# Let's investigate 2 – number talk (5 x 9) Stage 2

A thinking mathematically targeted teaching opportunity investigating strategies to solve 5 x 9

## 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, 2021

• MAO-WM-01
• MA2-MR-01

You will need:

• pencils

## Watch

Watch the Let's investigate 2 (5 x 9) Stage 2 video (4:05).

Explore and visualise strategies to solve 5 nines.

### Speaker

Alright mathematicians, let's investigate.

You might have been with us recently when we looked at this number talk of different ways that we could solve 5 nines and we know many of you know this number fact, but what we really wanted to explore were different ways of thinking and develop these ideas of different strategies so that you can apply them in much harder contexts later on.

[There is a large piece of paper on the screen with a sticky note at the top. The sticky note says ‘5 nines’ with the equation 5 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.

In the first left column ‘5 is half of 10’ is written, with 2 rectangular figures drawn underneath. The top rectangular figure is twice as long as the one underneath. The top rectangular figure has the number 10 on it, whilst the bottom rectangular figure has the number 5 on it. Underneath ’10 nines equals 9 tens equals 90’ is written. At the bottom 90 multiplied by 2 equals 45 is written.

In the middle column a range of equations are written. At the top 5 times 9 equals 5 times 10 minus 5 times 1 is written. Underneath it says 5 nines equals 5 tens minus 5 ones. There is a second equation underneath which reads 5 times 10 equals 50 with 5 tens equals 50 underneath. It then says 5 one equals 5 and finally 50 minus 5 equals 45.

The final column has the equation 9 equals 8 plus 1. Underneath 5 nines equals 5 eights plus 5 ones. There is then another equation which reads 5 times 8 equals 10 times 4 equals 20 times 2 equals 40 with 5 ones equals 5. The final equation in this column reads 40 plus 5 equals 45.]

And we looked at 3 different possible strategies, but we just wrote them down and so what we thought we might do today is look at different ways that we could imagine what's happening inside of someone’s head and with numbers when they are using these strategies.

So, let's look at the first one. Here's 5 nines. And here's the strategy.

[Screen shows the previous strategy from the first column. Underneath there is an array of 9 rows of dots arranged in 5 columns. Underneath these dots ‘5 nines’ is written with the equation 5 times 9 underneath.]

So, the first thing they said they know is at 5 is half of 10.

So, if we double 5 nines, we'd end up with 10 nines and then they used the commutative property to say, well actually, 10 nines can be renamed 9 tens.

[The array doubles, with the initial array being mirrored using green dots. Underneath the words ‘10 nines’ is written. The array flips so that there are 9 rows of 10 with ‘9 tens’ written underneath. The equation then changes to 9 tens = 90.]

And I know that is 90 because of place value I can rename it.

[Screen now shows the array changing into 2 vertical rectangles. The left rectangle is green and the right rectangle is blue. On the middle of the 2 rectangles the number 90 is written.]

And then they said, well, now we still have to halve that array, so have half of 90 is 45 and we remove that 45. And now we have... our answer. 5 nines is 45.

[The 2 rectangles are split into 2 equal parts, with each rectangle containing the number 45. The green rectangle is removed, only leaving the number 45.]

That's one strategy. Let's have a look at another one.

[The strategy from the middle column is now displayed on screen. At the top 5 times 9 equals 5 times 10 minus 5 times 1 is written. Underneath it says 5 nines equals 5 tens minus 5 ones. There is a second equation underneath which reads 5 times 10 equals 50 with 5 tens equals 50 underneath. It then says 5 one equals 5 and finally 50 minus 5 equals 45.

Underneath there is an array. In the array there is 5 rows of 9 dots, with the words ‘5 nines’ written underneath and the equation ‘5 times 9’ below it.]

So, these guys were also thinking about what they know, and they said, well, we know that we don't have to use 5 nines.

We could work with 5 tens. So, they made 5 tens.

And they did this because they know something about place value. That 5 tens is renamed as... 50 and then they just needed to remove the extra 5 ones they used. And they did 50 - 5 ones which left them with... 45.

