# Let's generalise (23-19)

Stage 1 to 3 – A thinking mathematically targeted teaching opportunity focussed on exploring and investigating how a strategy can be used to solve other problems

## 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
- MA1-RWN-01
- MA1-RWN-02
- MA1-CSQ-01

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

- MAO-WM-01
- MA3-AR-01

## Collect resources

You will need:

pencils or markers

something to write on.

## Watch

Watch Let's generalise 1 video (6:19).

[A title over a navy-blue background: Let’s generalise - 1. Small font text in the upper left-hand corner reads: NSW Department of Education. In the lower left-hand corner is the white waratah of the NSW Government logo.]

[Text over a blue background: Okay mathematicians… what do you notice?]

**Speaker**

OK mathematicians, are you ready for some sweaty brains? Good. What do you notice when you see this?

[On a white background on the left side is a pink box around two math problems: 23 – 9 = 14 and 24 – 10 = 14]

You might notice the pink outline and there's some numbers.

[A blue speech bubble in the top right corner appears with text: What do you notice?]

**Speaker**

Can you have a look at what you notice about the numbers in particular, the mathematical noticings?

[Below the speech bubble, another blue speech bubble appears with text: Take a close look at the relationship between the numbers. For example.]

**Speaker**

OK, let's take a close look at the relationship between the numbers. That sounds like a really fancy word, but don't worry too much.

[23 and 24 are highlighted. In the speech bubble, a point appears: 24 is one bigger than 23;.]

**Speaker**

Look, that just means, what do you see happening here between 23 and 24? 24 is one bigger than 23. What about with nine and 10?

[9 and 10 are highlighted. In the speech bubble, a point appears: 10 is one bigger than 9;.]

**Speaker**

Yeah, 10 is one bigger than nine.

[In the speech bubble, a point appears: But the difference is still the same.. 14!]

**Speaker**

And what do you notice about the difference? It's still the same. So, that's really interesting to me. Let's have a look at a few more number sentences or equations and see what happens. OK, what can you see happening here?

[In the pink box, two equations appear below the others: 25-11=14 and 26-12=14.]

**Speaker**

Yes. Like you, I can see the same thing happening. So, as one number gets bigger, the other number also gets bigger by the same amount. But the difference is still 14 and it looks like this happens for each pair of equations.

[The larger speech bubble disappears. The ‘What do you notice?’ speech bubble gets larger, moving to the centre top area. Under the speech bubble, a blue starburst appears with text: This happens for each pair of equations (number sentences). Under the starburst, the speech bubble from earlier with points:

- 24 is one bigger than 23;

- 10 is one bigger than 9;
- BUT the difference is still the same.. 14!

re-appears.]

**Speaker**

Yeah. Look.

[The pink box disappears. The bottom speech bubble morphs into a red box and moves to the left side. A pink box with equations 23-9=14 and 24-10=14 appears above it.]

**Speaker**

So, here we have 23 minus 9 is 14 and 24 minus 10 is 14. 24 is one bigger than 23, 10 is one bigger than nine, but the difference is still the same, still 14. Let's look at this pair of numbers.

[Next to the pink box with equations, another pink box with equations 24-10=14 and 25-11=14 appears.]

**Speaker**

What do you see? Yeah, the same pattern. To get from 24 to 25 I add one, to get from 10 to 11 I add one, but the difference is still 14.

[Under the second pink box, a red box with these points appear:

- 25 is one bigger than 24;

- 11 is one bigger than 10;
- BUT the difference is still the same.. 14!]

**Speaker**

OK, what about this pair of numbers?

[Next to the last pink box with equations, another pink box with equations 25-11=14 and 26-12=14 appears.]

**Speaker**

Yeah, the same thing is happening.

[Under the third pink box, a red box with these points appear:

- 26 is one bigger than 25;
- 12 is one bigger than 11
- BUT the difference is still the same.. 14!]

**Speaker**

25 is one less than 26, 11 is one less than 12, still 14. Or you could say it the other way. 26 is one more than 25, 12 is one more than 11, still 14. This is pretty awesome.

[The boxes are cleared, except for the left one. The bottom equation is replaced with 26-12=14.]

**Speaker**

What if, though, I don't look at just numbers in the counting sequence? What about 23 minus nine is 14 and 26 minus 12 is 14. Let's have a look together.

[Under the box, a row of 23 blue squares appears. The first 10 are blue, the second 10 are light blue, the last 3 are blue.]

**Speaker**

Here's 23…

[The 23 blue squares become one long rectangle with text: 23 inside it. 9 blue squares appear under the rectangle.]

**Speaker**

…and here's nine…

[The 9 blue squares become one long rectangle with text: 9 inside it.]

**Speaker**

…and the distance between them or the space between them…

[Next to the ‘9’ rectangle, a red rectangle that aligns with the ‘23’ rectangle above appears.]

