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 learn about a famous unsolved maths problem called Goldbach's conjecture.

Goldbach's conjecture

﻿Hi, I'm Miss Kidd-Rossiter, and I'm going to be taking this really exciting lesson on Goldbach's conjecture. This is an as yet unsolved maths problem, that's been causing mathematicians all over the world a bit of a headache for a long, long time. So I'm really excited to get started with it. Before we get going you're going to need a pen and something to write on. If you've got any large paper A3 or bigger, that might be helpful for today's lesson. Make sure you're in a nice quiet place and you're free from all distractions. If you need to pause the video now, to sort anything out, then please do. If not, let's get going. Right, let's get started. So with the try this activity, Zaki is thinking about prime numbers. He says any number can be written as the sum of two prime numbers. What does sum mean? Tell me now. Excellent. It means that we're adding doesn't it. So that's what he's saying. Well do you agree with Zaki? And can you find some examples that either show or don't show this to be true? Pause the video now and have a go at this task. Excellent. Did you agree? Tell me. Okay, brilliant. So can you tell me an example that shows that we can write a number of the sum of two prime numbers? Excellent. Okay, we could write seven as two plus five. Two is prime. Five is prime. So we can write seven as the sum of two prime numbers. Can we write two as the sum of two prime numbers? No, we can't. So that is one that doesn't show this to be true. What about one? Excellent. Another one that doesn't show this to be true. So Goldbach's conjecture is all to do with writing numbers as the sum of two primes. So Christian Goldbach was born in 1690 and he died in 1764 and he is most famous for this conjecture that he wrote in a letter to Leonard Euler. And he's a really, really famous mathematician. So on the 7th of June, 1742, he wrote this: Every even integer greater than two can be written as the sum of two prime numbers. So let me read that to you again: Every even integer greater than two, can be written as the sum of two prime numbers. So let's just, uh, pick this apart a little bit. So integer, what does that mean? Excellent, a whole number. So every whole number greater than two can be written as the sum. Remind me what that means again? Excellent. Adding two prime integers. And again, we've got the word integer, which means whole numbers. Let's just recap what's a prime? Excellent, a number with two factors. Exactly two factors I should say. And what are those two factors? Excellent. Itself and one. So every whole number greater than two that's even can be written by adding two prime numbers together. Prime whole numbers. So this is thought to be true. We've not managed to disprove this statement yet, but we've also not managed to prove this statement. So that's why it's one of the really troubling, unsolved problems in maths. And you can see how old it is. One way to represent this conjecture is using a very common diagram. So I'm going to draw an arrow or a line that would extend further. I'm going to put all my numbers on it greater than two. So greater than or equal to two, I should say. And then I'm going to do the same on the other side. And then from each of these numbers, draw a parallel line to the opposite line. So like this, and then also for the other one, like this. Now I want to remove all the numbers that are not prime. So two is prime. Three is prime. Four is not prime, so I'm going to remove that and its line. Six is not prime, so I'm going to remove that and its line. And you could continue this diagram even further. So that's why I've asked you to get some A3 paper, or even A4 would be fine, if you want to have a go at continuing this diagram. So let's remove the numbers that are not prime. And now we can see that any place that these two lines intersect, it is the sum of those two prime numbers. So for example, here, we've got two plus two. So this intersection represents four. So we can write four as the sum of two prime numbers. What does this intersection here represent? Tell me now. Excellent. Five. What about this intersection here? Excellent. Seven. So we've got some odd numbers at the moment. What about this intersection here? Excellent. Nine. And then we can see can't we, that we're going to have to repeat some of these going down this side. What would this intersection here be? Excellent. Six. What would this intersection here be? Eight. Good. What would this intersection here be? 10. Good. What would this intersection here be? 10 as well. So you can see