.

[Michelle tips the teddy bears out.]

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.

[Michelle places a teddy bear into each box of the 2 ten-frames.]

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.

[Michelle places one of the paper ten-frames back, and fills it with teddy bears, leaving one teddy bear left.]

Oh look. So how many ten-frames do we have? That's right, 3. 1, 2 and 3, and then one 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.

[Michelle now writes 82 above the ten-frames and places a pasta shell above it, she then writes 31 and places a teddy bear above it and places the empty cup beside it.]

I wonder if you've got a cup like this, or a different colour, 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 ones. 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.

[Screen shows 8 ten-frames with pasta shells in each box.]

Yeah, and something else that we really realised today is that you can use structures that you have, like pre-made ten-frames, you can also draw your own structures, or you can imagine them.

[screen shows a ten-frame with 10 pasta shells in each box.]

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.

Instructions

• Find a cup or container and find some collections you can quantify.
• Can you find 2 different collections of objects that your cup or container holds the same amount of?