Lesson video

Hi everyone! My name is Ms. Ku and I'm really happy to be learning with you today.

It's going to be fun and really interesting lesson and I'm so excited to be learning with you.

You'll come across some new keywords and maybe some keywords you've already come across before.

Now we're going to work really hard today, but I am here to help and we can learn together.

In today's lesson from the unit, arithmetic procedures with integers and decimals, we'll be looking at checking and securing understanding of commutative and associative laws.

And by the end of the lesson, you'll be able to state the commutative and associative laws and use them efficiently.

So today's lesson, we'll be looking at these key words.

Firstly, an operation is commutative if the values it is operating on can be written in either order without changing the calculation.

So what do I mean? Well, for example, three add four.

We know this is the same as four add three or seven times two, it's the same as two times seven.

A non example would be 10 subtract two.

This is not the same as two subtract 10 or 10 divided by two is not the same as two divided by 10.

So that's our first key word, commutative.

Next, let's have a look at the word associative.

Well, the associative law states that a repeated application of the operation produces the same results regardless of how the pairs of values are grouped and we group using brackets.

So what do I mean? Well an example would be three add four, group together with those brackets, add 10.

And this is the same as seven add 10 because we've grouped up three add four, which makes seven, and we're still adding our 10.

Looking at the same calculation and now going to group of four and 10.

So three add grouping of four add 10 is three add 14, which is still 17.

So that means we know addition is associative.

A non example would be division.

48 divided by two.

Look how we've grouped those together with brackets divided by six.

Well, 48 divided by two is 24, divided by six gives me four.

Now let's group in a different way.

I'm going to group the six and the two.

From a priority of operations, we do our brackets first.

So it's 48 divided by three gives me 16.

So this is not the same.

So division is not associative.

So let's see how our lesson is split.

We'll be splitting our lesson into three parts, looking at the commutative law first, then the associative law, and then looking at how to use these laws to calculate effectively.

So let's have a look at the commutative law first.

Now an operation is commutative if the values it's operating on can be written in either order without changing the calculation.

So let's identify the different representations of addition and show if addition is commutative.

Well, I'm going to start using dots.

So let's start with five dots and three dots, and we're going to add them together.

Well, you can see this gives us eight dots.

So, now let's add three dots with five dots.

Is that still the same? Does that still give us the eight dots? Yes, it does as they both equal eight dots.

This is a nice way to show commutativity with addition.

I'm going to show in a different way.

So let's have a look with bar models.

We're going to sum five and three and two.

Is this the same as summing three and five and two? Is it the same as summing two and three and five? Well hopefully you can see yes, they are exactly the same length and they all sum to 10.

So this is another nice diagrammatically way to show that addition does use the commutative law.

So let's have a look at a check.

I'm going to do the first part and I'd like you to do the second part.

At first, we're going to use the commutative laws with a bar model to show the summation of eight and two.

I'm going to show you eight add two using the bar models here, is the same as two add eight, and you can see the same as they're both sum to 10 and their lengths are the same.

So, commutativity is shown as they're the same length and the same values.

So I want you to have a think of this question.

I want you to use the commutative laws to write the equation, 10 add one add three equals 14, in a different sequence of the addends.

See if you can give it a go and press pause if you need.

Well done! So hopefully you can spot where we could write it as one add 10 add three is 14, or we could write as one add three at 10 is 14 or we could write as three add 10 add one is 14 or we could write as three add one add 10 is 14 or we could write it as 10 add three add one is equal to 14.

Commutativity is shown as they show addition with the same values but in a different order.

Now let's move on to another check.

I want you to use the commutative property by changing the addend.

See if you can fill in those missing numbers.

See if you can give it a go and press pause if you need.

Let's go through our answers.

Well hopefully you've spotted for A.

We have 34, add eight is exactly the same as eight add 34.

For B, 19 add seven is the same as seven add 19.

For C, we have 120 add 12 add three is the same as 12 add three add 120.

And for D, the sun add the flower add the heart is the same as the flower add the sun add the heart.

Really well done if you got that one right! So now we know addition is commutative.

Let's move on to multiplication.

Well firstly we'll look at different representations of multiplication to show if it is commutative.

First, we'll have a look at an array of two by four dots.

Well, does it have the same number of dots as an array of four by two dots? Well, yes it does.

They're both equal and they both have eight dots or about cuboids.

Well, I'm going to use a cuboid with a length five by two by three centimetres.

Is it the same volume as a cuboid with two by five by three centimetres? Is it the same volume as a five by three by two centimetres? Is it the same volume as a three by two by five centimetres? See if you can give this some thought and press pause if you need more time.

