• MAO-WM-01
• MA3-AR-01

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 five different ways than to solve 5 different problems.

[Screen shows quote.]

And so, to George’s point, we're going to think about how many different ways, in fact, can we think of five different strategies to solve our problem. 23 minus 19. Now I know what you're thinking, ha, 23 minus 19. This is not much of a brain sweat for me yet. Stick with me. Your challenge is coming.

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

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 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 still thinking. This means, I have one possible strategy of thinking through this problem. This means I have another strategy and so on.

[The presenter makes a fist to show they are thinking. Then they keep their hand in a fist and lift one thumb to show they have one strategy. They lift a second finger to show 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.

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

[Presenter shows a ‘strong man’ figurine and places it on the table. They also place an A4 piece of paper in portrait orientation on the table and get a blue marker. The presenter writes ‘23 minus 19’ at the top of the page. From underneath the number 19 they draw one line to the left and write 10 and one line to the right and write 9. The presenter writes ‘23 minus 10 equals 13’ on a new line and ‘13 minus 9’ on a new line under that. Presenter moves the paper with recordings and marker out of the screen.]

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.

And the first thing is, we've modelled our quantity so we have 23. The 2 long towers 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 some connected cubes in rows. There are 10 red connected cubes and 10 orange connected cubes placed in 2 rows, one underneath the other. The presenter points to the '2' digit in the number ‘23’ on the card at the top of the screen. Then 3 more connected cubes are added, one green, one blue and one black. The presenter points to the '3’ digit in the number ‘23’ on the card at the top of the screen.]

And I know these are 10 '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, so now I have my 2 tens, which is what this shows me and my 3 ones.

[Presenter snaps the orange tower of cubes in half, making 2 towers of 5 cubes. They point to chunks of 3 cubes and then 2 cubes on the tower of 5. The presenter then puts the 2 towers of 5 cubes back together to make a tower of 10 connected cubes again. They connect all the cubes together, making one large row with the 10 red, 10 orange and 3 coloured cubes.]

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 mark the end. That's where 23 goes 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 so.

[Presenter holds the horizontal row of cubes still with one hand and uses the marker to draw a straight line the length of the cubes. At the end of the line on the far right they mark a dash and write 23. The presenter extends the line and draws an arrow at the end to show it could keep going. They then mark a dash and write zero at the end of the line on the far left. The presenter extends the line and draws an arrow at the end to show it could keep going in the other direction also. They place the paper with recordings on it under the number line to refer to.]

So, I'm now thinking about where my 10 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 uses fingers to indicate the 3 coloured individual cubes on the end of the row. They then snap off the 3 coloured cubes and 7 red cubes from the row. The presenter draws a curved line or ‘jump’ on the number line, touching from the number 23 to the end of the cubes remaining. They then write ‘minus 10’ near the curved line. The presenter shows direct comparison by lining up the tower of 10 that was snapped off with the 10 orange cubes remaining in the row.]

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

[Presenter draws a smaller curved line on the number line, the size of each cube they take away, as the count back by ones 9 times. They remove the cubes one-by-one until they have drawn 9 small, curved lines or ‘jumps’. The presenter counts the number of cubes remaining and marks the number line with a dash and writes 4.]

So, the 13 minus 9 is 4 and so what we have here is the one 10 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.

So, like George Polya, though we're like, well, let's see what other strategies that we can come up with.

[Presenter refers to the mathematical recordings written on the paper earlier and next to ‘13 minus 9’ writes ‘equals 4’. They show the tower of 10 connected cubes and 9 more individual cubes that were taken away and point to the number 19 on the card at the top of the screen. The presenter writes ‘23 minus 19 equals 4’ to the right of their number line.]

And so, as I reassembled these cubes 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.

[Presenter connects the red and the orange cubes back together into towers of 10.]

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, really 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. 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.

[Presenter shows a ‘fancy robot dancing man’ figurine and places it on the table. They also place an A4 piece of paper in portrait orientation on the table and get a green marker. The presenter writes ‘19 plus something equals 23’ at the top of the page, the something indicated by an underlined blank space. They then write ‘19 plus 1 equals 20’ on a new line and ‘20 plus 3 equals 23’ on a new line under that. Presenter moves the paper with recordings and marker out of the screen.]

And what we wondered about is how we could record that on a number line. 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.

[Presenter connects all the cubes together, making one large row with the 10 red, 10 orange and 3 coloured cubes underneath the other number line. They hold the horizontal row of cubes still with one hand and use the marker to draw a straight line the length of the cubes.]

And here's my number line. 23 with my arrow 'cause that it extends in that direction and zero. And my arrow and what they were saying is that what we what we know is that 23 is here and we need to find a 19 to work out the space between the difference.

And they said, well, since we know this is one 10 and this is another 10, 19. must be here because 19 is one less than 20.

