# Let's investigate 1 – Stage 3 (additive thinking)

A thinking mathematically targeted teaching opportunity focussed on investigating one of the strategies to solve 230 - 190

## 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

## Outcomes

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

## Collect resources

You will need:

- pencils or markers
- something to write on.

## Watch

Watch Let's investigate 1 Stage 3 video (9:39)

### Michelle:

Welcome back mathematicians.

We thought today that we would dig into and investigate one of the strategies that we shared yesterday.

And we're thinking about the one that was shared by the green team, where they said they could think about working out 23 minus 19 by rethinking the problem as 24 minus 20.

[Screen shows 2 horizontal rows of connecting cubes, going from left to right, with the start of the rows aligned to a vertical orange line, the top row as 23 cubes and the bottom row has 24 connecting cubes.

The top row starts with 10 orange cubes, 10 red cubes and then one green cube, one blue cube and one black cube.

The second row starts with 10 blue cubes, then 10 green cubes and then one brown, one red, one orange, and one yellow cube at the end. On the right of these 2 structures, the equation 23 minus 19 equals 24 minus 20 equals 4 is written.]

And they said they would do this because they then know that the difference between 20 and 24 is 4.

And we drew a number line then, that looked like this, where we said 24 would be here, with an arrow to show that my number line continues.

And zero would be there. And they did a big jump to say, well then, I can just get rid of 20 and I know that that leaves 4 left.

[Michelle draws a number line with arrows on each end. On the left side of the number line she writes the number zero, and on the right she writes the number 24. She draws a jump from the number 24 which shows the subtraction strategy used, and labels it minus 20. Where the jump ends, she writes the number 4.]

So, I wanted to talk about this idea and spend some time investigating how does this strategy work.

[Michelle points to the number line with a marker.]

And in fact, we're using a few different strategies at the same time, but what we're essentially doing is this thing of keeping a constant difference.

[Underneath the strategy, Michelle writes the words constant difference.]

So, let's look at how that works. Because here is 23 and here's 24 and as you can see that not the same quantity, which is why it looks a little bit weird to get started with, and in fact 24 is one bigger than 23.

[Michelle points to the top row of 23 and then the bottom row of 24 showing that they are different quantities.]

Now what I'm going to do is just change my representations around a little bit so that we're looking at the same colours.

So, I've recreated 23. So that my tens are in blue and green, and my ones are in orange, and I've also recreated 24.

[Michelle now adds another row at the bottom with 24 connecting cubes composed of 10 blue, 10 green and 4 orange.]

So, I'm going to take off these cubes here.

[Michelle removes the top row and replaces it with her new row of cubes. On the row of 24 cubes, she removes the multicoloured single cubes and replaces them with 4 orange cubes.]

Ooops and I need that orange one. And I can have 24 and that makes it a little bit clearer for me to see.

And the other thing that I'm about to model, which we don't usually do, when we are doing subtraction, but we are in this case cause it will help us, is model the amount that we're taking away, which is 19, I think.

[Michelle now places another row of cubes on the bottom with 10 blue cubes and 9 orange cubes. She presses it against the top 2 rows to show the difference.]

Yes, and I can check its 19 cause there's one ten and this must be 9 cause it's one less than 10 here.

[Michelle points her finger over the 10 blue and 9 orange cubes.]

So, what I've done is represent the tens in blue and green and the ones are represented in orange.

And if I move this row into the middle and I'll align them carefully at the bottom.

[Michelle now moves the bottom row into the middle and aligns the 3 rows.]

I can see here that there's a difference of 4, so 23 minus 19 is in fact 4. It leaves a difference of 4.

[Michelle now points to the top row on the right, showing the difference of 4.]

But over here, where I've got 24, it actually leaves a difference of 5. So how does this work?

[Michelle now moves the middle row down to the bottom row and points her finger to show it now has a difference of 5. She then removes the last orange cube on the bottom row.]

Well, actually what happened is these guys rethought the numbers, and they said, well if I increase this number by one, I get 24, and which means I also have to increase this quantity by one.

[Michelle now adds a red cube to the start of the bottom and middle row.]

And now I'll keep a constant difference of 4.

Can you see that?

Yes! So, that's right.

[Michelle now removes the top row from the screen.]

So, what they did was they added on to one, one onto one number and added one onto the other number as well, and that kept the same difference.

Yeah, and so you're right, I could actually add on 2 to each number.

[Michelle now adds an extra red cube to the start of each row.]

And if I increase each by 2, I still keep a difference of 4.

So, actually over here I also know that 23 is equivalent, 23 minus 19 is equivalent to 24 minus 20 and now I can actually also see it's also equivalent to 25 minus 21.

[Michelle now writes under constant difference, 23 minus 19 equals 24 minus 20, and underneath she then writes equals 25 minus 21.]

Yeah, I know, and it, well, I could even go crazier if I wanted and I could say, well, I don't want to increase them by blocks.

I want to increase them by bananas, and if I increase and precariously balance this collection on a banana, at the moment, they are not the same, but if I also balance this one on a banana, very carefully, I still have a difference of 4.

Yes, so, so actually it's the same as 23 plus one banana, minus, yes, 19 plus one banana.