that some numbers we can write as the sum of two prime numbers in two different ways or more different ways. What about here? Exactly. This is eight. This one is 10. This one here is 12. Good. And this one here is 12. Excellent. Have I missed any out on that diagram? I can't see that I have. So we can see that we've got all the even numbers there greater than two and less than or equal to 12 represented by adding two prime numbers. So we've got four, six, eight, 10, and 12. And as I said, you can continue this diagram as far as you like and see how many different even integers you can write as the sum of two prime numbers. You're now going to apply your learning to the independent task. So pause the video now, navigate to the independent task and then have a go at answering the questions. When you're ready to go through some of them and discuss them, not all of them will have one correct answer, then resume the video. Good luck. How did you get on with that? Some tricky questions there and some really interesting things for you to think about as well. So list all the prime numbers up to a hundred. That one should have been fairly straight forward. There you go. Then I asked you to continue the diagram. What was the largest even number you could write as the sum of two prime numbers? Now, every single one of you is going to have a different answer for this, but let me tell you something really interesting. The Goldbach conjecture has been shown to hold, so that means it is true, for every number up to and including, every even number I should say, up to and including four times 10 to the power of 18. So that is four with 18 zeros. Can you tell me what that number is? Go and research it if you don't know. What do you notice? Did you make any conjectures? I'd be really interested to hear them if you did. Explain why the conjecture cannot hold for most odd numbers. So when we add two prime numbers together, why can it not hold for most odd numbers? Well, we know that for an odd number, if we're adding two numbers together, to get an odd number, we have to add an odd and an even don't we. If we add two even numbers together, we get an even number. If we add two odd numbers together, we get an even number. So the only even prime number is two. So if we cannot form the odd number by adding two to another prime, then it cannot hold can it? So for example, 19 we could write as a sum of two prime numbers, because two is prime and 17 is prime. However, 11 we couldn't write as a sum of two prime numbers because we would have to write it as two plus nine, but nine is not prime. Because it has factor of three as well, doesn't it? So that's why it doesn't hold. Thinking about the explorer task now, Yasmin is thinking about Goldbach's conjecture. She says, as even numbers get larger, then there is more ways to write them as a sum of two prime numbers. Do you agree with Yasmin? Can you find some examples that either show or don't show this to be true? So pause the video now and have a go at this task. How did you get on with that? Well, the thinking is that this is generally true. So we could think about, first of all, we can write 12 as the sum of seven and five, but we could write 10 as the sum of five and five. And we could write 10 as the sum of seven and three. So you can see that there's only one way for 12, but there's two ways for 10. This is interesting, isn't it? But as we get larger, are there more ways, for example, to write 1000 as the sum of two primes, then there are to write 100 as the sum of two primes. That's something interesting for me to leave you with, to think about, and if you're really interested in this, then it might be worth researching something called Goldbach's comment. And that's a really nice representation of this statement here. So that's the end of today's lesson. There is absolutely loads of research going on in the mathematical community around this conjecture. And so there are so many mathematicians that would absolutely love to be able to prove this. Other mathematicians have done some research around the conjecture and some different things. So for example, in 2013, there was a paper published that proves that every odd number can be written as the sum of ... I'm not telling you how many primes. I'll let you research it. But I hope that you've found this really interesting and that you have learned a lot about something quite tricky in the mathematical community that has stumped mathematicians for a long time. If you'd like to share your work with Oak National, then please ask your parent or carer to share your work on Twitter, tagging @OakNational and #LearnwithOak. Hopefully I'll see you again soon. Bye!

