Lesson video

Hi, my name's Ms. Lambell.

You've made an excellent choice deciding to stop by and do some math for me today.

I'm really pleased.

Let's get started.

Welcome to today's lesson.

The title of today's lesson is Check In and Further Securing Understanding of the Commutative Law and that's within the unit standard form.

By the end of this lesson, you'll be able to state the commutative law and use it to calculate efficiently.

Some keywords that we are going to be using in today's lesson.

We'll just quickly recap those because some of them you might not have seen for a while.

The first one is exponential form.

When a number is multiplied by itself multiple times, it can be more simply written in an exponent form.

2 multiplied by 2, multiplied by 2 we could write as 2 cubed and 10 multiplied by 10, multiplied by 10, multiplied by 10, multiplied by 10, multiplied by 10 is 10 to the power of 6.

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

So for example, 3, add 4 is equivalent to 4 add 3.

7 multiplied by 2 is equivalent to 2 multiplied by 7, but a non-example would be 10 subtract 2 that is not equal to 2, subtract 10 so the order cannot be changed when we're subtracting.

Primes.

A prime number is an integer greater than 1 with exactly 2 factors.

All integers greater than 1 are either composite or prime.

Prime factors.

Prime factors are the factors of a number that are themselves prime.

And prime Factorization is a method to find prime factors of a given integer.

Today's lesson, I've split into two separate cycles.

In the first 1 we'll look at reviewing product of prime factors.

This should be familiar to you, but we just need to review this before we progress through this unit and in the second learning cycle we'll look at how we can use those products of primes.

Let's get going with that first one.

Sophia and Alex have been practising writing numbers as a unique product of their prime factors.

"600 written as a product of its prime factors is 2, multiplied by 2, multiplied by 2, multiplied by 3, multiplied by 5, multiplied by 5," Sophia says.

Alex says "600 written as a product of its prime factors is 2 cubed multiplied by 3, multiplied by 5 squared." Who is right? They're actually both right, and I'm sure you said that the only difference is is that Alex has written his using index notation, so he's used that exponential form to write 2 multiplied by 2, multiplied by 2 as 2 cubed and 5 multiplied by 5 as 5 squared.

Index notation is when we write the repeated multiplication using its base and it's exponent.

For example, here we would read this as 7 to the power of 4, where 7 is the base and 4 is the exponent, and we know that this means a repeated multiplication of 7, 4 times, so 7 multiplied by 7, multiplied by 7 multiplied by 7.

Which of the following shows 2, multiplied by 2, multiplied by 3, multiplied by 3, multiplied by 3, multiplied by 5, multiplied by 7 written correctly using index notation? Pause the video and when you've got your answer come back and we'll check.

What did you decide A, B, or C? And it was C.

We had 2, a repeated multiplication of 2, twice, so 2 squared and a repeated multiplication of 3, 3 times giving us 3 cube.

We are now going to write 1,260 as a product of its prime factors.

We start with the number and we find two factors of 1,260 and because I can see there that in the ones column there is a zero, I know it's a multiple of 10, so I've chosen to choose the pair 126 multiplied by 10.

I could have chosen a different pair as long as their product was 1,260.

I'm now going to take each of those parts and write that as a factor pair.

126 I've decided to write as 2 multiplied by 63.

A good tip is to look and see if the number is even, you know it is a multiple of 2 and so therefore you can write it as a product of 2 and its other factor, so 2 multiplied by 63.

And 10 is the product of 2 and 5.

I need to repeat this now for all composite numbers.

So all numbers that are not prime.

2, 2 and 5 are prime.

63 is a composite number, so we are going to write that as a product of 2 factors.

63 is a product of 7 and 9.

I'm now going to repeat that process for any composite numbers that are left.

So 2, 7, 2 and 5 are all prime.

9 is composite and I can write that as a product of 3 and 3.

I now only have prime numbers in my product.

