Solve calculations mentally by relating them to known facts

Key Stage 2

Maths

Extending calculation strategies and additive reasoning

﻿Hello. My name is MR. Richard, and I am going to be your teacher for your math lesson today. Before we get started, you're going to need a few things, your practise activity from yesterday, a pen or pencil and some paper. And that's it. Why don't you press pause, go and fetch what you need and come back and press play again when you're ready. Welcome back. Did you remember everything? I hope so. Let's get going. Last lesson, Ms. Tartman set you this challenge to complete. Did you have a go? I did. Let's see how you got on. Question one. Can you read it aloud with me? The combined mass of two dogs is 12 kilogrammes. The mass of one dog is 5.6 kilogrammes and the mass of the other dog is 6.4 kilogrammes. One of the dogs gains 0.7 kilogrammes. What is their combined mass now? I had a read of the problem and I had a think about any drawings that might help me to see the maths within the problem. Now I chose to draw a bar model. Did anyone else choose to draw a bar model? Can I show you my one? I knew from the problem that the total, that the whole that my bar was representing was 12 kilogrammes. And I also knew from the problem that my total, that my whole was split into two parts, one part for each dog's mass. So I've got one part showing 5.6 and the other 6.4. When I looked back at the problem, I focused in on the sentence that says one of the dogs gains 0.7 kilogrammes. Does it matter that we don't know which dog has gained that amount? No, it doesn't matter. We know that combined mass, we know that one of them has increased by 0.7. So we can just increase the total, the combined mass by 0.7 to solve that problem. On my bar module, I chose to draw the extra parts on the right hand side. Could I have drawn it on the left-hand side? I could. It doesn't matter where I placed that part so long as I now know, I need to increase my total by that amounts. Take a look at this sentence with me. Let's read it aloud together again. And can you fill in the gaps as we go? I have added to one addend and kept the other addend the same. So I must add to the sum, well done. I also had a go at using some equations to represent the maths that I had just worked on. Now, when I was doing that, I realised, can you see the empty box? I hadn't worked out what 6.4 plus 0.7 is equal to, does it matter that I've left that blank? It doesn't matter. I have increased my total by 0.7 to solve the problem. And I know that 0.7 has been added on to one of those addends. I've still been able to solve it. Did anyone use any other representations to show how they solve this problem? Hold it up to your camera for me. Now, some of you might have recorded an equation with three addends, one addend for each dog's mass, one for the increase of 0.7 and then the sum to show what each of those three add ons total. Question two. Can you read it aloud with me again? The combined mass of the father and son is 121 kilogrammes. The mass of the father is 87.3 kilogrammes and the mass of the son is 33.7 kilogrammes. One of them loses 800 grammes. What is their combined mass? Now, when I looked at this question, I thought there were some similarities to the first question. And because I had used a bar model for that one, I wondered if it would help me again for this one. I still knew the whole from the problem. This time, 121 kilogrammes. And I still knew in a similar way that the whole was representing two parts 33.7, for the son and 87.3 for the father. But at this point I've got a bit stuck. When I read the sentence, one of them loses 800 grammes. I could understand that that meant one of them had lost some weight. And so I need to decrease my total, but my total is 121. I need to decrease it by 800. If I do that, I'm going to have a negative number which I know won't make sense in this problem. Did anyone else have the same problem as me or can you think of a way to help me with this problem? I spent a bit more time thinking. I looked at the question again and then I noticed some things that were the same and some things that were different. I noticed that three of the masses were recorded in kilogrammes and one of the masses was recorded in grammes. Then I remembered as a mathematician, it's best to work with units that are the same. So I used what I know about converting units and changed 800 grammes into 0.8 kilogrammes. Once I had done that, I was able to mark it on my bar module. I wanted to show that I'm going to be taking away, removing 0.8 from the whole. And once again, does it matter whether you show the 0.8 on the right hand side or the left-hand side, why doesn't it matter? We don't know whether it was the son or the father that lost the weight but we do know that it was one of them. Because of that, we know that we can subtract 0.8 from the whole. Have a go at reading the sentence with me again, filling the gaps as you go. I've subtracted from one addend and kept the other addend the same. So I must subtract from the sum. Well done. So then I had to go writing down some maths, some jottings, I wanted to show what the bar model had helped me to see. I decided to subtract 0.8 from the