Apply the generalisation about how the minuend and difference change to solve problems

Key Stage 2

Maths

Extending calculation strategies and additive reasoning

﻿Hi everyone, Mr Easter again I'm going to take you through today's lesson let's start by looking at the previous lesson's practise activity. Now you asked to look at both of these pairs of calculations and use the same sentences to describe what is happening. So the first one, 16 subtract 14 is equal to two and then 18 subtract 14 is equal to four. Does our minuend increase or decrease? It increases, doesn't it? So let's use the sentence down on the left. I've added two to the minuend and kept the subtrahend the same, so I must add two to the difference. Hope you said that one with me, make sure you say the second one with me. So let's think about it first, 172 subtract 138 is equal to 34 and then 152 subtract 138 is equal to 14. So does my minuend increase or decrease? It decreases this time. So we need to use a sentence stem on the right. I've subtracted, what's happened? Let's look at the digits. The one in the hundreds column has stayed the same or the tens digit has decreased from seven to five and the ones digit has stayed the same, so it's decreased. How much the decrease by? By two 10s, so that's 20. So let's say this together, I've subtracted 20 from the minuend and kept the subtrahend the same, so I must subtract 20 from the difference. Did you get that? Sure you did. Okay, so let's think about a math story, so a context that we can use to describe what's happening, and I suggested, think about children in a class. So the, for the first pair of calculations, let's think about children in a class you will have boys and girls, so maybe the sentence could be about this. It could be maybe there was 16 boys in the class, there were 14 girls in a class. So the difference would be two. And then maybe two more boys joined the class, the number of girls stayed the same, so the difference would also increase by two. Now let's look at the next one. We've got 172 subtract 138 is 34, and I suggested a length of ribbon, so if we're finding the difference, that would mean maybe we have two different pieces of ribbon, maybe two different colours, and if it's decreasing, if the minuend is decreasing, then maybe the length of ribbon would decrease in size, one of them would become smaller, the other one would stay the same size, so let's look at that. I've written this problem as a word problem. So here is the word problem I have written. A blue ribbon was one metre, 72 centimetres long, and the red ribbon was one metre, 38 centimetres long. So the blue ribbon was 34 centimetres longer than the red ribbon. We knew all of that information from the calculation that we were given. Then the blue ribbon was cut down to one metre, 52 centimetres. How much longer than the red ribbon is the blue ribbon? Now that's my word problem. Let me show you that as a representation. So we have our blue ribbon is one metre 72 centimetres. We had our red ribbon was one metre, 38 centimetres. Where are we going to put this bit of information? The blue ribbon was 34 centimetres longer than the red ribbon. It's a little bit like our bar chart in the previous lesson. It's going to be in the space between the end of the red bar and the end of the blue bar, it's going to go into this section and we're told it's 34 centimetres long. Now I'm going to show you the calculation you were given. Can you notice something slightly different? The calculation says 172, but I've written one metre 72. Am I wrong? Some of you might be saying that they are equivalent to each other 172 centimetres is the same as one metre 72. I've just chosen to represent it slightly differently. It's the same because in a metre there's a hundred centimetres. The next thing we're told is then the blue ribbon was cut down to one metre 52, so watch what happens to my blue ribbon. It becomes smaller and we now know one metre and 52 centimetres long. Does that difference stay the same? Is it still 34 centimetres? Look on our representation, 34 centimetres and the space that it shows on our representation was larger than it is now, so because we've decreased the minuend we also have to decrease the difference. So we need to show what that is. We know it's this smaller amount and can you remember what it was? Can you remember the previous slide? If you can't see if we can work it out, let's look at our minuends again, 172 and the first calculation, 152 in the second calculation. It's decreased by 20. The subtrahend has stayed the same, so our difference needs to decrease by 20, so it's 14 centimetres. That's a look at that, and one more way, let's look at it vertically like we did the four, that was our first calculation. We've just said that the minuend decreased by 20, the subtrahend stayed the same, so that means our difference must also decrease by 20, and that was a completed calculation that you were given in the practise activity. So in today's lesson, we are still using the same generalisation. If the minuend is changed by an amount and the subtrahend is kept the same, the difference changes by the same amount. Here's the first calculation, luckily there's nothing for us to solve cause we're given the complete calculation. 