[At the end of the line on the far right the presenter marks a dash and writes 23. They extend the line and draw an arrow at the end to show it could keep going. The presenter then marks a dash and writes zero at the end of the line on the far left. They extend the line and draw an arrow at the end to show it could keep going in the other direction also. Presenter points to the 10 orange cubes and then the 10 red cubes and moves their finger back one cube from 20. They mark a dash on the number line to indicate 19.]

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

[Presenter draws a small, curved line on the number line from the dash at 19 to number 20. They then write ‘plus 1’ above the curved line. The presenter draws another curved line from 20 to 23 on the number line. They write ‘plus 3’ above the curved line. The presenter snaps off the last 4 cubes and moves them to the start of the blue number line to show that it is the same amount. They refer to the mathematical recordings written on the paper earlier. The presenter writes ‘19 plus something equals 23’ to the right of their number line and writes ‘19 plus 4 equals 23’ underneath.]

That was their solution 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. 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 shows a ‘flamingo’ figurine and places it on the table. They also place an A4 piece of paper in portrait orientation on the table and get a purple marker. The presenter writes ‘23 minus 19 equals 24 minus 20’ at the top of the page. They then write ‘equals 4’ underneath on a new line.]

And we're 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 holds the flamingo and robot figurines and pretends to talk between them.]

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

[Presenter connects all the cubes together, making one large row with 23 cubes – 10 red, 10 orange and 3 coloured cubes. They then make another large row underneath with 24 cubes – 10 green, 10 blue and 4 coloured 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. 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 removes the tower of 23 cubes and holds the horizontal row of 24 cubes still with one hand. They use the marker to draw a straight line the length of the cubes underneath the other number lines. At the end of the line on the far right they mark a dash and write 24. The presenter extends the line and draws an arrow at the end to show it could keep going. They then mark a dash and write zero at the end of the line on the far left. The presenter extends the line and draws an arrow at the end to show it could keep going in the other direction also.]

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. And I can check by measuring.

[Presenter locates where the 2 towers of 10 cubes connect, moves their fingers along 4 cubes and snaps the tower off. They check there are 10 cubes by comparing it with the 10 green cubes at the start of the row.]

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

[The presenter then snaps the first 4 cubes off and connects the rest of the cubes together. They draw a curved line or ‘jump’ on the number line, touching from the number 24 to the end of the 20 cubes where they have snapped off the 4. They then write ‘minus 20’ near the curved line. The presenter marks a dash at the end of the curved line and writes 4.]

So, we thought this was really interesting the, the strong man, the robot guys and the flamingo team had come up with 3 different ways or different strategies to think about 23 minus 19.

[Screen reads – Where’s my challenge?]

And my challenge for you now mathematicians is how could you use these different strategies?

[Screen shows the 3 different figurines and their number lines with different recorded strategies for the problem 23 minus 19.]

The blue strategy, the purple strategy and the green strategy, or the green strategy to think about this problem instead: 2 and 3 tenths minus one and 9 tenths.

[Screen reads – 2.3 minus 1.9]

Over to you mathematicians. Plus, remember George Polya, 5 ways to solve one problem. You can try these 3 strategies, but you need another 2.

Over to you!

[Screen reads – Pause the video here whilst you do some thinking.]

Alright, welcome back mathematicians. How did you go?

[Screen reads – Let’s investigate this a little …]

I posed this problem to you is how could you use these strategies to think of solving this problem? And then I said as well, can you think of 2 other strategies to solve it?

Alright, so let's talk about how we can use thinking about 23 minus 19 to help us solve 2 and 3 tenths minus one and nine tenths.

[Screen shows the number lines with different recorded strategies for the problem 23 minus 19. Presenter points to the orange card at the top of the screen which reads ‘23 minus 19’. They then place the new problem ‘2.3 minus 1.9’ to the right of the other problem card.]

And the first thing that we need to think about for a moment is place value. So, I've got this little chart here to help us to start with.

[Screen shows a white A4 piece of paper in a landscape orientation. The page is divided up into different columns. They have drawn 2 vertical lines and one horizontal line to make 3 columns in the first row. The first column is labelled ‘thousands’, the next column is labelled ‘ones’, and the last column is left blank. Another horizontal line has been drawn to make a second row. They have labelled the column underneath the thousands as ‘ones’. The column underneath the ones in the first row has been divided up again by drawing another 2 vertical lines. They have labelled these columns ‘hundreds’, ‘tens’ and ‘ones.’ They have divided up the last blank column by drawing another vertical line. These last two columns are labelled ‘tenths’ and ‘hundredths.’ Between the ones column and the blank column there is a big black dot on the line which represents the decimal point.]

Well, I think what you might have like a strategy that you might have heard or someone might have said she before. Is this idea of well just move the decimal point. And so, the first thing to talk about is the fact that this decimal point never actually ever moves.