father's mass, 87.3, subtract 0.8. Once I had worked that out, I decided to add on the son's mass 33.7 to find their new combined mass, but totaling 86.5 and 33.7, found a little bit challenging to do without using a written method. So I've shown that, they're just at the bottom. Now, when I looked back at strategy one, I noticed there was a lot of writing, a lot of maths, a lot of jottings for that approach. So I had to think about whether or not there might be a more efficient way to work, perhaps a slightly smarter or quicker way. Strategy two shows you my thinking there. I started with the original combined mass, subtracted 0.8 by first subtracting one, and then adding on 0.2, you can see that there on my number line, and I've reached the same new combined mass 120.2 kilogrammes. I wonder which of my strategies you think was most efficient, strategy one or strategy two? Can you hold up your fingers to show me which of my two do you think was most efficient? I agree. Strategy two was the most efficient. I had to show the least jottings, the least amount of working out to show my thinking and to solve the problem. Problem three, create your own problem like the ones above, use your own jottings to show how you can solve the problem. Can you hold up if you had a go at this challenge, hold up the problems that you wrote for number three, here's my one. If you'd like to, you could press pause now, have a read of my problem, and have a go at solving it, using those sentences and those skills that we've just been recapping over the last two lessons, you've been using both of these sentences to describe the maths that you've been working on and the patterns that you've been spotting within the equations. At the end of your last lesson, you had combined to these two sentences with this one, give every with me because we're going to use it again today. Are you ready? If one addend is changed by an amount and the other addend is kept the same, the sum changes by the same amounts. Today, we're going to use our learning from recent lessons as we start to identify calculations that we can solve mentally by using our known and related facts. So what do I mean by known facts? Well, I'm thinking about things like days of the week, months of the year, our times tables, but for this session, we're thinking about addition facts that we can recall, things that we just know such as four plus five. I bet you are able to recall really quickly. Now related facts, let's stick with four and five. As an example, if I know four ones and five ones is equal to nine ones, then I also know that four tens plus five tens is equal to nine tens. Maybe you can think of a related fact to four and five. We're going to be thinking about calculations we can solve mentally instead of using a written methods. Now I wonder which calculations did you normally solve mentally? What is it about the calculation that tells you it's one to solve mentally? What do you notice? What do you see in the calculation? Maybe have a pause now and do think, when do I solve calculations mentally and when with a written method, and why do I make those choices? Here's our first problem. 0.29 plus 28.71. Now stop. Don't rush your heads and find the sum straight away. This isn't about getting the answer quickly using a written method. This is about looking carefully at the numbers we're adding in case there are any related or known facts within the calculation that can help us to solve it mentally. So look closely at those two numbers at the two add ons. Are there any facts in there that you can see? This is what I spotted. Did you spot it as well? Before we think about why those numbers have caught our attention, let's think about 28.71 as the whole and how that whole can be split into 28 and 0.71. Can you fill in the blanks in this addition equation for me, ready? 28.71 is equal to, Brilliant. So now let's think about why 0.29 and 0.71 caught our attention. We can use the sentence at the bottom to help us. If I know that 29 ones and 71 ones is equal to 100 ones or one hundreds, then I also know that 29 hundredths and 71 hundredths is equal to a hundredths, which we know is equal to one. So we've spotted in the original calculation, a fact that's helped us. Now we need to use our learning around keeping an addend the same and changing one of them and what we then need to do to our sum. So adding 28 to 0.71 gives us the number from our problem, 28.71. We need to add 28 to our one as well to complete the problem. Let's look at a second question. Now, again, stop slow down. Don't race ahead to find the sum, look at the calculation. Are there any known facts or related facts within those two numbers that can help us to solve this one mentally? This is what I've spotted. So again, before we think about why that has caught our attention let's think about what we know about 646,000. And the two parts that we can split that number into, are you ready? 