Practice: Transforming Calculations to Make Them Easier to Solve Mentally

Key Stage 2

Maths

Extending calculation strategies and additive reasoning

﻿Hi, my name's Mr. Coombs and I'm here to guide you through lesson 15. Just like in previous lessons, you're going to need a pen or a pencil and a piece of paper to complete some calculations. So if you haven't got one, just pause the video, go and grab one and come straight back. Let's start by looking at the practise activity that I sent you at the end of the previous lesson. Remember the stem sentence? Excellent, well, that's what we were trying to do. We were trying to find a way that we could change the minuend and subtrahend in order to make the calculation easier to complete. So let's have a look at this first one. It was asking us to find the difference between 35,621 and 25,478. So I was trying to change the subtrahend to a multiple of ten, or a multiple of a hundred, or even a multiple of thousands as it's a five digit number in order to make that calculation easier to complete. So I looked and I realised that actually I could easily change that into a multiple of 100 that I think I'd be able to calculate mentally with. So I added 22 to that to get to 25,500. Now you told me you could remember the stem sentence. So let's say it, "If the minuend and the subtrahend are changed by the same amount, then the difference stays the same." So remembering that, I know I need to do exactly the same to the minuend. So I need to add 22 and that creates my new calculation. And I can easily complete that new calculation because I can look at the 35,000 and subtract the 25,000 to give me my 10,000 and I can subtract the 500 from the 643 to give me my difference of 10,143. And because I changed the minuend and the subtrahend by the same amount, I'm confident that the difference is the same. So I can go back to that original calculation and confidently say that the difference between the two numbers is 10,143. Well done. Did you get it? Excellent. So I'm going to give you the answers for the next two. But actually, what I want you to see and think about is how did you do it? So 74,123 subtract 32,088, the difference between those two numbers is 42,035. Then the second one, the difference between the two numbers is 33,250. But remember math isn't all about the answers. It's about seeing the math. It's about understanding how can we make things easier for ourselves? So let's see how you did it compared to how I did it. Again, I just want to reinforce the importance of understanding that it's okay if you did it in a different way to how I've done it. If you made that calculation easy to complete, then well done. That's exactly what we were looking for. Now when I first looked at this, I thought, "Oh, I'll subtract 88 from the subtrahend to get down to 32,000." But then I thought, "If I subtract 88 from the minuend, it's not actually that easier of a calculation to complete instantly." So instead I added 12 to get to a multiple of a hundred. So once I've done that, I could really quickly and confidently add 12 to the minuend to find my new calculation. Now let's look at that calculation. Do you think you can solve that really simply? Yeah, we can. So we've got our difference between it and because we did the same to the minuend that we did to the subtrahend, the difference has stayed the same. So we can go back to that original calculation and confidently say that the difference is 42,035. How did you complete this final calculation? I looked at the subtrahend and I decided that I needed to get it to a multiple of ten or a hundred or a thousand to try and make it easier. And I quickly saw that if I added four to the subtrahend, it got me to 3,250. And I made sure that that did make this calculation easier to complete. So I looked at my minuend and saw the 496 and realised that I could mentally and quickly and confidently subtract that from it. So I added my four to minuend to create my new calculation. And because I did the same to the minuend, as I did to the subtrahend, I kept my difference the same, so could answer my original calculation. So it's really important you keep that generalisation, that stem sentence in mind for today's learning. Because, "If the minuend and subtrahend are changed by the same amount, the difference stays the same." What do you notice about the calculation on the screen? How's it different to the ones we've looked at? Yeah, we're now looking at numbers involving decimal fractions. But does that actually matter? Can we still use our understanding of the same difference to make this calculation easier to calculate? Well, how? What could we do to the minuend and the subtrahend to make them easier? Yeah, make them a whole number. That really would help us. Now, which one do you think would be the best? The subtrahend's the one we've been using most of the time. So let's explore that. What could I change 4.94 to, to make this question easier to calculate? I could add to make it five. I could add 0.06 to make my subtrahend five. I can subtract five from a number. So as long, as I do exactly the same to the minuend, I am able to create my new calculation. We look at that and we've got our 19 ones, I subtract my five ones to give me 14, and I've still got my point seven eight. 14.78 is the same difference. So I can bring that to my original calculation. And I do that by remembering that stem sentence, "If the minuend and the subtrahend are changed by the same amount, the difference stays the same." Here's another calculation involving our decimal fractions. Now, I'm sure lots of you, are seeing exactly what you're going to do straight away. Pause the video, complete the calculation, and then I'll show you how I did it. Remember, there's always more than one way, so it's absolutely fine if you do it in a different way. How did you get on? Let's see if you did it the way that I did it, because I saw that subtrahend being really easily changed to become an integer, to become a whole number. So I subtracted by 0.08 to get it back to four. And I remembered the stem sentence, "If I change the subtrahend, I need to change the minuend by the same amount." And that's exactly what I did to create that new calculation. And then I was confident in being able to subtract the four ones from the nine to get me my five. And I've still got my point six one, my 61 hundreds. And because I did the same to the minuend that I did to the subtrahend, I know the difference has stayed the same, so I'm able to answer it. Did you do it like that or did you do it a different way? It's absolutely fine. Whatever works for you. Because, as long as we remember that stem sentence, this generalisation works. A little challenge for you now. In a moment, I want you to pause the video and I'd like you to complete the calculation in at least two different ways. And after you've done it, I'd like you to reflect on which one was the easiest? Which one involved the easier calculations to get all the way through to the answer? How did you get on? Did you do it in more than one way? Okay, well, my first instinct was to look at the minuend this time, because if I subtracted 0.15, it gets me to 11 and dealing with whole numbers, as we discussed it earlier, makes it easier. So I needed to do the same to the subtrahend, which I did. And now I created my new calculation. But actually I looked at this and thought, "How easy is that?" Now for some of you, you'll be confident. You'll be able to find the difference very, very quickly. And that's great. So that would have been a really good way for you. But others might not be as confident. So they need to find a different way. But remember, it's not just the minuend we can change, it's the subtrahend. And actually for our finding the difference, our subtraction calculations, it's been the subtrahend we've been changing more often than not. So how could we change this subtrahend? Well, if we added 0.33, we get to eight. And because I've done that to the subtrahend, I need to do exactly the same to the minuend to get to 11.48. And now I looked at the new calculation that we created and felt really confident in being able to answer that. So therefore, really confident in being able to answer my original calculation. So remember, it might be that the first thing you try, isn't actually the most effective way and that's fine. But the more of these calculations you do, the better you become at them. And you start to see which way is the best way. On the screen now you'll see five related calculations. But before we start, I want you to pause and decide which one would you start with? Would it be the top one? Or is there actually another one that's going to make this whole process a lot easier to complete. Is one jumping out at you? I know it did me. And why is that? Well, it's a multiple of a thousand and subtracting multiples of thousands is a lot easier than looking at the other numbers. So, why? How would you complete that calculation? I would do it by looking at the 121,000 subtract 11,000. The 21,000 subtract the 11,000 gives me my 10,000. So, I now have 110,374. How would I now use this to complete the others? Yeah, I would use that stem sentence. That stem sentence we've been working on. "If the minuend and the subtrahend are changed by the same amount, the difference stays the same." Pause the video, have a look at how those minuend and subtrahend are changing from one calculation to the other. And use that to be able to answer them. I'm pretty sure you'll do that really quickly. Did you spot what was happening? Did you use the stem sentence? It's helping me a lot, so I'm sure it's helping you. Let's look at the one underneath the one we've already completed. You'll see that the minuend has increased by one and the subtrahend has increased by one. So because they both, changed by the same amount, I knew the difference was staying the same. Now let's look at the one above the one we completed first. You'll see, that actually, the minuend and the subtrahend decreased by one. And because it's the same amount, I know that the difference has stayed the same. And I can look at the top calculation now. I could use the one I've just completed, or I could go back to my original calculation. To go back to my original calculation, I can see it's decrease by two. I've subtracted two to the minuend, and I've subtracted two to the subtrahend. And because it's the same amount for both, I know that the difference has stayed the same. A little challenge for you to complete now. You see on the screen, there's two calculations. One, has got our decimal fractions, and the other is asking to find a difference between two, three digit numbers. I don't just want you to answer them though. I want you to complete each one in at least two different ways. And after doing so, I want you to reflect on which one made it easier to complete. Because remember, that's the whole idea of what we're doing. We're trying to make things easier for us. So, pause the video, complete each one, and then we'll come back and we'll have a look how you could have done it. Okay, so this first one, I think again, probably was jumping out at you. I know it did me. We try to change that subtrahend where we can to become a whole number. And most of us will have spotted, if we added 0.1, it's going to get us to our seven. And because I added 0.1 to the subtrahend, I need to do exactly the same to the minuend to make sure that the difference stayed the same. Now, look at that calculation. It's a nice and easy calculation to complete. And because we're confident with this, we're able to then take that forward and answer the original calculation. Now, the second one's a little bit different, because if you were just thinking about the minuend and the subtrahend, you could have done it in two different ways, but maybe some of you spotted that. Actually, I think I can do that without changing the minuend and the subtrahend. Let's just have a look. I think you'll all agree. The way we found it easiest is to change the subtrahend, and where possible to change it to a multiple of ten or a multiple of a hundred. So that's exactly what I did first time. I changed it to a multiple of ten. I subtracted one. And I did exactly the same to the minuend. And that left me with an easier calculation to complete. And I was confident that I could do that. And therefore, I was able to answer that original calculation. However, some of you might have seen that the minuend was able to be changed to something. Because remember, "If the minuend and the subtrahend are changed by the same amount, the different stays the same." It doesn't matter which one we change. On this one, I looked and I tried to change the minuend because if I just added 14, I got to 1,000. And actually, I know my number bonds to 1,000 really quickly, so therefore can find the difference between a number and a thousand confidently. So I did exactly the same to the subtrahend, added 14 to get me 275. And because I'm secure in my understanding of how to get to 1,000, I was able to confidently answer it. Therefore could answer the original calculation. But I'm sure some of you were looking at that calculation and thought, "I don't need to change the minuend. I don't need to change the subtrahend, because I can confidently do it already." Some of you will have looked at it and thought about partitioning that number. And using that understanding of place value to help you subtract the 200 and subtract the 68 and subtract the one. And be able to then, get your answer that way. Some of you will have preferred to use the written method. And that is absolutely fine as well. Because it's all about trying to find whatever works for you. The written method in this case, didn't involve any regrouping. So it was really straight forward to be able to get your answer. Remember, we're showing you this idea of changing the minuend or a subtrahend to make the calculation easiest to complete. If it doesn't make it easier to complete, then you don't need to do it. Between now and the next lesson, I'd like you to have a go at the practise activity. I'm sure you're now feeling really confident with using the idea of the same difference and the stem sentence. "If the minuend and subtrahend are changed by the same amount, then the different stays the same." Remember, it might be worth exploring more than one method for those first five calculations. I've then got a little challenge for you. Now, just look at that calculation. How are you going to change that, to make it easier to complete? I'd like you to try and do it as many different ways as you can think, because the teacher at the beginning of the next lesson, will explore all the different ways that you could look at that, and hopefully show you a way that's going to make it a lot easier. Thanks for all your hard work, you've done really well.

Practice: Transforming Calculations to Make Them Easier to Solve Mentally

Practice: Transforming Calculations to Make Them Easier to Solve Mentally

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UnitMaths / Extending calculation strategies and additive reasoning

Maths