Balancing equations using the 'same sum' strategy

Key Stage 2

Maths

Extending calculation strategies and additive reasoning

﻿Okay. So here's the practise that I left you with yesterday. Let's have a careful look at these. So we've got 7.1 add 1.98, and we needed to use those connections that we were looking at yesterday, so that's same sum calculation to help us. So you'll notice 7.1 as that first addend, has decreased by 1/10 to get seven. So 1.9 needs to increase by 1/10 to give us two, and seven add two is nine, therefore 7.1 add 1.9 is also a nine, cause the sum remains the same. The one underneath 2.91 add 3.24 has already been redistributed. And you can see that it's gone up by 9/100. That's right. So 2.91 has increased by 9/100 and 3.24 has decreased by 9/100. And now I know I'm going to get that equivalent sum, or that same sum, that sum remaining the same. So three add 3.15 is 6.15. And therefore the answer for both of those calculations is 6.15. Next one at the top on the right-hand side is 2.6 add 3.9. You can see that the 2.6 addend has decreased to 2.5. So that's 3.9 addend needs to increase over. First one is decreased by 1/10. So this one addend needs to increase by 1/10, 3.9 and 1/10 more is four. the 2.5 add four, I think you'd probably know 6.5. So that's what 2.6 add 3.9, is also, 6.5. Final one in section a, 175.7 add 24.86. You can see that the 175.7 has increased by 3/10 to give us 176. So the other number, the other addend needs to decrease by 3/10. So 24, sorry, 24.86. Will go down to 24.56, and now we've got a calculation to make of 176 add 24.56. I've still got to think quite hard to be honest, but I do know my number bonds, 176 add 24 is 200. And then we've got that .56, So it's 200.56, and therefore the other calculation has been factored by the same. Okay. So for the next challenge that I gave you, it was where Salvo says that the best way is to solve these calculations, using the same sum. And you can see that on here. I have got the answers for you in case you wanted to see them. But here we're saying that the best way is to solve them using the same sum of those equivalent calculations by redistributing the numbers to make it easier, and Mia disagrees and says that it's quicker to use a written method. And I wanted you to find out yesterday who was right. I wonder, did you actually get round to competing against anybody or did you just make a decision for yourselves? Well, anybody that knows me knows I am super competitive. So I decided to video myself, working both of these out in both ways to see which was the quickest. The only issue is that for the same sum calculations, normally I would do those in my head, but I thought it was a little bit difficult for me to be able to prove to you how quick that could be. So I had to write them down, which slowed me down a little, but we'll see which one was the fastest. Okay. So in a moment I'm going to play these two videos side-by-side of me calculating them. And it's going to be a little bit like a race and we'll see which one is the fastest, if any, we'll find out. Okay. So the top one is the same sum or the equivalent calculations. And the bottom video is me working it out, using a written method. And I just went for the column method, just so I had consistency with my written method. Oh at the moment I can see the same sum one is just pipping the column one to it. So that's quite exciting for me. You might also notice that on the same sum one I'm not bothering to write the totals both times because there wasn't any need. And I think the same sum just about won there. So we're going to get started with our learning for today. It's going to be a slightly different focus to the last three lessons. The last three lessons have really been looking at, ways to make those equivalent calculations or to redistribute the numbers in order to make it easier or quicker to solve something. But today that's not going to be our focus. Today, we're just looking at, how we can use that same sum to help us to solve and produce balanced equations. So in a moment, there's going to be a short video clip from my friend, Dr.Sharrock. And I asked her if she would do me a favour and show you how, when we redistribute something with a mass, that that mass stays the same. She's got some electronic scales to be able to show this by which you'll see in a moment.

Here you can see some weighing scales I use when I'm baking on top of the scales are two containers with some cereal inside, what do you notice? That's right. The total mass is 85 grammes, but we do not know the mass of each individual part. I'm going to redistribute some of the cereal. Watch what happens. What do you notice? And when i ask that. I mean, what has stayed the same? and what is different? This shows some cereal being redistributed from one container to the other, the amount of cereal in one container decreased, whereas the amount of cereal in the other container increased by the same amount. And because of this, the total mass has stayed the same.

