# How would you solve 25 x 8?

Stage 2 – A thinking mathematically targeted teaching opportunity exploring different strategies and ways of thinking to solve 25 x 8

## 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

## Outcomes

- MAO-WM-01
- MA2-MR-01

## Collect resources

You will need:

something to write on

something to write with

someone to talk with (if you can).

## Watch

Watch the How would you solve 25 eights (25 x 8) part 1 video (1:31).

[A title over a navy-blue background: How would you solve 25 eights (25 x 8)?. Below the title is text in smaller font: A number talk. Small font text in the lower left-hand corner reads: NSW Mathematics Strategy Professional Learning team (NSW MS PL team). In the lower right-hand corner is the red waratah of the NSW Government logo.

### Speaker

How would you solve 25 eights or 25*8? Let's enjoy a number talk.

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

Bullet points below read:

- Something to write on
- Something to write with
- If you can, someone to talk with too.]

### Speaker

You will need, something to write on, something to write with, and if you can, someone to talk to you.

[Text over a blue background: Let’s talk!]

### Speaker

Let's talk.

[There is a white sheet of paper over a large blue sheet of paper. In the top middle section of the white paper is a handwritten title: 25 x 8. Off the right-hand side of the white paper are markers.]

### Speaker

Hello there mathematicians. Welcome back. We're lucky enough to have one of our favorite mathematicians joining us today. Hi Sam.

### Sam

Hi

### Speaker

And we'd like you to also join us on exploring this problem, 25*8. Now as I was writing this, Sam said something really interesting. Sam, how do you like to think about 25*8?

### Sam

Well, I would it rather be 8*25.

### Speaker

So Sam that's really interesting. I might write that down here if that's OK.

[Below the title, the speaker writes: (8 twenty-fives).]

### Speaker

So you like to think about it as eight 25s. So over to you mathematicians. How would you solve 25*8, or as Sam likes to think about it, eight 25s. Think about that for a moment. Record your thinking and your ideas, and we'll come back and explore what Sam and I are thinking about when we want to solve 25*8. Over to you.

[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]

## Instructions

- Write down or draw how you would solve this problem.
See if you can think of more than one way!

## Watch

Watch the How would you solve 25 eights (25 x 8) part 2 video (5:57)

[There is a white sheet of paper over a large blue sheet of paper. Above the top middle section of the white paper is a label with handwritten text: 25 x 8 = 8 twenty-fives (8x25). Near the top left section of the white paper is a Lego figure of a red ninja.]

### Speaker

Welcome back, everybody. I hope you had a lot of fun exploring all the ways that you could solve 25 times eight or eight twenty-fives. Now, I asked ninja man here…

[The speaker points to the Lego.]

### Speaker

…what was one way that he solved eight twenty-fives? And he said, well, when I thought about eight twenty-fives, I thought about skip counting. So ninja man, let's try and get inside your brain and represent how you thought about eight twenty-fives. So ninja man, we're going to use a number line…

[The speaker draws a line from the Lego across the page.]

### Speaker

…to represent what your brain did.

[She draws arrows at the ends of the line.]

### Speaker

And we know that number lines extend this way…

[She points to the left.]

…and this way…

[She points to the left.]

### Speaker

…infinitely. And let's put a zero here.

[Below the left arrow, she writes: 0]

### Speaker

So ninja man told me that he first thought about a skip count…

[On the line, she draws a small semi-circle from 0. Above the semi-circle, she writes: +25.]

### Speaker

…or a jump of 25 and then another jump count of 25.

[Next to the semi-circle, she draws another small semi-circle. Above the semi-circle, she writes: +25.]

### Speaker

And his first jump was 25.

[Under the line, where the two semi-circles join, she writes: +25.]

### Speaker

And then he knew that when he doubled 25…

[Under the line, where the second semi-circle ends, she writes: 50.]

### Speaker

…that he would get to 50.

[Next to the last semi-circle, she draws another small semi-circle. Above the semi-circle, she writes: +25.]

### Speaker

And then his brain did another count of 25…

[Under the line, where the last semi-circle ends, she writes: 75.]

### Speaker

…which meant that he was on 75…

[Next to the last semi-circle, she draws another small semi-circle. Above the semi-circle, she writes: +25.]

### Speaker

…and then another count of 25, which meant that he was on 100.

[Under the line, where the last semi-circle ends, she writes: 100.]