In this lesson, we will learn about a famous unsolved maths problem called The Collatz conjecture.

The Collatz conjecture

﻿Hi, I'm Miss Kidd-Rossiter and I'm going to be taking today's lesson on the Collatz conjecture. This is an unsolved puzzle in maths that's been stumping mathematicians for generations. You might find it helpful for this lesson to have some A3 or big paper. If you don't have that, A4 will be fine. Make sure you've also got something to write with. Try and get yourself in a nice, quiet place where you're free from distractions so that you can fully concentrate. If you need to pause the video now to get any of that sorted, then please do. If not, let's get going. So then I'd like you to try this: Yasmin and Zaki are writing number sequences. To get the next number, they apply these rules to their sequence. Now the rules depend on the parity of the number. So parity might be a new word for you. That just means whether the number is odd or even. So if your number is even, you just divide it by two. If your number is odd, then you multiply it by three and add one. So you're going to have a go at this, choose a number between one and ten to start your sequence, continue the sequence, and what do you notice? I think you'll know when to stop and choose a different number and try again. So try this with three or four different numbers and see what you notice. When you're ready to go through it, resume the video. Excellent work. I'm actually not going to talk about this because it's going to form the connect part of the lesson. So Collatz, he was born in 1910 and he died in 1990, so not that long ago. He defined a sequence as follows, and this is similar or it's actually the same as what you did in the try this. So start with any positive integer value . So I just limited you to the ones between one and ten, but you could start with any positive integer. And you find each term from the previous term as follows. If the value is even, you divide it by two. And if the value is odd, you multiply it by three and then add one. So Collatz, in 1937, which was just a couple of years after he'd finished studying, came up with the collapse conjecture, which has got lots of different names in maths now. So, some people call it the three X plus one problem. Some people call it a hailstone sequence, but I'm going to go with the Collatz conjecture today and the Collatz conjecture states that no matter the start number, so whatever number you choose, it doesn't matter so long as it's a positive integer value, Mr. Collatz says that the sequence will always reach one. Did your sequence reach one? What happened when it reached one? Let's have a look at some together. So, I'm going to start my sequence with the number three. Three is odd, so I multiply it by three and add one and I get ten, Ten is even so I half it. And I get five. Five is odd, so multiply it by three and add one. And I get 16. 16 is a special kind of number and it's even. I get eight because I half it. Half eight gives me four, half four gives me two, and then half two gives me one. So I have got to one when I started with three. What's the first number between one and 10 that is not on our sequence here? So, we've got one, two, three, four, five. We haven't got six. So if we started with six, Ooh, that's even so I'd half it. And that gives me three. So I can just join that onto my sequence here. What's the next number then? Seven is not in my sequence yet. So seven is odd. So I would multiply it by three and add one to get 22. 22 is even so I half it and get 11. 11 is odd. So multiply that by three and add one to get 34. 34 is even so I half it and get 17. 17 is odd. So I multiply it by three and add one and get 52. 52 is even, so I half it and get 26. 26 is even so I half it and get 13. 13 is odd. So I multiply it by three and add one, which is 40, 40 is even so I half it and get 20. 20 is even so I half it and I get 10. So that means I can add this in now to my sequence here. So now I've got all the numbers from one to seven. I've also got eight. I haven't got nine yet. So let's add nine in. So, nine multiplied by three, add one gives me 28. 28 is even, so I half it, it gives me 14. 14 is even so I half it, and that gives me seven. So I can add it in here. So now you can see that my sequence has got all the terms in it from one to ten. This is like what we call a tree. So lots of mathematicians have drawn, really beautiful ones of these really big ones that go up to huge numbers. Now they haven't managed to prove that this is true for every positive integer yet. They believe that it's true, but they've not yet proven it. So you're now going to apply your learning to the independent tasks. So pause the video here, navigate to the independent tasks. And when you're ready to discuss it, come back and we'll talk about it together. Pause now. How did you get on with that? Did you have some fun drawing out those trees, which number between one and ten gives the longest sequence? Well, we already saw that in the connect, didn't we, that nine made the longest sequence. Which number between one and 20 gives the longest sequence? Oh this was a bit of a mean question because there are actually two answers. So both 18 and 19 give equally long sequences. Are there any shortcuts you could take with the conjecture? Well, I actually was thinking about this and when you think about three plus one, if you start with an odd number and do 3 plus one, it gives you what? Exactly, an even number. So what you could do to do two steps in one is to do three plus one divided by two. Because here you would divide by two, wouldn't you? And then you would get another odd or even, we're not sure. But you could do odd. So you could just combine these two steps. Now, this one I really liked because you have to think of it mathematically here, and it shows you how multiple numbers can link into the same thing in these trees. So for example, here, we know that 32 is even so we know that half of that must be 16 and that links in with the rest of our chain. What other number could give me 16? So I've done the divide by two, one here. So something let's call it X multiplied by three, then add one gives me 16. Let's just make that look a little bit more mathematical three X plus one equals 16. So we can solve this now by taking away one from both sides. So we get three X equals 15 and then dividing by three on both sides to get the X equals five. So we know that this one here must be five and so on, and we have to apply this process over and over and over again with our mathematical thinking to fill in this tree. So here are your answers. Hopefully you've got all those. Well done if you did. Let's look at the explore task now, then. Carla is thinking about the Collatz conjecture. She says that some sequences will only contain even numbers. Do you agree with her? And can you find some examples that either show or don't show this to be true? Pause the video now and have a go at this task. How did you do, what did you think? Hopefully you thought, Oh, mess, what you're talking about? All the sequences we've looked at so far have got one in them and one is an odd number. So if we take one out of the equation, can we think about any sequences that have all even turns apart from one? Excellent, anything where it's two to the power of something is your start number will give you all even terms apart from the last one, which is one. So for example, if we did two to the power of four, which we know is 16, then we go 16, eight, four, two, straight to one. If we did two to the power five, then we would just add 32 onto there, if we do two to the power of six, we'd add on 64 onto there. If we did two to the power of seven, we'd add on 128 and so on. So if we've got two to the power of anything, it will have all even terms apart from one. Hope you've enjoyed this lesson on the Collatz conjecture, I've really enjoyed teaching it to you. If you'd like to share your work with Oak National, then please ask your parent or carer to share your work on Twitter. Again, at Oak National and hashtag Learn with Oak. Don't forget to go and take the end of lesson quiz so that you can show me what you've learned and hopefully I'll see you again soon. Bye.

