# How would you solve 25 x 18?

Stage 3 – A thinking mathematically targeted teaching opportunity exploring different strategies and ways of thinking to solve 25 x 18

## 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
• MA3-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 eighteens (25 x 18) part 1 video (1:28).

Think of a strategy to solve 25x18.

### Transcript of How would you solve 25 eighteens (25 x 18) part 1

[A title over a navy-blue background: How Would You Solve 25 Eighteens (25x18). Below the title is text in large 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 eighteens? A number talk.

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

• something to write on
• something to write with
• if you can, someone to talk with.]

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

[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 18.]

Hello there mathematicians. Welcome back. We have a problem to get your brains sweaty, 25 x 18, or as my friend…

[Near the top left section of the white paper, the speaker drops a Lego figure of a red ninja.]

…Ninja Man likes to think about it, 18 twenty-fives.

[Next to the title, she writes: =.]

So thank you for that Ninja Man, we might write that in here…

[Next to the ‘=’, she writes: 18 twenty-fives.]

…18 twenty-fives. And Ninja Man is showing us his knowledge of the commutative property of multiplication.

[Next to ‘twenty-fives’, she writes: 18 x 25).]

So over to you mathematicians to think about how you would solve 25 x 18 or as Ninja Man likes to think about it, 18 twenty-fives. Come up with one strategy or way of solving. Then come up with another. And then see if you can come up with a third. Record your ideas and then come back and we'll explore some of the ways that we can solve 25 x 18. Over to you.

[Text over a blue background: 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 eighteens (25 x 18) part 2 video (6:29).

Investigate solving 25x18 using area models.

### Transcript of How would you solve 25 eighteens (25 x 18) part 2

[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 18 = 18 twenty-fives (18x25).

### Speaker

Welcome back, mathematicians. I hope you had lots of fun exploring all of the ways that you could solve 25 eighteens or 18 twenty-fives. I'm sure you got your brain nice and sweaty. Now, I have a friend with me who also got his brain very sweaty thinking about 18 twenty-fives.

[In the top left section of the white paper, the speaker drops a Lego figure of a red ninja.]

Ninjaman is very keen to share how he solved 18 twenty-fives. Now, Ninjaman said that he liked to use the area model to think about 18 twenty-fives, and we can represent…

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

…Ninjaman's way of thinking about the problem using the area model like this.

[On the left side of the rectangle, she writes: 25. Above the rectangle, she writes: 18.]

And Ninjaman said that he could then use what he knew about 18 being able to be partitioned into ten…

[Under ‘18’, she writes: 10. Next to 10, she writes: 8]

…and eight.

[Inside the rectangle, she draws a line down near 8, creating a small rectangle on the right and a large one on the left.]

And that helped him because then he could solve 25 tens using his knowledge of renaming with place value, in that 25 tens can be renamed as…

[Inside the large rectangle, she writes: 250].

…250. And then Ninjaman thought about, "OK, 25 eights." And it was at this point that he realised, oh, he doesn't know 25 eights, but what he does know is that 4 twenty-fives is equivalent in value to 100. So he decided that he's going to use that knowledge…

[Inside the small rectangle, she draws a line down the middle, creating 2 small rectangles.]

…to further partition eight into 2 fours.

So let's write a little four here.

[On both sides of ‘8’, she writes: 4.]

Hopefully you can see that. I hope I'm getting this right, Ninjaman. OK, into two fours. So then he could think about 100…

[Inside the small left rectangle, she writes: 100].

…as being equivalent to 4 twenty-fives, and then…

[Inside the small right rectangle, she writes: 100].

…again here. And so, then all he needed to do was now collect all of those. 250, 100 and 100, which he knew when combined…

[On the right side of the rectangle, she writes: = 450.]

…totalled 450. What a really nice way of thinking about 18 twenty-fives. Thank you, Ninjaman. Hmm, I wonder if there is another way we could solve 18 twenty-fives.

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

Now, we have another friend who'd like to share how he solved 25 times 18 or 18 twenty-fives.

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

Welcome, Policeman. Now, Policemen told me that he got a bit stumped when he saw 25 times 18. So what he did because he likes working with landmark numbers or benchmark numbers, which are typically numbers that end in five or zero, that he wanted to use that way of thinking about the problem, and instead of 18 twenty-fives, reimagined the problem as 25 twenties.

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

So, his thinking might look a little something like this.

[On the left side of the rectangle, she writes: 25. Above the rectangle, she writes: 20.]

And what we can do is we can put a dotted line…

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

…down here to see where the original problem of 25 eighteens is hiding inside of 25 twenties. And then Policeman thought, well, I know something about 20, and that inside of 20 is 2 tens.

[Inside the large rectangle, she draws a line down the middle, creating 2 medium rectangles. On both sides of ‘20’, she writes: 10.]

And then I can use my renaming knowledge, 25 tens…

[Inside the medium left rectangle, she writes: 250].

…as 250, and 25 tens again…

[Inside the medium right rectangle, she writes: 250].

…as 250, which he knew was…

[On the right side of the rectangle, she writes: = 500.]

…500. And then all he did was need to get…

[She shades in the small rectangle.]

…rid of the extra 25 twos that he added. So the problem no longer was…

[She crosses out 500.]

…500 but…

[Under ‘500’, she writes: 450.]

…450. Really nice thinking, Policeman. Hmm, I wonder if there is another way.

[A new piece of white paper covers the right side of the previous papers.]

We have somebody else who'd like to share their thinking.

[She places a Lego figure of a worker on the top left corner of the new paper.]

Welcome, Workerman, who wanted to share with us something, that he learnt from Policeman, and that is the doubling and halving strategy. He said that he thought about…

[On the right side of Workerman, she writes: 25 x 18.]

…25 times 18…

…as 50, where he doubled 25 to make 50…

[Under ’18’, she writes: 9.]

…and then he halved 18 which we know is nine.

[Below the text, she draws a rectangle.]

So if we were to represent how Workerman re-imagined 25 times 18…

[On the left side of the rectangle, she writes: 50. Above the rectangle, she writes: 9.]

…as 50 nines, it might look something like this. And Workerman said he wanted to do that because then that allowed him to use a known fact.

He knew what 5 nines was, that it was 45. So then, he knew that 50 nines could…

[Inside the rectangle, she writes: 450].

…then be renamed as 450. Wow, really nice thinking. So now, we have three different ways of thinking about 18 twenty-fives. I'm wondering if there is another way that you could have solved 18 twenty-fives. Perhaps ask somebody from home. And then once you've thought about that, think about how you could solve another problem…

[She puts down a square card with text: 50 x 19.]

…50 times 19 or 19 fifties. Perhaps you might want to give one of these strategies a go. Over to you, mathematicians. Have fun!

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

So what's some of the mathematics?

[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 18, 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 interesting!]

Well, as we saw, the same problem can be solved using lots of different strategies or ways of thinking. For example, in 25 eighteens, we shared three different strategies. Now, 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 interesting.

[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 eighteens?

• Can you use one of these strategies to solve 50 nines?