Numbers and numerals

Axioms and arrays

Factors and multiples

Order of operations

Positive and negative numbers

Expressions, equations and inequalities

Angles

Classifying 2-D shapes

Constructing triangles and quadrilaterals

Coordinates

Area of 2-D shapes

Transforming 2-D figures

Prime factor decomposition

Conceptualising and comparing fractions

Manipulating and calculating with fractions

Ratio

Percentages

Different number systems

Sequences

Forming and solving equations

Forming and solving inequalities

Linear graphs

Accuracy and estimation

Algebra and problem solving

Ratio (8.6a)

Ratio, real life graphs, and rates of change

Direct and indirect proportion

Univariate data

Bivariate data

Famous maths problems

Angles and parallel lines

Angles in polygons

Bearings

Volume and surface area of prisms

Area review

Circles

FDP review (9.1a)

Probability

Sets and Venn Diagrams

Solving linear simultaneous equations algebraically

Solving linear simultaneous equations graphically

Angle review (9.5a)

Constructions, congruence, and loci

Pythagoras's theorem

Ratio review (9.7a)

Similarity and enlargement

Surds and trigonometry

Quadratic expressions (9.9)

Quadratic equations (9.10)

Indices and standard form

Growth and decay

Finance

Maths

Y8 Maths

Key Stage 3

Year 8

UNIT

In this lesson, we will investigate some number problems, especially focusing on the number 9, and the rules of divisibility by 9.

