Lesson video

Hello, my name is Mrs. Behan, and in this lesson I will be your teacher.

In this lesson we will be using arrays to help us explore the commutative law and the associative law in multiplication.

Let's start by taking a look at the lesson agenda.

We will will recap commutativity and associativity.

We will explore a statement and find proof.

Then we will decide on efficient strategies to use.

And at the end of the lesson there will be an independent task for you to have a go at.

And I will go through the answers with you.

I know you'll be keen to find out how you got on.

There are a few things that will help you in this lesson.

You'll need something to write with, so a pencil or a pen, and something to write on, like some paper.

It might be useful to find some small objects that you could use as counters.

If you don't have those things to hand right now, just pause the video whilst you go and get them.

And remember, try to work in a quiet place where you won't be disturbed.

So let's remind ourselves of the two laws that we're going to use in this lesson, the law of commutativity and the law of associativity.

When multiplying, you can change the order of the factors and the product remains the same.

This in number is the commutative law.

We can use letters to express this law.

a multiplied by b will equal to c.

We know that b multiplied by a will also equal c.

So a multiplied by b is the same as b multiplied by a.

Now, if that doesn't make sense to you, I'll use some numbers.

Instead of a, let's imagine it says three.

And instead of b, we'll imagine it says two.

Three multiplied by two is equal to six.

Let's change the order now.

Two multiplied by three is equal to six.

So we know that three multiplied by two is the same or equal to two multiplied by three.

So we can use letters to represent numbers to help us understand things like this, like the commutative law.

The associative law is when we multiply three numbers, you can group the numbers in any combination.

We'll be trying to find some proof for this in this lesson.

Let's use letters again to express this law.

a multiplied by b multiplied by c is equal to d.

So when it's in brackets like this, this means we'll multiply those together.

So we'll multiply these two and then when we multiply it by a, we'll get d.

If we change the order, if we multiply a by b and then we multiply the product of a and b by c, we should still get the same product.

So a times b times c is the same as a times b times c.

And it's just the order in which you multiply first that will be different.

I'm now going to show you a calculation on the screen.

Four multiplied by five.

And if we have a look at the array, we can see that we've got four in a row and there are five rows of four, or there are five in a column and there are four columns, so four groups of five.

If we know four multiplied by five, what else do we know? Well, let's use the commutative law.

If we use the commutative law, which is you can change the order of the factors and the product will remain the same, we also know that five multiplied by four is equal to 20.

So the whole group is equal to 20, but what if I add in a third factor? My calculation is now four multiplied by five multiplied by two.

Well, for this we will use the associative law.

All we've done is we have our four multiplied by five and then we're going to double it, because I know doubling is the same as multiplying by two.

So I have used the associative law.

What is the whole now? Four multiplied by five multiplied by two is equal to 40.

I doubled the number of equal parts and the whole was doubled.

There are eight groups of five, or 10 groups of four.

Use your finger on the screen to identify those groups.

There are eight groups of five, or 10 groups of four.

Can you see them? So our eight groups of five are the columns.

This is a group of five and now we have eight groups.

And where do we find the 10 groups of four? That's right, there's 10 groups of four, because we've got five groups of four here and on this array we've got five groups of four as well.

Se we know that four multiplied by five multiplied by two is equal to 40.

So I wonder, if I double the value of the parts, will the whole still be doubled? So we're now going to explore instead of repeating or doubling the whole set, what if we changed the value of each counter so it doesn't mean one anymore? So now the value of each counter is? That's right, each counter now has a value of two.

So this is still the same, and I know it's still the same, because if I count in twos, how many is each row worth? Two, four, six, eight.

The value of one row is eight.

And how many rows do we have? We have five.

So it's like we've multiplied four and two to get eight and then we've multiplied by five.

So four multiplied by two multiplied by five still gave us the same product.

So I've multiplied in a different order and the product has stayed the same.

There you can see, eight multiplied by five is equal to 40.

Can you read the statement on the screen? The array now represents five groups of eight or four groups of 10.

Can you find those groups on the array? Use your finger on the screen to point them out.

There are five groups of eight.

Well, I can find eight in each row, because four lots of two is equal to eight.

And we have five groups of them.

So I now have five groups of eight, and what about the four groups of 10? Where do we find that? That's the columns, isn't it? Because each column is worth 10 and we can see there are four groups of 10.

No matter which way we look at it though, the whole is still 40.

Now it's your turn to explore.

I want to know, does multiplying three one-digit numbers in any order always give the same product? I'll show you what I mean.

I have rearranged four multiplied by five multiplied by two in all the different combinations for you.

And you can see them here on your screen.

I'd like you to multiply them in order.

So four multiplied by five multiplied by two gave us 40.

And we drew this part here as an array, didn't we? I'm going to show you what this one looks like.

Four multiplied by two multiplied by five.