Well done! So hopefully you've spotted, they all give a volume of 30 centimetres cubed.

So this means all the volumes of 30 centimetres cubed and it really does show that multiplication does use that commutative law.

Now let's have a look at another check.

We're asked to fill in the missing numbers using multiplication and that commutative law.

See if you can give this a go and press pause if you need.

Well done! Well, hopefully you spotted two times three is the same as three times two.

For B, six times seven times 10 is the same as 10 times seven times six.

For C, if we know the area of rectangle A is the same as area rectangle B or the area of rectangle A is five times two and the area of rectangle B is two times five.

Well done if you got that one right! So an operation is commutative if the values it is operating on can be written in either order without changing the calculation.

So let's have a look at subtraction.

Well using the counters, let's see if five subtract two is the same as two subtract five.

I'm going to start with five subtract two.

Using counters, you can see I have five positives and I have two negatives.

Now let's pair.

Well we know we have a zero pair here and we have a zero pair here, thus leaving us with a final answer of three.

So five subtract two is equal to three.

Now let's have a look at two subtract five.

Well, what does this mean? We have two positive counters and we have five negative counters.

Same.

Let's group up these zero pairs.

Zero pair here, zero pair here.

Thus leaving goes with negative three.

So is subtraction commutative? No, subtraction is not commutative and you can see that using our counters.

Now let's have a look at division.

Is division commutative? Well we're going to have a look at the calculation, eight divide by four and identify is it the same as four divide by eight.

While using the sharing approach, eight divided by four means we have eight and we're going to divide it into four different groups.

So in each group I get an answer of two.

Now I'm going to look at the calculation, four divided by eight.

Well I'll have my four.

I'm going to divide it into eight groups.

So that means if I divide it into eight groups, in each group, I have a half.

So this shows division is not commutative.

Great work so far.

So let's move on to your task.

For question one, I want you to pair up the commutative calculations and fill in the blanks.

Press pause if you need more time.

Great work.

So let's move on to question two.

Question two wants you to identify if the calculations are commutative or not.

Simply put a tick where you think it is commutative or not commutative.

See if you can give it a go and press pause if you need more time.

Great work! So let's move on to question three.

Question three wants you to fill in the numbers using the commutative laws.

See if you can give this a go and press pause if you need more time.

Great work.

So let's have a look at our fourth question.

Here, you you're asked to match and fill in the calculations using the commutative laws.

Give it a go and press pause if you need more time.

Really well done.

So let's go through these answers.

For question one, hopefully you've paired up the cat, the dog and the mouse is exactly the same as the dog and the cat and the mouse.

We have C multiply by D is the same as D multiply by C.

Something add nine add two is the same as nine add six add two.

A add B is the same as B add A.

11 multiply by three multiply by two is the same as two multiply by 11 multiply by three.

Four add five is the same as five add four.

Great work if you got this one right! Now, let's fill in our table.

Hopefully you've spotted with addition, it is commutative.

Division is not commutative, multiplication is commutative and subtraction is not.

Well done! So let's have a look at question three.

12 multiply by five is the same as five multiply by 12.

For B, nine add seven is the same as seven at nine.

For C, eight times 12 times two is the same as 12 times two times eight.

And for D, A times B times C is the same as B times A times C.

Really well done if you got that one right! For question four, well we had to pair up for calculation with the question and fill in the gaps.

Well hopefully you spotted five multiply by two gives us the area of this rectangle.

We have two multiply by five multiply by three gives us the volume of this cuboid.

We also have four add five add three is the same as three add four add five, which is the perimeter of this triangle.

Really well done if you got that one right! Great work so far on the commutative law.

So let's have a look at the associative law.

Well an operation is associative if the repeated application of the operation produces the same result regardless of how the pairs of values are grouped and we can group using brackets.

So let's identify if the different representations of addition show if addition is associative.

Let's use dots again and we're going to add five dots and three dots and four dots.

So you can see here, we have all these wonderful dots.

And what I'm going to do is I'm going to group five dots with three dots and then add the four dots.

So does this give us the same number of dots? Yes, it does.

So now I'm going to group three dots with four dots and then add the five dots.

Does that still give us our 12 dots? Well grouping our three dots on our four dots adding our five dots gives us exactly the same.

So you can see how addition uses the associative law using counters.

So let's try using bar models and sum five with three and two.

Well summing the five with the three and then adding the two.

Is it the same? Well that's eight add two.

Yes, it's the same.

What about if we sum the two and the three and then add the five? Well the two and the three is five.

Yes, it's the same.

They're all exactly the same length and they're all summed to the same value, which is 10.

So this is another nice way to show addition uses the associative law.

