Lesson video

Hi, my name is Mr. Peters and in this lesson today we're gonna be thinking about developing our understanding of the commutative and associative law and how we can apply these when multiplying three numbers together.

If you're ready let's get started.

So by the end of this lesson today you should be able to say that I can change the order of the factors or group them in different ways and the product each time would remain the same.

Throughout the session today we've got a key words we're gonna be referring to.

The first one is factor.

You have a go at saying them after me.

The first one factor.

Second one product.

The third one commutative.

And the fourth one associative.

Factors are whole numbers that exactly divide another whole number.

The product is the result of multiplying two or more values together.

The commutative law states that you can write the values in a calculation in a different order and the result will stay the same.

This applies for both addition and multiplication.

And the associative law states it doesn't matter how you group or pair values in a calculation.

It doesn't matter which pair we calculate first as the result will be the same throughout as well.

This also applies to both addition and multiplication.

So this lesson today is broken down into two cycles.

The first cycle will be to express and represent three factor problems and the second cycle will be manipulating three factors.

If you're ready let's get going.

Throughout the session today we're gonna have both Izzy and Alex to join us and they're gonna be sharing their thinking and their questions throughout to help us develop our understanding too.

So we can see here that in supermarkets strawberries are often displayed in open trays, and in each one of these trays we can see that the strawberry punnets are placed into rows and columns.

In these examples there are three rows and there are four columns.

If there were two trays of strawberries out on display how many punnets would there be altogether? How do you think we could represent this as a calculation? That's right we could say it's three multiplied by four multiplied by two.

Let's see where these numbers come from then.

Well the three is represented by the number of rows in a tray.

The four is represented by the number of columns in the tray.

And then finally the two represents the number of trays that we've got all together.

Let's think about this in a different context now.

In the dairy aisle boxes of eggs are stored on a shelf.

There are 4 boxes left on the shelf.

What calculation would help us to find the total number of eggs on the shelf? Here's an example of the boxes.

Take a moment for yourself to have a think.

That's right we could represent this as 2 multiplied by 3 multiplied by 4.

Let's again see where these numbers come from.

Well, in one tray there are two rows so the 2 represents the number of rows.

And then in one tray there's also three columns isn't there so the 3 represents the number of columns.

And as we know there are four boxes on the shelf aren't there so the 4 represents the number of boxes that there were.

Okay, time for you to check your understanding now.

Have a look at our picture.

What does each number represent? The 2 represents the number of rows in one tray doesn't it.

The 3 represents the number of columns in one tray and the 5 represents the number of trays that there are all together.

Well done if you got that.

Okay, let's revisit the context of the strawberry punnets.

We could use cubes to represent this, where one cube would represent one punnet of strawberries.

Let's see how we can build this up together.

Let's take a look at the blue layer on the top.

We know that this blue layer has three rows and it has four columns.

Then we also know that there are two layers here.

So we've got the top layer which would represent the first tray of strawberries and then we've got the bottom layer which represents the second tray of strawberries.

So we can represent this as 3 multiplied by 4 multiplied by 2.

The 3 represents the number of rows in one tray.

The 4 represents the number of columns in one tray and the 2 represents the number of layers or in this case the number of trays that we've got.

In terms of calculating this we can say that 3 multiplied by 4 multiplied by 2 is equal to 12 multiplied by 2 because 3 times 4 is equal to 12.

12 multiplied by two is 24 and therefore we can say that there are 24 punnets of strawberries on display.

Have a look at our cubes this time.

What's the same and what's different about them? Well, that's right.

It's exactly the same shape as what we've just used isn't it? However this has been separated into three different colours this time so each layer looks slightly different.

Let's work through this and think about how we could calculate it this time.

Hopefully we can see that in one layer for example the blue one there are two rows and there are four columns.

So we could represent this as 2 multiplied by 4 and how many layers are there altogether? That's right there's three layers this time aren't there? So we can represent this as 2 multiplied by 4 multiplied by 3.