62,865 subtract 41,294 is equal to 21,571. Now we're going to use that calculation to help us solve a different calculation. In this calculation. Do we know the minuend? Yeah, it's 63,865. Do we know the subtrahend? Yep. I can see that. We don't know the difference. So let's think about how the minuend or the subtrahend all the difference changes. Does the subtrahend change? No, the subtrahend we can see is 41,294 in both calculations. The minuend however, does change. Let's look at the digits in each of our place value columns to help us work out what is changed by. Let's start with a ones. We have the digit five in both columns that hasn't changed. What about our 10s? We have the digit six in both columns. What about the hundreds? We've got the digit eight in both columns. What about the thousands? Oh, I can notice a change. It's changed from the digit two in the thousands column to the digit three, so it's increased by one, 1000. What about our 10,000 column? That's the same as well, so that means our minuend has increased by 1000, we set our subtrahend to stay the same. So what happened to our difference? It will change by the same amount, I can see that in our generalisation, so that will also increase by a thousand. So let's think whether one's column change, nope, that will stay as a one or tens or hundreds. They'll both stay the same, our thousands column will increase by one, 1000, so the digit one will become the digit two, so our difference is 22,571. Okay, now we have another calculation. Now I've kept both of the calculations that we know on there, cause actually we can use either for our new calculation. Let's think what's changed. We've got 60,865 subtract 41,294. First thing I noticed is the subtrahend is the same again. That must mean the minuend has changed if there's a different difference. Okay, the minuend has changed. Let's look at the digits again. The ones digit is the same, the tens digit is the same, the hundreds digit is the same, the thousands digit is this. No, is not the same. It's changed. We'd had the digit two and the thousands column, we now have zero. So has the minuend increased or decreased? This time, it's decreased. We know that because the 10,000 column is also the same, so it's decreased by 2000. So what's going to happen to our difference that needs to also decrease by 2000. Let's think carefully about that. There's a digit one in the thousands column for us difference at the moment for instead, I'm going to think about this 21 thousands. The hundreds tens of ones is going to stay the same, that's still going to be 571, but what's 21,000 subtract 2000, 21 subtract two is 19. So if we subtract 2000, our difference will become 19,571. Okay, so I kept those three calculations now on the screen, now we've got a fourth calculation to solve again. What do you notice straight away? Or this time there's a different unknown, this time we don't know our minuend. We're told us after hand and we can see that stayed the same, we're told our difference. Our differences changed. Now you could use any of the other differences to help us, but, the one I can see the most things are similar, again is our top calculation, the bold calculation. I can see 21,571 and then our new calculation to solve, I've got 31,571 a hundreds, tens, and ones stayed the same, our thousands column is stayed the same. Our 10,000 column has changed, it's increased by one, so therefore, we've added 10,000 to our difference. To how's that going to affect our subtrahend. It's not, is it, we can see that's the same. What about our minuend? The minuend is going to change as well. It's going to change by the same amount, that's from our generalisation. So what is 62,865, add 10,000 the digit and the 10,000 column is six, that's going to increase by one, so it's going to be 72,865. Final calculation for us to think about, all four there see if they can help us. Let's see what's the same, what's different. Again, you've got a minuend that we don't know, the subtrahend stayed the same, the difference has changed. I could use any of them. I think about the calculation we've just done. The 10,000 columns increased by one, thousands is the same, the hundreds is the same, the tens is the same, the ones changed. So two columns have changed. I think I can see how they're both changed, but I wonder if there are fewer changes in any of the other differences. Can you see in the bold difference? 21,571 and now we've got 21,570. What's the difference between those two differences? The ones column has changed. The one's column has decreased by one. So what's going to happen to our minuend? That also has to decrease by one. So what is one less than five and one's column? It would be four, so our minuend would be 62,864. Okay, so another calculation with another unknown. We have our first calculation in bold 23.18 subtract 0.82 is equal to 22.36. Again, thinking about what stayed the same and what's changed for our calculation with the unknown. The minuend, I can see that's changed. The subtrahend hasn't changed how many to work out the difference. So let's think about how much our minuend has changed by. Let's look at our hundreds column, that's eight that stays the same, our tenths column has stayed the same, our tens and our one's columns have changed. So we can just think about 23 and 46 it's increased. How much has it increased by? Our ones column has increased by three, our tens column has increased by two, so that would mean it's increased