So, in mathematics the decimal point, this thing here, that separates the whole numbers from the fractional numbers is always between the ones and the tens, always and it's always there because it represents the shift from working with whole quantities to now having to work with fractional quantities.

[Presenter points to the decimal point on the chart. They then point to the 2 decimal points on the problem card at the top of the screen. The presenter shows that the decimal point separates the ‘whole numbers’, those to the left of the decimal point from the ‘fractional numbers’, those to the right of the decimal point. They point to the columns labelled ‘ones’ and ‘tenths’ showing that the decimal point lies between the 2 columns.]

So, for example, if I wanted to represent 2 and 3 tenths, I have a bit of a problem in that my paddle pop sticks are hard to re-partition into tenths, and so I'd like to invite you into my mathematical imagination and here's the one that I need to partition into 10 smaller parts or to tenth it.

[Presenter places 2 craft sticks in the 'ones’ column and is unable to re-partition the last craft stick into tenths.]

So, I'm going to make it bigger so that you can see what's happening, and then what I do is I imagine tenthing it. So, for me, the first thing I do is half. And then for each half. Half I've made. I then fifth each one and that now gives me 10 equal size parts, so I've tenthed it and I only need 3 of them for the number that I'm making and I'll shrink them back down to size and I can put them now back into my place value chart. And I can see here now that I have 3 tenths and my number is 2 and 3 tenths.

[The screen then changes as they move to the presenter’s mathematical imagination and shows a white screen with a single craft stick. A short, red vertical line appears halfway down the craft stick to show half. Then 4 short, blue vertical lines appear evenly spaced in each half, making 10 equal size parts on the craft stick. The presenter moves 3 of the tenth parts of the craft stick to the ‘tenths’ column on the place value chart, to the right of the decimal point. Presenter places 2 full craft sticks in the ‘ones’ column, to the left of the decimal point. The screen shows the number 2 and 3 tenths represented by craft sticks.]

And now what I want to do is actually multiply each of those by 10 so that I can get to a whole number. And I like whole numbers 'cause my brain feels more comfortable working with them. So, if I multiply each of those collections or each of those sections by 10. I end up with 2 tens and 3 ones, otherwise known as 23, and you'll notice that the decimal point has not moved.

[Presenter multiplies the number 2 and 3 tenths by 10 and the number becomes 23. The 2 craft sticks change to 20 craft sticks in 2 bundles of 10 in the ‘tens’ column. The 3 tenths change to 3 whole craft sticks in the ‘ones’ column. The ‘tenths’ column, after the decimal point, is now empty.]

And so, from this collection now I know that I need to remove 19 and so if I have 2 bundles of 10, that's 20 and if I get rid of 19. It means I just have one paddle pop stick left, which means I have 4 in my collection and back into my place value chart. I now have 4 ones and what I need to do is divide each of those by 10.

[The place value chart disappears, leaving 20 craft sticks in 2 bundles of 10 and 3 single craft sticks. The 20 craft sticks disappear, and one single craft stick joins the other 3, making 4 in total. The place value chart reappears, with the 4 craft sticks in the ‘ones’ column to the left of the decimal point.]

And what that leaves me with is 4 tenths, and as you'll notice, the decimal point has not shifted at all.

[Presenter divides the 4 whole craft sticks by 10 and the number changes to 4 tenths. The 4 whole craft sticks disappear and the 4 tenths parts appear in the ‘tenths’ column to the right of the decimal point.]

But what I've done is multiplied my collection to work with them.

[Screen reads – What was the mathematics?

On the left, the screen shows a screenshot of the number 2 and 3 tenths represented by craft sticks on a place value chart. Underneath is the number 2 and 3 tenths in written form and numeral form. On the right, the screen shows a screenshot of the number 23 represented by craft sticks on a place value chart. Underneath is the number 23 written as 2 tens and 3 ones and in numeral form. Between the two pictures are two big arrows pointing to the right, from the 2 and 3 tenths representation to the number 23 representation. One big arrow has ‘times 10’ written in it and the other has ‘make 10 times bigger’ in it.]

And then, when I finished doing my operating, I've divided the remaining quantity by 10 to make sure that I'm keeping on the same quantity. And it's a fair outcome.

[Screen reads – What was the mathematics?

On the left, the screen shows a screenshot of the number 4 represented by craft sticks on a place value chart. Underneath is the number 4 written as 4 ones and in numeral form. On the right, the screen shows a screenshot of the number 4 tenths represented by craft sticks on a place value chart. Underneath is the number 4 tenths in written form and numeral form. Between the two pictures are two big arrows pointing to the right, from the 4 ones representation to the 4 tenths representation. One big arrow has ‘divided by 10’ written in it and the other has ‘make 10 times smaller' in it.]

So, that's one way that you can use something like 23 minus 19 to help you work out 2 and 3 tenths minus one and 9 tenths.

And now what I wonder is how could you use those other strategies? And remember you still need 2 more.

So back to you mathematicians!

[End of transcript]