The two as we know represents the number of rows, the four represents the number of columns and the three represents the number of layers this time doesn't it? So we could say that 2 multiplied by 4 multiplied by 3 is equal to 8 multiplied by 3 and again that would be equal to 24.

Exactly the same answer as what we had before wasn't it? And let's have a look at this last example as well.

What do you notice this time? That's right the number of cubes again has stayed the same hasn't it? And also the number of layers has changed hasn't it? We've got four layers this time.

Let's see how we could represent that.

Let's have a look at the blue layer.

We can see that it has two rows and three columns and there are four layers with exactly the same layout as the blue layer.

So we could represent this again as 2 multiplied by 3 multiplied by 4.

The 2 represents the number of rows, the 3 this time represents the number of columns and the 4 represents the number of layers this time as well.

Again 2 multiplied by 3 multiplied by 4 is the same as saying 6 multiplied by 4 because the 2 times 5 over 3 makes a 6 and therefore 6 fours are 24.

So the total amount would be 24 cubes.

So let's look over these three again then.

Have a look at them.

What do you notice? That's right you may have noticed they've been coloured into different layers haven't they? But also the amount of cubes stayed the same each time and the calculations we did were reordered and actually the product remained the same each time didn't it? So we can say that when we're multiplying no matter what order we place the factors in, the product will always remain the same.

Okay, a quick check for understanding here then.

Say the sentences and write the equation to match the cubes.

One layer has 2 rows and 3 columns.

There are 5 layers altogether.

We can write this as 2 multiplied by 3 multiplied by 5.

Okay, time for you to have a go at practising now.

Can you look at each one of these cubes and fill in the stem sentences below including writing the equation at the bottom.

Once you've done that have a look at these different contexts and write the expression which would represent each one of these problems. Good luck with that and I'll see you back here shortly.

Okay, let's run through these together.

In the first cube there were 4 rows there were 2 columns and there were 3 layers.

We can write that as 4 multiplied by 2 multiplied by 3 and that was equal to 24.

In the second one there were 3 rows there were 2 columns and there were 4 layers.

Again we can represent this as 3 multiplied by 2 multiplied by 4 this time and that would still make 24.

And then the last one there were 4 rows 3 columns and 2 layers this time.

So 4 multiplied by 3 multiplied by 2 is also equal to 24.

Okay, and then for each one of these examples.

The first one there are 5 chocolate chips on each cookie and there are 6 cookies on each tray and there are 2 trays so we can represent this as 5 for each chocolate chip 6 for the number of cookies that we've got and then 2 for the number of trays that we've got all together.

In the second one there are 6 petals on each flower and there are 5 flowers in each vase and then there are 2 vases so we can represent this as 6 multiplied by 5 multiplied by 2.

The 6 represents the number of petals the 5 represents the number of flowers and the 2 represents the number of vases.

And finally if a horse is fed 6 carrots a day and there are 2 horses in each stable and there are 5 stables in the barn how many carrots were fed each day? Well, we could represent this as 6 for the number of carrots, 2 for the number of horses in each stable and 5 for the number of stables in each barn.

Well, done if you got all those.

Okay, on to cycle two now manipulating three factors.

Okay, so let's have a look here we've got a block here made up of cubes and if we wanted to find out the total number of cubes how could we represent this as an equation? Alex seems to think there's gonna be more than one way of doing this.

Take a moment to have a think for yourself.

Well, here's one way we could tackle it isn't there? Let's use our stem sentence to help us here maybe you could say it with me.

One layer has 3 rows and one layer has 5 columns.

There are 4 layers all together so we could write this as 3 multiplied by 5 multiplied by 4 couldn't we? This would be one way to find out total number of cubes which in this case would be 60.

Alex is saying we could say this as, 3 times 5, four times.

Let's have a look at a different way.

If I think this time how could you represent this? Again say the sentence then with me.