I'm going to rearrange the prime factors so that they are in numerical order and then I'm going to write those in exponent form using index notation.

2 squared multiplied by 3 squared multiplied by 5, multiplied by 7.

Let's recap what we did.

We started 1,260 and we found a pair of factors whose product was 1,260.

We then repeated that process for any composite number that was left in our product until we ended up with just prime numbers.

We then rearranged those prime numbers in numerical order and you may decide to skip that step.

And then the final step is to write it using index notation.

So in its exponential form.

Let's have a go at one more together and then as this is a review, I think you'll probably be ready to have a go at one of these independently.

I'm going to write 250 as a product of its prime factors.

Start with 250 again here I'm going to write it as 25 multiplied by 10 product of 25 and 10 is 250, spotting again, there was a zero in the ones column, so it must have been a multiple of 10.

Neither of those are prime, okay? They're both concept and so I'm going to repeat the process.

25 is 5 multiplied by 5 and 10 is 2 multiplied by 5.

These are all now prime numbers.

So they're all our prime factors of 250.

I'm now going to write that using index notation, so in its exponential form, so 2 multiply by 5 cubed.

Notice this time I didn't rearrange the prime factors into numerical order.

I just felt that I was able to do that because it was a fairly short list of prime factors.

Now I'd like you to have a go at this one.

I'd like you to write 360 as a product of its prime factors.

So you're gonna pause the video and then you're gonna come back when you've got an answer to that.

Good luck with that.

Okay, let's see how you got on.

So 360, I chose the factor pair 36 and 10.

Remember you may have chosen something else.

You may have chosen 2 and 180.

It doesn't matter.

As long as you followed the process free correctly, you'll end up with the same answer as me 'cause remember, any composite number can be written as a unique product of its prime factors.

There is only one answer.

I chose to do 36 multiplied by 10 and then I broke 36 down into 6 multiplied by 6 and 10 into 2 multiplied by 5, 6 was 2 multiplied by 3 and then another 6 2 multiplied by 3.

And then writing that in exponential form, we can see it's 2 cubed multiplied by 3 squared multiplied by 5.

How did it get on? Super work.

Well done.

Here, I already know that 360 is 2 cubed multiplied by 3 squared multiplied by 5 'cause you've just worked that out for me and we can use this to write other numbers as a product of their prime factors without repeating the process from the beginning.

720.

Well we know that 720 is 360 multiplied by 2.

We already know that 360 is 2 cubed multiplied by 3 squared multiplied by 5.

I don't need to go back to the beginning and work that out.

I know that from a question that I've already answered.

We can see here that those two things are equivalent.

The two bits in my purple boxes and then I'm going to multiply that by 2.

2 cubes multiplied by 2 would be a repeated multiplication of 2 4 times.

So that's 2 to power of 4.

And then I've got my 3 squared and my 5.

Let's take a look at another one.

3,600.

We know that that is 360 multiplied by 10.

And we know that 360 is 2 cube to multiply by 3 squared, multiply by 5.

Now we're going to write 10 using prime factors which is 2 multiplied by 5.

I'm going to tithe this up by collecting together all of my prime factors at the same 2 cubes multiplied by 2 is 2 to the power of 4.

3 squared is 3 squared and 5 multiplied by 5 is 5 squared.

So the key thing is here we do not need to go back and work out the prime factors right from the beginning.

We can use the prime factor knowledge we already have.

Which 3 numbers can be found more easily using 120 written as a product of its prime factors? Pause the video, there are 3 of these that are correct.

So you could think of this as finding the one wrong one.

When you've got your answer come back, you can pause the video now.

What did you decide? A was correct.

That would be 20 multiplied by 120.

B also correct because we could do 4 multiplied by 120 and C, sorry D because we could do 3 multiplied by 120.

So it was A, B and D.

We'll just try a couple more.

450 is 2 multiplied by 3 squared multiplied by 5 squared.

So I've given you that fact.

