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Speaker

Welcome back mathematicians. We hope you're having a really lovely day today.

Today we thought we would embrace our inner George Polya, who was a really famous mathematician who also once famously said this. That it's better to solve one problem in 5 different ways than to solve 5 different problems.

[Screen shows an image of the quote.]

And so, to Georges' point, we're going to think about how many different ways, in fact, can we think of 5 different strategies to solve our problem, 23 minus 19.

[Screen shows a blank piece of paper which fills the screen. There is an orange card at the top of the screen that reads, ‘23 minus 19’.]

Now I know what you thinking, oh, 23 minus 19. This is not much of a brain sweat for me yet. Stick with me. Your challenge is coming. Okay, so what I'd like you to think about is, what is one strategy that you could use to solve this problem? Okay, and once you've thought of one strategy, you might, you know, can you think of a second strategy that you could use?

Yeah, and for those of you that are familiar, we're sort of doing a number talk. Aren't we? Where in a classroom, we might use hand signals. Like this means I'm thinking.

[Presenter makes a fist to show that they are thinking.]

This means I have one possible strategy of thinking through this problem.

[Presenter keeps their hand in a fist, but holds up a thumb.]

This means I have another strategy and so on.

[Presenter holds up a second finger to show that they have 2 strategies.]

Okay, so hopefully you've got one way of thinking about this. We thought about this with some students too, they can't be here with us today, so we're going to represent their thinking. So the team represented by the ‘strong man’ suggested, well, you could think about 19 and partition it, into its parts, so to break it apart.

[Presenter shows a ‘strong man’ figurine and places it on the paper.]

And they said, really 19 is made up of 10 and 9.

[Presenter pulls out an A4 piece of paper and a blue marker. At the top of the paper, on the right-hand side of the page, they write the number ‘19’. They then draw 2 diagonal lines underneath the number to partition it into tens and ones. One diagonal line slants towards the left, and the other diagonal line slants towards the right. Underneath the left diagonal line, they write the number ‘10’. Underneath the right diagonal line, they write the number ‘9’.]

So we could think of 23 minus 19 as 23 minus 10. And they said that was 13 and then 13 - 9 and they said that what they would do is subtract the ones by using the jump strategy.

[Presenter goes back to where they have written the number ‘19’. They add more detail to create a full number sentence. The number sentence now reads, ‘23 minus 19’. Underneath all of their previous working out, the presenter writes, ‘23 minus 10 equals 13.’ Underneath this number sentence, they then write, ‘13 minus 9’.]

So let's have a look at what that looks like on a number line.

[Presenter moves the A4 piece of paper out of view.]

And we've been playing around with this idea of, you know, how do we record number lines and get our eye in to make them proportional. So we'll share with you a strategy that we've been using with these guys today. And the first thing is, we've modelled our quantity so we have 23.

[Presenter brings out some connected cubes and places them on the paper. There is one row of 10 red cubes, one row of 10 orange cubes and one shorter row of 3 cubes. In the shorter row of 3 cubes, there is one green cube, one blue cube and one black cube.]

The two long sticks here are each 10. That is what this number here represents.

[Presenter points to the 2 rows of 10 cubes. On the orange card at the top of the page, they then point to the ‘2’ digit in the number ‘23’.]

And the three here is what this number represents in the, in the numbers.

[Presenter points to the shorter row of 3 cubes. On the orange card at the top of the page, they then point to the ‘3’ digit in the number ‘23’.]

And I know these are ten 'cause I made them, but we could, I could prove to you it's 10 by snapping them in half. And what I know is that my brain and your brain has this capacity to subitise quantities, so without having to count, I can actually see this chunk of 3 and this chunk of 2, and I know 3 and 2 together is 5 and double 5 is 10. So that has to be 10 bricks high, and if I line that up. That's also 10.

[Presenter snaps the orange row of blocks in half, making 2 rows of 5. On one of the rows of 5, they circle their finger around 3 of the blocks. They then circle their finger around the remaining 2 blocks. Presenter puts the 2 rows of 5 back together, once again making a full row of 10 blocks. They then place the row of orange blocks next to the row of red blocks.]

So now I have my 2 tens, which is what this shows me, and my 3 ones.

[Presenter connects the orange row and the red row together to make one long row. On the orange card at the top of the page, they point to the ‘2’ digit in the number ‘23’. Presenter then connects the shorter row of 3 cubes to the end of the long row. They have now created a row of 23 cubes – the row of orange cubes, followed by the row of red cubes, followed by the short row of 3 differently-coloured cubes.]