One Thousand and Eighty Nine

﻿Hello, I'm Mr.Langton and today we are going to look at some number puzzles. All you're going to need is something to write with and something to write on. Try and find a quiet space with no distractions, and when you're ready, we'll begin. Okay. We're going to start with a try this activity. What I want you to do first is to think of a number between 1 and 20. Write it down. I want you to take that number and multiply it by 9. Now, if you're not very confident with the 9 your times table, maybe you want to pick a smaller number rather than a larger number. If you need a moment to just change you number and then multiply by nine, do it now. Okay. Once you go the answer, I want you to take the digits of that answer and add them together. So for example, if you have 36, you're going to add three and six together. Okay. Now you've got that answer. I want you to subtract five from it. Then you're going to take the answer that you've got now and convert it into a letter from the table. So if your answer with 17, your letter is Q. Okay so far? Right now, focusing on that letter, I want you to think of a country that begins with that letter. You got your country? Okay. Now I want you to do now, Think of how you spell that country, and write it down. Write down the name of the country on your paper. Okay, now the second letter of that country, I want you to think of an animal that begins with that letter. You got animal? Finally, what colour is that animal? Now, I'm going to go out on a limb and I'm going to guess that you're thinking of a grey elephant from Denmark. Was I right? Was I right? I was right. Wasn't I? You were thinking of a grey elephant from Denmark. Okay, so, let's have a think about this. Why does it work? Why was I right? And let's look at the maths that we had to do. Let's think of a number between one and 20. Let's do that together. Now let's say, for example, if I pick the number eight, alright. If I multiply that by nine, I get 72. I add those digits together. Seven plus two is nine. I subtract five. That gives me four. And its that point there that that four converts to letter D, and I can't think of another country beginning with D other than Denmark, there might be others, but Denmark is such an obvious one that I'm hoping that that's what you're going to pick. The second letter of Denmark, is E and I hope that the most obvious animal you can think of is an elephant, which of course is grey. And that's all great. If you come up with that answer of four, I've got to make sure that no matter which number you pick, you come up with four. And that's where the trick is. Now work here. If we're at four here, we've already subtracted five. That means that before that must have been nine. So I'm giving you a choice here of what number you want to pick. By the time we get to here, you must have nine, otherwise my tricks not going to work. So it's this bit here, whatever you pick, you multiply it by nine, and then you add those digits together. And that is a big, huge property of the nine times table. If I write up my nine times table here, can you see that if we have the two together each time, they always make nine. Oh, a little bit there that crept in and add those digits together to get 18. And if you add the digits of 18 together, it equals nine. Now I said between one and 20, because in almost every single case, we only we will only get digits that will add up to make nine. One and zero add eight, for example, is nine. Um it does work for any number in the nine times table. You could do 30 times nine. That's a other reason one to see, isn't it? Because that's going to be 270. And two add seven add zero is nine. And so on. Sometimes you might get 18 or 27, but if you notice that once again, it's going to be, a number that if you add those together, You still going to get nine. And that is a key feature of the nine times table. But every time you multiply a number by nine the digits will eventually add up to make nine. Okay, now I've got two more tricks for you. The one on the left you're going to think of a three digit number, reverse the digits, subtract one number from the other number, and then reverse the digits of your answer. Add those two answers together. And you're going to get a result. I want you to practise that one a few times. See what you can get up to. see what answers you get. On the right hand side, you're going to try and win a prize. You've got all those boxes out, 18 different prizes that you can win. Let's see which prize you win. And I promise you right now, I will pay you the money for the prize that you win. So I want you to try each game a couple of times, three or four times, give it a go, see what you can get to and see if you can spot any patterns. See if you see where it's working, where it's not working, on what you could, and can you tell me why it's working that way? Good luck. Right? How do you get on with this one? Let's work through it together. Let's come up with an example. Let's think of a three digit number. Let's go for 439. I'm going to reverse that number, which is 934. And I now need to subtract the smaller number from the larger number. It's going to be 934 take away 439. Four take away nine can't do. 14 take away nine is five. 12 take away three is nine, and eight take away four is four. So my A is 495. If I reversed those digits, I get 594. And now I'm going I've got to add those together. So 495 and 594. Five and four is nine, nine and nine is 18, four and five is nine, and on that one there is 10. And I've got 1089, at which point your brain might be clicking into gear just a little bit, because you might be saying, well hang on a minute, Mr. Langton, that's the name of the lesson. And do you know, what? I picked a different three digit number to you and I also got 1089. Well, there's a reason for that. Try it with a different number and a different number and a different number and you will always get 1089. There were a few things that you can do to trick it and make it fall apart. For example, if I think of a three digit number where all the digits are the same, and then I reverse it and take it away, then of course it's not going to work. Cause I'm just going to end up with zero. So we found an example where it doesn't work, but if you pick three digits that are all different, you will get 1089. Roll up, roll up! It's time to play a math game, win a prize every time, real money to be won. Let's go, let's see what we can do, right? Pick a number between one and nine, a number between one and nine. What you thinking of? what you thinking of? Let's go for seven. Let's start with seven. and let's multiply that seven by three. Seven times three is 21. We're going to add three to that 21. Add three makes 24. I've now got to multiply 24 by three. 