One layer has 4 rows, one layer has 5 columns and there are 3 layers all together, so we could represent this as 4 multiplied by 5 multiplied by 3 and again we know that's now gonna be 60 don't we? And again we could say this as, 4 multiplied by 5, three times.

4 times 5, three times.

Can you have a go at saying that? And here's one more example as well.

Have a look at each layer this time.

How many rows and are there? Okay, let's use our stem sentence then.

One layer has 4 rows, one layer has 3 columns and there are 5 layers all together so we can represent this as 4 multiplied by 3 multiplied by 5.

Again that would equal 60 altogether and we could also say this as 4 times 3, five times.

We've got 4 lots of 3 and we've got that five times.

So let's go back and look at all these three again together.

What was it that we noticed? Well hopefully you've noticed by now that the product was the same for each one of these equations wasn't it? The product each time was 60.

There were 60 cubes in total in each amount.

And Alex has noticed that the factors were placed in different arrangements weren't they? Again in the first one we have 3, 5, and 4 in that order.

The second one we had 4, 5, and then 3 and then the third one we had 4, 3 then 5.

So we can begin to generalise this.

We can say that if you change the order of the factors the product remains the same.

Could you have a go at saying that? Let's have a look.

3 multiplied by 5 multiplied by 4 is equal to 60.

4 multiplied by 5 multiplied by 3 is equal to 60 and 4 multiplied by 3 multiplied by 5 is equal to 60.

The factors keep changing position but the remains the same each time.

When the order of the factors change and the product remains the same we call this a special law.

The law is called the commutative law.

Can you say that? So the commutative law tells us that when the order of the factors change position the product remains the same.

Well done if you spotted that.

Okay, time for you to check your understanding now.

True or false? The commutative law means you can swap the position of the first two factors in an equation only.

Take a moment to have a think.

Okay, we know that's false don't we? Have a look at our justifications.

Which one of these helps to support your argument? That's right it's the second one isn't it? You can have any number of factors and swap the position of those and the product will remain the same.

It's not just the first two factors you're allowed to swap around.

So we've got a cuboid again and we've got three expressions which represent how we could find the total number of cubes in this cuboid.

Alex is saying did you know that we don't actually have to multiply from left to right each time either? So far we've always started with a number on the left and then multiplied it by the number in the middle and then multiplied that by the number on the end.

We don't actually have to work like this.

Izzy's saying what do you mean? Well let's have a look.

Alex says when multiplying you can choose a pair of numbers to multiply first.

So let's have a look.

So in this example we could multiply the 3 by the 5 first of all and then multiply that by the 4, and we can use a pair of brackets to show which pair we multiply together first.

So if we multiply the 3 by the 5 first of all, that would give us 15 and then we can multiply that by 4 which would give us the 60 wouldn't it? However we could also do this another way.

We could multiply the 5 by the 4 first of all and then multiply that by the 3.

So let's put some brackets around the 5 and the 4 first of all.

We know that 5 times 4 is equal to 20 and then if we multiply that by 3 that would be equal to 60 again.

Izzy's pointed out the product has remained the same no matter what pair of factors we multiplied first.

Hmm, that's really interesting.

Which pair did you prefer to multiply first? We're gonna look at another example now, but before we do that have a quick look at how the layers on this cuboid have been laid out.

Now have a look how the colours have been used to show the layers on this cuboid.

How could we work this one out? Well again we've got 4 multiplied by 5 multiplied by 3 this time.

The factors have changed position because the 4 represents the number of rows, the 5 represents the number of columns and the 3 represents the number of layers doesn't it? So as Alex is saying we could multiply the 4 by the 5 first of all and we'll put some brackets around that and then multiply that by the 3.

That's a similar calculation to what we did before and we still got the 60 didn't we? However using the same cuboid in the same layers it doesn't matter which order we put these factors in does it? So even though the 4 represents the rows, the 5 represents the columns and the 3 represents the layers, we could find a pair of these to multiply first and it won't matter we'll still get the same product.

