Lesson video

Hello, and welcome to today's lesson about associativity.

For today's lesson, all you'll need is a pen and paper, or something to write on and with.

Please take a moment to clear away any distractions including turning off any notifications.

Finally if you can try and find a quiet space to work, that would be brilliant.

Okay when you're ready, let's begin.

Okay, so it's time for the "Try this' task.

I'd like to pause the video and have a go.

Pause in three.

two.

One.

Okay welcome back, so let me go through some of the answers that I found.

And hopefully you've got some similar ones.

Okay, so now I've started off with this one.

Okay, people always forget about using ones, but it's perfectly valid.

And now I can write this in three different ways.

This one like this.

Okay, so it's three, and I've only used one multiplication sum really.

I could have this,.

And I can actually write this, and you can try this yourself, I can actually write this in six ways.

Okay.

See if you can work out all six, if you haven't found them.

I've got one, times three, times eight and I can write this in six ways by just changing the order so three, times one, times eight.

Three, times eight, times one.

One, times eight, times three.

Eight, times three, times one.

That kind of thing.

One, times four, times six.

Two, times three, times four.

And two, times two, times six.

Okay, and this one, this one, and this one have six ways.

But this one only has three ways.

Okay, that one is like the first one, because there's a repeated number, there's only actually three ways for that, which is quite interesting.

And you'll learn more about combinations in future years.

Okay so we've got that.

So, in total, I've got 30 ways there.

And if non-integers were allowed, well that is infinitely many, because I could have a half, times one, times 48.

So many different possible ways using all different decimals and fractions.

So there's actually infinitely many ways there.

Two students, both use the same array to show that 24 equals four, times three, times two.

Okay so what's that he's done? Well, he's split his array into two, four by three arrays.

So can you see the four by three array? And he has two lots of them, so he has a four by three array, twice.

And can you see how he's grouped the four and the three? So that's one group of a four by three.

Now, Yin has done it differently.

"I split up my array to show 24 is equal to four, three by two arrays." can you see the three by two array? And she's got four of them, so she's got four lots of a three by two array.

Can you see how she's grouped the threes and the twos? Now, these both give 24 cubes in total.

These are equal.

And this brings us to the associative property.

For multiplication, and it's actually true for addition as well, it doesn't matter which part of the multiplication you do first.

So if we do the four by three first, and then times by two, we get the same answer as if we did the three by the two first and then times by four.

Okay? And this is known as associativity, okay? Say that word, associativity.

It's a difficult word and it's a new word, so you need to practise saying it.

Associativity.

Okay, so can you show 24 equals two, times two, times six in two different ways by splitting the array into two equal groups? So how could we do this? Well one way could be, I'm going to have two lots of two by six.

So here.

is the two by six.

And here.

is a two by six.

So I have two lots of.

two by six, okay? Now I'm going to do it a different way.

I'm going to do.

I'm going to have two by two.

Six times.

That's a very nice circle isn't it? okay so you can see here, I've got two by two six times.

So what calculation have I done first there? What calculation would go in the brackets? Well the two by two, and I've got that six times.

So my answers will be the same, okay? And you can count and you can check, I have 24 in both of them.

So it doesn't matter what we do first, and.

This can make.

Calculations easier sometimes.

So here I'm doing two times twelve, which is 24.

Okay so do the two and the six first to get twelve, and here I'm doing four, two times two, times six, which is 24.

Okay? So it doesn't matter which way, we get the same answer.

But sometimes it's useful to do them in different ways.

And we'll see why shortly.

Okay, this is going to be a good example of why.

So I've got a rectangle, and a rectangle with length 12 and one length of 25.

No what I'm going to do, is I'm going to split that length of twelve up.

Like this.

Okay? Can you see that? That's twelve, three times four.

So four.

lengths of three.

So three times four.

Hopefully you can see that.

All right, now what I'm going to do is I'm going to rearrange this.

I'm going to make it into one big long rectangle.

Like this.