24 times three. 24 that's going to be 72, isn't it? 72. We're going to add those digits together. Seven plus two equals nine. Find a number in the grid. Win that prize. win that prize. Win that prize. Nine is zero. I've not won a thing. Rats! And I was so close to winning a thousand pounds, only two out. What if I would have gone two bigger? What if I would instead, I was only two out. What if I started with nine instead? Nine multiplied by three is 27. 27 at three is 30. 30 multiplied by three is 90. Add those digits together.You get nine and zero is still nine. We've got a problem. And I'm sure that you've figured out the moment that I said, add those digits together. You realised that you were being tricked. Can you explain why that has happened? How many times did you ever go before you realised hang on a minute, Mr. Langton, you are having me on. What you need to do is think about what we're doing here. Obviously, if we add the digits together and we get nine, we must be adding together a multiple of nine. This number here, once you multiply by three at this stage, you must be in the nine times table. And the reason for that, I'm going to do a tiny bit of algebra now. Let's just see if you can follow it. If I think of a number, let's call it N. If I multiply that by three, I've now got three N the next step is to add three. I get 3 N plus three, and the next step then is to multiply by three. So I've got three lots, the 3 N and plus three. And if I expand those brackets, that gives me 9 N plus nine. So I've got a number that's been multiplied by nine, and I've added nine to it. So that number must be in the nine times table. And as such, I've tricked you. I've made sure that every single time you play, you are going to get the answer of nine. You're going to win a value of zero, and my wallet can stay in my pocket not to be touched. Okay. Finally, we're going to finish with the Explore activity. Which of these numbers are multiples of nine? So, I want you to think about what we said about multiples of nine. What do you know about the digits of any number that is in the nine times table, but always add up to make nine? So your going to go through all these digits. You're going to see which of these are multiples of nine. And in that second box, each of the numbers that I've written down there, they are actually multiples of nine, but some of the digits are missing you to work out what that missing digit is going to be every time. Pause the video and have a go. when you're ready unpause it and go through it together. You can pause in three, two, one. Right, so, in that top one, there are five numbers that are multiples of nine. I'm just going to take, I'm just taking this one as an example, just to show you. So I would do one plus eight plus two plus seven plus three plus six plus four plus five. Now one and eight is nine, two and a seven is nine, three and six. Oh, so I've actually got nine of nine of nine of nine, which makes 36. Three and six makes nine. So that number is in the nine times table. Whereas if I were to take, for example, 89, let's put a square box around it because it's not a multiple of nine. Eight plus nine equals 17. One plus seven equals eight. So, nope, it's not in the nine times table. So we're going to use that trick in that second part. Each of these numbers are multiples of nine. What could the missing digits be? So in this case here, if I had the two digits, I know two and three makes five. Now I need to make nine altogether. So that number there must be a four, because two and four is six add three makes nine. Next one. 400, so many tens, and five ones at the end. So four and five already makes nine. There are two possible answers you can get there. You could either say there is 405, because that makes nine. You could say it's 495, because four and nine is 13. Add five is 18. And one and eight makes nine. So both of those numbers are multiples at nine Over the next bit. Eight, add something, add five, add one. Well eight and one makes nine. So five and four makes nine. Now, we come to hit eight and five is 13. Now, if I have those digits together, one plus three is four. That means that these two boxes here, the digits must add up to make five because four and five makes nine. So you could have had, in either order, one and four, you could have had two and three, and you could have had three and two. You could have had four and one. You also could have had you could have zero and five, or you could have had five and zero. So six possible things that you could have had for example, 8,055 is a multiple of nine. Let's move on to the next one. Four and six is 10. Add one is 11. Add five is 16. 16 add five is 21. Two plus one is three. So that number there is six. And finally, what could the missing digit be for this last one? One and seven is eight. Eight and four is 12. 12 and five, 12 and five is 17. 17 add eight is 25. So two plus five equals seven, which means that these two numbers here need to add up to make two, which is one and one. They could also add up to make you 11, because 11 plus seven is 18. So you could have had, uh you could have a two and nine. You could have had three and eight. You could have had four and seven. You could have had five and six. Six and five. I'm running out of space. Um six and five, seven and four, eight and three and nine and two. So it's one, two, three, four, five, six, seven, eight, plus this one here. Nine possible options. That's it for today's lesson. A little bit of a whirlwind, went through a lot of things really, really quickly there. I'll see you later. Goodbye.

