# Let's talk – how many ways?

Stage 2 – a thinking mathematically targeted teaching opportunity exploring halving and repeated halving strategies.

## Syllabus

Syllabus outcomes and content descriptors from Mathematics K–10 Syllabus (2022) © NSW Education Standards Authority (NESA) for and on behalf of the Crown in right of the State of New South Wales, 2024.

## Outcomes

- MAO-WM-01
- MA2-RN-01
- MA2-AR-01
- MA2-AR-02

## Collect resources

You will need:

- something to write with
- something to write on
- a calculator
- someone to talk to.

## Let's talk – how many ways

Watch the Let's talk – how many ways video (10:25).

[A title over a navy-blue background: Let’s talk – How many ways… Below the title is text in slightly smaller font: (Stage 2). Small font text in the lower left-hand corner reads: NSW Mathematics Strategy Professional Learning team (NSWMS PL team). In the lower left-hand corner is the white waratah of the NSW Government logo.]

#### Speaker

Hello mathematicians. Let's talk, how many ways.

[A title on a white background reads: You will need…

Bullet points below read:

- something to write with
- something to write on
- a calculator and someone to talk to.

Under the points is a quadrant of images: in the top left section, a cup of pencils and pen; in the top right, a boy holding a paper over his face; in the bottom left, a calculator; in the bottom right: a drawing of 2 people facing each other.]

#### Speaker

You will need something to write with, something to write on, a calculator and someone to talk to, to share your mathematical thinking with.

[Text over a blue background: Are you ready for your brain to get sweaty?]

#### Speaker

Are you ready for your brain to get sweaty? We have a question that we would like you to think about today and this question has more than one answer.

[A title on a white background reads: Number talk. Under the title in smaller red font is text: Record your thinking using numbers, pictures, symbols or words. Under the text, centred and in large font is text that reads: How many ways can you share 48 into 4 equal groups?]

#### Speaker

The question is, how many ways can you share 48 into four equal groups? Remember that mathematicians use numbers, pictures, symbols or words to record their thinking. So do this on what you have chosen to write on to answer this question. Pause this clip now to record as many ways as you can think of.

[Text over a blue background: A little while later…]

#### Speaker

How did you go?

[A title on a white background reads: Number talk. Under the title in smaller red font is text: Record your thinking using numbers, pictures, symbols or words. Under the text, centred and in large font is text that reads: How many ways can you share 48 into 4 equal groups?]

#### Speaker

How many ways did you share 48 into four equal groups. Were you able to think of more than one way to do this? Fantastic, well done.

[A title on a white background reads: How did you share 48? Under the title in smaller red font is text: This is how one student thought about it. Under the text, is a large image of a sheet of paper with 48 as heading. On the left side of the image is text: Ashley.]

#### Speaker

So we asked some students to tell us how they would share 48 equally into groups and this is what they said. We will first listen to Ashley explain her thinking.

#### Ashley

To share 48 into groups, I'm going to split it into its 10s and ones. 48 has four 10s…

[She points to the 4 of the 48.]

…and eight ones.

[She points to the 8 of the 48.]

#### Ashley

I can share 40 into four groups of 10.

[Under 48, she writes 10 four times.]

#### Ashley

I can share the eight ones…

[She points to the 8 of the 48.]

#### Ashley

…into these four groups equally so that there are two ones in each group.

[Under each 10s, she writes 2s.]

#### Ashley

And when I look at each group, they all have 12.

[She points to each pair of 10 and 2.]

#### Ashley

12 here, 12 here, 12 here, 12 here. So I can share 48 into four equal groups.

[She circles each pair of 10 and 2.]

#### Ashley

One, two, three and four. And each group has 12.

[A title on a white background reads: Ashley’s thinking. Under the text, centred and in larger font is a heading: 48]

#### Speaker

Let's have a closer look at Ashley's way of thinking.

[Under 48, a bracket appears: on the left side is text: 4 tens. Under the text is 4 rows of 10 red squares. On the right side is text: 8 ones. Under the text is 2 rows of 4 blue squares.]

#### Speaker

So Ashley thought about place value and used her knowledge of 10s and ones to first share the four 10s…

[Under the images, a row of 4 10 red squares appears.]

#### Speaker

…and then she shared the eight ones.

[Under each 10 red squares, 2 blue squares appear. A black box appears around each group of red and blue squares. A 12 appears in each box.

Under the boxes, text appears: 48 can be shared into 4 equal groups of 12.]