And we're going to represent their thinking using a number line and will use blue for the strong man.

[Presenter places the long row of cubes horizontally on the piece of paper. Using a blue marker, they trace a straight line that is the exact length of the row of cubes.]

And yeah, we've been using them almost like a measure, and if I come here and carefully mark the end, that's where 23 goes. And actually my number line could keep going if I wanted.

[At the end of the line, the presenter draws a small vertical marking and writes the number ‘23’. They then extend the line slightly and add an arrowhead that is pointing to the right. This shows that the number line could extend further.]

And this is where zero would be and also, my number line would keep going in the other direction.

[At the start of the line, the presenter draws a small vertical marking and writes the number ‘0’. They then extend the line slightly and add an arrowhead that is pointing to the left. This shows that the number line could extend further.]

And what the strongman team said they did, was the first thing was, they got rid of 1 jump of 10. So, so I'm now thinking about where my ten is and I know there's 3 here. So if I go with the 3 left behind strategy, that will be a jump of 10 and I can prove that by using direct comparison.

[Presenter brings back the A4 piece of paper and places it under the horizontal row of cubes. They point to the 3 differently-coloured cubes at the end of the long row. They then point to the first 3 red cubes in the middle of the row. Presenter holds down these 3 red cubes and breaks off the line of cubes that come after it. In total, they remove 10 cubes from the long row. Presenter takes this line of 10 cubes and holds it up against the line of orange cubes to prove that they are the same size. They then put the line of 10 cubes to the side. This leaves 13 cubes remaining under the number line – 10 orange cubes and 3 red cubes. Presenter draws a curved line above the number line. This line starts at the number ‘23’ and stops at the end of the row of 13 cubes. Underneath the curved line, they write ‘minus 10’.]

And then they said, now we would count back by ones 9 times. So can you help me keep track of the count? 1. 2. 3, whoops. 4.

[Presenter snaps off one block at a time from the right-hand end of the row, counting out loud as they do so. Each time they remove a cube, they draw a curved line on the number line to mark the cube that has just been removed. Presenter snaps off the first 4 blocks while counting out loud. After that, they continue snapping off the blocks silently. Presenter keeps removing blocks in this way until only 4 blocks are left.]

9, which leaves? 4. So the 13 minus 9 is 4.

[Presenter draws a small vertical marking on the number line at the end of the fourth block. They then write the number ‘4’ above it. On the A4 piece of paper, they complete the final number sentence that they started earlier. The number sentence now reads ‘13 minus 9 equals 4”.]

And so what we have here is the 1 ten and the 9 more of 19 and I can record the strong man's team's thinking over here as 23 minus 19 is equivalent in value to 4.

[Presenter holds up the line of 10 blocks that they removed earlier. They then gesture to the 9 individual blocks that they removed. Presenter moves their marker to the end of the blue number line that they have created. Next to it, they write, ‘23 minus 19 equals 4’.]

So like George Polya, though, we're like, well, let's see what other strategies that we can come up with. And so, as I reassemble these blocks, someone else in our group had a really interesting idea and they were thinking about, well, I know something about addition and subtraction and that is that they are related, and so I can use addition to solve subtraction problems. So enter in fancy robot dancing man, that's what we decided to call him.

[Presenter reassembles the cubes to how they were in the beginning. There is a row of 10 orange cubes, a row of 10 red cubes, and a smaller row of 3 differently-coloured cubes. Presenter picks up the ‘strong man’ figurine and places it out of view. They then bring in a robot figurine and place it on the paper. They also collect a green marker and put it underneath the figurine.]

And this team, the green team we will call them, thought, re-thought, about the problem and they said, well, actually, when you're solving subtraction, you can just think addition. So what I know is that 19 plus something is equivalent in value to 23 and we need to work out what the difference is.

[Presenter pulls out a new piece of A4 paper. Using the green marker, they write ‘19 plus a blank space equals 23’.]

They said then what they would do is 19 plus 1 is 20, because that gets them to a landmark number and then they said from 20, they know that just to add, 3 more is 23 because they would rename it. And what we wondered about, is how we could record that on a number line.

[Underneath the first number sentence, presenter writes ‘19 plus 1 equals 20’. They then write, ‘20 plus 3 equals 23’ under that. Presenter places the A4 piece of paper out of view.]

