# Handfuls (thinking multiplicatively)

Handfuls (thinking multiplicatively) is a thinking mathematically context for practise focussed on quantifying collections, multiplicative strategies and representing thinking.

## 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
• MAE-FG-02
• MAE-RWN-01
• MAE-RWN-02

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

## Collect resources

You will need a collection of items like bears, LEGO mini figs or an alternative like toy cars, trains, or other animal figurines.

## Watch

Watch Handfuls thinking multiplicatively video (11:04).

Organise a collection of items using structure.

### Transcript of Handfuls

(Duration: 11 minutes 5 seconds)

### Michelle

Hello there, mathematicians!

We hope you're having a really nice day today! We are black, black! Well, I am in a black shirt, but we're also back! Back in black.

[There is a plastic container with assorted LEGO mini figs.]

Back in black!

### Michelle

Like Batman!

Anyway, we could keep going but we, we mustn't!

Um we are back to talk about playing handfuls, which you might have seen us play before, using things like pasta, where we grab a handful of pasta or counters, and we have to try to visualize visualise about how many we have.

[Michelle picks up a handful of pasta shells with left hand and covers shells with right hand. The pasta shells are no longer visible because they are held within Michelle’s hands.]

### Barbara

Yep! I really like that game.

### Michelle

Looking and thinking and then I might estimate, I don't know, I reckon I might have about 15 pieces of pasta?

[Michelle holds the pasta shells in left hand, then puts the pasta shells in a pile on the table.]

Uh huh.

### Michelle

And then I could, I'm going to think of how I could arrange it, so that Barbara, you can work out how many there are by looking and thinking.

Sounds good!

### Michelle

So, I could make it into fives, because I think you would know fives on dice patterns.

I do.

### Michelle

And so, if I make these into the shapes on the same structure as a dice pattern or, and I said they were 15, but I'd now revise my estimate to be a bit more.

[Michelle arranges pasta shells into 3 groups of 5, arranging each group in a dice pattern of 5. There are 2 pasta shells left over.]

### Barbara

So, I think, um, I can see that there are 2 fives here. Okay. Like 2 hands, so I know that that's 10.

[Barbara outlines 2 groups of 5 pasta shells with right hand, showing 10. Barbara then displays her right hand to show 5.]

Yeah.

### Barbara

Ten and then another 5 is 15 and 2 more is 17.

[Barbara points to the third group of 5 pasta shells, then to the 2 pasta shells left over.]

### Michelle

Aha! So, you could use it by looking and thinking. If I put it like this, can you see how many there are by looking and thinking?

[Michelle gathers the pasta shells back together in one pile.]

No, not really.

### Michelle

What about if I like made fives, but push them like, really close together?

[Michelle begins to arrange the pasta shells in groups of 5. There is no space between the groups.]

### Barbara

Well, no, because then I can't really tell that it's 5.

### Michelle

Oh, so it's a bit like with the words, where you need space between them as you're writing?

### Barbara

Yeah, because otherwise, you know, with these I'd still have to count them to know that's 5.

[Barbara points to the pasta shells with her finger.]

### Michelle

Ok, I could have also made it, maybe like this, because I think you might know this, um, structure?

### Barbara

Oh yep, there's a ten frame!

### Michelle

Uh-huh and hold on, don't say it yet!

[Michelle arranges 10 pasta shells in a ten frame, and then the remaining shells in a dice pattern of 5 and a dice pattern of 2.]

### Barbara

Ah well, I like that actually!

### Michelle

Maybe I'll put it like this closer together so you can see the chunks.

[Michelle moves the shells in the dice pattern of 5 closer together and the shells in the dice pattern of 2 closer together.]

### Barbara

Yeah, that's better. So now I can see a ten frame and then I can see some dice patterns, so I can see a 5 and a 2.

[Barbara circles the ten frame and then each dice pattern with her finger.]

Uh huh.

### Barbara

So, 10, so, and 5, 5 and 2 is 7, and then another 10 so 17.

[Barbara points first to the ten frame. Then Barbara decides to count the dice patterns first. Barbara points to the dice pattern of 5, then the dice pattern of 2 and finally the ten frame.]

### Michelle

You know what else I could have done? Is made it like a 7 on a domino.

Oh yeah.

### Michelle

Like that.

[Michelle re-arranges the dice pattern of 5 and dice pattern of 2 into a 7 domino pattern.]

I like dominoes.

### Michelle

Because then it's 10 and 7.

[Michelle circles the ten frame and then the 7 domino pattern.]

