﻿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.

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?

One Thousand and Eighty Nine

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

Maths

Y8 Maths

Key Stage 3

Year 8

Algebra and problem solving

UNIT

Pick a number

Odd or even?

Algebraic Proof

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?

Lesson overview:One Thousand and Eighty Nine

Core Content

Algebra and problem solving: