Sharing collections

A thinking mathematically targeted teaching opportunity focussed on sharing a collection exploring the structure of arrays

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-FG-01

Collect resources

You will need:

• paper
• pencil or marker
• 24 objects like pegs or blocks
• someone to talk to that you can share your mathematical thinking with.

Watch

Watch the Sharing collections part 1 video (4:03).

Investigate sharing collections of fruit using arrays.

Transcript of Sharing collections part 1

[White text on a navy-blue background reads ‘Sharing collections (Stage 1)’. Small white text at the bottom reads ‘NSW Mathematics Strategy Professional Learning team (NSWMS PL team)’. In the bottom right corner, a white NSW Government ‘waratah’ logo.]

Female speaker

Hello mathematicians. Today we are going to be sharing collections.

[Black text on a white background reads ‘You will need…’ Black text bullet points below (read by speaker). On the right are 5 still colour images – a child holds a blank sheet of paper in front of their face; a cup of pens and pencils; some colourful clothes pegs; some building blocks; and a cartoon drawing of 2 stick figure children facing each other. In the bottom right, the NSW Government red ‘waratah’ logo.]

Female speaker

You will need something to write on, something to write with, 24 objects such as pegs or blocks, and someone to talk to, to share your mathematical thinking with.

[White text on a blue background reads ‘Let’s look and think!’]

Female speaker

Let's look and think.

[Black text on a white background reads ‘What is different?’ Below on the left, an array of 12 coloured images of kiwi fruit arranged in 4 rows of 3 columns. On the right, an array of strawberries arranged in 3 rows of 4 columns. Further explained by speaker.]

Female speaker

Look at these two collections, one is a collection or a group of kiwifruit, and the other a collection of strawberries. First of all, what do you notice about these 2 collections of fruit that is different? Yes, you've probably noticed that they are a different colour and that each collection is made up of a different type of fruit. Great. Think about what you know about mathematics. Can you see something else that is different? Write it down or draw it on your piece of paper.

If you've recorded one thing that is different, can you think of another? Look closely. Now, can you think of another difference? Write that down too.

[White text on a blue background reads ‘What is different?’ Pink text below reads ‘Students sharing their thinking’. Below, the 2 colour images of kiwi fruit and strawberries. Coloured speech bubbles appear when mentioned by speaker.]

Female speaker

Let's talk about what is different. Well, we thought about what we know about fractions and noticed that one difference is that each kiwifruit is cut in half, but the strawberries are whole. Did you notice that? We also noticed a difference in how the fruit was organised. Can you see it? Yes, the kiwifruit collection is organised into 4 rows of 3 or 4 3s. How are the strawberries organised? Yes. They are organised in 3 rows of 4 or 3 4s. We also noticed that the strawberries have one more in each row, but the kiwi fruits have one more in each column. Can you see that too?

[White text on a blue background reads ‘What is the same?’ Below, the 2 colour images of kiwi fruit and strawberries.]

Female speaker

Now, I have another question to ask you about these collections. What is the same? Write or draw what you notice.

[White text on a blue background reads ‘What is the same?’ Pink text below reads ‘Students sharing their thinking’. Below, the 2 colour images of kiwi fruit and strawberries. Coloured speech bubbles appear when mentioned by speaker.]

Female speaker

We noticed that both collections have 12 in them. Did you work that out too? Great. That's one thing that is the same. We also noticed that if you turn the strawberry collection on its side and you can kind of see it, if you look at the strawberries and turn your head sideways, it would become 4 rows of 3, just like the kiwifruit collection. We also noticed another thing that is the same is that both collections are organised in an array. Both collections are in rows that go across and columns that go up and down.

[The NSW Government waratah logo turns briefly in the middle of various circles coloured blue, red, white and black. A copyright symbol and small blue text below it reads ‘State of New South Wales (Department of Education), 2021.’]

[End of transcript]

Watch

Watch the Sharing collections part 2 video (5:43).

Investigate ways to share 12 and 24 objects.

Transcript of Sharing collections part 2

[A small whiteboard sits on a table. To the left of a whiteboard is a pile of 12 wooden clothes pegs. The Speaker gathers together the pegs and shows them to the viewer.]

Speaker

Hello, mathematicians. Let's explore the concept of arrays together. I'm going to use these 12 pegs to help us.

[The Speaker picks up three pegs and arranges them into a row above the pile. They draw a circle in the air above the group with their finger. They arrange another row of three pegs below, and circle the two groups of three. They arrange a third row of three pegs, and circle the three groups of three. They arrange a fourth row of three pegs, and circle the four groups of three.]

