Balancing equations: Does the order of addends matter?

Key Stage 2

Maths

Extending calculation strategies and additive reasoning

﻿Good morning, everybody, welcome to Lesson number six in this series of lessons on number addition and subtraction for Upper Key Stage 2. Let's start our learning today by looking at the review activity that I left for you yesterday. We had this scenario, didn't we? That dad was standing on the balancing scales, holding his cat. His daughter was standing on the other side of the balancing scales, holding a dog. And you were asked some questions regarding how the mass of the dad and the cat could be the same. I wonder what you came up with. There are lots of possibilities, but the key point would be, that the dad plus the cat, the sum of their masses must be the same as the sum of the mass of the daughter and the dog. So, that the scale remained balanced. One option, they could all weigh the same mass. Then you were asked if the mass of the pets is the same. So, if you've got a small cat, sorry, a large cat and a small dog, the mass might be the same. Oh yes, you remember, I sent you a little bit of extra homework to try and find out the approximate mass of the dog and the dad. Did you find out? Fantastic. So, on average, rough male adult would weigh about 70 kilogrammes, but obviously, this varies. An average mass of a cat, you're looking at about four kilogrammes and a dog anywhere between 15 and 20 kilogrammes for a medium sized dog. Okay, so, I digress again there, but hopefully you found that bit of information out for yourselves. So, if the mass of the pet is the same, so if you've got a dog and a cat that are the same mass, then the dad and the daughter must be the same as well. How is that possible? That's right, the daughter, we don't know how old she is, she could be grown up daughter, and they could weigh the same mass. What if dad weighs two kilogrammes more than the daughter? What does that mean about the mass of the pets? Well, to keep the balance level, that then means if the dad is two kilogrammes more than the daughter, the dog would have to be two kilogrammes less than the cat, so that the scale remains balanced. Then I gave you the information, I said that dad is actually 90 kilogrammes, so he's a bit more than the average weight of a man isn't he? The daughter weighs two kilogrammes less than that, so she must weigh 88. So the mass of the pets could be then anything you wanted but let's try and keep this realistic, but the dog must weigh two kilogrammes more than the cat, so that the scale remains balanced. Okay, how did you get on with those? Yeah, well done, brilliant, fantastic, right. Today's learning, we're going to look at some more balanced equations, where one of the addends is unknown and we're going to look at the relationship between the numbers to help us. So, if I start you off with this calculation, pause the video, see what you spot, about the addends. Did you spot that I got 340 here and we need to work out this missing addend, I've got 350 add 720 here. So, which addend do you think we should redistribute? That's right, I'm going to work with the 350, because that's nearer to 340, and I know that 350 is 10 more than 340. So if I subtract 10 from 350, what do I need to do? You've got this by now, that's right, we need to add 10 to the other addend, to get 730. So, without doing much of a calculation, just spotting the relationship between the numbers, spotting that 340 is related to 350 by 10, we can redistribute and subtract 10 from one addend and then we need to add 10 to the other addend. Okay, let's have a look and see if you spot something different about this. So, this is the one we've just done. Okay, we had our 340, it was related to the 350, so, the first addend here was related to the first addend here. Do you notice something different about this calculation? Pause the video if you need to. That's right, we've got 340 here but a 720 here. The addend I really want to work with, is not in the same place as the first addend here, but the second addend over here, does that matter? And we talked about this yesterday, didn't we? It doesn't matter which addend I use, as long as I get the the redistribution correct. So I'm actually going to look at now redistributing from the same number as above the 340 and 350. That they're just in different positions. So, I've got 350 is 10 more than 340, so if I subtract 10 from the 350, I will then need to add 10 to the 720, and we will get 730. So it didn't matter where the addend was, this 350 was the first addend, and it's the second addend up here, didn't matter where the addend was, we got the same result. What's different about this calculation? It looks a bit trickier to me, it got some decimals, hasn't it? Do you think it's any trickier? No, not really, when we give decimals, the same rules apply, what we've got to do is, look at the calculation really look at it. What do you spot? Have you spot anything, that's the same? Anything that's different? I Wonder if you spotted this, I've got 0.45, so four tenths and five hundredth, oh, look, it's the same here! So I think I might choose to redistribute from this addend. This addend is 16 more than this, so, if I'm going to subtract 16, what do I need to do? To the 30.25? That's right, I will need to add 16. So, it'll become 46.25. So this balancing equations and solving missing numbers problems, is all about spotting what's the same, what do you notice? So, always take a really careful look at the calculations. Just to show you this, in a different way, with a balanced equation, we looked at the 16, and we put point four, five, and we took away the 16, and then we had to add 16 to the other addend. So here, I've got some more calculations, I would like you to pause the video have a go, but remember, spot what you notice about the structure and the relationship between the numbers and use that to help you. Okay, what did you notice? 