Now let's use numbers and the grouping is going to be shown with brackets, for example.

There's lots of different ways to sum, five add eight add 12 add nine.

So here's one way.

Five add eight in brackets, add nine add 12.

Another way would be 13 add nine add 12.

Another way would be 22 add 12.

All of these add 34.

Now can you find another equivalent calculation using two brackets? See if you can give it a go and press pause if you need.

Great work.

Well, we can group like this.

The five add the eight add the nine add the 12.

This is the same as 13 add 21, which still gives us that 34 or you could have grouped it like eight add 12, add bracket five add nine.

This gives us 20 add 14, which still makes our 34.

This is a lovely way to really show that addition uses the associative law.

So you may have seen the associative operations before as it allows a calculation to be split sensibly to solve or calculate an answer quickly and efficiently.

For example, multiplication of integers.

300 times 7,000.

This is exactly the same as three multiply by seven times a hundred times a thousand.

So applying our associative law, three multiply by the seven is 21 and 21 times a hundred times a thousand.

We're going to group together that 21 in a hundred to give 2,100 times 1000, which gives us that big number, 2 million and 100,000.

So you may have seen how we've used the associative law before.

Let's look at an example using addition.

While 11 add three add 17 add 19 add 15 add six add five add four.

Let's see if we can group them together sensibly so we can complete this calculation.

Well I know this is the same as 11 add 19 add 17 add three, add 15 add five add six add four.

I'm going to now group using brackets.

The reason why grouped it is because it makes it a little bit easier.

11 add 19 is our 30.

17 add our three is our 20.

15 add our five is our 20 and six add four is our 10.

This makes the calculation much easier to work out, which is 80.

Another nice way how we've used the associative law is with multiplication of decimals.

This is the same as 246 multiplied by 0.

1 and 14 times by 0.

1.

Then we can group together those integers.

246 multiplied by 14 and then we're going to multiply the 0.

1 and the 0.

1.

This gives me 3,444 multiplied by the 0.

01 which gives us 34.

44.

So you may have come across the associative law already.

So now we now addition and multiplication is associative.

Let's have a look at subtraction and division.

Well we're going to use counters again and we're going to look at the numbers five, four and three and repeat the operation of subtraction to see if it's associative or not.

So five subtract using our brackets, four subtract three.

I'm going to start off with our five counters.

And remember we're going to subtract the four, subtract the three.

So let's identify our zero pairs.

We have three zero pairs here, thus leaving me with five subtract one.

Well five subtract one gives me a nice answer of four.

Let's see if it's the same as three subtract five subtract four.

So all I'm going to do here is start with my three counters.

I'm going to subtract the five and the minus four using our zero pairs again.

This is the same as three subtract one.

So hopefully you can spot subtraction is not associative.

So let's see if division is associative.

And we're going to look at the repeated operation of division on the numbers 24, 6 and two.

For example, do you think these calculations are associative? 24 divided by six, grouped together, divided by two.

Do you think that's the same as 24 divided by the grouping of six divided by two? And do you think it's the same as six divided by the grouping of 24 divided by two? See if you can give it a go.

Press pause if you need more time.

Great work! So hopefully you've spotted 24 divided by six, divided by two gives you an answer of 2.

24 divided by the grouping of six divided by two gives me eight and six divided by the grouping of 24 divided by two is 0.

5.

So this really does show division is not associative.

So now let's move on to a quick check.

Using the associative laws, fill in the blanks of the equations below.

See if you can give this a go and press pause if you need.

Great work.

So let's see how you got on with A.

Three add the grouping of 10 add eight is the same as 10 add eight group together add all three.

A plus bracket something add C close bracket is equal to B, add bracket A plus something.

Well, hopefully you can spot we have our B and C there.

For C, we have seven multiply by bracket nine multiply by two is the same as nine multiply by seven times two.

For D, we have A times B multiply by our C equals A multiplied by the C multiplied by the B.

And for E, nice little question where you have the sun multiplied by the moon multiplied by the star and the planet is exactly the same as the sun multiplied by the moon multiplied by the grouping of the star multiplied by the planet.

Really well done! Let's have a look another check question.

Here, you need to fill in the blanks to identify the sequence of working out and the final answer.

So you can give it a go and press pause if you need.

Great work.

So let's see how you did.

318, multiply by 30.

Well 380, we can apply the associative law here.

It's the same as 38 times 10 and the 30 applying the associative law is the same as three times 10.

Now the next line of working out shows the multiplication of 38 and three.

So that means we're multiplying it by the 10 and the 10.

Well we know 38 multiply by three is 114.