So this is what we came up with. We said, well, we could use our 23. And I'm going to try to line them up so that you can see them. And here's my number line. There's 23 with my arrow 'cause that it extends in that direction and zero and my arrow.

[Presenter picks up the long row of cubes and places it underneath the first number line. Using the green marker, they create a number line similar to the one that was drawn before – with the number ‘0’ on the left side, the number ‘23’ on the right side, and arrowheads pointing outwards on either end.]

And what they were saying is that what we what we know is that 23 is here and we need to find 19 to work out the space between the difference and they said, well, since we know this is 1 ten and this is another ten. 19 must be here because 19 is 1 less than 20, that's right. And then they added 1. And then they added 3 more.

[Presenter traces their finger along the 10 orange cubes, followed by the 10 red cubes. They point to the end of the ninth red cube, and place a little vertical marking on the number line above it. Presenter draws a small curve above the tenth red cube. Above the curve, they write, ‘plus 1’. They then draw another curve above the final 3 differently-coloured cubes. Above the curve, they write, ‘plus 3’.]

Yeah, so they still have, if I take this section of brick off, it's still a difference of 4. But they just thought about the problem differently.

[Presenter breaks off the last 4 cubes in the row. They then move the cubes up to the start of the blue number line to prove that the size is exactly the same.]

So in this case what they thought about was 19 plus something is 23 and they worked out that that means 19 plus 4 is 23. That was their solution.

[Presenter brings back the A4 piece of paper and places it under the horizontal row of cubes. They then move their marker to the end of the green number line that they created. Next to it, they write, ‘19 plus a blank space equals 23. 19 plus 4 equals 23.’]

And then we were having a really interesting conversation about how you can use addition to solve subtraction and in fact subtraction to solve addition, when along came the Flamingo team, and the Flamingos were like, well, hold on a second. We've got another way that we could think about this problem, and they said we would just rethink the problem altogether, where I don't want to deal with 23 minus 19 because 19 is not a landmark number.

[Presenter picks up the robot figurine and places it out of view. They then bring in a flamingo figurine and place it on the paper. Presenter reassembles the cubes to how they were in the beginning. There is a row of 10 orange cubes, a row of 10 red cubes, and a smaller row of 3 differently-coloured cubes.]

So, in actual fact, I can say this, 23 minus 19 is equivalent in value to 24 minus 20 and they said, and I immediately just know it in my head that that's a difference of 4.

[Presenter pulls out a new piece of A4 paper. Using a purple marker, they write ‘23 minus 19 equals 24 minus 20’. Underneath this, they also write, ‘equals 4’.]

And we were like, wow, can you explain your thinking more please? It was a bit like this. Can you explain your thinking more please, Flamingo? Of course I can, Robot.

[Presenter picks up the flamingo figurine and the robot figurine. They play with the 2 figurines on screen to make it look like they are talking to each other. Presenter then puts both figurines back where they were.]

So this is what happened. Because, because what the Robot team and the Strongman team were wondering about is that if this is 23, and if I now make a collection of 24, you know this, this tower is one block more than this one, so how does this work?

[Presenter picks up the long row of cubes and places it underneath the second number line. They then create a new row of cubes that is 24 blocks long. There are 10 green cubs, followed by 10 blue cubes, followed by a brown cube, a red cube, an orange cube and a yellow cube. Presenter places this new row of cubes underneath the original row.]

So let's have a look, so we'll use the 24 and I'll line this up as best as I can to create our number line. And this time we're starting at 24, but again, our number line can continue in this direction. And this is where zero is. And it continues in this direction.

[Presenter moves the row of 23 cubes off screen. Using the purple marker, they use the row of 24 cubes to create a number line that is similar to the ones that were previously drawn. This number line has the number ‘0’ on the left side, the number ‘24’ on the right side, and arrowheads pointing outwards on either end.]

And the first thing they did was to take a big jump to subtract 20. So to work out 20 what I'm going to think about is this section here. There's 4 more than the number of tens, and so I'm going to leave the same quantity behind, so that will give me 10. And I can check by measuring. And I'm going to do the same thing where there's 4 extra, so I'm going to do the 4 left behind strategy.