57, 24, 58, which of these addends, you think we should use, which is nearest, and which makes the calculation efficient? That's right, the 57. I'm going to add one to get to 58, which means I need to subtract one from 24 to get 23. Missing number solved? Hardly any calculating. What do you note about this calculation? Well, these addends are the same as the ones above, and here, though the missing box is in a different place. So, I know I need to look at the relationship and look at these numbers here, and what do you spot? Oh, yes, 24 and 22, 22 is only two less than 24. I could work from the 57 as well but that's a little bit more complicated. So, if I subtract two from 24, I need to add two, to the 57. and I get 59. The next one, oh, it's different again! What do you notice? That's right, yeah, we've got the two addends the two known addends on this side. And the unknown is on the left hand side. Still the same rule applies. Look for a relationship between this addend and one of the addends on this side. You could use either addend but I'm going to go for the one that I think is easiest. 34 and 24, I know are related by 10. 24, I need to sub, sorry, 34, we need to subtract 10 to get 24, so I need to add 10 here to find the missing number. 47 add 10 is 57. Brilliant, well done! Okay, last one. What do we notice here? Ah, yes, look, the known addends are on the right hand side, the unknown is on the left hand side, but the missing box is in a different position. Okay, so we're going to follow the same strategy, the same procedure. What do we notice? Which of these addends, would we like to use? Well, I've noticed 57 end in a seven, and so is 37. So that must mean they're multiples of 10 different from each other. So 37, I know I must add 20 to get to 57, so I need to subtract 20 from my 44, and I get 24. Okay, so it's really important, in math, that we look in the relationship between the numbers between the addends on one side and the addends on the other side. And usually then, you can just redistribute, and calculate, without having to do any more complicated calculations. Because as we said yesterday, there's always one more than one method to do this, but we want you to be using the most efficient method. Okay, so we have a look at a word problem. Sam and Eva spent the same total amount of time doing activities at a youth club. So that's key, isn't it? The key word there, the same total amount of time, the same. That means, Sam and Eva, if they were on balancing scales, the amount of time it got to be balanced, it's got to be the same for Sam and the same for Eva. Sam spent 75 minutes painting and 35 minutes playing tennis. So those are your numbers for his side of the balance. Eva spent 65 minutes doing portrait and the rest of the time playing football. This must be our missing box, the rest of the time. How long did she spend playing football? This is our missing box. I wonder if you can pause the video now, and have a go to either using a balance representation, or a calculation with a missing box and have a go at solving it just like we did. Press play when you're ready to through this. Okay, I wonder if anyone use, the balance representation like I've done. So I've got Sam's activities here, 75 minutes and 35 minutes, it doesn't really matter what they were doing, it's the numbers I'm interested in, and we know that Eva spent 65 minutes, and we don't know how long she spent playing football. So this is my representation of all those words, and in math, sometimes the words are confusing, it's really important if we can, to get our calculation from the words, to really help us see the relationship and the structure between the numbers. Okay, but we could also have represented this in a bar model. So, Sam's activities, he spent 75 minutes painting, and 35 minutes playing football so this is what his bar model would look like. We know Eva spent the same amount. So her bar model must be the same size. But what's different about her bar model? That's right, she only spent 65 minutes doing one activity. So this bar, this part of the bar is shorter, than the Sam's first part of the bar. It's 10 minutes shorter. So what does that tell us about the length of this bar? We know they spent the same amount of time, and you realise we've not actually worked out, the total amount of time, that's not important really, what we want to look at is the structure between the numbers. So, we know Eva has got 10 minutes left here, So if she spent the same amount of time she must have 10 minutes more there. Okay, so we can also represent this, using a balanced equation. Looking for the relationship between the numbers. 75 is 10 more than 65. So 75 subtract 10, what have we got to do to the other addend? That's right, I have to add 10. 35, sorry, I'm getting ahead of myself, so if I've subtracted 10 from one addend, I need to add 10 to the other end, so that means Eva has spent 45 minutes playing football. Not once did we work out the total time, spent by either of them. We could have subtracted 65. But it's much more efficient to look at the structure of the numbers and to form yourselves a balanced equation from the information that you have been given. Okay, fantastic work again today, folks, that's all, I'm going to leave you with, a couple of practise activities, here's your first one, I'd like you to read each calculation, and decide if each calculation is true or false, and put a tick or a cross. So 101 add 99, is that the same as 100 add 100? Now, as well as giving me a tick or a cross, have a think about, "Could you represent this? And why?" If you're telling me, "Oh, yes, that's correct," I want to hear that "Because," "Yes, that's correct, because". Okay, so that's what I'll be listening out for tomorrow. Your second practise activity, is looking at some mass of some fruit. But have a really, really close look at the numbers. How would you write the information here as an equation, or as a bar model? That might help you, might be a good place to start. And can you say what's going on using the stem sentence? If I have added, to one addend, so I need to subtract from the other addend to keep the sum the same. Have a close look at these numbers. Look, I noticed this has got three decimal places, two decimal places, one decimal place. I have also noticed that they're all in kilogrammes, so that's something, okay? But have a really careful think about that one, okay? And until next time, good luck! And see you all soon, bye bye.

Balancing equations: Does the order of addends matter?

Balancing equations: Does the order of addends matter?

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UnitMaths / Extending calculation strategies and additive reasoning

Maths