So we could multiply the 5 by the 3 first this time.

That would mean multiplying the number of columns by the number of layers first of all so let's put the brackets around that.

5 multiplied by 3 is equal to 15 and then we'd have to multiply that by the 4 which would give us the 60 again.

So again the product has remained the same even though we've multiplied a different pair of factors first.

We can see that more clearly here.

We can see how we've grouped my factors into pairs by using my brackets and the product remained the same each time.

Izzy's wondering does that always work for any three numbers that multiply together? Alex says yes, let's look at another example.

This time he's chosen 4 multiplied by 6 multiplied by 2 and again in the first example he's multiplied the 4 by the 6 first of all that gives us 24 and then multiply that by 2 to give us 48.

And in the second example he's multiplied the 6 by the 2 first of all which gives us 12.

Multiply that by the 4 that also gives us 48.

So to summarise, when you multiply any three numbers together the product will remain the same no matter what pair of factors you multiply together first.

This also has a special name.

This is called the associative law.

Could you say that? Well done.

So we can use the commutative law and the associative law to make our calculations easier and Izzy's saying that as well.

She's saying she can already see how this is gonna help her to calculate more efficiently when working with numbers in her head.

Okay, time for you to check your understanding now.

Look at the equation.

Which numbers would you multiply together first? A, B or C? Take a moment to have a think.

That's right, it'd be B wouldn't it? We'd need to multiply the 7 and the 5 together first 'cause it's surrounded by the brackets wouldn't it? The brackets are showing us which numbers we are gonna multiply first.

Okay, and onto our last task for today then.

What I'd like you to do is to use the commutative law to show as many different equations as you can to represent and calculate how to find the total number of cubes in this cuboid.

And then for task two what I'd like you to do, is match the expressions that are equal.

Good luck with those and I'll see you back here shortly.

Okay, let's see how you go along.

You may have represented it as 3 multiplied by 4 multiplied by 5.

The 3 in this case would represent the number of columns, the 4 would represent the number of rows and the 5 would represent the number of layers.

You may have done 3 multiplied by 5 multiplied by 4.

Again changing the order this time but getting the same product.

You may have done 4 multiplied by 3 multiplied by 5 or you may have done 4 multiplied by 5 multiplied by 3.

You could have done 5 multiplied by 4 multiplied by 3 and you could have also done 5 multiplied by 3 multiplied by 4.

Do you notice how I worked that out systematically? I started with the 3 being the first factor and then exhausted all the options I could have for that.

Then I used the 4 as the first factor and exhausted all the options and then I used the 5 as the first factor and exhausted all the options as well.

Hopefully you can manage to come up with those as well.

Okay, and then matching the expressions, here we go.

2 multiplied by 4 multiplied by 1 is equal to 2 multiplied by 4 because the brackets around the 4 multiply by 1.

4 multiplied by 2 is in brackets so that would be 8.

8 multiplied by 3, those two would match.

Here the brackets first around the 5 multiplied by the 8 so that would be the 40 and then we multiply that by 3.

Again 5 multiplied by 2 is in the brackets here so that would be equal to 10, so that would be equal to 10 times 8.

2 multiplied by 4 is equal to 8 that's in the brackets so 8 times 8.

20 multiplied by 4 well that again would be equal to 80 so 80 times by 8.

And then finally 200 multiplied by 4 that would be equal to 800, so the last one would be matching to 800 times by 8.

That's the end of our learning for today.

Hopefully you're feeling a lot more confident about what the commutative law is and what the associative law is and how you can apply these to different calculations.

So to summarise our learning today, if you change the order of factors, the product remains the same.

This is known as the commutative law and can be applied to both addition and multiplication.

You can also multiply different pairs of factors and the product would remain the same.

This is known as the associative law and again applies to both addition and multiplication.

Thanks for joining me today, hopefully you've got something to take away with and you can start applying that in your everyday maths.

Take care and I'll see you soon.