In this lesson, you will learn about a famous maths problem called Gabriel's problem.

Gabriel's problem

﻿Hello everyone, it's Mr Miller here, and in this lesson we going to be looking at Gabriel's problem. So first of all I hope that you are all doing well. And just to introduce myself in case you haven't seen any of my videos before on Oak. So my name is Mr Miller. I'm a maths teacher at a secondary school in central London, and I've really been enjoying doing these lessons on Oak, and really looking forward to doing this one, where we going to be looking at Gabriel's problem. And also the next one where we going to be looking at the happy caterpillar problem. So that's to look forward to, but first of all, this lesson, this is Gabriel's problem, so let's see how we are going to do this. So we going to have a grid which you can see on the right, and the numbers one through nine go in the white boxes. You can see that all the white boxes are complete already. That's not going to be the case for all problems, but in this problem all the white boxes are complete. Now each of the blue boxes is the product of the three numbers in that row or column. So for example, if I was looking at this first row here, a product means multiplication. So six times by eight times by three, gives me a 144. And it's the same thing for this second row here. And the third row and all three columns as well. So what I want you to do is, just to warm up, I want you to copy down this grid and complete it. You just need to complete the rest of these blue boxes. So pause the video for two or three minutes to complete the rest of this grid. Okay great, so hope that was nice and straight forward. And you should've got, seven times by one times by five, which is 35. Two times by nine, sorry, times by four, is 72. And in the columns, six times by seven, times by two, gives me 84. And then I've got 72 and finally 60. And there's nothing in this final box there. I can just leave it blank. So anyway that is quite straight forward, and the next problem we're going to have a look at is going to be a little more difficult so, let's have a look. Okay so, again same idea where, remember that in the white boxes we were putting the numbers one to nine. And the blue boxes is the product of the three numbers in that row or column. So here we've got another problem. So I want you to pause the video to copy down this grid and see how far you can get. I'd recommend looking for blue boxes that you can fill out first, and then look to see what other boxes you can fill out. So pause the video now and have a go at completing this grid here. Okay great, so you should've noticed that you can complete this first row and first column nice and easily. So first row, four times by three times by five, is going to give me 60. And the first column, four times by eight times by one, gives me 32. So well done if you found that. And then next you need to think of a slightly different tactic here. Because there's no more blue boxes that we can complete. So we going to need to look at one of the white boxes. So in case you didn't find this out, the box to look for is this one down here. And the reason is, well, I know that one times by something, times by nine gives me 18. So what needs to go in that box? Well if you're thinking a two then really well done. Because one times by two times by nine gives me 18. And then what's next? Well you can think about this middle box here, because now I know that three times by something times by two gives me 42. And I can think about this by thinking, well three times by two gives me six. So what do I need to multiply six by to get 42? Well seven, so there we go. And that just leaves one more white box and one more blue box to fill out. And there's a coupla ways that you can fill out this last white box. The first is by saying, well, eight times by seven is 56. And what is 336 divided by that? That will give you the answer. And the other way is to realise that, as I said the numbers one to nine go in those boxes, and we've only had eight of these numbers, so the last one must go in that box. So we can see that we don't have a six already. So a six must go here and you can also work that out by doing 336 divide by 56. Okay so just one more box to fill out. It's the final blue box which is nice and easy. Five times by six times by nine, gives me 270. And I'm done. Okay, the next one is the independent task. Let's have a look at another problem. Okay, again here is another grid. Same ideas apply. This one is again slightly more difficult. It may be okay if you're stuck to have guess, And see if that works out. So that might be a tactic. But it's really up to you. Pause the video now, copy down this grid, and see how far you can get and see if you can complete it as well. So pause the video now to have a go at this task. Okay great, so let's have a look through. So there's a coupla that we can complete pretty easily. First of all we can complete this white box in the top row here. We know that nine times by three is 27. And then 27 multiplied by six gives me 162. So that is nice and easy. And that then allows me to work out this white box down here. Because