[Michelle removes both red cubes at the start of the bottom row, and she adds one banana and indicates they are not the same space. Michelle removes the red cubes from the top row and adds a banana to the base of the top row. She points to having the same distance of 4. Underneath the list, she writes: equals 23 plus one banana and 19 plus one banana.]

Yeah, and so I think you're seeing this, that what really matters here, is, is keeping, is the constant difference.

So, whatever you do to one quantity, you have to do the same to the other.

Yeah, and so when the, when the green team was thinking about reworking 23 minus 19, what they were thinking about, were, was this number here. That, well, actually this would be better for me if it were a landmark number.

[Michelle removes bananas from the base of both rows.]

So, if I get my 19, and I increase it by one, I get to 20, which is a nicer number for my brain to work with, cause it's a multiple of 10, it's a landmark number, which means I also need to increase this number by one to keep a constant difference.

[Michelle adds a red block to the start of the top row and then the same for the bottom row.]

Mm hmm, and so you're right, they could also have thought, well, maybe it's not about 19, that I'm worried about, but I could increase 23 by 7, to get a landmark number.

[Michelle removes the red block from the start of each row and adds 7 red cubes to the bottom row and points to the end. She now adds 7 red cubes to the base of the top row and writes equals 30 minus 26.]

So, if I increase this collection 23 by 7 more, I end up with 30 or 3 tens.

Yeah, and so this distance is no longer for no longer for, so they have to add 7 more, increase it by 7.

And that will also leave a difference of 4, yeah, so in this case we've said that's also the same as 30 minus 19 plus 7 more, which is 26, which also equals 4.

Ah hah, and here's the other thing that's really cool about this, is you don't have to add, you can also subtract.

[Michelle removes the 7 red cubes from both rows and aligns both rows to the left.]

So, if I'm starting with 23, actually the closest landmark number would be 20.

So, if I remove 3 from this number and remove 3 from this number, I still have a difference of 4.

[Michelle removes 3 orange cubes from the top of each row and points and indicates she still has a difference of 4 cubes.]

But in this case, I'm saying 23 minus 19 is equivalent in value to 20 minus 16.

And that is how you can work with constant difference.

[Michelle writes in list equals 20 minus 16.

Screen reads Let's look at this with 2 point 3 minus 1 point 9...]

Alrighty mathematicians, let's talk about this in the context of 2 and 3 tenths minus one and 9 tenths.

[Screen shows 2 block rows laying horizontally. The top row has 10 blue cubes and 6 orange cubes and the bottom one has 10 blue and 10 green cubes. To the right written on a piece of paper is, 2 point 3 minus 1 point 9.]

Let's talk about how this idea can still work. And in this case, we're going to re-imagine our collections, so I'll put the 3 back on, that we just removed.

[Michelle places a piece of paper above the rows which reads 2 point 3 minus 1 point 9. She adds 3 orange cubes to each of the end of each row.]

And I'll start with our 2 point 3 and what I know is that, now we're going to use the same equipment but re-imagine it.

[Michelle removes the top row leaving the row with 23 cubes.]

So instead of each one of these cubes being worth one, it's now worth a tenth. So, if I have 10 tenths, that's equivalent to one.

[Michelle covers the first 10 cubes with a strip of pink paper with the word one written on it and does the same for the next ten cubes. She then writes the word tenth on each of the last 3 orange cubes.]

And another 10 tenths is equivalent to another one. And now what I have is a tenth, a tenth and a tenth. And so now I have 1, 2 and 3 tenths, and here's my one and 9 tenths, that I need to remove.

So, a tenth, tenth. And I'll keep going.

[Michelle now re-adds the other row and places a pink piece of paper the length of the 10 blue cubes, which has one written on it over them. She writes the word tenth on the rest of the 9 remaining cubes.]

OK, the same strategy can happen and, in this case, one and 9 tenths for me, isn't a particularly nice number to deal with, because it's fractional.

So, what I could do is, increase it by one more tenth, which would make the 9 tenths equivalent to 10 tenths, which is one.

[Michelle adds one red block to the end of the bottom row.]

And if I do this, I now have 2 and if I increase this one by one more tenth, I now have 2 and 4 tenths.

Yes, and 2 and 4 tenths, minus 2 tens leaves me with 4 tenths.

[Michelle then takes the red block for the end and adds to the beginning of the bottom row, and she also adds a red block to the beginning of the top row.]

Yeah, so I can use this same strategy, just in a very different context.

Back to you mathematicians!

OK mathematicians. What was the mathematics?

[Screen reads what’s was the mathematics? Screen shows models of the equivalent equations mentioned below.]

So, what we realized today is that when we're subtracting, one strategy that we can use to solve the problem is to adjust both of the numbers, so we keep a constant difference.

And we saw that today when we saw 23 minus 19 is equivalent in value to 24 minus 20, 25 minus 21, 20 minus 16.

And even at 23 and one banana minus 19 and one banana.

So over to you now mathematicians to have a go at using this strategy and see how it works for you.

Until next time.

[End of transcript]

## Instructions

- Have a go at adjusting both numbers so we keep a constant difference.
- How could you use this strategy to solve 7 and 3-tenths - 2 and 9-tenths (for example)?
- What about with 3-tenths - 12-hundredths (0.3 - 0.12)?
- Record your thinking in your student workbook.