A thinking mathematically context for practise focussed on quantifying collections, estimation and communicating and representing thinking
Adapted from Ann Gervasoni, Monash University which was published on reSolve
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, 2021
You will need:
- counters, lima beans or pasta
pencils or markers
something to write on.
Watch the Handfuls video (3:33).
Let's play a game called handfuls.
Handfuls is really powerful in helping students and young children understand the structure of numbers and to think about working out the answer to the question how many without having to count.
[There is a plastic container filled with pasta shells.]
So, you just need to use something that you have at home. You could use pasta, blocks, anything that isn't too small but also not too big.
[Presenter shakes the container of pasta shells and also shows a clear plastic bag filled with coloured blocks.]
So, you, like the name of the game suggests, grab a handful.
[Presenter grabs a handful of pasta shells and moves the container of pasta shells out of sight.]
And now that I have a handful, I have to try to imagine in my mind about how many pasta shells do I think I have?
I think I might have about 18.
So now what I need to do is to determine how many I have, but to try not to use counting as my first strategy.
[Presenter puts the handful of pasta shells on the desk, keeping them in a group.]
So, what I could do is think about what do I already know and think about how I can work out, how many I have here by looking and by thinking.
[Presenter points to the group of pasta shells.]
So, what I know are things like when I see something that looks like this.
I straight away know that that's 5. I don't have to count that because I've seen it lots of time on a dice pattern.
[Presenter separates 5 pasta shells from the group. and puts them in a dice pattern. The presenter points to the 5 pasta shells.]
So, I could keep making fives and see how many of those I have. Even though they are a bit wonky, I can still trust that it's 5.
[Presenter continues to group pasta shells in dice patterns of 5. There are 3 groups of 5 and 3 pasta shells left over.]
And so there we go. I have 3 fives at the top and I also have 3, I might just put them down the bottom, left over and so what I can see here is that I've got 3 fives and 3 more, and that's one way of explaining what I can see, but I also know something more about that, and that is that when I have 2 fives that actually combines to make 10.
[The presenter circles the 3 groups of 5 pasta shells with finger, then circles the 3 left over pasta shells. The presenter then circles 2 groups of 5 pasta shells, indicating how 2 groups of 5 make 10.]
So, I have 10 over here and then I have 5 and 3 more. And actually, I could re-form that so that I know that I have 8.
[The presenter circles 2 groups of 5 pasta shells. The presenter pats the third group of 5 pasta shells and moves the 3 left over pasta shells into this group to make 8. The presenter arranges this group of 8 pasta shells in 4 rows of 2.]
And so now what I can see is one 10 and 8 more and I know that that's just called 18. But what I've used here is looking and thinking to help me solve it.
[The presenter points to the 10 pasta shells in 2 groups and then to the group of 8 pasta shells.]
Now if I wasn't quite as confident yet in knowing things about numbers, I might have gotten to this point right now that this is 10 and I'm still not sure what to do with these and that's where you might use counting, so I know this is 10 and I can trust that and I go 10, 11, 12, 13, 14, 15, 16, 17, 18 and I can use counting to help me check.
[The presenter points to the 10 pasta shells in 2 groups and then waves hand over the group of 8 pasta shells. The presenter again points to the 10 pasta shells in 2 groups to show 10. The presenter then counts each of the shells in the group of 8, adding these to 10 to make 18.]
Since I'm a mathematician, I would also think about recording how I saw 18 and so I had 18 and I would say 18 is the same as 5 and 5 and 8.
[The presenter writes on a piece of paper ’18 is the same as 5 and 5 and 8’.]
Or originally, I had 5 and 5 and 5 and 3. I also thought about how I could make my 2 fives into 10 and my 5 and my 3 into 8.
And so, I know that 10 and 8 which we just rename as 18.
[The presenter continues writing on paper ‘5 and 5 and 5 and 3’. The presenter then links ‘5 and 5’ to the number 10 and links ‘5 and 3’ to the number 8.]
Then what I might do is say to someone else, hey, what's another way that you could organise these 18 pasta shells so that you can see how many just by looking and thinking.
[The presenter removes the piece of paper and shuffles the pasta shells into one group.]
And that's handfuls.
Over to you mathematicians.
[End of transcript]
Take a handful of counters (or lima beans or pasta).
Hold the objects in your hand and imagine how many you have.
Record your estimate.
Describe what that collection might look like by visualising and imagining.
Organise your collection so that someone can determine how many items there are by looking and thinking.
How many do you have altogether?
How have you organised your collection?
Did you have more or less than your estimation?
Can you organise them differently?
How many ways can you arrange your collection so that you can see how many there are by looking and thinking?