six times by two is 12, and 12 times by eight gives me 96. Now, what do I fill out next? Well it's not actually super straight forward to work out what I do next. So there's a number of ways you could've thought about this. But I'm just going to talk you through one of these ways which I think makes sense. Now, if you have a look at this second row here. You have got two missing white boxes. But have a think what two different numbers must go in those boxes. Well, you know that it's something times by two, gives me 14. So you know that these two boxes here have to multiply together to give me seven. Because seven times by two is 14. And how do you get seven by multiplying together two numbers. Well because seven is prime and it must be a seven and a one. So you must have a seven and a one, or a one and a seven in these two boxes. Now you could've tried out both of those combinations, and if you tried a seven in this box here, well you would've found out that it wouldn't work, because nine times by seven, looking at this first column here. Nine times by seven is 63, but I want to get 45, so that is not going to work. So the seven can't go in this box. It must go in this one. And one must go in this one, and that turns out to work and you can complete the rest of it really easily. So a five goes there, a four goes there, and finally, we've got three times by seven which is 21, times by four, so this is an 84. So again, work getting a little bit trickier, but well done if you managed to work that out. Well done if you did it with some clues, and for the final task, the explore task, it's going to get even trickier again. So let's have a look. Okay so for this explore task, again we have got the same problem. But this time you have noticed that we have all of the numbers in the blue boxes, but we need to find out what goes in the white boxes. And remember that in the white boxes we have the numbers one through to nine. So this is the trickiest one you've seen so far. So I've given you a clue in the speech bubble to help you get started. You know where the five must go, if you think about which two numbers five is a factor of. So anyway, pause the video now to copy down this grid. See if you can work out the first step, where the five goes. And even feel free to workout where all of the numbers go if you are feeling confident. Great so I hope that you managed to work out at least where the five went, and you know that the two numbers in the blue boxes that five is a factor of, 40 and 60 off course. So therefore where does the five go? Well the five has to go in this box here. Because you know that when you multiplying together different numbers, you then have to multiply a five by something to get 40 and a five by something to get 60, because five is a factor of both of those numbers. Now, you can actually do the same thing with some of the other numbers. So the number seven and nine will be useful ones to find out. So pause the video now if you haven't worked it out already, to find out where the seven and the nine must go using the same idea. Okay great, so you should've found that the seven must go here, the nine must go here, and this information actually allows you to work out the other numbers as well, because you can work that six goes here and you can work out the number in the top left as well. That's got to be a two. And then the rest of the numbers are quite straight forward. You could next look at this second column here, you know that the numbers three and one go in the two missing boxes, and you know by looking at the numbers at the end of the first two rows that the three goes here, the one goes here, and therefore the missing numbers are going to be eight there and four there. So that is that problem. And if you managed to do this, really really well done. And if you're thinking about why is this kind of important for maths, well really it's about this idea of factors and prime factors and prime factorization which you would've had a look at already, probably. So yeah, this task really shows that the factors of a number is really important and it's a nice way of seeing that in this task here. Anyway, that is it for this lesson. Apart from on the next slide, I'm going to show you some more problems that you could do if you find this interesting. So here are some additional problems. So again, here are some additional problems. The source here is nrich math, which have got some really nice problems, so that is where I got this from. So again, similar ideas here, but you looking at more different numbers and more setups of the grids here. Anyways that is it. So pause the video here if you want to have a go at these problems. And on this slide here, you can share your work if you have your parent or parents permission on twitter using the hashtag, sorry, the handle @OakNational and the #LearnwithOak, if you would like to show your completed grids. Anyway that is it for today's lesson. Hope you enjoyed it. And next time we going to be look at the happy caterpillar problem which I promised you is very exciting. So looking forward to that. Thanks so much for watching. Well done, have a great day and see you next time.