In this lesson, we will look at how different operations can affect the properties of a number. We will calculate a number of different operations using written worded instructions.

Pick a number

﻿Hello, I'm Mr. Langton and today we're going to be looking at some number patterns. All you're going to need is something to write with and something to write on. Try and find a quiet space where you won't be disturbed and when you're ready, we'll begin. We'll start with a try this activity. For each of these puzzles, you're going to start with a number in the beginning and you're going to follow all the rules all the way down. So for example, if I start with three, if I had two, I get five. If I times that by three, I get 15, then I subtract 12 and I get three and so on all the way down. Pause the video, have a go at all three of these. I want you to work out the answer for each one, make a note of anything that you find interesting. You can pause in three, two, one. How did you get on? Here are the answers. For the first one, when you started with three, you're going to end up with nine. In the middle one you started with four, you should end up with 12 and the last one, when you started with five, you should have got 15. So what about the patterns? Can you spot any links between what we started with and what we finished with? Is there anything else that you found interesting? So looking for connections between the starting number and the end number, when we started with three we finished with nine. We started with four, we finished with 12 and when we started with five, we finished at 15. Can you spot a pattern? Every single time it's being multiplied by three, isn't it? If we start it with four, four multiplied by three is 12. If we start it with five, multiply by three we get 15. We can predict that if we had to start number of six, then the final answer should be 18. So can you pick a different start number that will give an odd answer? So we've got some already. We started with three, that gave us an odd answer and five gave us an odd answer. So three and five, but four and six both gave even answers so I would predict that seven will be odd so I'm hoping it'll be 21 and nine would be odd and effectively if I start with an odd number, then my answer should be an odd number. If I start with an even number like four or six or eight, then my answer will be even... and that's my rule. Okay, now it's time for you to have another go. These are three completely different puzzles. You can pick any number that you like to start with, work your way through until you get to the end and you get an answer. Try each one and try each one with different numbers. See if you can spot any patterns or rules. Can you find a link between the start number and the finish number? Can you come up with any rules that say that if you know what the starting number is, you can make an odd answer or an even answer? Good luck. How did you get on? Let's look at a few answers together. I've no idea which numbers you picked, but I'm going to start with some simple ones. I'm going to start with one. If I started with one, my final answer was two. If I started with two, my final answer was five. If I started with three, my final answer was 10 and if I started with four, my final answer would be 17. So that's something, you might have picked different numbers, but there's enough here for me to start to look for a pattern. Now every time I start with an odd number, I end up with an even number and every time I start with an even number I end up with an odd number. So that's definitely one pattern that I can see. Could you spot a rule that links everything all together? So if I said, for example, if I started with 10, what would my answer be? Could you come up with a rule that would tell you exactly what it would be? There is a rule, a little bit difficult to spot with this one, it's a bit easier with the other two, actually but if you take the number and you square it and then add one, you get the answer. So if you start with three, three squared is nine, add one