Lesson video

Hello there everyone.

I'm Miss Brinkworth.

I'm going to go through this math lesson with you today.

So let's have a look at our learning objective.

What we're going to be doing is having a look at how multiplication can be completed in any order.

So this is a really nice little trick, and it will really help you in lots and lots of different calculations, I promise.

So let's get started, and let's look at today's lesson agenda.

So, we're going to recap a bit on arrays, 'cause that's a nice way of representing multiplication, making sure we're really clear what we're talking about.

We're then going to talk about this commutativity.

A lovely big word, and it's our key word for today's lesson.

We're then going to think about what your preferred order might be.

And then you're going to have a chance for that independent work to really hone in on what we've been talking about, and take as much time as you need to embed those skills.

And then that Exit quiz at the end.

It's just the nice bit of time to think about how much you've learned.

So, pause the video, if you need to go and get a pen or pencil and some paper.

Make sure you come back with that really, really positive attitude.

Fantastic.

Hopefully you've all got a smile as big as mine, and we can get started.

So, here's a nice warm up for you.

It's just some simple multiplication questions, which you should be quite confident with, for now, and hopefully you can do in your head.

So pause the video, I'd say take as long as you need, but hopefully it won't take you that long.

How did you get on? I wonder if there are some you feel very confident in answering and you could do straight away.

Others maybe, you left off completely, or you left until the end.

It's good to think about the ones that you do feel confident with, because you can use those to help you with the ones that you feel less confident with.

So, let's go through and as we go through, think, "Yeah, I knew that one straight away," or "Maybe I do need a bit more practise on that one." So, let's go through these answers.

So, three times two, maybe you want to think of it as two times three.

People know their two times table quite confidently.

Maybe they're not quite as confident with their three times tables, so you want to think of it as a two times table question.

That's absolutely fine.

Four times four.

Again, people not often remember these square ones, where it's a number times by itself.

So four times four, well done if you know that that one's 16.

Six times three, 18 Four times five.

People again are quite often comfortable with their five times tables.

So maybe you want to think of it as four, lots of five, rather than five, lots of four, that will get you the same answer.

One times three.

Most people are very confident with their one times table, but you might find that actually mistakes can creep in there.

If you've got one group of three, you've got three.

So well done, it's not one.

Sometimes people make that mistake.

Nine times four.

So, however you want to think about that one, you might want to make that nine times two and double it.

'cause that's a good way of thinking about your four times table.

So 36.

Or your nines, nine, 18, 27, 36.

Three times three.

Again, another square number there, where we have a number of times by itself, and people normally kind of remember those ones and they can be a good anchor.

So if you know your three times three, maybe that can help you with your four times three, for example.

Two times four.

Just a double for your two times table.

And then three times seven is 21.

Well done if you were able to get all of those done quite quickly and in your head.

That means you've got some really good knowledge about your times tables.

We're going to recap very quickly on what we mean by arrays.

Arrays are just a representation of multiplication.

That's because when we're talking about multiplication, what we mean is equal groups.

When we have more than one equal group we multiply it.

It's as simple as that.

So here, how many equal groups have we got? And how many are in this are equal groups? Well, if we count across the top, across our columns, we've got seven.

So we've got seven in each group.

How many groups have we got? We've got three groups of seven.

So we've got three times seven.

This array shows 21, split up into three groups of seven.

And they're just a really lovely way of making clear to us what we mean when we're talking about multiplication.

So there we have our 21.

Okay.

But what today's lesson is all about is commutativity.

Commutativity.

Commutativity.

It's a really fun word to say, makes it sound that you really know what you're talking about.

But all it means is that multiplication can be done in any order.

Let's have a look at what I mean.

We've just talked about three times seven is 21.

But if I split the array up, that's exactly the same array, I just switched it 90 degrees.

So it's got exactly the same number of dots in it, but I can see it as a different multiplication question.

Instead of seven times three, this can be, three times seven.

Sorry.

Seven times three, three times seven, they're both going to give us the same answer of 21.

So, here is another multiplication question, written out in an array.

I've got four across the top and five down the side.

So in total I've got 20.

What do you think might be the next one up here? If I flip that array over what might be the commutativity calculation? The one where I've switched around the numbers.

It's as simple as, four times five is 20, and five times four is 20.

Now this might seem like a really simple fact and one that you know very well, but it can be so useful when you move on to slightly trickier questions.

You might think, "Oh, I don't know my four times table or my nine times table or my 12 times table.

But I do know my twos or my threes or my fours." It'd be much easier if you can use those by switching your multiplications around.

You can answer slightly trickier questions.

So here we have three times seven is 21 and seven times three is 21.

So, here is your turn then.

Have a look at the array, think about what it's showing and how you can flip that multiplication around, answer it in any order.

So like, the answer should be the same, but I'd like the two numbers that you're multiplying to move around.

Pause the video here and take the thought you need.

How did you get on? Let's have a look.

So hopefully we're going to have a look at the arrays and see that we've got four downside and 7.

It doesn't matter which order you have these two in by the way.

If you've read across the top first, that's absolutely fine.

So, as long as you've got four and seven in each order.

So, four times seven and seven times four, they both give you 28.

So it's exactly the same array, exactly the same number of dots.

We've just moved it around, and that just shows that you can do this altercation in whichever order you feel comfortable with.