1,800.

Well that's 450 multiplied by 4.

We know 450 as a product of its prime factors is 2 multiplied by 3 squared multiplied by 5 squared.

And we're multiplying that by 4, which is 2 squared.

Here, I've decided to write it straight into its index notation.

So using that exponential form.

I'm now going to rearrange those so that they are in numerical order.

I end up with 2 multiplied by 2 squared multiplied by 3 squared multiplied by 5 squared.

And we know that 2 multiplied by 2 squared is 2 cubed and then 3 squared and 5 squared is the product of 2 cubed, 3 squared and 5 squared.

Therefore 1,800 written as a product of its prime factors is 2 cubed multiplied by 3 squared multiplied by 5 squared.

What about 150? I'm gonna give you a moment to think how we can use 450 to give us 150.

Yeah, you're right, we can divide by 3, can't we? So 450 divided by 3 is 150.

We know 450 as a product of its prime factors is 2 multiplied by 3 squared multiplied by 5 squared and we just divide that by 3.

Now let's rearrange our prime factors in numerical order.

We end up with 2 multiplied by 3 squared, divided by 3, multiplied by 5 squared.

And 3 squared divided by 3 is 3.

So 150 written as a product of it's prime factors is 2 multiplied by 3, multiplied by 5 squared.

Your turn now to have a go at this check for understanding, I'd like you to use the fact that 225 is 3 squared multiplied by 5 squared and I'd like you to write 900 as a product of its prime factors.

You can pause the video and then when you've got your answer come back.

How did you get on? Super work.

Well done.

900 is 225 multiplied by 4.

We know that 225 is 3 squared multiplied by 5 squared and we are gonna multiply that by 4, which is 2 squared.

And then I always like to write my prime factors in numerical order.

So I end up with 2 squared, 3 squared and 5 squared.

So 900 written as a product of its prime factors is 2 squared multiplied by 3 squared multiplied by 5 squared.

How did you get on with that one? You got it right.

Amazing.

Of course you did.

Now let's quickly remind ourselves of how we use our calculators to check our answers.

And here I'm making sure that we're just checking our answers.

I don't want you to actually use the calculator to work out the answers.

Now this is the calculator that I'm gonna be using.

You may need to look at your calculator.

It may do this in a slightly different way, but if you've got 1 of these calculators, then this is the buttons that you need to press.

So this is a Casio fx-991EX.

Firstly, we need to ensure that calculate is highlighted and then we press the EXE button.

We are going to check the prime factor decomposition of 360.

So we type in 360 and we press the EXE button.

Your screen should now look like this.

We are going to change that into a product of its prime factors.

We press the FORMAT button, so we're gonna select that FORMAT button and then you will end up with this screen.

We then are going to use the scroll down button until we see Prime Factor highlighted, we then press the EXE button again and we can see now that 360 is 2 cubed multiplied by 3 squared multiplied by 5.

It's definitely worth checking with on your calculator how to do this if your calculator's not the same as mine.

And for each of the questions you are gonna do in a moment, it would be useful if at the end of each question you check your answer using your calculator to give you lots of practise at using those particular buttons on your calculator.

Now we're ready or you are ready to have a go at task A.

Without a calculator, I'd like you to write the following as a product of their prime factors, remembering to give your answer using index notation.

So your product will be written using exponential form.

Good luck with these.

Pause the video and then when you come back we'll check those answers.

But of course you will have checked them on your calculator, won't have worked them out on your calculator, but you will have checked that you've got some practise at using that button.

Okay, you can pause the video now.

Great work.

And question number 2, so this time I don't want you to go back to the beginning.

I want you to use those second examples that we were looking at together.

I would like you to use the fact that 60 equals 2 squared multiplied by 3, multiply by 5 to write the following and again without a calculator, be useful though to practise using the calculator to find the answers that you check your answers before you come back.

Anyway, pause the video now and then come back and I will check your answers with you anyway.