is 10. So that's one method that you can use. Let's look at the second one. If I started with one, my final answer was two. If I started with two, my final answer was four. If I started with three, I got all the way through to six and I'm betting you could spot a pattern straight away, couldn't you? You can tell me, even if you've not worked out four, you can see what's going on here. It's going to be eight. No matter what number I start with, I'm always getting an even answer and that answer is actually double what I started with and if I told you what would happen if I started with a hundred, rather than going through all these steps to work it out, now that we've looked at these simpler ones and we've got a rule, we can say, "Bet that's going to be 200." If you want to pause it and check that I'm right, you can, there's a lot of maths to do just to get to the answer though. Okay, what about third one? If I start with one, my final answer is three. Two gives me five. Three gives me seven. Four gives me nine and so on. So once again, we've got a pattern and that pattern, they're all odd numbers going up by two each time and so the rule is whatever number you start with, you double it and add one and that gives you your answer. So if you've started with 200, you put 200 in here and go through all the steps and you're going to find the answer is 401. We'll finish with the explore activity. Look at these last two puzzles again. Can you explain why the one on the left always gives an even answer, and can you explain why the one on the right always gives an odd answer? Pause the video and have a go. When you're done un-pause it and we can discuss it together. You can pause in three, two, one. So how did you get on? I'm going to do a little bit of algebra with you now, it might be a little bit tricky. All I want you to do is follow what I'm doing and when we discuss it, hopefully it'll be a little bit clearer. If I start this one by picking the number N, any number at all and I add 12 I get N plus 12. Multiply that by four, I've got four Ns and I've got 48. Take away 36, I've got 4N plus 12. I add nine, I've now got 4N plus 21 and if I multiply by two, I've got 8N plus 42. Adding 14 gives me 8N plus 56 and then dividing by four, gives me 2N plus 14. Yep and then if I take away 14, I get 2N. Now, I'm not sure if you feel confident to do every single one of those steps yourself, but hopefully you can see how I've done it and you can see how it links all the way from that start number. Whatever I start with, I end up with 2N and if I take a number and I multiply it by two, then that number must be even because every even number can be divided by two. So 2N is actually a formula or nth term that would give you all the even numbers, wouldn't it? We can use that to generate a sequence. If we said N is zero then obviously two times zero is zero. If N is one, we get two. If N is two, we get four and six and eight and so on. Now what about the one on the right? Again, if we start with N, for any number at all that we start with multiply it by 10 and we've got 10N. Add 30, 10N plus 30. Subtract five is 10N plus 25. Double it and we get 20Ns and double 25 is 50. Divide by five so divide 20N by five, we get 4N and 50 divided by five is 10. Add 17, 4N plus 27. Subtract 25, we get 4N plus two and if we divide that by two, we get 2N plus one. Now, as we said, any time we times a number by two we get an even number. If we take that even number, and then we add one, it must be an odd number because every number that follows an even number is an odd number, isn't it? So we've also got a formula here, 2N plus one, for any odd number that you like. If we start with N is zero, we get one. If N is one, two times one is two, add one is three. If N equals two, we get two times two is four, add one is five and seven and then nine and so on. So these formulas here, for odd and even numbers can be quite important later on when you're looking at algebra and algebraic proofs. We'll have a look at some of that in another lesson. That's it for today. Goodbye.