600,000 and 46,000. So what are the missing parts from my equation here? Read it out to me. Well done. Now we can think about why 46,000 and 54,000 caught our attention. Let's use the sentence at the bottom again, to help us. If I know that 46 ones and 54 ones is equal to 100 ones or 100, then I know that 46,000 plus 54,000 is equal to 100,000 or 100,000. So this time it was number bonds to hundred that really helped me because I spotted, I recalled 46 plus 54. So here's where we use our learning from our recent lessons. If one addend is changed, I'm going to change my 46,000, back to 646,000, I need to do the same to my sum, adding on 600,000 to those parts of the equation to help me solve my problem. Here's a chance for you to have a look at some expressions, not equations because you're not going to be thinking about the totals, the sums, what I want you to do is look at the expressions and decide whether you would solve them mentally like we've just been doing or the written methods. Now, while you're making your decisions, think, are there any known facts or related facts within the expressions as parts of those numbers that you would use to help you solve them mentally? If there are then pop them into the mental methods side, if there aren't, then you probably would use a written method to solve them to pop them into the written methods sides of your table. Here they are. Why don't you press pause, pop the letters into a table of your own and then we can compare, press pause now. How did you get on? Did you manage to solve all six expressions? Can you hold up your paper so I can see your tables and where you popped the letters? Let me show you how I sorted mine. Ready? Does this match your piece of paper? It does, brilliant. Let's have a think about why we've sorted them like this. If you haven't matched them in the same way as I have, that's okay too. Let's have a think about why I've popped the C and F into the mental methods part of the table. So here's the first one that I chose to solve mentally. And if you chose to solve this one mentally as well what was it that caught your attention? Is it the same as what caught my eye? These two parts. Now I know 529.25 is the whole, it can be split in to help me and excellent. So what is it about 0.25 and 0.75. That caught our attention? And as I looked at them for me I recalled a number factor, number bonds to 100, 25 and 75. So using the sentence at the bottom, this is how it can help us with 0.25. We get with me. If I know that 25 ones and 75 ones is equal to 100 ones, then I know that 25 hundredths and 75 hundredths is equal to one hundredths or one. So once we've spotted a known fact and then used a related fact to help us with this problem, we need to use our learning from recent days to change one of the addends so it matches the original problem. 0.25 needs to increase by 529. And so we need to do the same to our sum. One needs to increase by 529 to help us with that original problem and find the missing sum. How about for C? So again, if you put it into the mental method side, what did you spot in the calculation? What caught your eye? Was it this? It might not have been, maybe you saw 3.555 plus 0.445. Maybe you didn't spot either of them. Let me tell you why this caught my attention. I noticed that I could make a whole number from either the 3.555 or the 4.445 if I used part of the other addends. So with this example, it's knowing 3.555 can be split into and brilliant. So once I know that, I'm looking really closely and I can see that 0.555 and 4.445, which total five a whole number, that's really easy to work with now. Using our learning from recent lessons, I now need to make a change to my addend, and then to my sum as well. I'm adding on three to those parts of the equation. Five plus three is equal to, and that's a calculation that maybe when you first look at it it's looking a bit tricky, but when you look really closely and start using some things that you know or making some changes, using parts of the numbers and it's made it really quite simple to complete. Okay, what did you spot this time? This is what I noticed. And I know that I can use part of 6,400,000 if I split it into and, but why did it catch my eye 400,000 and 600,000, this time number bonds to 10, four and six. Let's use the sentence together. If I know that four ones and six ones is equal to 10 ones or 10, then I know that 400,000 and 600,000 is equal to 10 hundred thousands using my place value work. I know that that is 1 million. Now, with our sentence, I need to make a change to of my addends. I need to increase it by 6 million. So I need to do the same to my sum. 1 million and 6 million is equal to seven million. We've worked really hard today, focusing on slowing down and not racing ahead to use a written method to solve an addition equation but instead to look closely at the numbers and see if there are any related facts, anythings that we know already, that can help us to solve them mentally instead. Now I'm going to leave you here with a practise activity and a challenge for you to have a go at between now and your next lesson. On the next page, you could press pause and take a picture, ready for you to have a go at the problems in your own time. Enjoy the rest of your day, see you again soon.

Solve calculations mentally by relating them to known facts

Solve calculations mentally by relating them to known facts

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UnitMaths / Extending calculation strategies and additive reasoning

Maths