Number sequences (Part 2)

Key Stage 2

Maths

Reasoning with patterns and sequences

﻿Hello there everyone. It's me, Mr. C. How are we all doing today? I'm hoping that you're well and I'm hoping that you're excited for some more of our number sequences learning. We're going to continue today with reasoning with pattern and sequences, and we're going to look at further investigation of number patterns. So let's take a little look, shall we? Let's skip ahead. Well, here's a little starter just to get our brains warming up even more. I've got some anagrams here and if you're unsure, an anagram is a word that you take and you jumble up the letters so that they're in a different order to how they should be. Now, there are five anagrams here that all link to maths. Can you use your powers of deduction to figure out what these words all are? Now remember their all maths words. So have a go and I'm going to give you, oh, 30 seconds. So maybe you can pause the video, have a go and see what you can come up with. Best of luck. Think you got them? All right, I'll put you out of your misery then. I'm going to share with you all of the answers to those anagram teasers. Here they come. So rembun, great word, was actually a number. I'm sure you've got that one. Train leg, triangle. Limb lion was million. Moler kite was kilometre and Bart suction was subtraction. So fantastic job if you managed to get all of those. You are now officially warmed up and ready to learn. So for today's lesson, don't forget you need your equipment which would be the pencil, your ruler, paper or a book that school has given to you, or maybe you're working on printouts. That's absolutely fine. And somewhere quiet to work with no distractions. So our agenda for today, we're going to look in a moment at our key learning and vocab, and then we're going to do our what-goes-where warm up. A new one. We're going to recap our number sequences and patterns, and how to work out what's going on. We're going to look at sequence rules, then a main pattern investigation and a sequence challenge. Quite a lot to get through today. So our key learning then is to investigate number patterns. And our key vocabulary, my turn, your turn. Sequence, increase, decrease, ascend, descend, rule, term, and investigate. Now remember, just to recap on a couple of those words, the rule is basically what you're doing to get from one number to the next. So it could be doubling or it could be halving or it could be adding two. The term is what we would refer to each number. So each number in that sequence is one term, okay? Increase and ascend both mean that the number sequence gets bigger. Decrease in descend means that it gets smaller in value. All right. So let's have a look at our new warmup. So look here. We've got like a little web of numbers. Well, not yet we haven't, but we've got a little web of circles all joined up. Now, for each of these circles, I'm going to be asking you to make sure that you put the numbers one, two, three, four, five, six, or seven in there. So just to remind you, you're using the numbers one, two, three, four, five, six, and seven, and you can only use each number once. I would say write them out and cross them off as you go. Now, the idea is that in each of these circles, you're going to put one of those numbers and then you've got to try and fit the magically so that each set of three circles joined up, make the same total. So when I add these three together, they'll give me the same total as when I add these three together, they'll also give me the same total as when I add these three together. Now, when we're finished, I'll show you one way of doing it. There are multiple ways, okay? So there's no one right answer for where all of those numbers go. So think to yourself, "What could I use to help me? What skills do I have in my maths toolkit? What skills do I have that will help me to figure this out?" I'm going to suggest you think a little bit about number bonds with those numbers there. Could be useful. How will that help you? See if you can get each of those rows of three circles joined by lines to make the same total. What I would do is, if I for example, put the number three in here, this is just random, it doesn't mean that that's where the three goes. I then cross it out so I know I've used it. If I put the number four here, what am I going to do? Ya got it. Cross it out. Cross them out as you go along so you can keep tabs of what you've used. So spend a few minutes seeing how well you can do. Give it a go and come back when you're finished. Almost there, guys? Welcome back. All right. Should we see one of the many solutions that you could have used? So it's not always going to be the same positioning of all the numbers, but the sequence and placing that I came up with was this. So each of my sets of three numbers have added up to make 14. Let's just double check that, shall we? So I've used one. Yup. Two, three, four, five, six, and I've used seven and I've used none of them more than once. Seven add six, well, seven add three is 10. So the other three would be 13, add the one, that's 14. Seven add three, that's 10. That's 14. Five and two is seven, seven and seven is 14. So if you managed to get that, and it may be that all of your numbers have shifted one place round the outside, so you might have had one, four, five, six, and so on. They may have been in the next circle along if you've done that. And they all add up to the same, then well done to you. Just like I said, it's just one way of working out the answer. There's no specific right way that's the only way. And that's quite often the thing in mathematical investigations. Lots of different solutions for the same thing. Well done, guys. So let's just recap, shall we, on sequences. If you remember in one of our previous sessions, I said, "You need to ask a few questions about what's happening in a sequence." So just take a few seconds here to look at the numbers I've got. I've got 23, 28, 33, and I've got three questions. Is there an increasing or decreasing pattern? How do we get from one term to the next? And what are the missing terms? So when I'm talking about number two here, that says, "How do I get from one term to the next," that basically can translate to us, "What is the rule?" That's exactly what that means. The rule is what we're doing to get from one number to the next. So just take a second to read over those numbers and the questions and see if you can think of how to answer them. How do you think you did? Well let's go through them, shall we? So we need to understand the rule from getting from one to the next. And hopefully you have spotted that the rule in this sequence is that we're adding five. 23 add five gives us 28, 28 add five gives us 33. Okay? So now we can think about adding in the rest of the terms, because we know that it's going to be adding five each time. Yeah? So give it a go. And also be thinking, "Is it increasing or decreasing?" Well, it is an increasing sequence. It's getting bigger. 