In this lesson, we will investigate the algebraic representations of odd and even numbers.

Odd or even?

﻿Hello, I'm Mr. Langton. And today we're going to be investigating the algebraic representation of odd and even numbers. All you're going to need is something to write with and something to write on. Make sure you're in a quiet place with no distractions and when you're ready, we'll begin. We'll start with the try this activity. Choose different values for n, and substitute them into these expressions. Can you organise the expressions into different groups? There's always odd, always even, or it could be odd or even. And can you explain why they always give these results? Pause the video and have a go. When you're ready we'll go through it together. Pausing in three, two, one. How did you get on? This is how I grouped them. Compare these with your answers and in a moment we're going discuss why they gave the results that they do. So we'll start off by looking at 11. How do you know that 11 is an odd number? I'm going to show you two ways that you can show that 11 is an odd number. I've represented 11 here as 11 dots. And I can split 11 into groups of two. Now if it goes exactly into two, there won't be any left over, but the fact that one there is leftover tells me 11 must be odd. Another way I can do it is split 11 into two equal groups. I've got five in that one and I've got five in that one. And once again I've got this one left over. So that's two different ways to show that 11 is odd. I'm going to look at that same thing now, algebraically. So 2n + 1 is two lots of something. So I don't know what it is, but I've got two of these Ns. And I've got two of them, then they must be even. I then got that one that's added on afterwards, and that's what makes 2n + 1 odd. I can split it into two even parts, but there's still one leftover. Now 2n + 6, I'm going to represent this like this- I've got an n and I've got another n, that's my 2 Ns. Plus six, now I'm going to do my individual numbers as dots or circles like I did before. And can you see that I can make two equal groups, each one of them is n + 3, but because I can make exactly equal groups with nothing leftover, then that means the value of 2n + 6 is always even. Finally we've got 4n + 3. So this time I've got four Ns. Now I can come up with two pairs of Ns can't I? And then I can separate my three into a dot, the dot, and there's another dot leftover. So I can't make two equal groups, I almost can. I've got two equal groups, but I've got that one leftover again. So 4n + 3 must be odd. So what we're going to do now, is apply that knowledge to these number grids. If I start with n and if I add two, I get n + 2. Now multiply that by three, I've now got three lots of n + 2, which is equal to 3n + 6. If I subtract four from 3n + 6, I've still got my three Ns, but now I've only got two. And when I multiply all of that by four- Oh, sorry, I'm getting ahead of myself. That's four lots of 3n + 2, which is equal to 12n + 8. One half it. Now we need to half my 12n and I need to half my eight. I'm going to subtract one, which gives me 6n + 3. And then I'm going to divide by three. So 6n divided by 3 is 2n. 3 divided by 3 is 1. So this is my final answer. 2n + 1. So let's see if if try and represent that as a diagram, I can split the Ns into two equal groups, I've got one leftover, can't be split up, and as such, 2n + 1, must be odd. So now it's your turn to have a go. I want you to work through each puzzle starting with n. So you try and develop a formula to link the input with the output, with that final answer. And can you say if it will always be odd, always be even, or if it could be neither of them. Pause the video and have a go, access the worksheet, and when your ready un-pause it and we can go over through it together. Good luck. How did you get on? Let's go through them together. So the first one n multiplied by 4 is 4n. If I add an eight, I get 4n + 8. If I half that I get 2n + 4. And if I add 10, I'm going to get 2n + 14. Now I can split my 2n into n and n, can't I? And I can split 14 into two groups of seven, which means I've got two equal groups, which means that this one is always even. Looks at the next one. Start with n and multiply by 10 and we get 10n. Now subtract five and we've got 10n takeaway 5. Now I'm going to divide by five, so divide that first one by five and I'm going to get 2n, divide that by five and I get takeaway one. If I add two I get 2n + 1. Which we've show before is always odd. So whatever number I start with on this one, I'll always get an odd number. Right finally. This one's much longer. I'm going to start by adding four. I'm going to multiply by two, to get 2n + 8. Now subtract eight and I've got 2n. So at this stage, this number here will always be even. Let's keep going. Multiply by three, and I've got 6n. Add 12, 6n + 12. Half it, 3n + 6. Add nine, I'm going to get 3n + 15. Divide that by three and I'm getting n + 5. Subtract five and I get n. Add one and once again I've got n + 1. Now n + 1, that's a might tricky one. Because I can't draw that very easily. I can't split it into two equal groups, but it's totally going to depend on whether n is odd or even. If n is odd, and then I add one, an odd number plus one must be even. If instead, we started with an even number, and we add one, then the answer would be odd. So we can't say for this one whether it would be odd or even, it could be either. Totally depends on what number we start with. Now this is a rather tricky explore task. What you need to do is fill in the blanks to make each one work. Sometimes there's more than one possible answer that you could use. Good luck trying to work them all out. On that last one you've got to make sure the final answer is an even number, so that's going to be really tricky. Pause the video and have a go. When you're ready, we'll go through it together. You can pause in three, two, one. So how did you get on? Here are some possible answers. Certainly in the case of the first one, you might have come up with some slightly different ones, and in particular I'm thinking here to get from 10 to 20, you may have multiplied by two, you may have added 10 instead, you may have done something slightly different. So there's more than one possible answer. It's the same here as well. Instead of subtracting 18, you might have divided by two. So there's more than one possible answer but here is something you've got that you could use. Looking at this last one here, I've left that last one blank. If you follow it all the way through to n + 4, we've got all of these steps done on the left hand side, but this last bit here, if we need to make this even, then we're going to need to multiply it by an even number. So in this box as long as you've written an even number- oh sorry I'm in the wrong box, here. You've written an even number here, then that will make your answer an even number here. That was a really tricky task. I hope you got through it. That's the end for this lesson. See you later. Good bye.

In this lesson, we will look at how we can use algebra to prove what happens to calculations with odd and even numbers.