So what did that 10 multiply by 10 give? We gave a hundred so therefore our final answer is 11,400.

This was a great question.

Well done if you got this one right.

Now, let's move on to your task.

Using the associative law, fill in the blanks for the equations below.

So you can give it a go and press pause if you need.

Great work! So let's move on to question two.

Question two wants you to identify if the calculations are correct or incorrect.

See if you can give it a go.

Press pause if you need more time.

Great work.

So let's go through our answers.

Well, for question one, hopefully you've spotted six was the missing number for A.

We had X and Z, the missing letters for B.

Seven and two were the missing numbers for C and W and P were the missing numbers for D.

Well done.

And for question two, hopefully you've worked out.

The first calculation is incorrect.

The second calculation is incorrect.

The third one was correct and the fourth one is incorrect.

Fantastic work so far, everybody.

Now we're going to move on to the last part of our lesson, which is using the laws to calculate efficiently.

Having a real good understanding of the commutative and associated laws enables us to write a calculation so to solve more efficiently.

And spotting when we have the commutative or associative law can strengthen ability to use these laws.

For example, if we were looking at the product of its prime factors of 840.

We know 840 can be split into 10 multiply by 84 using the associative law.

Then two multiply by five, that's the same as our 10.

And two multiply by 42 is the same as the 84.

Same again, which law do you think we've applied here? It's the associative law.

Then we can apply it again.

Two multiply by five multiply by two, multiply by six, multiply by seven.

What do you think we did here? Well, we apply the associative law again because 42 can be split into six multiply by seven.

Then we have two, multiply by five multiply by two.

Then we used two multiply by three multiply by seven.

So which law do you think we used here? It was the associative law again.

This gives us our final answer of two multiply by two, multiply by two, multiply by three multiply by five, multiply by seven.

But what law did we apply in this line with the previous one? Well, we use the commutative law because we've put on multipliers in ascending order.

Using our exponents, we have two to the power three, multiply by three, multiply by five, multiply by seven.

Great work if you spotted those.

What we're going to do now is have a quick check and I want you to identify which law has been applied.

See if you can give this a go and press pause if you need more time.

Great work! So for A, hopefully you spotted 19 add 80, add one add 20.

Well, we've applied the commutative law here.

Then we applied the associative law.

For B, which law did we apply first? Well, we applied the commutative law first and then we applied the associative law.

Now let's move on to our task.

Question one wants you to use the laws efficiently to sum the first a hundred integer digits, one plus two plus three plus four, et cetera, et cetera, all the way to a hundred.

This is a great question.

See if you can give it a go.

Press pause if you need more time.

Now let's move on to question two.

Jun and Lucas both try the same set of four calculations.

Jun does the calculation in red first and Lucas does the calculation in blue first.

Without calculating, predict whether Jun and Lucas will get the same answer for all their questions and explain why or why not.

So even can give this a go and press pause if you need.

Great work.

So let's go through our answers.

Question one is a fantastic question and it's quite an old question as well.

This was a problem first anecdotally solved by a mathematician called Carl Friedrich Gauss.

And what Carl Friedrich Gauss did, is he used the commutative law and he ordered the numbers in this particular way.

One add 99, add two add 98, add three add 97, so on and so forth.

In doing this, then he's grouped those numbers using the associative law.

One at 99 is 100.

Two add 98 is 100.

Three add 97 is a 100.

So on and so forth.

Then we get to our 50 add a hundred.

And he was able to work out the answer to be 5,050 very quickly.

This is a great story and when you get a chance, press pause and have a read.

But the answer was found by Carl Friedrich Gauss when he was a child.

And the story anecdotally states that his school teacher gave all the students in the class this task to do, just to keep them quiet in the hope it would keep them occupied for a long time.

And it only took Gauss a few seconds to work out the answer.

This is a great question and a nice little story to follow.

Now let's have a look at question two.

Jun and Lucas both tried the same set of four calculations and we need to identify without calculating to predict if Jun or Lucas will get the same answer.

Well, Jun's approach of six add four add two is exactly the same as the approach of six add grouping those four and two together.

Addition and multiplication are associative while subtraction and division are not.

So hopefully you've spotted these answers.

Really great work today and we've gone through quite a few keywords.

An operation is commutative if the values it is operating on can be written in either order without changing the calculation.

And the associative law states that a repeated application of the operation produces the same result regardless of how the pairs of values are grouped.

Remember, we can use that grouping with brackets.

We know addition and multiplication are both associative and commutative and subtraction and division are not associative and commutative.

And we can also use the laws to evaluate calculations quickly and efficiently just like that child mathematician, Carl Friedrich Gauss.

Great work! Well done today!.