In this lesson, you will complete Happy Caterpillars - a visual game of repeated operations.

The Happy Caterpillar Problem

﻿Hello everyone, it's Mr. Millar here. In this lesson, we're going to be looking at the happy caterpillar problem. So, first of all, I hope that you are all doing well and in this lesson we're going to be looking at the happy caterpillar problem, which hopefully will be a lot of fun for you. So without further ado, let's check out how this one is going to work. So, a caterpillar starts with a number between one and 100. So you can see that this first caterpillar has started off with a 7 and the second one has started off with a 24. So, if that number is odd, the caterpillar will add one. If it's even, the caterpillar will divide two. It will keep on doing this until it gets to one. So let's have a look and go through the first caterpillar which has been completed for us already. So we can see that it starts off with a seven. And because seven is odd, we're going to add one, and seven plus one is eight. Now eight is even, and if it's even we divide by two. So eight divided by two is going to give me four. Four is even, so we divide by two and we get two. And two is even, so two divided by two gives me one. And then the caterpillar stops. So because we've got one, two, three, four, five numbers in our caterpillar, we can say that the caterpillar has got a length of five. Now how about a caterpillar which starts off with 24? What happens here? What is the first step first of all? Okay, well, the first step is divide by two because 24 is even of course. So if we divide by two, we get a 12. Okay, so pause the video now and see if you can complete the rest of this caterpillar by yourself. Great. So you should have divided by two as the next step to get six. And then we can keep them going. So six divide by two is equal to three. But when you get to three, three is odd. So if I continue with down here, we're going to have to add one to get four. And now four is even, so we're going to have to divide by two to get two. And then finally divide by two and we get one. So, hopefully your caterpillar looks a little bit better than mine, but anyway, the point is that to find out what length it is, we have to just count up all the numbers and I can see that we've got seven numbers here. So it's length is seven. So that is the idea of how this works. But all you need to do is remember that if a number is odd, we add one, if a number is even we divide by two. Let's move now to the connect task. Okay. So here, I've got three caterpillars here. One starts at 50. One with 32 and one with 10. And what I want you to do is to complete each of these caterpillars and then find out which one is the longest and which one is the shortest. So you will start off with the 50 caterpillar by first of all, dividing by two and then seeing how that works out. So anyway, pause the video now and see what you get. Great. So let's go through these. And I'm not going to draw out pretty caterpillars, but I'm just going to do it in a shorthand way which you could do as well, if you want to. So anyway, the 50 caterpillar starts off with 50 and then we divide by two to get 25. 25 is odd, so we add one to get 26. Divide by two to get 13. And then we're going to add one to get 14. And then divide by two to get seven. Add one to get eight. And then divide by two, divide by two and then divide by two to get one. We stop at one and then we can count up the number of, numbers that we have. So we have the length of 10. Next one, 32. So we just get 32 divide that by two, 16. Then we get eight. Then we get four, then two and then one. So that is a length of six. And finally 10, we do 10 goes to five. Add one to get to six. Divide by two to get three. Add one to get four. And then two and one, that's over. And this has a length of seven. And so we can see that the caterpillar of length 50 is the longest and caterpillar of length 32, the shortest, which is actually quite interesting to note that even though 32 is a bigger number than 10, it ends up giving a smaller caterpillar. Anyway, interesting idea. We're going to come back to this idea in the next slide. So let's have a look at the independent task. Okay. Let's look at the independent task. So for this task, you need to work out the shortest caterpillar which starts off with the length between 10 and 20. And explain why this number gives the shortest. So, what you need to do is you need to look at different numbers. So starting off with a caterpillar of length 10, and a length 11 and 12, et cetera. You need to find out how long each of these are and which number will give you the shortest link. So for example, if you are going through number 11 'cause we've done 10 already. We start off with 11, we add one we get to 12. And then six and then three. And then four, two and one. So therefore we know that if it starts off with 11, it is length seven. And we can do it the same for the other ones. So that is what you need to do now and see if you can find which number gives you the shortest length. Pause the video for a couple of minutes and have a go at this problem. Okay, great. So I hope that you work this out and well