Speaker

So, we saw in the kiwifruit collection that it was organised into four rows of three, or four threes. That's one three, two threes, three threes and four threes. I'm going to record and draw what this looks like to show my mathematical thinking.

[The Speaker uses a black marker to draw four rows of three ovals on the whiteboard. Below the drawing, the Speaker writes “4 threes”.]

Speaker

One row of three, or one three. Two threes, three threes, and four threes. I'm going to use numbers and words to record that as well. Four threes.

[The Speaker gathers the pegs into a pile at the bottom of the table. They arrange the pegs into three rows of four, circling each group of four in the air with their finger.]

Speaker

We also saw that the strawberry collection was organised into three fours. One four, two fours, three fours.

[The Speaker draws three rows of four oval shapes on the whiteboard. Below the drawing, they write “3 fours”.]

Speaker

Let's record that as well. One four.

[The Speaker gathers the pegs into a pile at the bottom of the table. They arrange the pegs into six rows of two pegs, and circle the air to indicate each group of two.]

Speaker

I wonder if there is another way that I can share 12. Hmm. What if I had twos? Let's try that together. One two, two twos, three twos, four twos, five twos, six twos. Well, that works as well and I don't have any left over. I've shared them evenly.

[The Speaker draws six rows of two oval shapes on the whiteboard. Below the drawing, they write “6 twos”.]

Speaker

OK, let's record that. One two.

[The Speaker arranges the pegs into two rows of five. Two pegs are left over at the bottom of the table. They hold the remaining two pegs up, then place them below the rows of five. They indicate the empty space in the row with a finger.]

Speaker

Is there another way? Hmm. Did somebody say fives? Well, let's try it. One five, two fives. Uh-oh, I only have two left. If I put them here, my row won't be complete. So, that tells me that I can't share 12 by fives.

[The Speaker gathers the pegs into a pile. They arrange the pegs into two rows of six. They indicate each group of six, and that there are no remaining pegs left over.]

Speaker

Let's try another number. Let's try six. One six, two sixes. And I have none left, I have shared them. So, I have one six, two sixes. Let's record that thinking as well.

[The Speaker draws two rows of six oval shapes on the whiteboard. Below the drawing, they write “2 sixes”. They point to each drawing on the board.]

Speaker

Well, look at that. We were able to share our collection of 12 pegs into four different arrays.

[Blue text over a white background: How many different ways can you share 24 objects equally?]

Speaker

It's over to you, mathematicians. How many different ways can you share 24 objects equally? Remember, share your objects, draw to record your thinking and write using numbers and words what your array is showing you. Have fun.

[White text over a blue background: What’s (some of) the mathematics?]

Speaker

What's some of the mathematics?

[Title on a white background: What’s (some of) the mathematics?

• Some collections of objects can be shared equally in different ways

For example, 12 can be shared into: 4 threes, 3 fours, 2 sixes, 6 twos.

Below the text are photographs of the wooden pegs in their corresponding arrays.]

Speaker

Some collections of objects can be shared equally in different ways. For example, 12 can be shared into four threes, three fours, two sixes, and six twos.

[Title: What’s (some of) the mathematics?

• Some collections of objects can’t be shared equally in some ways

For example, 12 can’t be shared equally into fives because if we made 2 fives, we would have 2 left over, but we don’t have enough to make 3 fives.

To the right of the text is a photograph that shows two rows of five pegs, with two left over pegs below.]

Speaker

Some collections of objects can't be shared equally in some ways. For example, 12 can't be shared equally into fives, because if we made two fives, we would have two left over. But we don't have enough to make three fives.

[Title: What’s (some of) the mathematics?

• Mathematicians use objects to help them solve mathematical problems and record their thinking using pictures, numbers and words.

Below the text is a photograph of the whiteboard, on top of which are three rows of four pegs. Next to the pegs is a drawing of three rows of four ovals. Writing reads: 12 can be shared into 3 fours.]

Speaker

Mathematicians use objects to help them solve mathematical problems and record their thinking using pictures, numbers and words.

[Over a grey background, the red waratah of the NSW Government logo appears amongst red, white and blue circles. Text: Copyright State of New South Wales (Department of Education), 2021.]

[End of transcript]

Discuss

• How many different ways can you share 24 objects equally?
• How do you know if you have found all of the ways to share 24 objects equally?
• Is there a way that you can keep track to check that you have found all of the ways?