Deriving addition and subtraction facts

Key Stage 2

Maths

Addition and subtraction

﻿Hello, everybody, I'm Mrs. Crane, and welcome to today's lesson. In today's lesson, we're going to be deriving addition and subtraction facts. Now, don't worry, the word derive sounds a little bit tricky, doesn't it? But the word derive actually just means taking something or getting something from something else. So it will become a bit clearer in a minute when we go through the lesson together. I'll go through all the equipment we'll need in a moment. So don't worry about that just now. What I would like you to do though, please, is if you could, turn off any notifications on your phone, tablet, or whatever device you're using for today's lesson, and then if you can, try and find somewhere nice and quiet in your house or your home that you won't get distracted in, so that we can get on with today's lesson together. When you're ready, let's begin. Okay then, let's start off by talking through today's lesson agenda. So we're going to just start off by finding out how to devise related facts by looking at 10, 100 or 1,000 times greater. Then we're going to move on to Let's Explore, and we're going to be thinking about why some of the answers are wrong. Then we're going to look at what happens when you change one part. And then for our Independent Task today, we're going to be deriving facts, and I'll go through the answers with you. So for today's lesson, if you don't already, please could you get yourselves a pencil and some paper. If you don't have them, pause the video now to go and get those things, and don't forget to resume it when you're back. Welcome back, let's get started then. So, as I said, we're going to start off by looking at 10, 100 and 1,000 times greater. Before we do that, we're going to think about what calculations this part-whole model shows here. So have a really close look at this part-part-whole model. What is it showing us so far? Well, I can see it's showing me three ones here, and I can see it's showing me four ones here. Let's add something else to it then. Ooh, I can see my whole becomes seven ones. Can you think of any calculations that this part-part-whole model represents? Let's go through them then. So it represents the three plus four is equal to seven. So I can put that one down here. Or I could say it represents four plus three is equal to seven. Then I can use it to show some subtractions as well. I could say seven, my whole, and if I wanted to take away four from there, I'd be left with three. This time, I'll go back to seven again, so it looks like this here. Go back to my seven again. This time I want to take away three, and I'm left with four. So these are what we call our derived facts that we can make from this first equation here, and we can use what's called the commutative and the inverse to find these. Now, what's commutative? What might commutative mean? Commutative means we can switch around three and four and four and three to get our answer seven. When we're thinking about commutativity within subtraction, we're always subtracting from our whole. Our whole here is seven. So I can do seven subtract four is equal to three, or seven subtract three is equal to four. I can't do four subtract three is equal to seven, because my whole must be the number I'm subtracting from when I've used the inverse. Oh, did you notice I used that other word there? The inverse, we know that the inverse operation from addition is the operation subtraction. So these are the four calculations that I can derive, I can take from that first calculation there, and from our, our, that's not a bar model, what am I talking about, from our part-part-whole model here. So let's have a look then if the parts were 10 times greater. So this time I'm not using my deans like I did before. I'm using my place value counters. Here, I've got three tens, and here I've got four tens. What's changed? Absolutely, they are 10 times greater, and I'm showing that using my place value counters. Now, can you think of any calculations I can derive from this part-part-whole model? Thinking about those facts we've already practised just now, well, not practised, looked at, I should say. Let's have a look then. So if I know that three plus four is seven, I actually know that 30 plus 40 is 70. They're 10 times greater. I've made sure each part is 10 times greater. Therefore, I can use that related fact to three add four is equal to seven to help me add some numbers that are greater than three and four. Because I know 30 add 40 is equal to 70, I also know 40 add 30 is equal to 70. Then using that whole I can then take away 40 from it to give me 30. And again, using that whole, I can take 30 from it to give me 40, using the inverse operation, and using the laws of commutativity. Now, remembering I cannot just take 30 and say 30 take away 40 is equal to 70, I must start if I'm doing my subtraction with the whole, and taking away one of the parts. Then start with the whole, and take away the other part, okay? This time then what's going to happen if my parts are 100 times greater? Have to think, what's going to happen? My parts are 100 times greater. Well, I can see here that I have 300 and 400. I know, if I know three add four is seven, I know 30 add 40 is 70, then I know 300 add 400 is? Well done, 700. So let's have a look at the related facts we can get when our parts are 100 times greater. So we can get 300 and 400 is equal to 700. Again, we can use that commutativity and do 400 plus 300 is equal to 700. Then we're going to use our whole. We're going to do our whole subtract 400 is equal to 300. Going to do our whole again, and this time we're going to subtract 300, and it's equal to 400. Fantastic. Lastly then, we're going to think about what if our parts were 1,000 times greater. Here, you can see 3,000. Here, you can see 4,000. Have a think. What do you think my calculations are going to be that are related to that original calculation, which was three add four? Fantastic if you've got this. So let's see, 3,000 plus 4,000 is equal to 7,000. 