Algebraic Proof

﻿Hello, I'm Mr. Langton and today we're going to look at some algebraic proofs. All you're going to need is something to write with and something to write on. Try and find a quiet space where you won't be disturbed and when you're ready, we'll begin. We'll start with a try this activity. Fill in the blanks to each one with either odd or even. Pause the video and have a go. When you're ready, un-pause it. We can go through it together. You can start in three, two, one. Here are the answers. How did you get on? It's rather easy to show that adding two odd numbers together gives you an even number. We can do it by example. Three plus five equals eight. There's an example of two odd numbers add together to make an even number. Now what I'd like to do is prove that's always the case and I'd like to do that with some algebra. Now, in a previous lesson, we looked at how we can represent odd and even numbers. Now an example of an even number is any number that can be divided by two. So we can represent an even number algebraically as two times by a number. So for example, four is two times two. 10 is two times five. 50 is two times 25. Every even number can be written like that. Now odd numbers are always either one bigger or one smaller than an even number. So if an even number is two lots of something, then an odd number can be written as two lots of something plus one. For example, seven is two lots of three, add one. Nine is two lots of four, add one. Now, what we're going to do is use these algebraic expressions with these words here. So if an odd number can be written as two n plus one, and we're going to add to that another odd number. That gives us four n plus two. Now four, we can split that up into two equal parts. We can write four n like this. And we can write two, two ones like that. So that's four n plus two. And I can split that into two equal groups with nothing left over. Which means that this bit here, four n plus two, must be even. So we've now got a formula. We've proven that any time we add together two odd numbers the answer must be even. Zacky's made a statement. He says, "If I add together four consecutive numbers "then the answer will be even." Do you agree or do you disagree? Try it for yourself. Pick four consecutive numbers. For example, three, four, five and six. Let's add those together. Three and four is seven. Seven and five is 12. 12 and six is 18. That's even. So pause the video, try it for yourself. Pick some different numbers, see if you can come up with the same answer. Can you always get an even number if you add four consecutive numbers? You can pause in three, two, one. Could you find anything where it doesn't work? It seems to me like I'm going to have to agree. Seems like it always works. But what I'd like to do is prove it algebraically. Now, my four numbers, each one is one bigger than the last, isn't it? So whatever I pick as my first number, I'm going to call that n. And the number that comes after that, the next consecutive number, is going to be one bigger. So that next number would be n plus one. The number that comes after that will be one bigger than that, n plus one plus one. Or n plus two. And the fourth number that comes after that is going to be one bigger than that, n plus three. So we've got here a formula that will list any four consecutive numbers if we want. So, for example, if n was three, we'd add one to that to get four or three plus two is five or three plus three is six. So what happens when we add together these consecutive numbers? We're going to do n, add n plus one, add n plus two, add n plus three. So in this case, I've got one, two, three, four ns. And I've got one, two, three, four, five, six. So six. Now I can split that into two equal groups, can't I? So we can show that this must always be even. I'll draw this diagram just to help. That's four n. That's one, two, three, four, five, six. And I've got two equal groups. So I've shown I can pick any four numbers, any four consecutive numbers, and they will always add up to make an even number. This has been really tricky so far, but now it's your turn to have a go. Access the worksheet, see how many you can do and then we'll go over them together. If you're not very confident, just leave the video going a little bit longer and I'll give you some hints on our first one. Good luck. I'm going to start off by giving you some hints. First question is an even number adding an even number. We can represent an even number, we've already said, as two n. So can you show me what happens when we add those together? What about an even number plus an odd number? An even number can be shown like that, and an odd number can be shown like that. Now, what about question two. Use algebra to show what happens when you add three consecutive numbers together. What do you notice about your answer? So here's a hint. That's the first number, the next number is n plus one and the next number is n plus two. I'm going to leave that there. If you've already finished, just leave the video going. But if this is your hint, if you were using this as a hint, pause the video now, see how many you can do. Pause in three, two, one. Okay, let's go over the answers. So two n plus two n would make four n, which we can easily show splits into two equal groups. And if it splits into two equal groups, then it must be even. Question B, two n plus two n plus one equals four n plus one. Now follow from what we just said, if four n is even, then four n plus one is one bigger. It must be odd. We can draw it. Like so. So we can see we can't get two equal groups, can we? That and that are not the same. So in this case, even plus odd is odd and we've proven it. Now an even number multiplied by an even number. That's going to be two n multiplied by two n, which is four n squared. So what would