# Let's investigate 2 – number talk (15 x 9) Stage 3

A thinking mathematically targeted teaching number talk focussed on developing flexible multiplicative strategies, reasoning and communicating

## 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

You will need:

• pencils

## Watch

Watch the Let's investigate 2 (15 x 9) Stage 3 video (2:17).

Explore and visualise strategies to solve 15 nines.

### Speaker

Ok mathematicians let's investigate.

So yesterday you joined us for a number talk on different ways that we could think about to solve 15 nines, 15 times 9 and what we thought is we might come back today and investigate different ways of representing 2 of the 3 ideas and that would help us investigate how these strategies work so that you can then apply them.

[There is a large piece of paper on the screen with a sticky note at the top. The sticky note says ‘15 nines’ with the equation 15 times 9 underneath.

There are 3 columns that represent the 3 teams or 3 strategies used. At the top of each column is a LEGO minifig to represent each team.

The first column on the left has 5 rows. Row 1 reads: 15 equals 10 plus 5. Row 2 reads: 15 nines equals 10 nines plus 5 nines. Row 3 reads: 10 nines equals 9 tens equals 90. Row 4 reads: 5 nines equals 45 and row 5 reads: 90 plus 45 equals 100 plus 35 equals 135.

The middle column has 4 rows. Row 1 reads: 15 times 10 minus 15 times 1 equals 15 tens minus 15 one. Row 2 reads: 15 tens equals 150. Row 3 reads: 15 times 1 equals 15. Row 4 reads: 150 minus 15 – 135.

The final column has 6 rows. Row 1 reads: 9 equals 8 plus 1. Row 2 reads: 15 nines equals 15 eights plus 15 ones. Row 3 reads: 15 times 9 equals 15 times 8 plus 15 times 1. Row 4 reads: 15 times 8 equals 30 times 4 equals 60 times 2 equals 120. Row 5 reads: 15 times 1 equals 15. Row 6 reads: 120 plus 15 equals 135.]

So, here's a representation of 15 nines.

[Screen shows the first strategy with 9 vertical rows and 15 horizontals with blue dots and 15 nines written underneath it. On the top left, the first team’s strategy is shown. Row 1 reads: 15 equals 10 plus 5. Row 2 reads: 15 nines equals 10 nines plus 5 nines. Row 3 reads: 10 nines equals 9 tens equals 90. Row 4 reads: 5 nines equals 45 and row 5 reads: 90 plus 45 equals 100 plus 35 equals 135.]

And now let's look at the first team’s way of thinking.

So, they said, well 15 nines we know that we can partition the 15 so that we have 10 nines and 5 nines.

[The screen partitions the array into 2. The top array has 10 rows of 9. Underneath this array the words ’10 nines’ is written. The bottom array has 5 rows of nine. Underneath this array the words ‘5 nines is written.]

Then they said we know something about 10 nines that we could actually use the commutative property to re-imagine that as 9 tens.

[The top part of the array turns vertically to create 9 rows of 10. Underneath the array, the text changes to say ‘9 tens equals 90’. The text under the bottom array also changes to say ‘5 nines equals 45.]

And that would give us 90 and we also know 5 nines is 45. And then we could turn it back together. Re-join our array. And find the total of 135.

[Screen shows the arrays being pushed together to create one whole array. On it the number 135 is written.]

And that's one way of thinking about it. But then another team said, well, we have a different strategy.

[Screen displays the original array of 15 rows of 8 once more. On the top left, the second strategy is shown. It has 4 rows. Row 1 reads: 15 times 10 minus 15 times 1 equals 15 tens minus 15 one. Row 2 reads: 15 tens equals 150. Row 3 reads: 15 times 1 equals 15. Row 4 reads: 150 minus 15 – 135.]

Look, here's 15 nines and I can think of 15 nines as 15 tens. And this is helpful for me 'cause I can rename it in place value and then I just have to remove the 15 ones that I borrowed.

[An additional column gets added to the array to make it 15 tens. The original array transforms into a red rectangle with the equation 150 minus 15 ones written on it.]

So, once I've got 150, I then need to remove the 15 ones and that leaves us with 135.

[The additional row is removed from the array. The equation on the red rectangle changes to the number 135.]

And that was a second way of thinking through the problem.

So now we have 2 strategies that we've explored, but the 3 one we've left some mystery around.

[The screen shows the 3 strategies from the beginning of the video.

The first column on the left has 5 rows. Row 1 reads: 15 equals 10 plus 5. Row 2 reads: 15 nines equals 10 nines plus 5 nines. Row 3 reads: 10 nines equals 9 tens equals 90. Row 4 reads: 5 nines equals 45 and row 5 reads: 90 plus 45 equals 100 plus 35 equals 135. Underneath there is a visual representation of the 2 filled-in arrays with the number 135 within it.

The middle column has 4 rows. Row 1 reads: 15 times 10 minus 15 times 1 equals 15 tens minus 15 one. Row 2 reads: 15 tens equals 150. Row 3 reads: 15 times 1 equals 15. Row 4 reads: 150 minus 15 – 135. Underneath it is a filled in representation of the arrays with the number 135 within it.

The final column has 6 rows. Row 1 reads: 9 equals 8 plus 1. Row 2 reads: 15 nines equals 15 eights plus 15 ones. Row 3 reads: 15 times 9 equals 15 times 8 plus 15 times 1. Row 4 reads: 15 times 8 equals 30 times 4 equals 60 times 2 equals 120. Row 5 reads: 15 times 1 equals 15. Row 6 reads: 120 plus 15 equals 135.]

So, mathematicians it's back to you now to see how you could use these ideas and representations to help you solve or to record your thinking around 16 twenty-fives.

Ok, over to you.

[End of transcript]

## Instructions

• Use diagrams, drawings and/ or materials to represent how you might use these strategies to think about for 16 twenty-fives (16×25)?