### Barbara

And you can tell that it's 7, because it looks like a 6 but with just one more in it.

[Barbara points to 6 shells in the 7 domino pattern and then points to the one shell in the middle of the domino pattern.]

### Michelle

Yeah, that's true, so if you know number after then you know 7.

Okay so we're gonna play handfuls, but this time, thinking about not just how many ones things that we have, but how many legs!

[Michelle gathers the pasta shells and moves them away. Michelle introduces the plastic container of LEGO mini figs.]

### Barbara

Oh okay! So not arms though, just legs?

### Michelle

No, legs. So, we're gonna have to think multiplicatively now.

Okay.

### Michelle

So, if I take a handful of my mini figs, oh, it's a shame I didn't pick up that one because we could break out into karaoke!

Look! This guy's got a spider for a hat, and he looks like he's having a very bad day!

[Michelle picks up a handful of LEGO mini figs from the container, and holds them in left hand. Michelle picks out one more LEGO mini fig from the container and puts it back. Michelle picks out another LEGO mini fig and again puts it back in the container. Michelle moves the container away.]

He does!

### Michelle

Anyway, um, so now I've got to think about about how many mini figs I think I'm holding um, and they're bigger than the pasta shells, so if I had 17 pasta shells, I definitely don't think I have 17 mini figs so I think I might have about maybe 9 mini figs, and so that would.

[Michelle moves hand holding the LEGO mini figs trying to estimate how many.]

### Barbara

Your hand looks quite full, as well.

### Michelle

Yeah, and I think that would mean that I have, um, 18 legs because 9 twos is 18.

Double 9 is 18.

So, um, I'm going to challenge myself to try to arrange them in a way that you could work out how many there are by looking and thinking.

[Michelle puts the LEGO mini figs on the table.]

Yep.

### Michelle

So, I'm going to try a few different things because if it was 9, I might make it like, like, I see the symbol on TV.

Like a 3 by 3. But it's not 9.

[Michelle arranges the LEGO mini figs in 3 rows. There are 2 rows of 3 LEGO mini figs. The third row has 2 LEGO mini figs, with one empty space on the bottom right.]

### Barbara

But that's still useful.

### Michelle

Yeah, but what I think I might do is, I could do it like this and if you know dominoes, you might know what that is. Do you know what that is when you see it straight away?

[Michelle moves the LEGO mini figs into an 8 domino pattern.]

### Barbara

I do, but I also knew it before because it looks like 9 with one missing.

[Michelle moves the bottom right LEGO mini fig to the empty space in the middle of the 8 domino pattern. The LEGO mini figs are back in the arrangement of 3 rows. There are 2 rows of 3 LEGO mini figs. The third row has 2 LEGO mini figs, with one empty space on the bottom right. Barbara circles this arrangement with finger.]

### Michelle

Ah, so you can see that's 8.

[Michelle points to the arrangement of LEGO mini figs.]

### Barbara

Yeah, because it was one less than 9, so even though this doesn't, this is not a pattern, that I, that I see often.

[Michelle moves one pasta shell into the bottom right corner of the pattern. There is now a 3 × 3 pattern.]

Yeah.

### Barbara

I could see that, you know, that you were trying to make um, a 9.

Yes.

### Barbara

And then I just thought well one's gone.

### Michelle

And that one was missing, and that's actually why they make 8 looking like this.

[Michelle removes the pasta shell from the 3 × 3 pattern. Michelle moves the LEGO mini fig from the middle to the bottom right position. The pattern is now the 8 domino pattern.

### Barbara

And it's easy to see which one's missing.

[Michelle puts the pasta shell in the empty space in the middle of the 8 domino pattern and then removes it.]

### Michelle

The one is missing, there in the middle. So, so you can see that's 8 and then, 8 twos would mean that there's 16.

[Michelle moves the bottom right LEGO mini fig to the empty space in the middle of the 8 domino pattern. The LEGO mini figs are back in the arrangement of 3 rows. There are 2 rows of 3 LEGO mini figs. The third row has 2 LEGO mini figs, with one empty space on the bottom right. Michelle circles this arrangement with finger.]

Sixteen legs. I'm gonna try to make it another way. What about like this? What can you see now?

[Michelle moves LEGO mini figs into 2 groups. Each group is arranged in a dice pattern of 4. Michelle moves the pasta shell away.]

### Barbara

Well now I can see 2 fours.

Uh huh.

Um.

### Michelle

So how many legs?

### Barbara

Uh, well, 8 and 8. Uh huh.