53 is definitely bigger than 23. And we can see that we've added five each time. 23, 28, 33, 38, 43, 48, 53 and so on. Now just spend a moment though, here now, having a look at this sequence of numbers. Is there anything else you spot to that? Is there anything else? And can you spot my deliberate mistake on this page somewhere? What is my mistake? Few seconds. Well, those major detectives amongst you will notice that I have said here that I'm adding three each time, but we know that's not true. We know that we're actually adding five. Well done. And you'll notice something about the ones column. They alternate between eight and three. Eight and three, just like in the five times table where it alternates between zero and five. So whenever you count in steps of five, that final digit, your ones digit, will always alternate between two numbers. Is there something else we can state about this sequence? There is that the ones column alternates between three and eight, three and eight. So always look for things that you can talk about in a sequence. What do you notice? Okay? So have a look at these and just put the same questions through your mind. What's happening each time to get from one term to the next? So what's the rule? Is it increasing or decreasing? And are there any other things that you notice? So spend a couple of seconds now, just taking a look at that. I'm going to suggest that you pause and just have a look over those sequences. Have a go. Alright. So what did we notice? Well, if you remember another time when we've looked at sequences that have got decimal points in them, I've always looked at it by partitioning and I've looked at what's happening before the decimal point and what's happening afterwards. So let's just focus at the after the decimal point. I had gone from 0.6 to 0.3. So I could say to myself, "What do I know that will help me with this?" Use what you know to work out what you don't know. So if I just imagine that I'm looking at six take away something to get to three, I can use that knowledge to help me. I know that I take three away from six to get to three, but I'm not taking it from six, I'm taking it from something that's 10 times smaller. So instead of saying, I'm taking three away, I need to be 10 times smaller. That is in fact 0.3. So each time I'm taking away 0.3. So 6.6, 6.3, 6. If I were then to take 0.3 away from that, I would move to 5.7 and then taking away my 0.3 again, can you work out what would be next? Yeah? And then taking away another 0.3, what would be next? Yeah? 5.1. And then taking away another 0.3. Tricky. Yeah. 4.8. Well done. If you've got that, fantastic. So 4.8. Is that bigger or smaller than the number we started with? It's smaller. So this must be a decreasing sequence. And on this one, hopefully you'll have noticed what's going on here. What do I do from 99 to get to 90? What would the rule be? I suggest just taking away nine. Let's just check. 90 take away nine, would that give me 81? Yes, it would. So if I'm taking nine away each time, oh look, it's just our nine times table. Yup. You got it? You guys are amazing. Very well done. Is this an ascending, so getting larger, or descending sequence? Getting smaller so it must be a descending sequence. Brilliant! Okay. I think you're well and truly ready for a challenge. Just to recap though, before we do, negative numbers are important. When we're looking at number sequences, we will sometimes cross into those negative numbers. So my top tip from previously is don't forget, sometimes zero will count as a number when we're passing from one side of zero to the other. Don't forget to count it. Okay? We wouldn't go from one straight to minus one if we're counting back in steps of one, we would also land on zero as we went. So have a look at these. You've got five sequences there, but each of those sequences, you need to work out the rule and just tell me what that is. So it could be that we're adding, we could be subtracting, multiplying, or dividing for each of those. If you want to push yourself even further, just in front, you might want to say, is it an ascending or descending sequence? So working out the rule and then just popping it in there. Spend just a couple of minutes with that. Remember, the best way to work out the rule is just to think what's happened between this number and this number. When you think you've spotted it, check it with the next couple just to make sure that your rule works. Alright. I think I've talked enough. Off you go. I'm sure you smashed it with that one. So shall we take a little look at what those answers are? Let's do that, shall we? So in the first one we were doubling it or you could say multiplying by two each time. On the second one we're halving it or dividing by two. On the third one we're doubling, multiplying by two. And then we move on to subtracting 10 each time and then adding 500 each time. So is this getting bigger or smaller? Ascending or descending? Yup. Look at the first, look at the last and see. Ascending or descending? How about here? Yeah. What about this one? Oh, but this starts on 20 and this ends on 20. Oh, but it's a minus 20. Okay. So it must be descending. And here... Yeah. Sorry frog. All right. Not bad, hey? I'm sure you did fantastically on that. So very, very well done. Alright. Let's take a look at the next thing. Sometimes in a sequence, you might have two steps. Okay? So I might not just be doing one thing to this number before I get to this one. So I'm going to ask myself a few questions again. First of all, is it getting bigger or smaller? Well, it's getting bigger. From two to five, that's increased. From five to 11, that's increased. So I think I can probably rule out the fact that the first thing I'm doing is dividing or subtracting, because that would mean that number was getting