Lesson video

Hi.

My name's Mr. Peters.

Welcome to today's lesson.

In this lesson today, we're going to be thinking about how we can explain what a factor is and how we can find these factors using either arrays or multiplication and division facts.

If you're ready, let's get started.

So, by the end of this lesson today, you should be able to explain what a factor is as well as how to find factors either using arrays or multiplication and division facts.

Throughout this lesson today, we've got a number of keywords we're gonna be referring to.

I'll have a go at saying them first, and then you can repeat them after me.

The first one is row.

Your turn.

The second one is columns.

Your turn.

The third one is array.

Your turn.

The next one is factor.

Your turn.

After that, we've got product.

Your turn.

And finally, positive integer.

Your turn.

Let's have a think about what these mean.

So, objects lying in a horizontal line, side by side, is known as a row.

A column is an arrangement of objects that are one above the other in a vertical line.

An array is when items are arranged in rows and/or columns.

Factors are whole numbers that divide exactly another whole number.

The product is the result of two numbers or factors being multiplied together.

And finally, a positive integer is a whole number that is greater than zero.

Look out for these words throughout our lesson today and see if you can use them in any of your reasoning.

So our lesson today is broken down into two cycles.

The first cycle, we'll be thinking about finding factors in arrays, and in the second cycle, we'll be looking at linking these factors to multiplication facts.

If you're ready, let's get started.

In this lesson today, we're also going to have Izzy and Sam joining us.

They'll be keen to share their thinking as well as any of their questions throughout the lesson as we go.

Okay, so we're gonna start our lesson here today.

You can see on the screen that we've got 12 tiles, and I'm wondering, how many different rectangular shapes could we make using these square tiles? If you have some square tiles to hand, you might like to use these.

However, if you don't, you could just draw square tiles on a piece of paper to show them.

Take a moment to have a think for yourself.

You might like to pause the video here and have a little go and see what you can come up with.

Okay, let's have a look at what Sam and Izzy came up with, shall we? Sam says, "I made one like this." And Izzy's saying, "I found a different way of doing this." Let's have a look at these rectangles in a little bit more detail.

So we can use these rectangles to represent numbers, and when numbers are represented as a rectangular arrangement, we call these these an array.

We can describe these arrays by the number of rows and the number of columns that these arrangements have.

Let's have a look at Sam's example.

There are 12 tiles altogether in Sam's example.

Sam's rectangle has 2 rows and 6 columns.

Let's have a think about Izzy's then.

Izzy also has 12 tiles in her array.

She has 3 rows, and she has 4 columns.

We can also represent rows and columns like this.

So we can see the 2 represents the number of rows and the 6 represents the number of columns.

Let's have a look at Izzy's.

The 3 represents the number of rows, and the 4 represents the number of columns.

An array can also represent the multiplication equation.

So we can say that there are 12 tiles altogether.

There are 2 rows, there are 6 columns, so we can represent this as 12 is equal to 2 multiplied by 6.

Let's have a look at Izzy's, for example.

You might like to say the stem sentences with me.

There are 12 tiles altogether.

There are 3 rows, and there are 4 columns.

We can represent this as 12 is equal to 3 multiplied by 4.

Well done if you managed to say that with me.

Of course, with our understanding of how we can order the multiplication equations, we could also represent 12 is equal to 2 multiplied by 6 as 2 multiplied by 6 is equal to 12, where we swap the position of the factors and the product in the equation.

Similarly, we can do the same for Izzy's equation.

12 is equal to 3 multiplied by 4.

This is also the same as 3 multiplied by 4 is equal to 12.

So, when thinking about what the numbers in these equations represent, we can say that the 12 represents the product.

We can then say that the 2 and the 6 are the factors in our equation.

Take a quick look at Izzy's.

Which number represents the product, and which numbers are the factors? That's right, the 12 would also be the product again, and this time, 3 and 4 would be the factors in this equation.

Izzy's carried on playing around with the tiles and found another way of doing it.

Let's have a look at this example.

There are 12 tiles altogether.

There is 1 row, and there are 12 columns.

We can represent this as 12 is equal to 1 multiplied by 12.

Or we could represent this as 1 multiplied by 12 is equal to 12.

Take a moment again.

Which number is the product here, and which numbers are the factors? That's right, the product is 12, and the factors are both 1 and 12 themselves.

So we can say that 1 and 12 are factors of 12.

Okay, take a moment to check your understanding.

Have a look at the array here.

Can you use the stem sentence below and fill in the missing gaps? Good luck.

Okay, let's have a look.

Well, we know that there are 14 tiles altogether.

There are 2 rows, and there are 7 columns.

And we know we can represent this as 14 is equal to 2 multiplied by 7, or we can represent this as 2 multiplied by 7 is equal to 14.

So, 2 and 7 are factors of 14.

Well done if you got that.

Andeep's been thinking about how he could represent his 12 tiles.

Have a look at what he's done.

What do you think? Sam's saying, "I'm not quite sure that works as an array." Sam's saying that each column must have the same number of objects in it as the other columns.

Or we could say that each row must also have the same number of objects in it as the other rows.