**Speaker**

…is 14. OK. Here's 26…

[Under the rectangles, a row of 26 blue squares appears. The first 10 are blue, the second 10 are light blue, the last 6 are blue.]

**Speaker**

…here's 12…

[The 26 blue squares turn into a long rectangle with text: 26. Under the rectangles, a row of 12 blue squares appears. The first 10 are blue, the last 2 are light blue.]

**Speaker**

…and the difference between them is 14.

[Next to the ‘12’ rectangle, a red rectangle that aligns with the ‘26’ rectangle above appears.]

**Speaker**

Look, they are the same.

[The other rectangles disappear, while the bottom red rectangle move directly below the other.

The red rectangles disappear. The red box with the equations 25-11=14 and 26-12=14 re-appear.

Below the pink box, a red box with these points:

- 26 is 3 bigger than 23;

- 12 is 3 bigger than 9;
- BUT the difference is still the same.. 14!

Under the speech bubble, is the blue starburst with text: This happens for each pair of equations (number sentences). Under the starburst, is another starburst with text: It doesn’t matter how much you increase, or decrease the numbers by, the difference between them seems to stay the same.]

So, in this case, 26 is three bigger than 23, 12 is three bigger than nine, but the difference is still the same, it's 14.

So, I think we're noticing something, that it doesn't matter how much you increase or decrease, so go up or go down, that you do to the numbers, the difference between them seems to stay the same.

[The red boxes disappear. The speech bubble text is replaced with: Does this always work? It moves slightly lower.]

So, now I'm really curious, does this always work? 'Cause this would be an awesome strategy for us to know about. Let's have a look what happens with addition. Ready?

[Under the speech bubble, an orange box appears with equations: 23+9=32, 22+10=32.]

**Speaker**

Here's two number sentences or equations, what do you notice? I see that too. Look, this time, yeah.

[23 and 22 are highlighted. Below the orange box, a smaller orange box appears with points:

- 23 is 1 more than 22;
- 9 is less than 10;
- BUT the difference is still the same.. 32!]

23 is one more than 22. Nine is one less than 10. So, 23 is one more, nine is one less, the sum is still 32.

[The smaller orange box disappears. Under the equations, 21+11=32 and 20+12=32 are added.]

Let's have a look at a few more number sentences or equations with addition. What can you see happening here? Oh, I see that too.

[23, 22, 21 and 20 are highlighted.]

Look, as each of these numbers goes down by one, these ones…

[9, 10, 11 and 12 are highlighted.]

….increase by one, but the sum stays the same.

[All of the 32s are highlighted.]

**Speaker**

So, I think we need to revise what we were thinking before, 'cause it doesn't seem to work, this strategy doesn't seem to work in the same way with addition, does it?

[The boxes are cleared. The text in the bottom starburst is replaced by: Our working conjecture is: WHEN WE'RE WORKING WITH SUBTRACTION, it doesn't matter how much you increase, or decrease the numbers by, the difference between them seems to stay the same, so long as you do the same thing to both numbers!]

So, maybe what we need to say is that when our working conjecture at the moment is that when we're working with subtraction, it doesn't matter how much you increase or decrease the numbers by, the difference between them seems to stay the same, so long as you do the same thing to both numbers. I think that could be a really important idea. OK, so here's your challenge now mathematicians.

[Text over a blue background: Here’s your challenge, mathematicians…

On a white background, on the left side, a blue starburst with text: Our working conjecture is: WHEN WE'RE WORKING WITH SUBTRACTION, it doesn't matter how much you increase, or decrease the numbers by, the difference between them seems to stay the same, so long as you do the same thing to both numbers!]

Here's our working conjecture. And what does happen with addition?

[On the right side, a blue speech bubble appears with text: What happens with addition?]

This is what we'd like you to explore now.

[Below the blue speech bubble, another blue speech bubble appears with text: Use any numbers you feel comfortable working with to investigate this idea. Record your thinking using pictures, as well as words and symbols.]

**Speaker**

So, you can use any numbers you feel comfortable working with to investigate this idea. So, you might work with numbers between one and 10, or you might work with numbers with fractional quantities. The choice is yours. But make sure you record your thinking using pictures as well as words and symbols and see if you can come up with a conjecture for what happens in addition. OK mathematicians, over to you.

[Text over a blue background: Over to you!]

Enjoy investigating.

[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]

## Reflection

**What happens with addition?**

Use any numbers you feel comfortable working with to investigate this idea. For example, look at these numbers and see if you can work out a strategy that you could use:

7 + 14 = 21

8 + 13 = 21

9 + 12 = 21

10 + 11 = 21

Or:

5 + 16 = 21

7 + 14 = 21

9 + 12 = 21

11 + 10 = 21

Note down your ideas in your student workbook.