Two handfuls (organising collections)
Stage 2 – A thinking mathematically context for practise focused on quantifying collections, estimating and communicating ideas.
Adapted from Ann Gervasoni, Monash University – reSolve - Counting handfuls
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:
your mathematics workbook
pasta, counters, marbles or dried beans.
Watch Two handfuls part 1 video (2:56).
OK, let's play handfuls, so I'm going to take any items that we have at home that we could use. In this case I'm using pasta.
I'm going to grab 2 handfuls...and move this out of the way. I'm going to think about and estimate, about how many do I think I have in my hands. So, I can feel that this hand actually has a few more than this one. So, I think in this hand I might have about 14 and in this hand, I think I might have about 20. So, I think I have somewhere between maybe about 35 to 40 pieces of pasta.
[The presenter grabs 2 handfuls of pasta shells from a container and moves the container out of the way. She moves hands up and down to gauge the weight of the shells to help her estimation. The presenter places the pasta from her hands down onto the table.]
She takes 5 pieces of pasta from her pile. The pieces of pasta are then arranged in a dice pattern. The presenter circles the pattern to signify that they are comprised of 5 pieces.
She begins to arrange the rest of her shells, using 5 shells in a group to create more of the dice pattern. The presenter creates 7 groups of 5 shells and places the 4 remaining pieces of pasta in a small group towards the right-hand corner.]
So now what I'm going to think about is how I can determine how many I have without having to count. So, we know that counting is one way that we can work out an answer the question ‘How many do you have?’ But I can also use other things that I know to help me work out ‘how many?’
So, for example, I know things like when I see this arrangement of items or objects or dots, I always know that it's 5 because it looks like 5 on a dice. So, I could see how many 5’s I have as it's a familiar structure. So, let's have a go at that.
So, now I have... 1 five, 2 fives, 3 fives, 4 fives, [The video speeds up to show the presenter arranging the shells.] ...8 fives.... and not quite 9 fives, so I'll rearrange those in a different way so that I can see that they're different. So, what I can see here in my collection is that I have 1 five, 2 fives, 3 fives, 4 fives, 5 fives, 6 fives, 7 fives...and 4 more.
[The presenter circles each group of 5 as she counts them, before circling the smaller group of 4.]
Now, I happen to know that 7 fives is 35. And if I have 4 more, I know 35 and 4 more is 39. But, if I wasn't sure of that, I could say, well, I know 7 fives is 35 and I could count on the rest... 36, 37, 38, 39.
[The presenter uses one-to-one correspondence to count the 4 remaining pieces of pasta.]
I could also think about it in a different way because what I know about when I have 5 of something and 5 of something and I join that together I have 10.
So, in this case, I have 1 ten, 2 tens, 3 tens... and 5...6, 7, 8, 9 more. And 3 tens and 9 more is renamed as 39.
[The presenter circles the 2 groups of 5 shells vertically as she counts her groups of 10. She does this 3 times before circling the remaining group of 5 shells and the 4 shells underneath.]
I wonder if you could arrange your counters in a different way to think about how you could work out how many you have by looking and thinking.
[End of transcript]
Take 36 objects.
Form them into a rectangular structure so that you have equal rows and columns. We call this an array. Draw and describe your array using words and symbols.
Re-organise your pasta pieces so it forms a different rectangle. Draw and describe your array using words and symbols.
Keep re-organising your pasta pieces until there are no more arrays you can make. Draw and record all of your arrays.
Pick your favourite array and describe it using both multiplication and division. Record your thinking in your notebook.
Post the pictures of your arrays and share your thinking on our digital platform.
Watch Two handfuls part 2 video (1:14).
[On the table there are 7 groups of 5 pasta pieces arranged in a dice pattern. On the bottom right-hand side there is a group of 4 pasta shells.]
I can reorganise my pasta pieces too. So, I said before that I can re imagine 2 fives as 1 ten. So, I might re-organise these into a different structure.