### Speaker

Now, it was at this point that ninja man said that he used his knowledge of 100 being half of 200 and then could see that he wouldn't need to count 25 four more times, but that he could just do…

[On the line, from where the last semi-circle ends, she draws a curvy line to the right arrow.]

### Speaker

…one big jump of 100…

[Above the curvy line, she writes: +25.]

### Speaker

…which meant that he got to 200.

[Under the line, at the right arrow, she writes: 200.]

### Speaker

Wow, thanks ninja man for sharing that way of thinking. I wonder if there's another way?

[Below the number line, a new piece of white paper covers the previous paper. There is writing on it that reads: 8 x 25 = 4 x 50. On the left side of the writing is a Lego figure of a policeman.]

### Speaker

We have another friend policeman…

[She points to the policeman.]

### Speaker

…who would also like to share how he thought about 25 times eight or eight twenty-fives. Policeman told me that he liked to use what he knew about the relationship of doubling and halving. He knew that if he halved eight…

[She points to 8.]

### Speaker

…that that would equal four.

[She points to 4.]

### Speaker

And then he used his knowledge of double 25…

[She points to 25.]

### Speaker

…being 50…

[She points to 50.]

### Speaker

…to rethink about the problem as four fifties.

[She points to 4, then 50.]

### Speaker

And then he said that he even went one step further, where he halved four…

[She points to 4. Under the ‘=’, she writes: = 2.]

### Speaker

…which he knew was two, and then doubled 50…

[Under the ‘x 50’, she writes: = 100.]

### Speaker

…which he knew was 100, and then could rename that…

[Under the ‘=’, she writes: = 200.]

### Speaker

…as 200. Wow. What a really interesting strategy. Doubling and halving. Thank you policeman. I wonder if there's another way?

[Below the text, a new piece of white paper covers previous paper.]

### Speaker

So we have another friend who wanted to join the party, the eight twenty-fives party.

[She places a Lego figure of a worker on the new paper, below the policeman.]

### Speaker

Workerman, now workerman had another way of thinking about eight twenty-fives. He said that he liked to use what he knew about numbers hiding inside of other numbers and that he could use the area model of multiplication to help him use this knowledge.

So let's get inside how workerman thought about this problem…

[On the right side of workerman, she draws a rectangle.]

### Speaker

…using the area model of multiplication.

[On the left side of the rectangle, she writes: 8. Above the rectangle, she writes: 25.]

### Speaker

And then we can see how workerman thought about what he knew about 25. And that inside of 25 is…

[Under ‘25’, she writes: 20.]

### Speaker

…two 10s…

[Inside the rectangle, she draws a line down near the right edge, creating a small rectangle on the right and a large one on the left.]

### Speaker

…and also five ones.

[Above the small rectangle, she writes: 5].

### Speaker

And then this allowed him to use what he knew about eight 20s…

[She points to 8 and 20.]

### Speaker

…being 160.

[Inside the large rectangle, she writes: 160].

### Speaker

And then eight five, which he knew was…

[Inside the small rectangle, she writes: 40].

### Speaker

…40, and then could combine…

[She points to 160 and 40.]

### Speaker

…160 and 40 to be…

[On the right side of the rectangle, she writes: = 200.]

### Speaker

…200. Wow. Another great way of thinking about solving eight twenty-fives. We have three ways that we explored this morning. Over to you now mathematicians to think about how else you could solve eight twenty-fives. Perhaps you might like to ask somebody at home to see how they solved it. And then when you've done that, have a go at solving another problem. And perhaps try using one of these strategies either workerman, policeman, or ninja man's strategy to solve…

[She drops a square card with text: 15 x 8.]

### Speaker

…15 times eight or 15 eights. Over to you mathematicians.

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

### Speaker

So, what's some of the mathematics here?

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

Text below reads: The same problem can be solved using lots of different strategies (ways of thinking). For example in 25 x 8, we shared 3 different strategies:

Below the text is an image of the previous strategies written on paper. Below the image is text that reads: This reminds us that even though the answer might be the same, the ways of thinking can be different. That's one of the things we find makes maths really interesting!]

### Speaker

Well. This number talk reminds us that the same problem can be solved using lots of different strategies or ways of thinking.

For example, in twenty-five eights or 25 times eight, we shared three different strategies and you might also have come up with some more. And this reminds us that even though the answer might be the same, the ways of thinking can be different. And that's one of the things we find makes maths really interesting. Back to you to go solving.

[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/reflect

Can you find another way to solve 25 eights?

Can you use one of these strategies to solve 15 eights?