To develop strategies to plan and solve problems (Part 1)

Key Stage 2

Maths

Reasoning with patterns and sequences

﻿Hi there everyone. It's me, Mr. C with your next math lesson. So let's, without further ado, get straight on to see what's happening today when we continue with our reasoning with patterns and sequences. So let's take a look. So here's a little riddle to get us going today. Listen carefully, finding the key information while I read this riddle to you. If it takes one man six hours to dig a hole in the ground and it takes two men three hours to dig a hole in the ground, how long would it take for four men to dig half a hole? A little bit of thinking time there, one man six hours, two men, three hours, okay, so that's twice as fast. So how long for four men to dig half a hole? 10 seconds of little bit of thinking time. See if you can figure out the answer to that riddle, 10 seconds. I'm going to think too, five seconds, hmm, I got it, did you? Confession time again, it's another one of those trick questions. Ladies and gentlemen, it's just another trick question because you can't actually dig half a hole. Half a hole it's actually still just a hole after all. So no matter how big a hole is, it's still a hole, it can never be half a hole. So the answer to that is they can't, they can't dig half a hole because half a hole doesn't exist. I know it's a rubbish one, my dad used to be that one to me and I fell for it pretty much every time. All right, well, did I catch you out or are you a bit of a smart cookie? I'm sure that's you. Let's take a look, shall we? Enough of this nonsense, let's have a look at what's coming up. Well, you're going to make sure that you've got your equipment, your pencil, your ruler, paper, or printouts of this if you need them or a book to work in and somewhere quiet with no distractions. I would suggest actually sitting in a hole somewhere in the middle of nowhere, that might be quite nice and without distractions, but would you be able to get back out again? So maybe just do this from home. Alright, so our lesson ahead, here's our agenda. You've done it already. Then we're going to look it up. Key learning and our key vocabulary, a bit of a multiplication warm up, not our speedy tables, something different. Then we're going to look at a blocky dog investigation, and I'm going to introduce that to you, then your main activity will be to solve the blocky dog problem. And then a challenge will be, can you find out dog 100? Interesting stuff. And then you final knowledge quiz with a chance for you to figure out what you've remembered. So, key learning, let's start off with that. Today we're going to be looking to develop strategies, to plan and solve a problem. And our key vocab, my turn, your turn, sequence, patterns, similarities, differences, increasing, decreasing, term and rule, brilliant. And let's just focus in on a couple of words on that little list there, increasing and decreasing, do you remember? Increasing means that the pattern is getting bigger and decreasing must mean that it's getting smaller, fantastic. And also actually another couple of important words are similarities and differences. Now they're really important when you're thinking about patterns and sequences to find out what's the same between each step, between each term and what's different, two very key words for you there. So, this is how we're starting off, this is our to start. We're looking at calculations here, all to do with multiplication. Now, I wonder if any of you will do it in a very similar way to me where I just do one little adjustment at the end. My little adjustment is when I looking at this, I can say to myself, if I know one thing times this, then I must know something else times this. So if I know, then I must know, I wonder how you'll deal with that. So I think about partitioning, how can that help you? What ways could you solve this problem? I'll show you my way at the end, but for now have a look and it's time to solve. Welcome back everybody. Hopefully you did all right with that. So let's take a look at some of those answers, shall we? So here we are, here are our first couple of answers and I wanted to just look in on one of those questions. Let's look in on this one. Now, I said to you use partitioning. I'm also going to say let's think of it as seven times and then we can at the end, adjust by timesing everything at the end by 10, 'cause timesing about tens of really simple skill, isn't it? So I'm going to look at 143 times seven. And so to do that, I'm going to have 100 times seven equals 40 times seven equals and three times seven equals. And I'm going to put my thousands, my hundreds, my tens and my ones just to help remind me. So I'm going to start down here three times seven. Well, we know that, seven, 14, 21, so two tens and a one. Now I'm going to look at 40 times seven. If I don't know, 40 times seven, I can do four times seven and then make my answer 10 times bigger. Four times seven, I know is 28, 10 times bigger is 280. So I'm going to pop that in like so. And then I'm going to look at 100 times seven, nice and easy, 100 times seven must be 700. Now that I've got these, I'm going to find the total. So, and add them together. Am I right to say, zero, zero add one? Yeah, I am because it's my one's column. What about if I said eight, add two? It's eight tens and two tens, so that's 80 add 20, 80 add 20 gives me 100. So 700 add 200 gives me 900, add the other hundred, 1,000, you mean a 1000? Yeah, so there we have 1001. However, that's only 143 times seven. So I need to make it 10 times bigger. So 1,001 times 10, nice and easy. We already know, everything shifts one place this way, and then we hold the spare place with the zero, we're making it 10 times bigger. So it's one zero, zero, one, zero, 10,010. Always use what you already know to help you, see how you can make easier. I knew that multiplying by 10 was really simple. So I could think about multiplying this by the seven and then making it 10 times bigger, so it's like I've multiplied by 70. And that would work for any of them. Take a look at the next set of answers and see how you did with those, here they come. And just in the same way, for this one, I could have done 342 times