#### Speaker

That's one way of sharing 48 into four equal groups of 12. Is this how you shared 48 into groups? Maybe you used a strategy like our next student Steffanie.

[A title on a white background reads: How did you share 48? Under the title in smaller red font is text: This is how another student thought about it. Under the text is a large image of a blank sheet of paper. On the left side of the image is text: Steffanie.]

#### Steffanie

I used halving to help me work out how to share 48 into equal groups. I know that 48…

[Steffanie writes 48 on the top centre of the paper.]

#### Steffanie

…is an even number, so that means it can be divided into two groups. When you divide 48 by two, you get two groups of 24.

[Under the 48, she draws two lines pointing down. Under these lines, she writes: 24.]

#### Steffanie

So that tells us that when 48 is divided by two, it equals two even groups of 24.

[On the right side of the 24, she writes: 48 ÷ 2 = 24

#### Steffanie

I know that 24 is also an even number, so it can be divided by two.

[Under the 24 on the left, she draws two lines pointing down. Under these lines, she writes: 12.]

#### Steffanie

So when 24 is divided by two it makes two even groups of 12.

[Under the 24 on the right, she draws two lines pointing down. Under these lines, she writes: 12.]

#### Steffanie

And 24 again divided by two, we get another two groups of even 12.

[On the right side of the 12, she writes: 48 ÷ 4 = 12

#### Steffanie

So when 48 is divided into four, it is equal to 12 in each group. And if I go back and check, because that is what mathematicians do, I can see that my thinking was right. I started with 48 and halved it to make two groups of 24.

[With a marker, she circles the 24s.]

#### Steffanie

I halved that again to make four groups of 12.

[With a marker, she circles the 12s.]

#### Steffanie

And this is how I shared 48 into two equal groups and four equal groups.

[A title on a white background reads: Steffanie’s thinking. Under the text, centred is a text box. Inside the box is a heading in larger font: 48. Under 48, on the left side is 4 rows of 10 red squares. On the right side is 2 rows of 4 blue squares.]

#### Speaker

Let's take a closer look at Steffanie's thinking. Steffanie started with 48 and she halved 48 into two groups.

[A green line appears across the 4 rows of red squares (leaving 2 rows above it) and the 2 rows of blue squares (leaving 1 row above it).

Arrows pointing down from each end of the green line appear. At the end of these arrows are boxes containing 2 rows of 10 red squares next to 2 rows of 2 blue squares. Above each box is text: 24.]

#### Speaker

And each group had 24.

[A yellow text box appears in between the ‘24’ boxes. Inside is text: 48 can be shared into 2 equal groups of 24.]

#### Speaker

Steffanie discovered that 48 can be shared into two equal groups of 24.

[A green line appears across the 2 rows of red squares and blue squares on the left (leaving 1 row above it).]

#### Speaker

She then halved the 24s again.

[A green line appears across the 2 rows of red squares and blue squares on the right (leaving 1 row above it).]

#### Speaker

Each 24 was halved into two equal groups of 12.

[Under each box of 24, 2 arrows pointing down appears. Under each arrow is a box with text above it that reads: 12. Inside the box is 10 red squares with 2 blue squares .]

#### Speaker

Steffanie discovered that another way that 48 can be shared is into four equal groups of 12.

[A yellow text box appears below the ‘12’ boxes. Inside is text: 48 can be shared into 4 equal groups of 12.]

#### Speaker

What Steffanie has done here, is used a strategy called halving, which is like dividing by two and halving again, which is like dividing by four.

[Text over a blue background: I’m curious…]

#### Speaker

I'm curious. I wonder if the repeated halving strategy will help me divide other numbers by four.

[A title on a white background reads: Let’s try repeated halving. Under the text in large red font is text: What is 64 divided by 4?]

#### Speaker

What is 64 divided by four? Let's use the repeated halving strategy by starting with 64 to halve it and then halve it again to work out the answer to this question.

[A title on a white background reads: Let’s try repeated halving. Under the text, centred is a box with 6 rows of 10 red squares next to 2 rows of 2 blue squares. On the left side of the box is text: Start with 54. Under the box are 2 arrows pointing down. Each arrow ends with a box containing 3 rows of 10 red squares next to 2 blue squares. Above these boxes is text: 32. Next to the left ‘32’ box is a yellow text box with text: Halve it.]

#### Speaker

Let's try repeated halving together. If we start with 64 and we halve it, we get two groups of 32. If we now halve each group of 32…

[Under each ‘32’ box, 2 arrows pointing down appears. Each arrow ends with a box containing 1 row of 10 red squares above 2 rows of 3 blue squares. Next to the blue is text: 16. Next to the left ‘16’ box is a yellow text box with text: Halve it again.]