Okay? Can you see that calculation? Three times four lots of 25.

Four lots of 25.

So I've got three times that big length there.

That would give me my area.

My area has not changed from the start, my area stayed the same.

I've just rearranged it.

So that means I can do this calculation, because four lots of 25 is 100.

So I can three times 100, and that is much easier than doing 12 times 25.

Or at least it is in my opinion.

So we can use associativity to make calculations easier.

Okay let's just do another one with numbers first.

I've got 16 times 75, now I know that 16 is equal to four times four.

So I can split that up like that.

Okay? Now, what are four lots of 75? Can I work that out? Hm, you know what I'm going to split this up again.

I'm going to split this up like that.

So you can split it up more than once.

So, what would I have here? Well that means I've got four, times two, times.

Okay what is two times 75? That is 150.

And now I'm going to do 150 times two.

300, and then four times three is twelve.

So 1,200.

Okay.

So let's talk through that, because there was a lot going on there.

All right, so what did I do? I split my 16 up into four by four.

Okay? And then I split one of my fours up into two by two.

Because I find times-ing by two quite easy compared to some of the other multiplications.

So four times two, times two times 75.

So I can do this two times 75 first to get 150.

And then I can do the two times the 150, to get 300.

And now that's quite a nice number, and I know how to do four times 300.

So I can do.

In fact, actually you can view this line as splitting up the.

Into like this.

So we do the four times three is twelve, and then we get the twelve times 100.

So if you want, you can even add in this extra step.

Okay? We can do this all because of associativity.

So I'd like you to now try and apply what we've learned, okay? And just on this last one here, they may be a few different ways you can get it, okay? There may be more than one answer for this one.

So see if you can find more than one.

Okay, so pause the video to complete your task, resume once you've finished.

Okay, and here are my answers.

Here are all the different ways that I managed to find calculations for this one.

Can you see them all? Can you see two- one lot, two lots of three by eight? Can you see three lots of two by eight? Can you see eight lots of three by two? It took me a while to see this one.

We've got eight, eight, eight, eight, eight, eight, three times two times.

Hm, very nice.

And what about six, times two, times four? Where's the six lots coming from? Oh, well that's four, and there's two of them.

That's a two lot of four.

Two lot of four, two lot of four, two lot of four, two lot of four, six times.

Very nice.

Did you manage to find all of them? If you didn't, can you see them now? It's a bit tricky to see them sometimes.

Okay next one, pause if you need to.

And finally.

Okay so now it's time for the explore task.

Which of these calculations could be transferred to 100 times a, where a is an integer.

Okay integer means whole number, and you need to remember that it's a quite important one.

So what does that mean? Well, I don't want you to be confused by the "a" here.

"a" just means it could be any number.

So it could be 100 times three, 100 times seven, 100 times nine, 100 times anything okay? But it couldn't be a decimal, it has to be a whole number.

So what I want, I want you to see which ones can be easily transferred into 100 times a.

Okay so I'm going to ask you to pause the video in a second, and I'm going to offer some hints as well.

Okay, so I'd like you to pause the video to complete your task, but if you've had a bit of a go and you're struggling a little bit, come back for some hints.

Okay so here are my hints.

I've said which ones are easily transferrable and which ones aren't.

And you're going to have to fill in the blanks.

And see if you can find a reason why the ones I've said no to aren't easy to do.

So pause in three.

Two.

One.

Okay so here are my answers.

Hopefully you've got the same.

We're just going to talk about these two quickly.

Why could this not be transferred easily into 100 times something? Well.

There needs to be two factors of five and two factors of two to get 100.

There's a factor of two and a five in there, but here there's a factor of three and a five.

There's no other factor of two.

So we can't easily make that into a calculation of something times 100.

And with this one, there's only one factor of five, okay? There's no factor of five in the twelve, so we can't easily make that into a calculation of something times 100.

Okay, and that is it for today's lesson.

Thank you very much for all your hard work, and I look forward to seeing you next time.

Thank you.