• MAO-WM-01
• MA1-CSQ-01
• MA1-RWN-01
• MA1-RWN-02

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 reads: It is better to solve one problem 5 different ways than to solve 5 different problems.]

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.

[Screens shows a card which says 23 minus 19.]

OK, so what I'd like you to think about is what is one strategy that you could use to solve this problem?

OK, 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?

In a classroom we might use hand signals like this a fist, [DESCRIPTION: Presenter makes a fist.]

This means I'm still thinking thumbs up. [DESCRIPTION: Presenter shows a thumbs up.]

This means I have one possible strategy of thinking through this problem 2 fingers up and thumbs up [DESCRIPTION: Presenter shows 2 fingers up and a thumbs up.] This means I have another strategy and so on.

OK, 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.

[Screen shows the equation 23 minus 19 at the top of the screen. The presenter begins by partitioning the number 19 into tens and ones. She writes the number 19 and draws two lines separating from the number. Underneath the left line she writes 10 to represent the tens value and underneath the right line she writes 9 to represent the ones value in the digit 19.]

And they said really, 19 is made up of 10 and 9. So we could think of 23 minus 19 as 23 minus 10.

[Screen shows presenter writing the number 23 and the minus symbol next to her partitioning of the number 19. Underneath this she writes the equation 23 minus 10 equals 13. Underneath the first equation she writes 13 minus 9.]

And they said that was 13. And then 13 minus 9 and they said that what they would do is subtract the ones by using the jump strategy.

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

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.

[Presenter places a tower of 10 red connecting cubes, a tower of 10 orange connecting cubes and a tower of 3 connecting cubes comprised of a green, a blue and a black cube horizontally on the table.]

And the first thing is, we've modelled our quantity, so we have 23. The two long sticks here are each 10.

That is what this number here represents and the 3 here is what this number represents, in the, in the number.

[Screen shows the presenter moving the 2 towers made of 10 cubes into a vertical position and places them side by side. She traces along them before pointing to the number 2 in the tens place of the number 23, indicating that the two rows add up to the number 20. The presenter then picks up the tower of 3 before pointing to the number 3 in the ones place of the number 23 to indicate that the tower represents the number 3.]

And I know there's a 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.

[Screen shows the presenter snapping the tower of 10 connecting cubes in half. She places the tower made of 5 onto the table. The presenter circles a chunk of 3 bricks within the tower, then the chunk of 2 bricks within the tower. She holds up 2 fingers, and then 3 fingers and shows her hand to signal the digits adding to 5. The presenter then connects the two smaller towers together to signify them adding up to 10.]

So that has to be 10 bricks high, and if I line that up. That's also 10, so now I have my 2 tens, which is what this shows me and my 3 ones.

[Presenter now joins all three towers of bricks together in one horizontal line.]

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

And yeah, we've been using them almost like a measure, and if I come here and carefully marked the end, that's where 23 goes.

[Screen shows the presenter creating a number line. She uses the length of the bricks as a ruler to guide how long her number line should be. The presenter writes the number 23 at the right end of her number line, along with an arrow pointing to the right, and then the number zero on the left end of her number line with an arrow pointing to the left.]

And actually, my number line could keep going if I wanted. And this is where zero would be and also, my number line would keep going in the other direction, and what the strongman team said that they did was the first thing was, they got rid of one jump of 10.

[Presenter points to the first chunk of 10 bricks on the right side.]

So, so I'm now thinking about where my ten is and I know there's three here.

So, if I go with the three left behind strategy. That will be a jump of 10 and I can prove that by using direct comparison.

[Screen shows the presenter breaking off the first 10 bricks from the right side of her connecting cubes. On her number line, she craws a curved line to signify the jump or subtraction of 10. Underneath the curved line, minus 10 is written. The presenter moves the connecting cubes underneath the first set of 10 connecting cubes on the left hand side. She compares their measurement to prove that she has taken away 10. The presenter then moves the connecting cubes to the right.]

And then they said, now we would count back by ones, 9 times.

So, can you help me keep track of the count?

Okay, that's 1. 2. Three, whoops. 4. 9, which leaves? 4, so the 13 minus 9 is 4.

[Screen shows the presenter removing one cube at a time. As each cube is removed, the presenter draws a curved line to represent the subtraction of the cube. When 9 cubes have been subtracted, a line is drawn where it stops and the number 4 is written to signify how many cubes are left over.]

