Lesson video

Hiya, my name is Ms. Lambell.

Really pleased that you've decided to pop along and do some maths with me today.

Welcome to today's lesson in the properties is number unit.

The type of today's lesson is listing factors.

By the end of this lesson, we will be able to state what factor is, and we will be able to identify factors of positive integers.

So here, we've got a list of keywords that we'll be referring to throughout this lesson.

These should be familiar to you.

So today's lesson is gonna be split into three learning cycles, systematic approach to finding factor pairs, then we'll move on to common factors, and then simplify fractions using common factors.

So let's start with the first one, a systematic approach to find factor pairs.

"Lucas has 12 counters.

He decides to make as many arrays as he can.

How many unique arrays can he make?" What I'd like you to do is to pause the video and have a think about this, and when you're ready, come back.

Brilliant.

Okay.

So let's have a look at what Lucas has done.

So he's got an array, which is one by 12.

He's got an array which is two by six, an array which is three by four, an array which is four by three.

Has Lucas made four unique arrays? Have you made more than Lucas or less than Lucas? Actually, he's not made four unique arrays because if we look at the last two, they're actually the same array.

Just one of them has been rotated.

So remember, unique means totally different to anything else, and actually the last two are the same.

They're just shown in a slightly different way.

So by drawing these arrays, Lucas has found the factor pairs.

So he's found the factor pairs of 12.

All right, what I want you to do now is to imagine that we wanted to find the factor pairs of a thousand.

Would it be sensible to draw the arrays to find them? I don't know about you, but I don't think it would.

I think it would take a long time.

But what we can do is we can visualise those arrays, and systematically list them.

So remember, this learning cycle is about using a system to list factors.

So let's have a go at that with 54.

So we need to remember that a factor multiplied by a factor gives us a product.

So we are going to find factors that have a product of 54.

So we start with one.

So what do we pair with one so that the product is 54? That's a nice easy one.

That's 54.

Okay, so now we need to think about two.

Is two a factor of 54? So is 54 divisible by two? Well, 54 is even.

So yes, it is divisible by two.

And then we're gonna do 54 divided by two.

And I would do that by having to give us the number that is the factor pair.

So two and 27.

So now we need to think about three.

Is 54 divisible by three? So with this, what I would do is I would say, well, I know that 10 multiplied by three is 30.

So if my 54, there's 24 left, is 24 divisible by three, yes it is.

So therefore three will be a factor and its pair will be 18.

Now we need to think about four.

Is four a factor of 54? So is 54 divisible by four? It's not.

So we'll move on to five.

Is five a factor of 54? It doesn't end in zero or five, so therefore it can't be divisible by five.

Let's move on to six.

Is 54 divisible by six? Yes it is.

What do we multiply six by to get 54? That's nine.

So we have found now the factor pairs of 54.

These are all of the pairs.

And notice we found them in pairs and we used a system to find them.

We started with one and then we worked through, Let's give that a go with another number.

So we're gonna use the number 80, so we're gonna list all of the factor pairs of 80, and remember, we're going to use a system to do that.

So we'll start off.

Nice easy pair with 1 and 80.

Moving on, two.

Is 2 a factor of 80? Yes, because 80 is even.

So 2 multiplied by 40 gives us the next factor pair.

Is 80 divisible by 3? No, but 80 is divisible by 4.

That's going to be our next number.

What do we pair with that to make 80? That's 20.

Okay, it ends in a zero.

So therefore, 5 must be a factor of it.

What do we pair with that? We pair 16.

80 is not divisible by 6.

80 is not divisible by 7.

80 is divisible by 8, and our pair of that one is going to be 10.

We're then going to have a look at 9.

Is 9 a factor of 80? No, it is not a factor of 80 because of some of the digits is not 9, so remember the visibility tests, if we sum the digits and it get 9, we know it's a factor of 9, and then we wouldn't need to carry on because we get the next number is 10 and we've already got 10 in our list.

And it would just mean that we would be repeating those pairs, but just in the opposite order.

Okay, what I'd like you to do now is to pause the video, and have a go at listing all of the factor pairs of 200, and remember to use that systematic approach.

When you're done, unpause the video and come back.

Good luck.

Great work.

Let's have a look and see how you got on.

I'm sure you did really well.

So these are our factor pairs.

We've got 1 and 200, 2 and 100, 4 and 50, 5 and 40, 8 and 25, and 10 and 20.

Well done if you've got all of those right.

Maybe give them a tick.

If you missed any out, maybe just have a little think about why you might have missed out that pair.

So just another quick check here.

So we've got Laura, she's finding factor pairs of 60.

So again, what I'd like you to do is to pause the video and have a think about this question.

Has Laura managed to find all of the factor pairs? When you are done, come back.

Hopefully you said no.

She's actually missed out one of the factor pairs, which is 1 and 60.

This is a common one to miss out, the number 1 and the number itself that you are finding factors of.