[Barbara points to one group of 4 LEGO mini figs, then the next group of 4 LEGO mini figs.]

### Barbara

Which is also 16.

### Michelle

Try another way by looking and thinking.

[Michelle re-arranges the LEGO mini figs into a ten frame. There are 5 LEGO mini figs on the top row and 3 LEGO mini figs on the bottom row. The 2 spaces on the bottom right of the ten frame are empty.]

### Barbara

Oh, I like that one. So that's like a, like a ten frame and I love working with ten frames. Um and I can see that 2 are missing.

### Michelle

Oh yeah, like this. Oh, I like your 2 left behind strategy, or 2 missing strategy. So, it would be 10, if there were men here.

[Michelle adds 2 pasta shells to the bottom row of the ten frame. Michelle points to the 2 pasta shells to show that 2 LEGO mini figs are missing from the ten frame.]

### Barbara

But because they're onthere aren’t any there...

### Michelle

But there's no figs there, so it must be 8.

[Michelle moves the 2 pasta shells away.]

### Barbara

And that's an interesting way to think about it because if you had a full ten frame of figs you would have 20.

[Barbara circles the ten frame with her finger.]

Legs.

### Barbara

Twenty legs, right. So, then you can go well actually, I need to take 4 legs away.

[Barbara points to the 2 empty spaces on the bottom right of the ten frame.]

### Michelle

Oh nice, yeah, I like that and that helps you see like, you know, how we talk about numbers that combine to make 10 like 6 and 4?

Yeah.

### Michelle

It helps you see numbers that combine to make 20, like one, 10 and 6 which is 16 and 4 more is 20.

Yeah!

### Michelle

Okay, so that's another way to play handfuls, and we could play with mini figs where we say how many legs or how many hands for example or how many eyes?

[Michelle gathers the LEGO mini figs together in one pile. Michelle picks up one LEGO mini fig and points to the legs, hands, and eyes.]

### Barbara

Oh, I like this guy! He's dirty.

### Michelle

Yeah, he looks like he's escaped from something.

But she looks very happy about his escape, so maybe she helped him escape out of prison, I don't know. Anyway, and Batman's coming to chase them.

But we could also play with, um, bears and we could say, how many paws.

[Michelle moves the container of LEGO mini figs away and introduces a container of mini bears of assorted colours.]

### Barbara

Oh, okay.

But now we're talking about 4.

### Michelle

Yeah, so for each bear there's 4 paws, whereas here for each mini fig there's 2 legs or 2 arms or 2 eyes.

[Michelle picks up one blue mini bear with 4 paws. Michelle then picks up a LEGO mini fig and shows it has 2 legs, 2 arms and 2 eyes.]

So that's another way that we can play. Do you wanna have a go? Grab a handful.

### Barbara

Oh, okay.

[Barbara grabs a handful of the bears.]

### Michelle

Okay and about how many do you estimate?

### Barbara

I think, I've got about, oh, they're quite chubby these little bears. I think about 8.

[Barbara puts the mini bears on the table in a pile.]

### Michelle

Okay, so arrange them so that I can see how many by looking and thinking.

### Barbara

Okay.

Now I've put them down and I think that I've underestimated already.

### Barbara

I'd revise my thinking, yeah.

[Barbara touches her pile of mini bears.]

### Michelle

What would you think now that you see it like that?

### Barbara

Uh well I think probably 10.

Okay.

### Barbara

Maybe a little bit more. I think I have to stand them up. What do you think?

[Barbara arranges the mini bears into a ten frame, placing each mini bear upright. There are 5 mini bears on the top row and 5 mini bears on the bottom row. There is one mini bear left over which Barbara puts to the right of the ten frame.]

### Michelle

Oh yeah, I can see, um, the structure of a ten frame here and one more, so I would say there's 11, um, bears and if I have to work out the number of paws.

[Michelle circles the ten frame with her finger and points to the one mini bear left over.]

I know that each bear represents 4, so that's 4 tens and I know that that's called 40. And then 4 more is 44.

[Barbara waves hand over the ten frame, then points to the one mini bear left over.]

44.

### Michelle

So, I can see like the big rectangle around the outside, here, I can imagine it there. I can imagine the 4 internal lines and I can imagine the one line down the middle.

[Michelle traces the rectangle of the ten frame, then the 4 internal vertical lines, and finally the internal horizontal line.]

### Barbara

And I think that's what, what you said, space is really important because when I first did the ten frame and it was a bit bunched up, it was when they were lying down, it wasn't as easy.