So, what we have here is the one 10 and the 9 more of 19 and I can record the strong man's teams thinking over here as 23 minus 19 is equivalent in value to 4.

[Presenter gathers the 19 cubes. She pushes the tower of 10 cubes to the left and the 9 single cubes to the right. At the end of the number line she writes the equation 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 reassembled 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're 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. And this team, the green team, we'll call them, thought, re-thought about the problem and they said, well, actually, when you're solving subtraction, you can just think addition.

[Presenter reassembles bricks as they were at the beginning of the video, and picks up a toy roboman. She places the toy at the top of the page with a green marker.]

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 writes the number 19 on a new piece of paper, a plus symbol, a blank space, an equals sign and then the number 23.]

They said then what they would do is 19 plus one 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.

[Screen shows the presenter writing 19 plus one equals 20. Underneath she writes 20 plus 3 equals 23.]

So, this is what we came up with and we said, “Well, we could use our 23. And I'm going to try to line them up so that you can see them.”

[Screen shows the presenter reassembling the cubes together. She lays them down horizontally and uses them as a ruler to create her number line. On the right end of the number line, she writes the number 23 and an arrow pointing to the right. On the left end of the number line, she writes the number zero and draws an arrow pointing to the left.]

And here's my number line. There's 23 with my arrow 'cause it extends in that direction and zero and my arrow.

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.

[Presenter points to the end of the cubes.]

And they said, well, since we know this is 1 ten and this is another ten, 19 must be here because 19 is one less than 20.

[Presenter points to first chunk of 10 and then the second chunk of 10.]

That's right, and then they added one. And then they added 3 more.

[Screen shows the presenter measuring the blocks. When she gets to the 19th block, she draws a small, curved line to signify the addition of one more to create twenty. Above this line she writes plus one. She then draws a bigger curved line to show the addition of 3 more. Above this line she writes plus three.]

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.

[Screen shows the presenter breaking off 4 cubes. She moves the cubes against the first number line to show that the cubes take over the numbers zero to 4 and that they are equivalent amounts.]

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.

[Screen shows the presenter writing the equation 19 plus a blank digit equals 23 next to the second number line. Underneath the first equation she writes 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.

[Presenter removes Roboman and replaces him with Flamingo.]

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.

[Screen shows the presenter reassembling the cubes into 2 towers of 10 cubes and a smaller tower of 3 cubes. She uses a new piece of paper to write the equation 23 minus 19 is equivalent to 24 minus 20. Underneath the presenter draws an equals symbol followed by the number 4.]

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.

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 Robot and Flamingo and implies they are having a discussion.]

So, this is what happened. Because, because what the robot team and the strong man team were wondering about is that if this is 23 and if I now make a collection of 24.

[Presenter reassembles cubes into one horizontal row.]

You know this, this tower is one block more than this one, so how does this work?

[Screen shows the presenter assembling a horizontal row of 24 cubes which she places underneath the horizontal row of 23 cubes.]

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.

[Screen shows the presenter removing the tower of 23 off screen.]

And this time we're starting at 24, but again, our number line can continue in this direction. And this is where zero is.

[Screen shows the presenter creating a new number line using the tower of 24 cubes. On the right end of the number line, she writes the number 24 and draws an arrow pointing to the right. On the left end of the number line, she writes the number zero and draws and arrow pointing to the left.]

And it continues in this direction. 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 10s, and so I'm going to leave the same quantity behind. So that will give me 10.

[Screen shows the presenter pointing to the last 4 cubes on the right then points to the next 10 cubes. She removes 10 cubes and moves them under the first 10 cubes to show that they are equal.]

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 removes the next 10 cubes from the right side, leaving 4 cubes on the left side.]

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

[Screen shows the presenter drawing a large, curved line from the start of the subtracted bricks to the end. At the end of the curved line, she writes the number four. Underneath the curved line she writes minus 20.]

So, we thought this was really interesting.

The, 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.

[Presenter removes bricks and places Flamingo, Roboman and Strongman in the center.]

But if we're going to embrace our inner George Polya, I wonder if there's another two strategies that you guys could come up with.

[Screen reads: What was the mathematics?

We found 3 different strategies we could us to solve the same problem.. can you find another 2?

This is important because as mathematicians, we need to be able to look to the context of the problem and make decisions about which strategy to use (as Cathy Fosnot said).]

Over to you mathematicians.

[End of transcript]