#### Speaker

…altogether we have four groups of 16. So by repeatedly halving 64…

[The ‘32’ boxes disappears. 4 arrows from the ‘64’ box appears. They point to each of the 4 ‘16’ boxes’ below.]

#### Speaker

…we can see that 64 divided by four is 16.

[Under the ’16’ boxes, an image of a calculator appears. Next to the image is text: Now try this… Next to the text are boxes containing 6, 4, ÷, 4 and =.]

#### Speaker

If you have a calculator, let's check this together. Enter into the calculator, 64 divided by four, equals? And the answer is…

[Next to the = box, a yellow box with 16 appears.]

#### Speaker

…16. So the repeated halving strategy helped us to solve 64 divided by four.

[A title on a white background reads: Let’s try repeated halving. Under the text, in large blue font is text: What is 88 divided by 4?]

#### Speaker

Let's now use the repeated halving strategy to answer the question, what is 88 divided by four?

[A title on a white background reads: Let’s try repeated halving. Under the text: Start with 88. On the right side of the text is a text box with 88 inside.]

#### Speaker

So if we start with 88 and halve it…

[Under the text Start with 88, a yellow text box with ‘Halve it’ appears. Under the ‘88’ text box, 2 text boxes of 44 appears.]

#### Speaker

…we get two groups of 44. If we halve again, by halving the two groups of 44…

[Under the yellow text box with ‘Halve it’, another yellow text box with ‘Halve it again’ appears. Under the 2 text boxes of 44, 4 yellow text boxes of 22 appears.]

#### Speaker

…we get four groups of 22.

[Under the text boxes, an image of a calculator appears. Next to the image is text: Now try this… Next to the text are boxes containing 8, 8, ÷, 4 and =.]

#### Speaker

Use your calculator to check. Enter into the calculator, 88 divided by four, and the answer is…

[Next to the = box, a yellow box with 16 appears.]

#### Speaker

…22. We were right. So the repeated halving strategy helped us to work out this too.

[A title on a white background reads: Over to you mathematicians… Under the title in smaller red font is text: Time to explore. Below in black font is text that reads: Use the repeated halving strategy to solve these questions. Start with a number, halve it and halve the answer you get again. Below are formulas: 68÷4, 96÷4, 120÷4, 84÷4 – alternating in blue and red fonts. Under the formula is an image with a speech bubble that reads: Think about it! Over a yellow background.]

#### Speaker

Over to you mathematicians. Use the repeated halving strategy to solve these questions. Start with a number, halve it and halve the answer you get again to divide each one by four. You might like to use a picture or model like I did in the examples to help you work this out. Then use a calculator to check your thinking. Once you have completed this task, share how you use the repeated halving strategy with someone that you know.

[A title on a white background reads: Further investigation. Under the title in smaller red font is text: Time to explore. Under the text, in large blue font is text that reads: What other numbers can you halve and halve again to work out if they can be shared equally into 4 groups?]

#### Speaker

As a further investigation, spend some time exploring what other numbers can you halve and halve again to work out if they can be shared equally into four groups.

[Text over a blue background: What's some of the mathematics?]

#### Speaker

What's some of the mathematics?

[A title on a white background reads: What's some of the mathematics? Bullet points below read:

- (In bold) Mathematicians use different ways to share numbers into groups.
- For example, the number 48 can be shared into four groups of 12 by sharing the 10s and then the ones.

Under the points are 4 boxes in a row containing 10 red squares, 2 blue squares and a 12.

Under the boxes is an image of Ashley’s working paper: a handwritten text with heading 48. Below are 4 sets of written 10 and 2, encircled.]

#### Speaker

Mathematicians use different ways to share numbers into groups. For example, the number 48 can be shared into four groups of 12 by sharing the 10s and then the ones.

[A title on a white background reads: What's some of the mathematics? Bullet points below read:

- Halving is a strategy that can be used to divide numbers by two.
- Repeated halving can be used to divide numbers by four and also by eight.

Next to the points are 2 images. The image above shows Steffanie’s working to divide 48 by 2 and 4. The image below is the speaker’s working of repeated halving of 88.]

#### Speaker

Halving is a strategy that can be used to divide numbers by two. Repeated halving can be used to divide numbers by four and also by eight.

[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

What other numbers can you halve and halve again to work out if they can be shared equally into 4 groups?