done for finding out the correct answer. The length, which gives you the shortest is actually 16, because in this case we get 16, then eight and then four, then two then one. So it is a length of five, which is the shortest length you can get if you start between 10 and 20. Which is again really interesting as we saw on the previous slides, that 16 is a bigger number than 10 and 11 and 12, et cetera, but it gives us a shorter line. So can you think why that is? Well, the reason is that as you can see, each time we are dividing by two, so divide by two, divide by two divide by two divide by two. So we're getting smaller and smaller as quickly as we can. And no point in this chain, are we adding one. And well done for spotting that this works for 16 because 16 is a power of two. In case you spotted that, well done. Anyways, so you found out that 16 is the shortest, and we're going to keep this idea in mind when we have a look at the next task. Okay. So let's have a look. There are rumours of a caterpillar with length 14. What number could produce such a beast? So here we have a huge caterpillar of length of 14. And just remember that it has to start off with a number between one and 100. So we can't have any more than 100. So this is a very tricky problem. So if you're feeling like you don't need any clues at all and you can have a go yourself, then pause the video now, otherwise I'll give you a clue in a couple of seconds. Okay, great. So if you need a clue, here is a clue. And if you read what's in the speech bubble, I'm going to be working backwards here. So I know that I have to finish off with a one. And the only way to get one is to have a two before that. There's no other way to get a one. You have to have the two before that. So you're going to be dividing by two and have the last step there. So how do we get the two? Well, we know that to get from here to here, we can either add one or we can divide by two. Those are the two options. But actually we can't add one because that would mean having a one here. So that is not going to work. So therefore we have to have a four here and we have to be dividing by two again. But now, if you think about going up from here to four, again, there are two possibilities. Divide by two or add one. And now we do have a choice. And the choice here is between, of course, add one, divide by two. And to get a longer chain, it turns out that we're going to need to add one. So we're going to have a three here. So see if you can use this idea to see if you can get the longest, to get the chain, a caterpillar with like 14. So pause the video now, and see if you can work this out. Great. So well done if you did manage to work that out. And I'm just going to go through a few more steps with you. So we need to have a six here. So this would be divide by two. And then the next one, well, this time again, we're going to be adding one here. So we're going to have a five here. And you should be spotting a pattern now that we are going to be dividing by two, then adding one and then alternating around. So if we just continue this pattern, we get the longest possible chain. So we're going to have a 10 here, a nine here and then an 18. Then a 17, 34, 33, 66 and 65 here. And that gives you the, that is the starting number that gives you a length of 14. So really well done if you managed to work out that this was a 65 here, cause that was a tricky question. And the final question I have for you is, there are six different numbers between one and 100, which gives you a length of 13. So six different numbers for you to find out. So see if you can find out those numbers. Pause the video now, before I give you the answers. So if you managed to work this out, really well done. It's six numbers between one and 100 which gives you a length of 13. Well done, if you managed to find those. Right, that is it for today. Hope you enjoyed these two lessons. Thanks very much for watching and I'll see you next time. Thanks very much. Have a great day. Bye, bye.

Fill the gap: Every even integer greater than two can be written as the sum of two __________ integers.

Goldbach wrote his conjecture in a letter. Who was the letter to?

What was Mr Golbach's first name?

True or false: Goldbach's conjecture is proven to be true.

Write 7 as a sum of two prime numbers.

A sequence is defined as follows: Start with any positive integer value (𝑛). Each term is found from the previous term as follows: If the value is even, divide it by 2 (𝑛/2). If the value is odd, multiply it by 3 and add 1 (3𝑛+1).

If the second term of the sequence is 30, what would could the first term be?

Fill in the gap: The Collatz conjecture states: no matter the start number (𝑛), the sequence will always reach __________.

What is the first name of Mr Collatz?

In what year did Collatz make his conjecture?

Each blue box is the product of the 3 numbers in that row or column. All the questions refer to this picture below. First, what is the value in box A?

Look at this caterpillar. When it gets to 8, what would the next step be?

Unit Overview: Famous maths problems

Lessons:

Goldbach's conjecture

The Collatz conjecture

Gabriel's problem

The Happy Caterpillar Problem