Let's have a look again, at a similar situation, but this time just using a representation that we've got on here. So here you can see that there are two bags on this scale. In total, the mass of the two bags is 45 grammes. Let's see what happens. A sweet comes out, of one bag and it goes into the other bag. So the mass of the left-hand bag of sweets must have decreased because one sweet was removed. But the mass of the right-hand bag must've increased because another sweet was put in. But as you can see our mass, our total mass remains the same. It's been conserved. So this time we're just going to look at this image one more time and think about this generalised statement. When an amount is redistributed from one part to another part, like so, the whole quantity remains the same, or we can say, as one part increases, the other part decreases by the same amount. You might just have spotted. That there's a change of scenery behind me. That's because my kitchen was needed by my family. Anyway, let's get back to the lesson. So let's have a look at this diagram this time. What do you notice? Ah! it's changed, hasn't it? Yes. We've now got values on our bags. We've got 35 gramme bag and a 10 gramme back of sweets. What happens? Watch carefully. Our sweet moves from one bag to the other. But, did you spot five grammes at the top? I wonder if that means our sweet weighs five grammes, or has a mass of five grammes. I think it must. So that means that from the left-hand bag of sweets, five grammes has been removed. So it's been reduced by five and the right-hand bag of sweets, five grammes has been added to it. So it's been increased by five, but you notice that that mass has been conserved. It stayed exactly the same. So even though we've redistributed those sweets, our mass has remained the same. one, went down by five grammes. The other, went up by five grammes, but our total or our sum remain the same. And that's really important. Has it made the calculation easier? 35 add 10 ? It's pretty easy in the first place? Wasn't it? It's not made it easier. It's just given us a new distribution of the numbers. I wonder if you could join in and say these sentences with me. So 35 add 10 has the same sum as 30 add 15. 35 add 10 is equivalent to 30 add 15, or finally we could say, 35 add 10 is equal to 30 add 15. Okay. So let's have a look at this diagram, we've got potatoes this time, we've got two bags of potatoes and you might be able to see their total mass is 10,000 grammes. That's quite a lot, isn't it? So I think this probably hasn't been weighed on kitchen scales, perhaps a different kinds of scales. Let's have a careful look at the numbers. The left-hand bag has 5,300 grammes. And the right-hand bag has 4,700 grammes, which is represented in our calculation. It's quite a straightforward calculations and tech because you can spot that number bond at the 300 and the 700 making that extra thousand to give us 10,000 altogether. Let's see what happens. One potato comes out, goes into the other bag and that potato weighs 340 grammes. What do you think our new equation is going to be? I'd like you to pause the video here. See if you can say our stem sentence. So, one addend has something by a certain amount, the other addend has done something else by a certain amount. And our sum remains the same. And then, can you have a go at writing our new equation? Remember it misses 340 grammes that moved from the left-hand bag to the right-hand bag. Pause the video here and try. Alright. Have you had to go? So the 5,300 gramme bag reduced by 340 grammes because the potato came out. And the other bag increased by 340 grammes. Cause that same potato went into that different bag. What were your new totals? Ah, 5,300 subtract 340. I'm going to subtract 300 first to get me to 5,000. Then I'm going to subtract 40. And that then gives us of 4,960. And on the other one, increasing it by 340 will get us 5,040. Did you get the same? And we still have a total of 10,000. So we've got our two calculations, 5,300 plus 4,700 is equivalent to 4,960 add 5,040 because we've subtracted from addend and added to the other addend to keep that sum the same. And it was that same amount that we subtracted and that we added. Fantastic. Can you join in with me on the stem sentences now. 5,300 plus 4,700 has the same sum as 4,960 add 5,040. I'll let you read the rest. I hope you said all of the numbers correctly, especially at the bottom 5,300 add 4,700 equals 4,960 add 5,040. And all of those relate back to that potatoes picture that we've just been looking at. Ah, we've got the potatoes again. Do you notice anything this time? The numbers have changed, haven't they? We've got 10 kilogrammes. I can imagine that a little bit more easily than I can 10,000 grammes. I wonder why? I'll let you think about that. Okay. So in our left-hand bag this time, we have 5.3 kilogrammes. And in our right-hand bag, we have 4.7 kilogrammes. Some of you might have spotted something about the numbers, but we're going to talk about that in a moment. Okay. Let's see what happens. So 5.3 kilogrammes plus 4.7 kilogrammes is equal to 10. And we can see, can't we? That 3/10 here and 7/10 here make one whole don't they? So therefore we have got 10 altogether. Let's see what happens. Oh, a potato is coming out again, from that left-hand back, going into the right-hand bag. This time it weighs 0.34 kilogrammes or 34/100 of a kilogramme. Okay. Can you write the new calculation? So remember the left-hand addend has just decreased by 0.34 kilogrammes