by 23. So we know by now that means we have to increase our difference by the same amount. So 22.36, if we increase that by 23, our hundredths and our tenths column will stay the same,` our tens column will become two greater because we're adding two more, one 10s, and our ones column will increase by three cause we're adding on three more ones. So it will be, 45.36. Okay, now we know our generalisation. We've got good at that and we know that in a sequence of calculations that contain an unknown, we can use previous known values to help us work out the unknowns. I think you can have a good go of this. So I'm going to pause the video and see if you can work how the three unknowns, not just by using a written method, but by thinking about our generalisation of if I changed the minuend by an amount, and if the subtrahend stays the same, then we will change our difference by the same amount. So have a go at that for me. So let's start with our first calculation, we got 23.28 subtract 0.82 is equal to something. We could use our first calculation that we know all the values for, or our second one. My choice would be using the first calculation. Let's look at the minuend. This data a hundreds column, eight hundreds has stayed the same, one 10th has changed a bit to tenths, and then my ones and my tens has stayed the same with the number 23. That means, I know that my minuend has increased by one 10th, so my difference needs to increase by one 10th. That will be 22.46. Let's look at my next one. Okay, 123.58, 0.82. Okay, I still have to state the same. What about our minuend? More than one things changed. So I've got some different choices. I could use the calculation I've just done cause I can through my tens and my ones are the same and the tens and ones the same here is increased by a hundred. So does that mean my difference increases by a hundred? No. Did someone say no? What else has changed? The tenths has also changed. It's increased from two tenths to five tenths. So it's increased by 103 tenths, so that would be 100.3. So what's 100.3 more than 22.46? Let's think. Our hundredths digit will stay the same as six, my 10th digit will increase from four to seven, my ones will stay the same, my tens will stay the same, my hundreds were increased by one, so 122.67. Some of you might've used this first calculation. That's fine as well. Let's think about what's changed. Our hundredths digit hasn't changed, our 10th digit has increased by four tenths, our tens and our ones have stayed the same and our hundreds column has increased by 100. So that would mean my difference would increase by 100.4 and yet we've got one in our hundreds column and our 10th digit is increased from three to seven. So we can use either of those calculations to help us. Okay, let's think about this one, 3.18, subtract 0.82. Which calculation do we want to go for? Did you go for the top calculation? Maybe some hands up at home. The second calculation, probably not, quite a lot has changed there. This calculation, possibly, this calculation, maybe quite a lot has changed. I think again, actually our top calculation is the most useful. Let's think about what stayed the same in our minuend and what's changed. My tenths and hundredths has stayed the same because we've got 18 hundredths here. Let's look at our tens and ones we've gone from 23, oh there's no tens here, just three. So our tens digit has decreased from two to zero to not be in there, that means it's decreased by two tens, which is 20, that means our difference needs to decrease by two tens. So that means the difference would be 2.36. Again, you could have used the second, third or fourth calculations to help you, that would mean the minuend difference would change by different amounts of what we just said, but as long as they both changed, by the same amount, you should still get the right solution. Okay, so your independent work is still using that same generalisation. Let's read it one more time together. Come on with me. If the minuend is changed by an amount and the subtrahend is kept the same, the difference changes by the same amount. You've got one known calculation at the top, which is 52.13 subtract 3.76 is equal to 48.37. Think about what's changed and what stayed the same, and then think about in the other calculations. Are you going to work from the bowl calculation or you're going to work out work from, sorry, one of the ones you've just worked out? And then your final challenge, which is shown in the red box. Can you see at the bottom here? We've got three unknowns. Now this doesn't mean just make up your own calculation. I want you to see if you can continue the pattern, continue the sequence of calculations. Think about in all of them. Is there anything that stayed the same? And then think about, maybe you can choose what you will change the other two things by. So it's not just a totally free question, there's some form of sequence. So think about our generalisation and what you know needs to stay the same. Good luck and we'll go through those in the next lesson.

Apply the generalisation about how the minuend and difference change to solve problems

Apply the generalisation about how the minuend and difference change to solve problems

Lesson details

Key learning points

Licence

Video

Lesson appears in

UnitMaths / Extending calculation strategies and additive reasoning

Maths