Lesson video

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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.