smaller. So it must be multiplying or adding at first. So I always look here for times table clues first to help me and think, "Well, if I use a times table I know are my answers ever any nearer to these things here. So for example, two times one would give me two. Okay. That's just stayed the same. So two times two, well that would give me four. Four is nearly five. So let me just say, I'm going to start off by saying, okay let's try times two. So two times two is four. Four is almost five. What do I add to four to get to five? Yeah, I add one. Let's see if that works here then. So five times two is, yep, 10 and then let's try adding the one. Add one is... Oh, well what do you know? We've just worked it out. So our rule is now times two, add one. So 11 times two is, mm-hm, 22. Add the one, 23. Brilliant. So times two add one. 23 times two, well double it and that's... Yeah, 46. Add one. Brilliant. And you get the picture, right? So 47 times two, work that out if you're not sure. Partition it. Two lots of seven is 14, two lots of 40 is 80, 80 add 14 is 94. Add the one. And see if you can work out this last one by yourself. I'm going to give you 10 seconds. Can you work out 95 doubled and then add one? Got it? Sure? It is... Brilliant. Guys, it's as simple as that. Just look for things that might be familiar to you and see how they'd been tweaked. All right. Come back to this one another time and have a go. Just refreshing. Okay, this is the one we've just looked at, but refresh yourself on how it worked. I looked for closest multiples and then saw what I could do. So your main task, this is what you're going to be doing. We're going to be creating and exploring your own number sequences. You're going to be choosing a starting number. So for example, I've chosen seven, and then you're going to decide on an operation. So I've done seven and I've done times. Then I'm going to decide on the number that I'm going to multiply it by. So I've done seven times three. Okay? And you can see that here. I've just decided that's what I want to do. And then I'm going to pick a different operation. So this time, instead of multiply, I've chosen subtract, and then again, decide what number to use. So I thought I'm going to stick with the three. So I'm going to do seven times three, take away three. So seven times three is 21. Take away three is 18. There's the first term in my sequence. And then I'm going to start with 18. So this then becomes my new starting number and follow the same procedure. 18 times three and take three from my answer, which then gives me 51. That's term one, term two, and then see what I'm doing? Popping that in, timesing it by three, taking three away. 150. So my next set would be, yeah. Oh, so I'm going to have to write on this chap's face. 150 times three, take away three, equals, and so on. Can you work out the first 10 numbers in your sequence? The first 10 terms and write them down. When you've written them down, we're then going to be looking at what we can explore within those numbers. So take a look. Here I've written out the first 10 terms of my sequence. Now guys, we got into some pretty big numbers here, so I'm going to have to think to myself, "Well, what did I notice about my sequence?" Well, if I look in my ones column, I can see that it goes eight, one, zero, seven. Eight, one, zero, seven. Eight, one, and I'm assuming it would go zero, seven. So there's a pattern already working out there. I can then look at them and think, "Okay, my answers go... Oh, that's an even number, odd, even, odd, even, odd, even, odd, even, odd, even, odd, even, odd, even, odd, and so on." So look for that. Are they odd? Are they even? Is there a pattern? Except for my first one, I can see that my digit total, remember the digit total is when you add them all together, so five plus one equals six, one plus five plus zero equals six, four plus four equals... Eight plus seven equals 15 but then if I had the one of the five together, I get six. So each time, apart from this first one, my digit total works out at six. I also know that all the numbers are divisible by three. So this is really for you now to explore. It's free reign. Work out the terms in your sequence and just explore. Every little thing that you notice, jot it down. I've got four things there. You could come up with 10 things that you've spotted. Any little thing you find that you think is, "Oh, that's interesting." Give it a go. Okay? So pop off, try that main activity. Come back when you're done. Welcome back, everyone. Hopefully you had a lot of fun investigating there. So one final thing, just to round off this session. Take a look here. Here's your challenge. I've got two boxes for you, the pink and the purple. And in the pink box, we've got minus three and one. And in the purple box, we've got 3.4 and 2.9. For each of those, can you create a sequence that has those numbers in it? So four sequences here and each of those sequences will have minus three and one in them. Four sequences here, and each of those sequences will have 3.4 and 2.9. You might want to increase or decrease. You might want to do one step in your rule. You might want to do two steps in your rule. Again, it's free reign. It's entirely up to you. Think, be smart and see what you can come up with. Really enjoy it, play around with it. That's the beauty of things like maths, you can just play with those numbers. So creating eight sequences in total, four for each pair of numbers. Have a look, give it a go, and come back when you're ready. Wow, ladies and gentlemen. Very, very well done. It was tricky. I will share with you one of the sequences that I came up with for each. So have a look here. I came up with these for each of them. I was really cowardly on the first one. I just went really step by step, but it's still a number sequence. It still has a rule. It still has five terms and it contains minus three and one. Brilliant. And the same on the other side. Well folks, because it's been such tough work, We have no final knowledge quiz today. Yes! So very well done for all of your hard work. You have been brilliant today. It's been a tricky old lesson in parts, but very well done. Great job. And I'll see you in our next session. So from me, Mr. C., goodbye.

Number sequences (Part 2)

Number sequences (Part 2)

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UnitMaths / Reasoning with patterns and sequences

Maths