### Michelle

Yeah. Can you arrange it in another way so I can see how many by looking and thinking?

### Barbara

Yeah, let's do it. I'm gonna put this one over here and this one over there. What do you think now?

[Barbara arranges the bears into 2 groups. Each group is arranged in a dice pattern of 5. There is one mini bear to the right of the dice patterns.]

### Michelle

Oh okay, I can see, um, 5 on some dice.

So, one 5 and 2 fives and one more, so, so 5 fours, I actually know that's 20 because I just, I just, it's something I know so that would be 20 and another 20, and, um, and I know that if I rename, I can go 2 tens and 2 tens, which is 4 tens, we call that 40, and then 4 more would be 44.

[Michelle traces around the first dice pattern of 5 mini bears. Michelle repeats tracing around this dice pattern of 5, and then traces around the second dice pattern of 5 mini bears. Michelle points to the one mini bear to the right. Michelle then points to the first dice pattern of 5 mini bears, each with 4 paws. Michelle then points to the second dice pattern of 5 mini bears, each with 4 paws. Michelle points to the first dice pattern when saying ‘2 tens’ and the second dice pattern when repeating ‘2 tens’. Finally, Michelle points to the one mini bear to show 4 paws.]

### Barbara

44. You want one more way?

### Michelle

Yeah, do it one more way.

### Barbara

Okay. Oh, I don't know if this is going to be useful or not. I'm not sure if it will help with thinking.

[Barbara arranges the mini bears into 2 groups. The first group contains 6 mini bears, 3 on the top row and 3 on the bottom row. The second group contains 5 mini bears, 3 on the top row and 2 on the bottom row. The bottom right corner in this bottom row is empty.]

### Michelle

Oh okay, so I can see a chunk of 6 because it's like a dice pattern, turned sideways and I can see a chunk of 5 because it's almost a 6.

[Michelle points to the first group of 6 mini bears. Michelle then moves a pasta shell into the empty space in the second group of mini bears and then removes it.]

### Barbara

Yeah, so I thought...

### Michelle

With one missing.

### Barbara

I could have put it this way, so you'd know it was 5 straight away, but then I thought maybe, because it looks the same, you could see.

[Barbara moves the mini bear from the top row of the second group into the middle to form a dice pattern of 5. Barbara then puts this mini bear back in the top right-hand corner. Barbara circles the empty space on the bottom row. Michelle places one pasta shell in this empty space.]

### Michelle

That's right, that's how I thought about it. So, actually this is a near double now.

[Michelle removes the pasta shell from the second group.]

Yeah, it is.

### Michelle

Because it's double 6, double 6 minus one or double 5 plus one. And so, I could think about it like that.

[Michelle moves the pasta shell into the blank space in the second group to show 2 groups of 6. Michelle removes the pasta shell to show minus one from the second group. Michelle then removes one mini bear from the first group to show 2 groups of 5. Michelle puts the mini bear back to show plus one in the first group.

Yeah.

### Michelle

So, because double 5 is 10 and.

[Michelle again removes one mini bear from the first group to show 2 groups of 5 mini bears.]

Then the one.

### Michelle

And then one more group of legs so, double 5, double 5 is 10, double 5 fours is 10 fours because each of those is worth 4 paws, and that's 40 and then one more set of 4 paws is 44.

[Michelle briefly puts the mini bear in the first group and removes it again. Michelle points to the 10 mini bears currently in the 2 groups and calculates 40 paws. Michelle returns the mini bear to the first group to add another 4 paws.]

So, 44 paws.

### Michelle

Three different ways of thinking about that, I really like it.

All right let's look at how we could record some of those ideas so with the bears we just realised that for each bear there are 4 paws.

[There is a red mini bear with 4 paws.]

So, 44 is 10 fours and one 4. We also saw that 44 is 5 fours and 5 fours and one 4. And we also saw that 44 is double 5 fours and one 4 more.

[There are 3 images, each showing 11 mini bears. The first image shows 10 mini bears in a ten frame and one mini bear to the right.

The second image shows 2 groups. Each group has 5 mini bears, shown in a dice pattern of 5. There is one mini bear to the right.

The third group shows 2 groups. The first group is in a 6 domino pattern. The second group is in a 5 domino pattern.]

All right over to you, mathematicians to play handfuls, thinking about groups of fours like our teddy bears or groups of 2 like our mini figs.

[Michelle points to 2 mini bears and points towards the LEGO mini figs.]

You choose! Over to you!

So, what's some of the mathematics here?