[An additional column of 5 green dots is added to the array. The text below transforms to say ‘5 tens’. The whole array is transformed into a red rectangle with the number ‘50’ inside. The shape changes once more to show the final column of 5 green dots, symbolising the ‘5 ones’ being taken away. The red rectangle changes, with the number inside becoming 45.]

Yes, so that was a second strategy and let's have a look at a third strategy.

Here's 5 nines again. And this team also said we can rethink the numbers.

[The screen shows the final column which has the equation 9 equals 8 plus 1. Underneath 5 nines equals 5 eights plus 5 ones. There is then another equation which reads 5 times 8 equals 10 times 4 equals 20 times 2 equals 40 with 5 ones equals 5. The final equation in this column reads 40 plus 5 equals 45.

Underneath there is an array showing 5 rows of 9 dots. Underneath ‘5 nines’ is written with the equation 5 times 9 underneath.]

So, what we know about 5 nines is that it's made up of 5 eights and 5 ones.

[Screen shows last vertical row moving apart to the right of the array, with 5 eights written under the left group of the array and 5 ones written underneath the single row of 5.]

And then they said we could do some doubling and halving of 5 eights.

So, if you double 10, double 5 you get 10 and if you have eights, you get fours to get 10 fours. Mm.

[The array on the left begins to transform to create an array of 10 rows containing 4 dots. Underneath ’10 fours’ is written.]

But now I have to shrink this so that you can see it on the screen.

So, let's shrink this down and then they did the same thing just for fun.

They said cause they could have from here said then 10 fours is 40 and five more is 45 but they said they wanted to play so they doubled and halved again.

So, they doubled 10 to get 20 and halved fours to get twos. And that's what that looks like.

[Screen shows the array transforming once more to create an array of 20 rows containing 2 dots. Underneath ’20 twos’ is written.]

Yes, and I should I just shrink it down so I can fit on my screen, and they doubled and halved again, look. 20 twos became... 40 ones.

[Screen shows dots shrinking down in size again to an array of 40 rows containing one dot, with ’40 ones’ written below.]

Uh-huh. And then they joined it together to find 45. And here's what that would look like as an area model.

[The third strategy is now shown written down. At the top the equation 9 equals 8 plus 1 is written. Underneath ‘5 nines plus 5 eigts plus 5 ones’ is written. Below that ‘5 times 8 equals 10 times 4 equals 20 times 2 equals 40’ is written, with ‘5 ones equals 5’. The final line has the equation ’40 plus 5 equals 45.’]

So, mathematicians that's 3 different strategies that we just investigated to think about 5 nines.

[Screen displays the 3 strategies shown at the beginning of the video.

In the first left column ‘5 is half of 10’ is written, with 2 rectangular figures drawn underneath. The top rectangular figure is twice as long as the one underneath. The top rectangular figure has the number 10 on it, whilst the bottom rectangular figure has the number 5 on it. Underneath ’10 nines equals 9 tens equals 90’ is written. At the bottom 90 multiplied by 2 equals 45 is written.

In the middle column a range of equations are written. At the top 5 times 9 equals 5 times 10 minus 5 times 1 is written. Underneath it says 5 nines equals 5 tens minus 5 ones. There is a second equation underneath which reads 5 times 10 equals 50 with 5 tens equals 50 underneath. It then says 5 one equals 5 and finally 50 minus 5 equals 45.

The final column has the equation 9 equals 8 plus 1. Underneath 5 nines equals 5 eights plus 5 ones. There is then another equation which reads 5 times 8 equals 10 times 4 equals 20 times 2 equals 40 with 5 ones equals 5. The final equation in this column reads 40 plus 5 equals 45.]

So, I'm going to get you to refocus back on a problem we posed yesterday about 6 nines. And don't worry if you know the number fact, what we're really interested in is how could you apply some of these strategies into that context of 6 nines?

And see how you could use diagrams, drawings, materials to represent your ideas.

Ok, back to you mathematicians.

[End of transcript]

## Instructions

• How might you use strategies to solve 5 nines (5 x 9) to think about 6 nines (6 x 9)?
• Use diagrams, drawings or other materials to represent your thinking.