4,000 plus 3,000 is equal to 7,000. Remember, we're starting with our whole when we are subtracting. So we've got 7,000 and we're going to subtract 4,000 and it gives us 3,000. Again, we're starting with our whole, 7,000. We're subtracting 3,000, and it gives us 4,000. So what we're going to do is look at all of those calculations together, oh, sorry, so that you can see them all together. So I want you to think what is the same and what is different about these four different parts of my table? Have a little look. So if I had to say what was the same, well, I know that when the digit three is added to the digit four it always gives us seven. Whether that three, that four represent three or four ones, three or four tens, three or four hundreds, or three or four thousands doesn't matter. What matters is that the zeros are there to show the place holders when they represent other values. Now, just from that simple base calculation, the first one here, three plus four is equal to seven, I've worked out lots and lots of different calculations that are related to that. Using that fact, that could really, really help me. If I looked at a question and it said 7,000 subtract 4,000, I might think, oh, that's quite tricky. But actually if I just stop and think, I know seven take away four is three. I have to remember my place value. So I have to remember my zeros as my placeholders, but using that related fact that I found out from just over here from the simple fact of three plus four is seven can really help me when it comes to more complicated calculations. And that's what's going to really be helpful for us today. So, what we're going to do next is we're going to look at Let's Explore, and we're going to have a think about why these statements are wrong. So in a moment, I'll ask you to pause the video, because you're going to do some thinking and some working out. But for the minute, I'm going to explain it to you first, so don't pause just yet. So my number, my whole is 16. One of my parts is nine. One of my parts are seven. If you want to, on your paper, you can draw it out using either deans or place value counters, whichever you feel confident with, or you can just write it out using the part-part-whole model. It's up to you. Now, I'm thinking, and I think I know that 16 subtract nine is equal to seven. So 900 subtract 1,600 is equal to 700, and I also think I know that nine plus seven is equal to 16. So I know 90 to 70 is equal to 1,600. What we need to do is explain why Mrs. Crane has got this wrong, Where's she gone wrong, what has she done? So to do that, what I would do is pause the video, write down the related facts, and have a think about why I have got my answers a bit muddled. Pause the video now to have a go at today's Let's Explore. Okay, welcome back. What we're going to do now is go through both of those two examples one at a time to work out why Mrs. Crane has got so confused. So first thing that I thought was I've said I know that 16 subtract nine is equal to seven. Is that correct? There's my whole, 16. If I took away nine, then it would give me seven. Absolutely, that's correct. So 900 subtract 1,600 is equal to 7,000. Oh, I've spotted where I've gone wrong here. I wanted to do a subtraction. Which number out of my part-part-whole model do I need to start with when I'm subtracting? Fantastic, I always need to start with my whole. So here I've started with one of my parts. If I'm subtracting a part, if I'm subtracting, sorry, the whole from one of the parts, I'm going to get very confused. It's going to give me completely the wrong answer, and I'm going to get myself in a bit of a muddle. So what I've done wrong is I needed to start with 16, sorry, not with 16, that was 16, with 1,600, and then either subtract 900 to equal 700, or subtract 700 to equal 900. Let's see what I mean by that. So this here, the 1,600, is our whole from our part-part-whole model. And as we've explained, we need to use that whole as the number that we're subtracting from. So we can subtract 900 from it to give us 700, or we can subtract 700 to give 900, but we cannot choose 900, and put it here and our subtraction equation, and then start subtracting from it. It's not going to give us the right answer. We've not used the rule of commutativity correctly in that example, okay? Let's have a look then at the second reason I am wrong today. Dear me. I know that nine plus seven is equal to 16. Well, let's check that part. Nine and seven are our two parts. 16 is our whole, absolutely. That's part's correct. So 90 plus 70 is equal to 1,600, oh. Where have I gone wrong there? Well, I know that nine and the seven and the 16 part's correct, but where I've gone wrong is I've added, or placed, I should say, sorry, not added, I've placed two zeros as placeholders after my 16, but that's not correct, because I've made my answer 1,000 times greater, not 10 times greater. So my answer should be 90 plus 70 is 160. So imagine if that zero wasn't there. 