So, which order do you feel most comfortable with? Do you prefer thinking of it as four times seven? Or do you prefer thinking of it as seven times four? You might feel more confident with your four times table than you do with your seven times table.

So that might make you want to think of it as four times seven, or you might think, "Four is quite a small number, and so I only need to count my sevens," if that's the way that you want to do it.

I only need to count seven, four times to get to seven times four.

It's completely up to you.

There's no right answer.

Whichever one you feel most confident with that's what's great about commutativity.

You can answer it in the way you prefer.

Okay.

Here's another way of representing this.

What question do you think this part-whole model shows? We've got the answer, we've got the whole is 12 and we know that one of the parts is three.

So each equal group there has got three in it.

The thing that's missing is how many groups of three do we have? How many groups of three make 12? So we count those threes.

How many times is three written? One, two, three, four.

So, the fact that's being shown here is four times three is 12.

Here's exactly the same number, we've got twelve, but instead of having four groups of three, we've got three groups of four.

But it still gives us the same answer.

So it's just a different way of showing that commutativity.

Okay.

How about this one then? What do you think is being shown here? Pause the video and have a go at telling me what this part-whole model is showing, and can you switch it around for that commutativity? Well, in each of those parts, there are five dots.

How many parts have I got? I've got three parts.

I've got three times five, and my answer is 15.

I can switch that around for five times three, it's 15.

Well then.

Okay.

A slightly different way of drawing it out.

We've got six, lots of three, three, lots of six.

So the equal, exactly the same amount.

We know that three times six is 18, but it's sometimes useful to think about numbers might appear bigger or smaller, because they're written out differently or they're shown differently in arrays or in groups, but it's really important to work out exactly what's being shown.

So when we're go back to the part-whole models or arrays, these just draw out the multiplication for us.

They don't give us the answer, they just make it clearer what the question is.

So be careful when you look at an array or if you're comparing an array in a bar model and you're thinking, "Oh, that one looks bigger or that one looks smaller." Do work out the math before you decide.

Here, for example, we've got six, lots of threes, and three, lots of six.

One might look bigger than the other to you, but they do actually represent exactly the same thing, which is 18.

Okay.

So, pause the video here and answer these questions in whichever order you prefer.

You know you're going to get the same answer in the two green boxes and in the two purple boxes, but which order you prefer, I wonder? So, we've got three eights or eight threes.

Which one did you prefer? It's completely up to you.

I would probably go for three eights, 'cause I just need to count my eight three times.

But maybe you prefer to stick with your three times table.

It doesn't matter.

You're going to get the same answer and well then if you saw that was 24.

Again, for your 11s, quite a lot of people feel quite confident with their 11 times table and so they don't want to count 11.

They want to count four 11 times.

They can just go 11, 22, 33, 44, or maybe they just know 11 times four equals 44.

So again, it's really good to say that you can answer that multiplication in whichever order you like.

Okay.

Time to pause the video and have a go at some of this work on your own.

Come back with the answers.

Let's see how you did.

I hope you're getting really confident with this snack of commutativity.

Okay.

So we just need some actual arrays, some of the arrays show exactly the same multiplication, just switched around for commutativity and they are like this.

We've got three fours and four threes.

And then we've got the other ones as well, which is, four fives and five fours.

Okay.

For that question, five people want three apples each, how many apples are needed? You could do five times three or you could do three times five.

It doesn't matter.

Whichever one you feel most confident with, which is 15 is the answer.

Okay.

When you answering these questions, hopefully you could see that once you'd answered one, there was another one for you to fill in.

Each question has it's corresponding commutativity question as well.

So you only kind of need to answer half the questions and then there's another answer which you can fill in.

So three times seven is 21, and then that gives you seven times three as well.

So each question does that.

So, you've got eight times five is 40 and five times eight is 40.

And then we've got nine times three is 27 and three times nine is 27.

Eight times four is 32, and four times eight is 32.

10 times seven, seven times 10 still is 70, and 10 times seven is 70.

And then we've got five, seven fives are 35, five sevens are 35.

Really, really well done if you got all of those right.

Especially if you only did half the work and moved each question over when you could see that it had it's corresponding question on that.

Okay.

This challenge kind of moves us away from the learning objective a little, but the thing about commutativity is, it allows you to do less work.

It allows you to move questions around, so that you can answer them in the way that you want.

It's about knowing facts and applying them when you can.

So, you probably all feel quite confident with the fact, two times four is eight.

You can switch that round.

We know four times two is eight.

What about 40 times two then? Well, 40 is 10 times bigger than four.

So, we can use two times four is eight to help us with 40 times two.

We just need to make the answer 10 times bigger.

Also, it can help us with division questions.

Now actually, people say that they feel more confident with multiplication than they do with division.

But if you know your multiplication facts, that's division.

So, if you know that two times four is eight, you know that eight divided by four is two.

And if you got all of those right, really well done.

You must be someone who feels very confident with their multiplications.

I would love to see your work.

If you'd like to share it with us, please ask a parent or carer to show your work on Instagram, Facebook, or Twitter, tagging @OakNational and #LearnwithOak.

But before you go, please do complete that final knowledge quiz.

It's just a nice way to see how much of today's learning has gone in.

Really good work today everybody.

I'm really proud of you.

Have a great day.

Bye bye.