What could Laura have done to make it easier to ensure that she had listed all of the factor pairs? Okay, hopefully you said that she should have used the systematic approach, she should have listed them systematically.

We can then be pretty certain we've got them all rather than just doing them randomly.

Okay, so first learning cycle done.

What we're gonna do now is we're gonna have a go at some questions.

So you're gonna pause the video, and you're gonna have a go at listing the factor pairs of these numbers.

Remember to use that systematic approach so that you can be pretty certain that you've got all of them.

Okay, good luck with that.

And then when you're ready, come back.

Super, right, well done.

So next, we're now gonna have a look at the second question.

So again, after I've read the question, pause the video and then come back when you're done.

So Sam makes two rectangles with an area of 10 centimetre squared.

What I'd like you to do is to explain how these rectangles show all of the factors of 10.

So how are these showing all of the factors of 10? So pause the video and then come back when you're ready.

Brilliant, great work.

Let's check those answers now.

So hopefully, you've got all of these.

You've used the systematic approach.

So, A, factors of 18 are 1 and 18, 2 and 9, 3 and 6.

B, factors of 28 are 1 and 28, 2 and 14, 4 and 7.

Factors of 36, 1 and 36, 2 and 18, 3 and 12, 4 and 9, and 6 and 6.

Obviously there, if you've just written 6, that's absolutely fine.

Factors of 32 are 1 and 32, 2 and 16, 4 and 8.

Factors of a 100.

1 and 100, 2 and 50, 4 and 25, 5 and 20.

And then the factors of 67 are just 1 and 67.

So give those a mark, I'm sure you've got those right and well done.

Now we can have a look at the answer to question two.

So in this question, we were looking at why these rectangles show all of the factors of 10.

And the reason for that is because you cannot make a rectangle with a height of 3, 4, 6, 7, 8, or 9.

So 1 and 10 and 2 and 5 are the only factor pairs.

Great work everyone.

So we're gonna move on now to common factors.

So second learning cycle, we're gonna be looking at common factors.

So here we've got Jun and Izzy.

So Jun says, "I found all of the factor pairs of 20." And Izzy, "I found all of the factor pairs of 30." Can you think of any factors that both Jun and Izzy have written? So Jun has written down the factors of 20.

So 1 and 20, 2 and 10, 4 and 5.

And Izzy has written the factors of 30.

So 1 and 30, 2 and 15, 3 and 10, and 5 and 6.

So we were looking and we were thinking about any factors that they've both written, so any factors that are common to both 20 and 30.

So you can see that one is common to both.

2 is common to both, 10 is common to both, and 5 is common to both.

So they say we both have 1, 2, 10, and 5.

So that means that 1, 2, 5 and 10 are the common factors of 20 and 30.

Maybe you came up with all of those yourself.

Well done if you did.

The check for understanding.

So this is true or false.

"There are no numbers that are common factors of all integers." Pause the video, have a think, and come back when you're ready.

So do we think that is true or false? There are no numbers that are a common factor of all integers.

Okay, so hopefully you came up with false.

It's false, but what I'd like you to do now is to actually justify that answer.

So here are two justifications.

So A, you cannot make an array of a height 2 for odd numbers.

You can always make an array of height or width of 1.

So which of those do you think justifies the fact that that statement is false? Okay, it's actually the bottom one, it's B, you can always make an array of height or width of 1.

We can also put factors into a Venn diagram and that sometimes helps us to identify common factors.

So we're going to look at the factors of 36 and 54.

Here's our Venn diagram.

I've listed them for you here so that you don't need to actually work them out yourself.

And we're going to put these into the Venn diagram.

Let's start with 1.

We can see that 1, and we've just done that in our check for understanding.

that 1 is actually a common factor of every number.

So that is going to go in the intersection of the two sets, so the middle part, because it is common to both.

But then we've got 2 and 3, they're also common to both.

We're then looking at 4.

So 4 is in the list for 36, but it's not in the list for 54, so it goes just in factors of 36, 6 and 9 are common to both, and then we put the rest of the factors in.

There's actually one factor, I've missed one factor from that Venn diagram.

So what I'd like you to do now is to pause the video and have a think if you can spot which one is missing and where it should go.

Great work.

Let's have a look.

Let's see whether you've got the same answer as me.

I'm sure you did.

We go through the list of all of those, we can see that 18 is missing.

18 is not in our Venn diagram, but we can see that it's in both our lists so it's a factor of 36 and 54, so it should therefore go in the intersection.

So remember the intersection is the middle part of the diagrams, the bit that is common to both.

So there's 18 in its correct place.

All right, we want to find all the common factors of those numbers.

So here are our factor pairs.

Are those all of the factors of each of those integers? What did you think? No, it's not.

For large integers, it's really difficult to list them all.

I'd like you now to write down a pair which has been left out for each.

So have a go at finding at least one pair that has been missed out for each of those numbers.

So pause the video and then pop back when you're ready.

Okay, so for example, remember these are just examples so you might have something different.

