Let's investigate 1 – number talk (exploring constant difference)
Stage 1 and 2 – A thinking mathematically targeted teaching opportunity focussed on investigating one of the strategies to solve 230 - 190
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
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Watch Let's investigate 1 video (7:36).
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 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.
[On the screen there are 2 rows of interlocking cubes. The first row has 23 cubes and the second row has 24 cubes. On the right of the screen there is a piece of paper with the equation 23 minus 19 equals 24 minus 20 equals 4. The presenter creates a number line underneath the equation. She marks and writes the number 24 on the right side and draws a large jump to signify the subtraction of 20. She marks where the jump lands and writes the number 4.]
So, I wanted to talk about this idea and spend some time investigating, well how does this strategy work?
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.
So, let's look at how that works. Because here is 23 and here is 24 and as you can see that's 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.
[The presenter writes the words ‘constant difference’ underneath the number line. She then gestures to the 2 rows of connecting cubes, pointing to the first row to show that there are 23 and the second row to show that there are 24.]
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.
[The presenter shows new representations of the numbers. In each row, the tens are represented by the colours blue and green and the ones are represented by orange.]
So that my 10s are in blue and green and my ones are in orange, and I've also recreated 24.
[Presenter shows her 2 rows. The first row has 10 blue cubes, 10 green cubes and 3 orange cubes whilst the second row has 10 blue cubes, 10 green cubes and 4 orange cubes.]
So, I'm going to take off these cubes here, oops 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.
[The presenter shows another row of cubes 10 blue and 9 orange cubes. She compares it to the row of 24 cubes and shows the matching tens as well as the 9 ones to make the number 19.]
Yes, and I can check it's 19 'cause there’s 1 ten and this must be 9 'cause it's one less than 10 here.
[The presenter aligns the rows so that they are all even.]
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 tower into the middle and I'll align them carefully at the bottom.
[She moves the block of 19 cubes in between the rows of 23 and 34 cubes. She gestures to the difference of 4 cubes.]
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. But over here, where I've got 24, it actually leaves a difference of 5.
[The presenter points to the row of 24 cubes and points out the difference of 5.]
So how does this work? 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. And now I'll keep a constant difference of 4.
[The presenter removes an orange cube from the row of 24. She adds a red cube to the left side of the row. The presenter also adds a red cube to the row of 19.]
Can you see that?
Yes so. That's right, so what they did was they added onto, 1 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. And if I increase each by 2. I still keep a difference of 4.
[The presenter adds an additional red cube to the left side of the row of 24 and 20, making each row 25 and 21.]
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.
[On the paper, presenter writes: 23 minus 19 equals 24 minus 20. Underneath she writes: ‘equals 25 minus 21’ to show that all of the equations are equivalent.]
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're not the same, but if I also balance this one on a banana, very carefully, I still have a difference of 4.
[The presenter removes the 2 red blocks on each row. She replaces them with a small, plastic banana which she joins onto the 2 rows.]
Yes, so, so actually is the same as 23 plus one banana minus, yes 19 plus 1 banana.
[Underneath the other equivalent equations she writes: ‘equals 23 plus one banana minus 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 was this number here.
They were like, well, actually, this would be better for me if it were a landmark number. 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 ten.
[The presenter adds an extra cube to her row of 19, making it 20.]
It is a landmark number, which means I also need to increase this number by one to keep a constant difference.
[She adds a red cube to the row of 23, making it 24.]
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.
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 4, no longer 4, so they have to add 7 more, increase it by 7.
[The presenter removes the 2 red cubes from each row. She adds 7 cubes to each row, making the first row 30 and the second row 26.]
And that will also leave a difference of 4. Four, 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.
[On the piece of paper, the presenter writes the new equation: equals 30 minus 26.]
And here's the other thing that's really cool about this is, you don't have to add. You can also subtract.
[Presenter removes the 7 red cubes from both towers.]
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, 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.
[She removes 3 blocks from each row, making each row 20 and 16. She adds this new equation to her list: equals 20 minus 16.]
OK mathematicians, what was the mathematics?
So, what we realised 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 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]
- How could you use this strategy to solve 13 - 9?
- How could you use this strategy to solve 73 - 29?
- Record your thinking in your student workbook.