eight equals, and then I could have done my answer times by 10, 425 times nine equals. And then whatever that answer was multiplied by 10. A really simple little trick, nice and easy, nice and straightforward. I think that we're now ready to move along, so let's do just that. So like here we are, here are blocky dogs. Now you're going to have to use your imagination and imagine they do look like dogs. To me, I'm not so sure and I love dogs, looking at these, I'm thinking if my dog look like that, I might take them to the vets, 'cause they're not looking that healthy. However, if this, imagining is a dog and this is a dog, what would the next dog in our sequence look like? Now, I want you to see if you can complete the sentence by filling in a word. So here's our first dog, here's our second dog. The second dog is, than the first dog. What word could go in there? The second dog is, than the first dog. And there's a couple of words I'm wondering that you might have said. You could have said the second dog is bigger than the first dog. You could have said that the second dog is taller than the first dog. So if this were a sequence, we would see that the next one would probably be bigger again. So it would be an increasing sequence, it would be ascending. But what do we need to know to be able to work at what comes next? So we need to think about these two important words, the similarities and the differences. What's the same between these two dogs and what's different. Now we could say there are a few things that are the same. For example, both dogs have got a tail, ears and a nose. Both dogs have got legs, both dogs have got a body, that's a similarity. The differences are the sizes of certain bits. So this one, the legs are bigger than this one, the body is longer than this one. Parts of the body will have stayed the same, and we'll come on to that later, but I'm just giving you a little hint here and here, here and here, a little hint. So what are the general statements we could make about that pattern? Well, we could say in general, the dogs are getting bigger. We could say that the legs are getting longer, we could say that the body is getting longer. General statements, we don't know if that's going to be true every time, because at the moment we only have two terms in our sequence. You really need three or more to be able to make sure that something is happening every single time. But from what we can see, our general statements would be that the dog is getting bigger. I'd be surprised if the next dog was teeny tiny, yup, makes sense, hopefully it does. Well, let's look then, here's our original blocky dog. One, two, three, four, five, six, seven, eight, nine, it's made up of nine cubes. Here is our second in the sequence. One, two, three, four, five, six, seven, eight, nine, those are our original nine cubes, but we've added these ones in. And then we've got one, two, three, four, five, six, seven, eight, nine, were originals, with some extras again. So each time we're adding some new blocks and if you look, you can see already what the pattern is starting to build up by how many we're adding each time. So how could we present this in a table? Because quite often in tables that really helps a pattern to pop, it pops out, it stands out more clearly in a table. When we had dog number one, the total number of cubes we used were nine. When we had dog number two, the total number of cubes we used were 12, what's changed? Remember, when we've talked about patterns before, the key thing is this bit in between, it's the bit between, it's the bit you can't see that's often the important bit. So what's happened? And then if it repeats, what would happen here? Well, that's going to be your first job. What do we know first of all? And how have we worked it out? What happens to get from nine to 12? You add, yeah, nine, 10, 11, 12, we've added three. So if that's the case, what would our next term be? For dog number three, what would our total number of cubes be? If we're saying that here, the question mark means that we are adding three, what would come next? How simple was that? Yeah, we're thinking 15. Well, I wonder if you're right. I'd like you to try the first few terms in the sequence. So for dog number three, four, five, and six, what do we need to know to work out the total number of cubes? Now, when you've done that, can you try and put the rule into a sentence here? So the rule for this pattern is you, every time, what happens to the total number of cubes? What kind of a sequence is it? Remember, key words that you might want to think about, increase or an increasing pattern, that means it's getting bigger and decrease means it's getting smaller. So which of those two do you think would best describe this pattern? So fill in the information on here and I'll see you in a moment. It may help just to draw out a table because that's when the sequences pop. Off you go. How did you do? Let's see, shall we? If you manage to get all of those sorted and correct, and then I'll just go through what I'd said in terms of what I'd spotted with the pattern. So total number of cubes, we've got nine, 12, 15, 18, 21, 24, that sounds like something to me. So the number of cubes that we use each time goes up by three. So it's an increasing sequence and the rule is add three. So did you see how I've used some key words? It's increasing and the rule is add three. Then that's really easy, isn't it? If we've got it in order, that's really easy. We can work out one, two, three, four, five, six, seven. We could work at number eight dog and so on, because we're looking at it in order and that's when it's easiest, when we're looking in order. I wonder what happens if we're looking out of order? We'll take a dip into that pool very shortly. Now, we can look at this more closely because there are some things that have not changed between the dogs. Now, our dog will always have one block for tail, one block for ears, one block for head and one block for nose, these bits never, ever change. If you think back to the first few, the tail was always one long, the head was always the same size, so were the ears, so were the noses, they don't change. So we could add this information to our table and it might look, concentrate