that look like? I've got for lots of n squared. So I've got n squared, n squared, n squared and another n squared. They can be split into two equal groups. So we've proven that when we multiply together two even numbers, our answer is even. Finally, even times odd. That means we need to do two n multiplied by two n plus one. I'll put some brackets around that. Two n times two n is four n squared. Two n times one is two n. Now, I'm just going to pick a different colour. Just make some space over here where we can draw. Four n squared plus two n. So I've got four n squareds, one, two, three, four. And I've got two ns. So we'll just add those, so that's an n and that's an n. And we can see we've got two equally sized groups. Each of those groups contains two n squared plus n, which means if we do an even number multiplied by an odd number, the answer is always even. And finally, question two. What happens when we add these three consecutive numbers together? So we have n, we've added n plus one and n plus two. So we've got three n plus three. Now, is it odd? Is it even? Not quite sure. I'll tell you one thing, though. It's going to be a multiple of three. If you add together three consecutive numbers, it will always be in the three times table. And I can show you that like this. N, n, n, that's my three n. And that's my three. And they can be split into three equal groups, can't they? So you can pick any three consecutive numbers you'd like and when they add together, they will always make a number in the three times table. You can do a quick example and just check. What about 10, 11 and 12. If I add those numbers together, that makes 33, doesn't it? And 33, obviously, is in the three times table. Finally, we've got the explore activity. And this is really, really tricky. You can certainly have a go at it, but if it comes to using algebra, it's going to get a little bit nasty. Binh has got an idea. She says, "If you pick any five numbers, "she can chose three of those five, add them together "and they will always make a multiple of three." So, Antony had a go. He says, "What about eight, seven, four, five and three?" Have a look at those numbers and from those five that Antony's picked, Binh's picked out seven, three and five, which add together to make 15. And 15 is in the three times table. So why don't you investigate Binh's idea and see if she's always right. Have a go by yourself and when you're done, un-pause it and we'll have a look at it algebraically. See if we can figure out what's going on there. You can pause in three, two, one. How did you get on? Hopefully, you found that Binh was correct. Any five numbers that you choose, three of them can be added to make a multiple of three. And we'll start to show you how that works and I'm not going to go all the way into proving it, because it's going to get quite tricky. But I just want to show you just an idea of how we start off, just by doing that. I've listed the first numbers from zero to 12 and obviously, this list goes on and on forever. All the integers keep going on and on and on and on. I'm going to start off just by looking at the numbers in the three times table. Three, six, nine, 12 and so on. Each of these numbers can be made by multiplying a different number by three. Three times something will give me any of those numbers. And in fact, three times zero is zero, isn't it? So I've got a formula that can make an awful lot of the integers that exist. Next up. Let's look at four. It's one bigger than three. Seven is one bigger than six. 10 is one bigger than nine. So I'll come up with a formula for those. Those numbers are one bigger than a number in the three times table. That's going to include one, because three times zero is zero, add one is one. And there's only one gap between each bit now. So let's have a look there. Five is two bigger than three. Eight is two bigger than six. 11 is two bigger than nine. So I've got a new formula now. Those numbers are all two bigger than a number in the three times table. And actually two goes with that as well, doesn't it? Which is two bigger than zero. Now, this one is already a multiple of three. These two aren't, but we can do some maths to show that whichever combo, you can pick any sorts of combinations that make multiples of three. For example, if I add them together, one and two is three. Three n, add three n, add three n is nine n plus three. And that is a multiple of three, isn't it? Because three goes into nine and three goes into three. And whichever combination of five numbers you pick, you must pick three numbers that will make a multiple of three. The are other ways that you can do it. There's lots more ways, I'm not going to show you them all now, because it will take quite a while but that's how you'd start to prove it. That's it for today's lesson. I'd love to see any ideas and any work that you've done so far. If you'd like to, please ask your parent or carer to show your work on Twitter, tagging @OakNational and #LearnwithOak. See you later, goodbye.

Is the value of 2n always odd, always even, or either?

What operation could I perform to 4n + 1 to make it always even?

What operation could I perform to n + 4 to make it always even?

What is the result of odd x even x odd?

What is the result of (odd x even) + (odd + even)

What is the result of (odd + odd) + (even x odd) + (even + odd + odd)

Which of these numbers is a multiple of 9?

This number is a multiple of 9. What is the missing digit?

Which of these numbers is a multiple of 3?

What is the final answer if I start with 6?

Start with 6. What is the final answer?

Will the answers to the puzzle above always be odd, even, or does it depend on the starting number?

Unit Overview: Algebra and problem solving

Lessons:

One Thousand and Eighty Nine

Pick a number

Odd or even?

Algebraic Proof