So 8 multiply by 31 for 248, 30 multiplied by 10 for 300, okay? But you may have something different.

You can just actually check your answer on the calculator, multiply your two numbers together and check if you get the correct product.

"Which of the following is not a common factor of 80 and 56?" Pause the video and have a go at this question.

Okay, so the question was focusing on not a common factor.

7 is not in the intersection, so that was our correct answer.

So 7, it's a factor of 56, but it's not a factor of 80, which is why it's just in the right hand side of our Venn diagram.

Okay, now it's your turn to have a go.

So use the Venn diagram to fill in the gaps.

You need to fill numbers in the gaps.

So pause the video now and then come back when you're ready.

Great, super.

Well done.

Now as a second question, find the common factors for each of these following pairs of numbers.

So you're gonna list the factor pairs, remembering to use that systematic approach that we talked about in the first learning cycle, and then I'd like you to identify the common factors.

So pause the video and then come back when you're ready to check your answers.

Super work.

Well done.

Okay, so question number one then, the missing number was 2, B was 6, C was 4, and D was 1, 3, 5, and 15.

And remember if you've got those in a different order, that doesn't matter.

And then question two, I'm not gonna read all of those out.

So if you need to pause the video to give yourself time to mark those, you can.

But all of the ones in the green boxes are all of the common factors.

So pause the video and check your answers.

We're gonna move on now to our final learning cycle, and we're gonna be looking at simplifying fractions.

So Aisha and Sam are simplifying fractions.

They both simplify 8/20, to its lowest terms using factor pairs.

They've noticed that all of this work that we've been doing on factor pairs we can use to help us to simplify fractions.

Now we're gonna take a look at what they've done.

So Aisha decides to use these factor pairs.

So she uses 4 and 2 for 8, and 4 and 5 for 20.

She then notices that actually we could just write that as 2/5.

Now let's take a look at what Sam did.

So Sam, again, started with 8/20, but decided that actually to use the factor pair of 2 and 4 for 8, and 2 and 10 for 20.

You've noticed that this is just 4/10, but this can be further simplified using the factor pairs of 2 and 2 for 4, and 2 and 5 for 10, given the same answer, 2/5, Whose method is more efficient? I'm sure you said Aisha's was more efficient.

You know as mathematicians, we like to do things in the most efficient way.

So Aisha's is more efficient, but why? What makes their method more efficient? Okay, the reason Aisha's method is more efficient is because she's done it in one step.

They've used the highest common factor of 8 and 20.

Let's practise that one more time.

So on the left hand side we're gonna do it together, and then on the right hand side, you're gonna have an opportunity to try one for yourself.

So we're gonna simplify 12/18, and we're going to use the highest common factor.

So I've listed here all of the factors of 12, and all of the factors of 18.

So which is the highest? The number that appears in both lists, that's 6.

So we're going to write 12 as 6 multiplied by 2, and we're gonna write 18 as 6 multiplied by 3.

And then we can see that this then simplifies to 2/3.

Now I'd like you to have a go at simplifying 32 over 36.

So pause the video, good luck, come back when you're ready.

Super.

Well done.

So let's have a look.

I'm sure you've got that right.

Let's just double check.

So here, our highest common factor was 4.

So we wrote 32 using the factor pair 4 and 8, and 36 using the factor pair 4 and 9, meaning 32 over 36 can simplify to 8/9.

Have a go at this one now.

So pause the video and just have a go at simplifying this one, and then pop back when you're ready.

Great work.

So the correct answer was B, 3/4, that actually all equivalent to 36/48 but actually 3/4 is in its simplest form.

It's in its lowest form.

Okay, back over to you now again.

So in this task of question one, you're gonna simplify fully the following fractions.

And then in question two, you're going to say which of those are in their simplest form.

And any that are not in their simplest form, you're gonna have a go at simplifying.

So pause the video now and then when you're ready and you're done, come back and check in your answers.

Wow, that was quick.

Well done.

So let's have a check of those answers then.

So A was 1/9, and I've shown you there how I've broken that down.

B is 4/5, C, 5/6, D, 5/9, and E, 2/3.

Well done if you've got those right.

Remember, you may have not done it in one step, that doesn't matter.

But for efficiency, if you've chosen the highest common factor, then you would've done it in exactly the same steps as I did.

And then in number two, the ones that were not fully simplified were 14 over 35.

And when we simplify that, that becomes 2/5, and 12 over 33, and that one when we simplify, it is 4/11.

So let's just recap what we did in this lesson.

So we started off and we looked at listing factors in a systematic way.

So remember the reason for that was we want to use a system so that we can be pretty certain that we've got all of them.

So there's an example there of how we list them in a systematic way.

We then moved on and we looked at common factors of integer.

So we listed them using the systematic way, and then we identified any common ones, any that were both in both lists.

And then finally, we used our factor pairs, and in particular we tried to use the highest common factor to simplify fractions.

Thank you so much for joining me today and working on listing factors.