guys, 'cause it's going to be a lot of numbers coming along, but it might look something like this. And can you see how the tail, ears, nose and head stays as one each time? Now in dog number one, the body is three long, one, two, three, the legs take up two cubes, one, two. In dog number two, the body is four cubes long, one, two, three, four, and the legs take up four cubes, one, two, three, four. In the next one, the body would be five cubes long, so one, two, three, four, five, and the legs would take up six cubes, that's one, two, three, four, five, six, and so on. But this section never changes. Now, if we look specifically down this column and then down this column, we might start to spot a pattern. Body, three, four, five, six, seven, eight, the legs, two, four, six, eight, 10, 12, ah, now I'm starting to see a pattern. Now with that in mind, I'm going to set you off on another task now, and hopefully you're ready for this, I'm sure you are, take a look. Have a quick look over this one again, because we're going to continue this piece of information and changing slides and three, two, one. So now look, dog number six, the body is eight, the legs are 12, tail, ears and nose are one each time, and our total number of cubes would be 24. Can you fill in the rest of the information for this table? And remember, some things, and I'm flashing back to here. Some things will not change. As you're filling in this information. Some places will have the same information written every single time, because some things never change. That was tail, wagging the dog tail, I was dreadful, I know. So can you fill in the information on this slide? And come back when you're ready. Off you go guys and best of luck. How did you do? Do you think you got all that information in? Let's take a look at how that might have looked when you'd completed it then. So taking a look then, remember this section here, the tail, ears, nose and head, that never changes. The only bits to do with the body, legs, and then of course the total. So our body here for six was eight cubes. The dog number six, the legs were 12 and then eight at 12 gives us 20 add the one, two, three, four, that gives us 24. I purposely didn't tell you this section how to work it out, 'cause it should have made sense. So taking a look 24, 27, 30, 33 36, 39, 42, and all of this information must make a general statement. Now in our body section, what's happening? What's the rule? Yeah, each time we're adding one. In our leg section, what was the rule? And in our total section, add one, add two, add three, and it makes sense because if in total adding three each time, that's because we've added one and another two here, one and two is three, so in general, we're adding three. Dead easy, right? Well, what happens if I want you to tell me straight away, how many cubes you need for dog 24 or dog 29 or a dog 120? How can we just go straight to a number? It's possible, shall we take a look? Now, there's what we call a formula. And I'm going to explain that to you now, because your challenge today is to work out how many keeps for dog 100. Now, to work out how many cubes we need, we just need to work out the body first. Now the body here is the number of the dog plus two. Six plus two is eight, seven plus two is nine, eight plus two is 10. And we know that's right, 'cause eight, nine, 10, this was adding one each time. So it's the number of dog plus two. So for example, if I wanted to work out dog, I don't know, 15, let's do 15. I would do 15 add two, gives me 17. Now that's the body. Then we've got the legs, the legs is the dog number times two, six times two is 12, seven times two is 14. So this time I'll do it times two, that's for the legs, 15 times two is 30. Now, I've got one piece of information left before I can work out the total. What has never, ever changed? The tail, ears and nose and head each time all add up to how much? Yeah, four, so I'm going to add these two together. So 30 add 17, but then I've got to have the tail, ears, nose and head, that's another four. So 47 plus four, that's how many cubes in total I would need to make dog number 15? I would need 51 cubes. I've done this add two to give me the body, this times two to give me the legs, and then added those two with the tail, ears, nose and head, which never change. Does that make sense? It's all written here for you. So can you now work out how many cubes I would need for dog 100? Simple, you can do it, I know you can do it. So working that out, very best of luck, see you shortly. Well, how did you do? Did you manage to work out that missing term just from one piece of information and our tiny little formula? Let's see how we did. So let me zoom in on here for you and just talk you through how we did it. So we had to remember that to find the total number, we would multiply the dog number by two to give us the legs, we'd add two to the dog number to give us the body. So here, if we were looking for dog number 100, for the body, 100 add two is 102, for the legs, a 100 times two is 200 and add them together, that's 302. And then the bit that never ever changes was the tail, ears, nose and head, which totaled four. So 302 add the remaining four. So dog number 100 had 306 cubes in total, 306 cubes. Did you get it? Oh, I bet you did. Well, if you did not have that on the back, big congrats, well done to you because that's brilliant. So, practise that and see if you can do it with different numbers. See if you can do it find dog number 84 or dog number 236. Give it a go, see if you can use that formula again. Well, you need to pop off and take our final knowledge quiz now. To make sure that you've done that and that you are ready to stop with our maths for the day and that your brain is all rounded off and prepared for the next time. Well, what a brilliant job you've done today. As usual, worked really hard, focused hard and actually did some quite sophisticated things there. So well done, I'm really proud of you. So from me, Mr. C, until next time, goodbye.

To develop strategies to plan and solve problems (Part 1)

To develop strategies to plan and solve problems (Part 1)

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UnitMaths / Reasoning with patterns and sequences

Maths