and the right-hand addend has increased by 0.34 kilogrammes. Can you pause the video here and have a think about what the new equation is going to be? Have you done it? Let's see whether you got the same as me. So the first addend decreased by 0.34, and the other increased by 0.34. Okay. Oh. A little bit harder the numbers this the time. Aren't they? I wonder how you got to that. Remember 5.3. If we subtract the 3/10 from the 0.34 that would get us to five, and then we still got to subtract the 4/100 from five. subtract 4/100. Hmm, a bit hard, but we know that a hundred 1/100 and make a whole. So 4/100 less than that would be 96/100. So we've got 4.96. And on the other one,the 4.7, we needed to increase it by 0.34. So again if we add the 0.3, the 3/10 of 0.34 to the 4.7. We'd get five wouldn't we? And then we've got to add on a further 4/100 to get this 5.04. Definitely hasn't made the calculation easier, but they are still balanced and they are still equivalent. So 5.34 add 4.7 is equal to 4.96 add 5.04. We have a balanced equation. Let's have a look at these two diagrams side by side now. I want you to make some notes about what you notice is the same, and what you notice is different between the two representations. So if you could pause the video here and write some of those down or discuss them with someone in your house. And then we'll come back in a moment. Did you manage it? Okay. What did you notice? Yes, you're right. There are two sacks of potatoes, on both sets of scales on there. Did you notice anything else? Ah, some of the details are the same. So for example, did you notice that here on this left-hand side, our 5,300 has the same digits as the five and the three, five in the thousands columns and the three in the hundred columns as we have here, five in the ones column and the three in the tenths column. You might have spotted it with the 4,700. We've got four in the thousands column and the seven in the hundreds column. And over here, we've got the four in the ones column and the seven in the tenths column. Same digits, same order. There's a relationship isn't there? Hmm. Remember what it is? Ah, something that's different, you might've spotted is that one's in grammes and one's in kilogrammes. Do you know that's significant? How many grammes are there in a kilogramme? Yes, there's a thousand. So if you think there's 1000 grammes in one kilogramme, then, how many grammes are there going to be in five kilogrammes? That's right. 5,000. So 5,000 grammes is five kilogrammes. So we can that relationship here. 5000 grammes is five kilogrammes, but it's not just 5,000 grammes. It's also got 300 grammes, 300 grammes is 3/10 of a kilogramme. So actually, our sacks of potatoes have the same mass. It's just one's represented in grammes and one's represented in kilogrammes. That's also true of our 4,700 grammes being represented in kilogrammes over here. Same mass. And of our 10,000 grammes, that's equivalent to 10 kilogrammes. So we could actually put thought, another equivalent calculation here. 5300 grammes plus 4,700 grammes is the same as 5.3 kilogrammes add 4.7 kilogrammes. But I've not redistributed the numbers here. It's just that we've had a change of units from grammes to kilogrammes. Okay. So now it's time for us to have a go at one together, before you do some independent practise. So I'd like you to have a look at these equations here. We've got 4.2 plus 5.8 is equal to 10. Have a look at what's changed. What's been redistributed? Did you spot it? So it's showing here that we're increasing this addend by 0.24. So have a think about what we've got in here. That's right. It would decrease by 0.24. Because if I have subtracted 0.24 from one addend, I need to add 0.24 to the other addend to keep the sum the same. Hmm. Can you work out for me, what would go in the box down here? I'll give you a moment to think about that. So yes. We need to increase 5.8 by 0.24, which would give us 6.04 because remember, 5.8 add on the 2/10 from the 0.24 will get us to six and then we've still got our 4/100 to make 6.04. So I have subtracted, remember, 0.24 from one addend. So I need to add 0.24 to the other addend to keep the sum the same. So now we have a balanced equation, 4.2 add 5.8. Are you saying it with me? is equal to 3.96 add 6.04. Okay. So now it's time for your independent practise. You're going to start with having a look at these equations at the top and filling in all of these empty boxes. Just like we did on the previous slide and saying here, so 6.5 and 3.5 and fill in all of those empty boxes. Carefully consider our sentence, which was, If we increase one addend by one amount, we need to decrease the other addend by the same amount to keep our sum the same. And then you're challenged down here, it says to use the same original, that means, use your original 4.2 add 5.8. Or use your original 6.5 add 3.5. And then redistribute them differently. So, instead of, if we were on this one, instead of increasing one by 0.24, and the other decreasing by 0.24 Choose your own increases or decreases. You can make them as challenging as you like. Okay. So some of you, you might want to make them really, really hard. Others might just want to explore those numbers and find some new balanced equations. I hope you enjoy it. Take care, everybody.

Balancing equations using the 'same sum' strategy

Balancing equations using the 'same sum' strategy

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UnitMaths / Extending calculation strategies and additive reasoning

Maths