The last factor is the one that we can play around with.

So four multiplied by two, I have drawn as an array.

Four multiplied by two.

Now I'm going to multiply that array by five.

So what do you think it's going to look like? Well, I now have five groups of four multiplied by two.

I can find the whole and that will tell me what the product is.

But like I said, you can change the way you use this last factor if you want to.

So why not have a go at changing the value of the factor? Of each counter, sorry.

So each counter isn't worth one anymore, it's now worth five.

So that's how we can find the five in our calculation in the array.

We know that five multiplied by eight, because there are eight counters, gives us 40.

So pause the video here whilst you explore the other calculations and the arrays.

Use drawings, pictures, maybe you can use your counters that you've found in your house.

And when you're ready, come back to me and I'll show you my ideas.

So what did you learn? Well, this is what I learned.

The first one I had a look at was five multiplied by two multiplied by four.

It still gave me a value of 40.

The product was still 40.

So here's the array, five multiplied by two, and I've drawn it out four times.

Now actually, I know five multiplied by two is 10 and four groups of 10 is 40.

So that's really easy for me to calculate.

And this one, this was the second, or it might be the fourth one on the list, five multiplied by four multiplied by two.

So here, what have I changed? Well, instead of drawing out five multiplied by four two times, I've just changed the value of each counter.

So each counter is now worth two.

So I now have 20 groups of two, which is 40.

20 groups of two is 40.

The next calculation, two multiplied by four multiplied by five also equals 40, because I've got my array two multiplied by four, I have two groups of four here, and I gave each counter the value of five.

I know eight lots of five is equal to 40.

And for the last one, I did it this way.

Two multiplied by five multiplied by four, so I've drawn my arrays horizontally.

There's 10 in each group, so I know 10 multiplied by four is equal to 40.

So does multiplying three one-digit numbers in any order always give the same product? Well, read this statement on the screen with me.

We have proven that it does not matter which order you multiply in, the product will still be the same.

Why is it useful to use the associative law? Well, it will make calculating much faster for us, and we'll see in this next example.

Have a look at this calculation.

Two times eight times six.

Earlier, we've been multiplying the first two factors together, then we've multiplied it by the third factor.

On this occasion, I think I want to avoid calculating two times eight first.

Can you think why? Well, if I calculate two multiplied by eight, I would have to multiply 16 by six.

Now, that could be pretty tricky to do in my head.

So multiplying by six is not always the easiest thing to do.

So what should I use to solve eight times six? These are the two factors I'm going to choose to multiply together, because I know my eights and I know my sixes, so that's what I'll use.

I can use my eight times table facts or my six times tables facts.

Now I've multiplied these together, what is the next step? If I know eight multiplied by six is equal to 48, how can I solve two multiplied by eight multiplied by six? I'm sure you're all calling out doubling at the screen to me.

Multiplying by two is the same as doubling.

Well done.

You can partition 48 into 40 and eight.

Double both parts and add them together to find the whole.

It's now time for you to have a go at your independent task.

For each of the calculations, think carefully about which known facts will help you and which mental calculating strategies will be most efficient.

Write only the answer for the calculation after you have calculated mentally.

So here are the calculations.

Four multiplied by two multiplied by seven equals, this is where you will write the answer or the product.

And then I want to see some reasoning.

So first I, what did you multiply together first, which two factors? Then I, and here you should be saying you multiplied the product by the last factor, or one of the factors that you've not used yet.

And then because, why did you do that? Is it because of facts that you know, or a strategy that you feel comfortable using? Pause the video here whilst you complete your task.

When you're ready, come back to me and we will have a look at the answers together.

Okay then, so you've now had time to complete your independent task.

Well done for giving it a go.

This was my answer for four multiplied by two multiplied by seven.

I got the product 56.

First I multiplied four and two, which equals eight.

Then I multiplied eight by seven, and that's because I know my eight times table.

So I could do eight times seven and the product is 56.

Three multiplied by six multiplied by five, the product is 90.

First I multiplied six and five, which equals 30.

So I've multiplied the second and third factors first.

Then I multiplied 30 by three, because I know that three times three is equal to nine, so 30 times three is equal to 90.

And for the last calculation, nine multiplied by four multiplied by five is equal to 180.

First I multiplied nine and five, which equals 45.

Then I multiplied 45 by four, because I know that double 45 is 90, then if I double again, 90 doubled is 180.

You might have used a different strategy.

You might have preferred a different order, but remember, we're looking for the most efficient calculations, not the most long-winded way, but the fastest way to get the answer.

If you'd like to, please ask your parent or carer to share your work on Instagram, Facebook, or Twitter, tagging @OakNational, @LauraBehan21, and #LearnwithOak.

Thanks for joining me in this lesson.

We've now proven how the law of associativity works in multiplication.

Don't forget to take the quiz to test out your new learning.

See you then soon, bye-bye.