90 plus 70 would be 160, not 1,600. I've got confused with my place value there. So I can show it like this. Here we have that 90 plus 70 is equal to 160. Now, what we're going to do is we're going to look at what happens when we change one of those parts. So we're going to keep continuing with the set, the three plus four is equal to seven. This time something has changed. What have you noticed has changed in my part-part-whole model? Absolutely, I have my four here at the top, my three here in this part. I also have 500 in here. So I know that four plus three is equal to seven. Can I use that to help me answer four plus 503? Absolutely, I can. I need to put in 507. So I can just imagine that I'm just doing four add three here. My 500 stays the same. The zero is there as a placeholder in the tens column. So my number has changed to 507. This time, this part has changed. What has happened? Why has it changed? Well done, it has now got my four still here. So that stayed the same. I've now got 1,104, plus my three. Can I use my related fact here, my derived facts to help me solve this equation? Absolutely, I can. I know that here I can just do 1,004 plus three is going to be 1,000 and, not 1,000, 1,104 plus three is going to be 1,107. So using that knowledge about the four and the three really helped me. When I'm looking at other equations, I can still think back to that original number fact. Might sound tricky when we first look at it, but actually, if we use our related facts to help us, it's not as tricky as it first looks. Okay, this time if you're feeling confident, I want you to add really quickly these two numbers together, and if you can, explain how you did it using these related number facts. If you're not feeling so confident, don't worry, we're going to go through this example together now. So what's changed? Well, we have our ones that have stayed the same. We have four ones here. We have three ones here. This time, we have three thousands of four thousands. So we can use these related facts twice, 'cause I know that three plus four is equal to seven. So that's 7,000 this time. And I know that four plus three is equal to seven. That's seven ones this time. We must keep our zeros as placeholders in the hundreds and tens column. So my new number would be 7,007. When I first looked at that, I thought that looks quite tricky, but using these facts, I could solve that really, really, in a really, really straightforward way. So last thing before you are going to have a go at your Independent Task today, what is the missing part and how do you know? This time, I've written it as numbers, not showing you as counters or deans. So I want you to have a think. How can using this information here help us solve this question here? Well, absolutely three, three, zero, 330, I know that three and four makes seven. So I need to have four, four, zero, or the number 440 in this box here, because my number here is seven, seven, zero, 770. So 300 plus 400 is 700. 30 plus 40 is 70. And absolutely I've used all of these numbers in my box here to help me to solve this part here. Now, your Independent Task today is going to be you writing your own derived facts. Let's have a look at what I mean by that. So you're going to have a look at these part-whole models here that you can see on the screen. Then what I'd like you to do is can you write the related facts for each part-whole model? My challenge today is what derived facts can you write from those? So when I'm talking about the related facts I'm talking about where you'd relate nine subtract three is six or three add six is nine. When I'm talking about the derived facts, I'm talking about how you could use these to help you in other calculations. That's why it's the challenge today. Please pause the video now to have a go at today's task. Okay, then. As I said, we're going to go through the answers together. So let's look at this first part-part-whole model. So I know three plus six is equal to nine. I also know that six plus three is equal to nine. I know that nine, my whole, take away three is equal to six. And I know that nine take away six is equal to three. Now, my challenge was what derived facts can we write for this part-part-whole model? I could write 1,003 plus six is equal to 1,009 using that knowledge with the three and the six to help me. Or I could say 33 plus 66 is 99, using that knowledge that three add six is equal to nine, whether that's in the tens or the ones column. Next part-whole model, we have 17 as our whole. 11 and six are our parts. So I could say 11 plus six is equal to 17. 11 plus six is 17. Or I could say six plus 11 is also equal to 17, or I could do 17 as my whole, say 17 subtracted, subtract, sorry, 11 is equal to six. Or I could say 17 subtract six is equal to 11. I could then use those facts to help me with a derived, some derived statements. So I could say 1,100 plus 600 is equal to 1,700. Using that place value knowledge, I've made them 100 times bigger and added them together. Then here I could say 110 plus 62 is equal to 172. This time I've made them both 10 times bigger, and I've put a two in my ones column here. So if you imagine there's no two there, you can still see 11 plus six is 17, but there's, this time there's a two in my ones column, so I have to put that two my ones column here. And using the knowledge that I've made them 10 times bigger, I've made sure my place value is correct. Lastly then, we've got 14 as our whole, five and nine as our parts. Let's see which facts we could write for this. I could say five plus nine is equal to 14. Or I could say nine plus five is equal to 14. Or I could say 14 subtract nine, subtract five, sorry is equal to nine. Or 14 subtract nine is equal to five. Then I could use those numbers to help me here. I've got 1,400 and I'm subtracting 900. It would give me 500. I've made them all 100 times bigger and used my whole to subtract one of my parts. Then I could say 105 plus 109 is equal to 214, still using that five and that nine that equaled 14, but this time I've got two hundreds as well. If you'd like to, please share your, please ask, sorry, your parent or carer before to share your work on Twitter, by tagging @OakNational and the #LearnwithOak. Been really impressed today with all of your hard work with deriving all of those facts. Now, using all of that hard work, have a go at completing today's quiz. Hopefully, I'll see you again soon for some more math. Thank you, and bye bye.

Deriving addition and subtraction facts

Deriving addition and subtraction facts

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UnitMaths / Addition and subtraction

Maths