Brilliant, well done.

Okay, let's now check those answers.

1A is 2 cubed multiplied by 5 squared B, 2 squared multiplied by 7 squared C, 2 multiplied by 3 cubed multiplied by 7 D, 2 squared multiplied by 7 multiplied by 11 and E, 2 cubed multiplied by 3 squared multiplied by 5 squared multiplied by 13.

And question 2.

We should have started with 60 multiplied by 2.

We knew what 60 was as a product of its primes.

We were just multiplying that by 2, giving us a final answer of 2 cubed multiplied by 3, multiplied by 5.

B 180 is 100, sorry, is 60 multiplied by 3.

So we've got our product of prime factors for 60 and multiply that by 3, given a final answer of 2 squared multiplied by 3 squared multiplied by 5.

From now on I'm just going to read out the answers, but if you need to, you can pause the video and you can look at how I've worked out those answers.

But from see onwards, I'm just going to give you the very final answer.

C is 2 squared multiplied by 3, multiplied by 5 squared D, 2 cubed multiplied by 3, multiplied by 5 squared E, 2 to the power of 4 multiplied by 3, multiplied by 5 cubed F, 2 cubed multiplied by 3, multiplied by 5 squared multiplied by 7, and G, is 2 multiplied by 3, multiplied by 5.

How did you get on with those? Wonderful.

Well done.

Now we'll move on to the second learning cycle and we're going to look at how we can use those product to prime factors.

They're really useful.

Laura and Jacob are talking about how to calculate 35 multiplied by 22.

Laura says, "Can we use products of prime factors to make calculating this easier?" Jacob says, "We could as they're both products, but does it make it easier?" What do you think? Let's take a look.

We're going to write 35 and 22 as a product of their prime factors.

So 35 is 5 multiplied by 7, and 22 is 2 multiplied by 11.

We are finding the product of 35 and 22.

So that's going to be the product of 5 multiplied by 7 and 2 multiplied by 11.

Laura says, "We can rearrange this calculation using the commutative law," and we can because we're multiplying so we can change the order.

Jacob says, "It's going to be best to pair the 2 and the 5 as a product of those is 10." We've rearranged them using the commutative law and I've put the 5 with the 2.

We then know that 5 multiplied by 2 is 10, 7 multiplied by 11 is 77 and then 10 multiply by 77 is 770.

Laura says, "That definitely made that calculation easier." Jacob says, "I wonder if it will always make the calculation easier." Will writing 105 and 231 as a product of prime factors make this following calculation easier.

So 105 multiplied by 231.

105 we'll write that as a product of its prime factors.

The digit in the ones column is a 5, so we know it's a multiple of 5.

5 multiplied by 21 is 105, then 21 is the product of 3 and 7.

231 is 3 multiplied by 77, and then breaking 77 down into a factor pair is 7 multiplied by 11.

We're then going to multiply those together and Jacob says, "I can't see any pairs that are going to make this calculation easier." Can you see any pairs that are going to make this calculation easier? Hmm, I wonder.

Well, let's take a look at the prime numbers we are most likely to see and decide which pairs will be useful for making a calculation easier.

Can you remember the first 8 prime numbers? How did you get on? Did you come up with the first 8? Well check them with me now they are 2, 3, 5, 7, 11, 13, 17, and 19.

Laura says, "It looks to me like 2 and 5 are the only useful pair." We know the product of 2 and 5 is 10, so if we can use that, that's going to save us time.

Jacob says, "Maybe if there were 2 threes, that could be useful as the product of those is 9." And Laura says, "Oh yes, and an easy way to multiply by 9 is to multiply by 10 and subtract one of the values." So Laura's remembering 1 of those mental arithmetic ways of calculating something multiplied by 9.

We'll take a look at this one.

33 multiplied by 21.

33 written as a product of his prime factors is 3 multiplied by 11 and 21 is 3 multiplied by 7.