[Presenter points to the 4 differently-coloured cubes at the end of the long row. They then point to the first 4 blue cubes in the middle of the row. Presenter holds down these 4 blue cubes and breaks off the line of cubes that comes after it. In total, they remove 10 cubes from the long row. Presenter takes this line of 10 cubes and holds it up against the line of green cubes to prove that they are the same size. They then put the line of 10 cubes back where they were, but slightly separated from the main row. Presenter creates another group of 10 cubes by holding down the first 4 green cubes and breaking off the line of cubes that comes after it. The presenter now has 4 cubes in a row, followed by 10 cubes in a row, followed by another 10 cubes in a row.]

And that's going to give me a really big mega jump of minus 20. And as you'll see, it leaves 4.

[Presenter draws a large, curved line above the number line. The curved line starts at the number ‘24’ and finishes at the end of the first 4 cubes. Underneath the curved line, they write ‘minus 20’. Presenter makes a small marking on the number line at the end of the curve. Above it, they write the number, ‘4’.]

So we thought this was really interesting. The Strongman, the Robot guys and the Flamingo team had come up with 3 different ways, or different strategies, to think about 23 minus 19 and my challenge for you now mathematicians is how could you use these different strategies?

[Presenter brings the 3 figurines back into view. Text on screen then reads, ‘Where’s my challenge?’]

The blue strategy, the purple strategy and the green strategy, or the green strategy to think about this problem instead.

[Text on screen reads, ‘230 minus 190’.]

Ah, told you it was gonna get a bit more sweaty! Over to you to think about that, mathematicians.

[Text on screen reads, ‘Pause the video here whilst you do some thinking’.]

Welcome back mathematicians. How did you go?

[Text on screen reads, ‘Let’s investigate this a little…’]

Okay, so let's debrief this idea of how we can use what we solved here with these guys, I'll just move them out of the way.

[Screen once again shows the 3 figurines and the big sheet of paper that has all of the working out on it. The orange card at the top of the screen now reads, ‘230 minus 190’. The presenter picks up this card and puts it to the side. Underneath it, the original card that read ‘23 minus 19’ is still there. Presenter picks up the 3 figurines and moves them off screen.]

What we did with the blue team, the green team, oh sorry, the Strongman, the Robot and the Flamingo. And how we could use those strategies in this context, to help us think about this context.

[Presenter points to the original orange card, followed by the new orange card.]

And it really comes down to this idea of renaming numbers. So I know some of you will be looking at this and you'll maybe have thought about a trick that you might have been told once about adding zeros and subtracting zeroes. So let's clarify that for a moment.

Here I've got some paddle pop sticks. I think you can see that. I'll put them on there so it's easier. Okay, I've got 3 paddle pop sticks. Okay, now add zero more paddle pop sticks to my 3. Yes, I still have 3. Now, take, take away zero paddle pop sticks.

[Presenter places three craft sticks on top of a red piece of paper. While the presenter is talking, the amount of craft sticks never changes.]

This is a really cool thing about mathematics. This law actually, that when you add zero or subtract zero it doesn't change what happens. So when we learn this trick of adding a zero, it's mathematically incorrect. What actually you are using here, is this knowledge of place value and renaming. So let's have a look at that.

[Presenter pulls out a piece of paper that has 6 columns on it. From left to right, the columns are labelled, ‘hundreds’, ‘tens’, ‘ones’, ‘hundreds’, ‘tens’, and ‘ones’.]

I've got this little part portion odds chart for you to look at. And look if I have 3 ones over here, we would write the number 3. 1, 2, 3, so we would write 3 here.

[Presenter places the 3 craft sticks into the right-most ‘ones’ column. They then remove the craft sticks to write the numeral ‘3’ in that same column.]

If I move them into here, I actually still only have 3 ones and if I add zero or take zero away, I still just have 3 ones. What I know is that if I want to make them move across into tens where a zero will appear when I write it, like this, then what I need to do to each of these is multiply them by 10, which means these 3 ones become 3 tens.

[Presenter moves the 3 craft sticks into the right-most ‘tens’ column. They then remove the craft sticks to write the numeral ‘3’ in the tens column, and the numeral ‘0’ in the ones column. The presenter puts the 3 craft sticks back on to the red piece of paper. One by one, they trade each of the individual craft sticks for bundles of craft sticks. There are 10 craft sticks in each bundle. The bundles are tied together with an elastic. By the end of this process, there are 3 bundles of craft sticks on the red piece of paper, making 30 craft sticks in total.]