[The presenter rearranges the pasta pieces into an array. She creates 2 vertical columns, with each column containing 5 pieces of pasta. She arranges the pieces of pasta so that there are 3 groups of the 2 vertical columns. Next to these, there is a group of 5 pieces of pasta arranged in a dice pattern and a group of 4 pieces of pasta clustered together.]
And what I know about 10 frames is that they have a rectangle around the outside. A line that divides the 2 rectangles in the rectangle into a skinnier rectangle, and for internal lines that will make 5 in each. And in this case, my 10 frame is oriented so it's standing on a skinny edge, so will have 5 boxes here, 1, 2, 3, 4, 5 and 5 on this side, 1, 2, 3, 4, 5.
[The presenter draws as she explains the features of the 10 frame. She creates a large rectangle that contains the 2 vertical rows of pasta. The presenter then draws a vertical line to halve the larger rectangle, with each half of the rectangle containing 5 pieces of pasta. The presenter then draws 4horizontal lines to split the rectangles into 10 sections, with each sections containing one piece of pasta.]
And I know that it is 10. I could draw around each of these, but I can also just imagine that structure is sitting there too. So now I have 1 ten, 2 tens or...1 ten, 2 tens, 3 tens, and over here I have eight more or nine more.
[The presenter uses her pencil to hover over each group of 10 as she counts, before rearranging the 9 remaining pieces of pasta.]
And I know that we call 3 tens and 9 more, 39.
[End of transcript]
Share your work with your class on your digital platform. You may like to:
- Post your pictures or drawings.
Comment on the arrangements of others.
Watch Two handfuls part 3 video (1:14).
[On the screen there is a pile of 36 pieces of pasta placed in the top-left corner.]
OK so I have my 36 pieces of pasta and what I'm interested in this time is thinking about those same ideas of how I can work out how many I have. But in this case, I know so we're working backwards.
Now with my 36 pieces, I can think about how I can arrange them into a particular form of structure that we call an array. So, it'll form a rectangle where it will have equal rows and equal columns.
So, I could use what I know about 36 already to form an array. Or I could use some trial and error. I know that 36 is a square number and so to make 36 I can make 6 sixes and so I might start by making that rectangular array, which also happens to be a square.
[The presenter rearranges the pasta pieces to form an array. She moves each pasta piece to create 6 rows and 6 columns. They each contain 6 pasta pieces.]
OK, and so now I can see I have my collection of 36 and I can see that it's formed into 6 rows with 6 in each row.
[The presenter motions to the collection of 36 pasta pieces. She then gestures to each vertical and horizontal row.]
So, there's one row, the next row, the third row, the fourth row, the fifth row and the sixth row. And in fact, each of my columns also has 6. So, I can record 36 over here as 36 is 6 sixes.
[To the right of the array, the presenter records her observation. Presenter writes: 36 is 6 sixes. She motions to her array of pasta pieces.]
And that describes my array. But what I wonder is, is there another way to organise my structure?
So, I could try arranging them by fives now because I know I have 5 in each row and if I make another row down here... I can't quite make it when there's one left over so that way won't work for me.
[The presenter moves the last column of pasta pieces and arranges them at the bottom of the array to create a new row of 5. She then gestures to each row, indicating that there are 5 pasta pieces within each one. The presenter shows one leftover piece of pasta and indicates that in this instance, an array can’t be made.]
But I could maybe try with threes, because 6 and 3. 3 is half of 6.
[Presenter gestures to show her thinking, symbolising that 3 is half of 6. She rearranges the array of pasta pieces into 12 rows and 3 columns.]
OK, and now I've gone a bit wonky, but I can still see that I have 3 in each row, so that means I can also say that 36 is 12 threes.
[The presenter records her new observation underneath her previous one. Presenter writes: 36 is 12 threes.]
I wonder how many other ways you can find to organise 36.
[End of transcript]