So 33 multiplied by 21 is 3 multiplied by 11, multiplied by 3, multiplied by 7.

We're going to rearrange this using the commutative law.

So I've rearranged it, so I've got 3 multiplied by 3, multiplied by 11 multiplied by 7.

3 multiplied by 3 is 9 and 11 multiplied by 7 is 77.

To multiply by 9, we can multiply by 10 and then subtract 1 the values.

So 10 multiplied by 77, subtract 1 multiplied by 77.

10 multiplied by 77 is 770 and we're subtracting 1 multiplied by 77 and then 770 subtract 77 is 693.

We can see here that actually 9 may be an easier way to go.

You may be looking at this and thinking, actually I would've much rather done 33 multiplied by 21, which is also fine.

And we'll now take a look at this one.

We're gonna calculate 42 multiplied by 75 using prime factors.

So this time the question states that we have to use prime factors.

42 is 2 multiplied by 21, which is 2 multiplied by 3 multiplied by 7.

The product of 3 and 7 is 21.

75 is 3 multiplied by 25, which is 3 multiplied by 5, multiplied by 5.

Product of 5 and 5 is 25.

We're then going to multiply all those together.

I'm going to rearrange them using the commutative law, rearranging them so that I'm pairing up useful fact pairs.

So we've got 2 multiplied by 5, multiplied by 3, multiplied by 3, multiplied by 7 multiplied by 5.

2 multiplied by 5 is 10.

3 multiplied by 3 is 9 and 7 multiplied by 5 is 35.

We're now going to calculate 10 multiplied by 9, multiplied by 35, 9 lots of 35.

So remember we can do 10 lots of 35 and then subtract 1 lot of 35.

Those 2 are equivalent.

The bottom 1 is a useful way of calculating, multiplying by 9.

We can then move on and we can calculate what we know.

So 10 multiplied by 35 is 350 1 multiplied by 35 is 35.

We're then going to calculate 10 multiplied by 315 because the result of the bracket is 315 and we know that that is 3,150.

Now you are going to have a go at using prime factors to calculate 36 multiplied by 175.

Good luck, pause the video, come back when you are ready.

Remember, no cheating, no, work it out on the calculator.

I need you to show me that you really understand how to use the prime factors to calculate this.

Good luck, pause the video now and I'll be here waiting when you get back.

Great work.

Now let's check.

36 multiplied by 175 it's 2 squared multiplied by 3 squared multiplied by 5 squared multiplied by 7.

I've then just taken that out of exponent form.

I'm gonna use the commutative law to rearrange.

So I've put my factor pairs that are useful together.

So my 2 and 5, 2 and 5 and 3 and 3.

2 multiply by 5 is 10.

2 multiplied by 5 is 10.

3 multiplied by 3 is 9.

And then I've got my multiplied by 7 on the end, which is a hundred multiplied by 63, which is 6,300.

Did you get that? Superb work.

Now we can have a go at task B.

So you're going to use the prime factors above to calculate each of those products.

Pause the video and when you are ready, come back.

Well done.

Let's check those.

So A 3,780 B, 8,316 C, 6,930 D, 27,720 and E 38,610.

How did you get on? Great work, well done.

Now we'll summarise what we've done during today's lesson.

We know that all composite numbers can be written as a unique product of their prime factors.

So for example, 52, we can break that down to 2 multiplied by 26.

26 is a composite number so we can break that down again, so we end up with 2, multiplied by 2, multiplied by 13.

Product to prime factors can be written more efficiently using index notation.

So for example, we could write 52 as 2 squared multiplied by 13.

The 2 multiplied by 2 becoming the 2 squared.

Commutative law and prime factors can be used to make finding products easier when calculating mentally.

Well done on today's lesson.

You've done really well and I've really enjoyed working through these product prime factor problems with you.

Thanks very much again for joining me and I look forward to seeing you really soon.

Goodbye.