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Hi everyone.

Ms. Jones here.

Today we are going to be looking at something called counting strategies.

So we're going to be looking at ways to count things a bit more efficiently, and eventually will lead to generalising some kind of patterns and that kind of thing, and it's really interesting stuff so, please make sure before we begin, you've got some paper and a pen, you've removed any distractions, and you can try and find a nice quiet place to work.

Pause the video to make sure you've got all of that sorted, so that we can start.

So the first thing we're going to look at is what's on the screen now.

Two students have used this dot pattern, so this one here, to create chains.

By essentially linking and overlapping this pattern here.

How many dots are in each of their chains? What is your counting strategy? So I would like you to think of way of counting these dots, maybe grouping them, probably not just counting each dot one by one, because when you get bigger chains, that's going to be really hard to do.

So try to find some kind of counting strategy.

Once you've had a go at that, try and draw your own chain diagrams of different lengths.

How many dots do those ones have? So pause the video here to have a go at counting those dots, and trying to group them together so it's a bit more efficient in counting them.

So hopefully you managed to get those 17 dots in the first one, and 25 dots in the second pattern.

Now there are lots of different ways you could've counted it.

For example, you could have seen that there were groups of three here, so you got four groups of three, and then five dots at the top, or maybe you saw that this is the pattern that is actually repeating, with a dot on the end, or maybe you wanted to use the original pattern, and see that yes it does overlap, but that's okay because you could just count these groups of five, and then just subtract the ones that overlap.

So well done if you use any of those strategies, or if you found your own strategy.

You can link together patterns of dots to form chains, which we've just seen.

Counting the dots in a strategic way can help you to find the number of dots in any length chain.

The students used a grouping strategy to count the number of dots.

So we've got exactly the same number of dots as we had before, same patterns, and they've just used their specific strategy to count them more efficiently.

So we got groups of three here, you got a group of four and a group of one, and we've got six groups of three here, and a group of six and a group of one.

Using this grouping, write different tracking calculations expressing the number of dots in each chain.

So as an example, for the first one we have got groups of three here as we said, we got four groups of three, so how can I write four groups of three? Four multiplied by three, good.

And a group of four, which I've added on, and a group of one, which I've added on, so this would be my tracking calculation.

Four lots of three, add four, add one.

Can you pause the video now to do the same for the six chain? And you should have got something that looks like this, six groups of three this time, add six, add one, and hopefully you can see the link between what chain it is, and how many chains there are, and the tracking calculation, how did these link? And thinking about that link, use this strategy to express the number of dots in a five-chain, an eight-chain, and an m-chain.

So you might draw out the five-chain, and have a go that way, or you might recognise the pattern because you're going to need the pattern for the eight-chain and the m-chain.

Pause the video to have a go at writing those tracking calculations.

So well done if you got this, and actually worked out the number of dots as well, that would be great.

So for the five-chain, instead of having four lots of three, add four, add one, we're going to have five lots of three, add five, add one.

For the eight-chain, eight lots of three, add eight, add one.

And for the m-chain, m lots of three, add m, add one.

And this is where we are generalising.

You can find any number of dots, any part of the chain.

And well done if you worked out what the answers were.

So we've got 15, add five, 20, add one is 21.

Eight multiplied by three is 24, add eights is 32, add one is 33.

M multiplied by three is 3m, add m is 4m, add one is not 5m remember? Our ones and all m's are separate.

So those were the number of dots in those chains.

Well done if you got those answers.

Pause the video now to complete your task.

In your independent task, the first question is asking you to substitute m=5 into the following.

So again, a reminder that 3m would not just mean three five, 35, it means three lots of m, so you should've done three multiplied by five, to get 15.

M squared, again reminder, m squared does not mean m multiplied by two.

It shouldn't be five times two, it should be five multiplied by itself.

So five multiplied by five.

Which is 25.

For question two, You are asked to complete the tracking calculation for the number of dots.

So I can see here that I've got one group of one, two, three, four, five, and I've got two groups of four.

So my multiplication is going to be my two groups of four, and then I'm going to add on my five at the end.

And we can see here that we're going to have a very similar tracking calculation, this time I got three groups of four, add five, and this time I got four groups of four, add five.

So using that strategy, write a tracking calculation to express the number of dots in a 300 chain.

Again, we're not going to draw out all of those dots for a 300 chain.

We're going to use the pattern that we have seen.

So each time, we had a different amount of groups of four.

So we had one less than the chain, lots of four.

So, we're going to have 299 lots of four, not 300 lots of four, and then every time we're adding the five in the end.

So really well done if you managed to get that last one, 'cause that was a little bit tricky spotting it was 299, not 300, and here are the rest of your answers.

Well done if you got some or all of those correct.

Now a student has used a different strategy to count the dots in this six-chain.

They have done six multiplied by four, 'cause we have got six groups of four, add one.

Use this grouping and write a calculation to express how many dots there would be in a three-chain, a 15-chain, a 200-chain, and an m-chain.

So just using what you've learnt today, to try and generalise, and find patterns.

Pause the video to have a go at that now.

So really well done if you got these answers.

So we should have got four for A.

Three lots of four, add one.

It's a three-chain rather than a six-chain, so that is what's going to change.

For B, 15 lots of four, add one, which is 61.

For C, 200 lots of four add one, which is 801.

And finally the m-chain, m lots of four add one, which is 4m, add one.

And if we think back to the rest of the lesson, where we have seen this actually exact same pattern, that they've just grouped it differently, we will have got exactly the same answer.

So no matter how you group the dots, we still get the same solution, still get the same tracking, but well not the same tracking, a different tracking, but that is the point, we are grouping it differently, but we have the same number of dots.

So that's why we end up with the same expressions, and the same number of dots in the end.

Really well done if you managed to have a go at that, and hopefully you find this useful in the future, in terms of counting things quickly and generalising things, and you've done a really really good job today.

Make sure you have a go at the quiz at the end, to check your understanding, and I'll see you next time!.