And they would then go into here and so I still have a 3 in my tens column and now a zero in my ones place because the zeros letting me know hey, these 3 are worth 3 tens, not just 3 ones.

[Presenter stacks the 3 bundles of craft sticks into the right-most tens column. Once they are done, they move the bundles back to the red piece of paper. They also bring back the 3 individual craft sticks. Presenter gestures to show the difference in size between the bundles and the individual sticks. They then move the craft sticks and the 2 pieces of paper off screen.]

Okay, so you hopefully were using some sort of strategy around renaming to help you here, and we could do the same with each of these strategies. So let's have a look at the first one. Because what you might have then thought about is, we had these blocks and each one of these blocks was representing 1.

[Presenter brings back the row of blocks that was used for the first solution. There are 10 orange cubes, followed by 10 red cubes, followed by 3 differently-coloured cubes. They place these cubes directly underneath the blue number line.]

You know, we had 23 and we have 23 blocks. What we can think of is that maybe each one of these blocks instead of now being worth 1 is actually worth a 10.

[Presenter begins writing the word ‘ten’ on each cube in black marker. They start with the differently-coloured cubes, and then repeat the process with all of the red cubes.]

Which means these are also all worth 10s. There's 1 of them, 2 of them, a third 10, a fourth 10, a fifth 10, a sixth 10, a 7th 10, an 8th 10, a 9th 10 and a tenth 10. And we do this really cool thing with place value that when we get to 10 tens, we regroup, and we rename it. And so actually, this becomes 100 or is representing 100 and the same, yeah, is happening over here.

[Presenter pulls out a strip of red paper that is the exact same size as the 10 red cubes. On the strip of paper, the presenter has written, ‘1 hundred’. The presenter places this strip of paper on top of the red cubes, covering them entirely. They then do the same thing with the orange cubes – this time, with an orange strip of paper.]

And so now what I'm thinking about is 230 minus 190 or 23 tens minus 19 tens, and we're going to use the same strategy. So the first thing these guys did was think about, you could think about, was instead of 23 minus 10. It's now 22 tens minus 10 tens, which is equivalent to 100.

[Presenter removes the orange card that reads, ‘23 minus 19’. They then point to the card that reads ‘230 minus 190’, and gesture to the parts of each number that match what they’re saying.]

And we could still use the same left behind strategy. With my 3 left behind, but I'm just now saying that's 100 gone and so actually this is now 23 tens, and that's 10 tens, which is 100 and that's gone.

[Presenter begins repeating the same process that they followed for the ‘strong man’ solution – but this time, using hundreds instead of ones. They move the red ‘1 hundred’ strip of paper to the final 10 cubes in the row, and also separate these cubes from the rest of the row. This line of cubes is moved to the bottom of the screen. They also take a yellow sticky note and place it on top of the number ‘23’ on the number line. The sticky note reads, ‘23 tens’. They then add another sticky note next to where ‘minus 10’ was written on the original number line. This sticky note reads, ‘tens’. As a result, the text on screen now reads, ‘minus 10 tens’.]

And then the next thing they did was go by ones, which in this case is 10. So that's 1 ten, another 10, a thrid ten and now I have to, yeah, repartition my 100, a third ten, a fourth ten, a fifth ten, a sixth ten, a seventh ten, an eighth ten and a ninth ten.

[The presenter begins breaking off one cube at a time, just like they did earlier. They count out loud as they do so. When they reach the orange cubes, they remove the orange strip of paper and continue breaking off cubes. They stop when there are 4 cubes left. The presenter writes the word ‘ten’ on top of each of the remaining cubes.

And because these are all worth ten, that means this is worth 4 tens, which means 23 minus 19 equals 4.

[The presenter adds a sticky note next to where the number ‘4’ was written on the original number line. This sticky note reads, ‘tens’. As a result, the text on screen now reads, ‘4 tens’. The presenter then points to the number sentence that was written at the end of the blue number line. Underneath it, they write, ‘23 ten minus 19 ten equals 4 ten’. They then write ‘equals 40’ below this.]

I could say 23 tens minus 19 tens is 4 tens. Which is 40 and we just rename it. So that's one way that we could think about this problem.

[Text on screen reads, ‘Back to you, mathematicians!’]

All right back over to you mathematicians, to think about how you can use renaming to adjust, or think about, your other strategies. Back to you.

[End of transcript]