Andeep's realised his mistake now.

I wonder what he could have done to his array to make it work.

Let's have a look at this example here.

Describe a way to make this array correct.

Well, there we go.

We can see that the number of columns don't have equal amounts in them, do they? And the number of rows don't have equal amounts in them.

So what could we do to make that happen? Well, Sam's suggesting that we could move two tiles from the top row and place them into the third column instead.

So we can now see that we've got three rows that have all the equal amount of objects in it, and we've also got three columns with the same equal amount of objects in them.

Okay, Sam's up for a bit of an investigation now.

He wants to investigate all of the numbers up to 20 to find out the factors that each of these numbers have.

Let's start with number 1.

Why don't you have a go at saying the stem sentences with me? We can see that there is 1 tile altogether.

There is 1 row, and there is 1 column.

So we can represent this as 1 is equal to 1 times 1 or 1 times 1 is equal to 1.

So we can say that 1 is a factor of 1.

Let's have a look at number 2.

What do you think we're going to be saying? Well, we can say that there are 2 tiles altogether this time.

There is 1 row, and there are 2 columns.

So we can represent this as 2 is equal to 1 multiplied by 2, or we could represent this as 1 multiplied by 2 is equal to 2.

For this example, we can say that both 1 and 2 are factors of 2.

Let's have a look at number 3 now.

So we know that there are 3 tiles altogether here.

There is 1 row, and there are 3 columns.

We can represent this as 3 is equal to 1 multiplied by 3 or 1 multiplied by 3 is equal to 3.

And we can say that both 1 and 3 are factors of 3.

You might be beginning to spot a bit of a pattern so far.

Let's have a look at number 4 then.

We know that there's 4 tiles altogether.

There's 1 row, and there's 4 columns.

We can represent this as an equation.

4 is equal to 1 multiplied by 4 or 1 multiplied by 4 is equal to 4.

And of course, we can say that both 1 and 4 are factors of 4.

Hmm, I think there might be another way we can represent number 4, though, and so does Izzy.

Let's have a look.

That's right, we could represent 4 like this, couldn't we? We could say that there are 4 tiles altogether again.

We've got 2 rows this time, and we've got 2 columns this time.

So we could say that 4 is equal to 2 multiplied by 2 or 2 multiplied by 2 is equal to 4.

And therefore, we could say that 2 is a factor of 4 as well.

Well, so have a look at all these examples we found so far.

What do you notice? Take a moment to have a think.

You may have noticed, just like Izzy has, that each number has itself as a factor as well as the number 1 as a factor.

And now she's beginning to wonder, is this true for every number? Okay, time for you to have a go at practising for yourself.

What I'd like you to do is carry on this investigation from number 5 up to number 20, using some square tiles or drawing them if you need to.

What I'd like you to do is jot down all the different arrangements that you can find for each number, and therefore jot down the factors that you were able to find.

Good luck with that, and I'll see you again shortly.

Okay, welcome back.

Let's see how you got on compared to how we did.

So we found that we could only make 1 arrangement for 5, so therefore the factors would be 1 and 5.

We found 2 arrangements for 6, where the factors would be 1 and 6, and 2 and 3.

For 7, we found 1 arrangement, which would be 1 and 7 as factors.

For 8, we found 2 arrangements.

We found 1 and 8, and 2 and 4.

For 9, we found 2 arrangements.

We found 1 and 9 as factors, and we also found 3 to be a factor.

For 10, we found 1 and 10, and 2 and 5, so another 2 arrangements.

For 11, we only found 1, which was 1 and 11.

For 12, we found 3 arrangements.

We found 1 and 12 could be factors, we found that 2 and 6 could be factors, and we found that 3 and 4 could be factors.

For 13, we found 1 arrangement, which obviously was 1 and 13 for its factors.

For 14, we again found 2 arrangements.

1 and 14 could be factors, or 2 and 7 could be factors.

For 15, we found 2 arrangements.

We found 1 and 15 could be factors or 3 and 5 could be factors.

For 16, we found 3 arrangements again.

We could have 1 and 16 as factors, we could have 2 and 8 as factors, or we could have 4 as a factor.

For 17, there was 1 arrangement, 1 and 17.

For 18, we found another 3 arrangements, 1 and 18, 2 and 9, and 3 and 6.

And then for 19, there was just one arrangement again, 1 and 19.

And then for 20, we actually managed to find another 3 arrangements, 1 and 20, 2 and 10, and 4 and 5.

Well done if you managed to find all of those different arrangements and those different factors for each of those numbers.

When you were doing that, did you notice anything in particular? Well, Izzy's saying that she noticed that it's mainly the even numbers that have the most number of factors, isn't it? Sam also noticed that every single positive integer here has 1 and itself as a factor.

And that's right, Izzy, number 1 is always the smallest possible factor that a positive integer could have, and itself is always the largest factor that a positive integer could have.

Sam also noticed that we were able to make a couple of squares, weren't we? He realised that 9 had a square arrangement and 16 had a square arrangement.

That's interesting, isn't it? That's the end of our first cycle.

Let's have a look at cycle two now, linking factors with multiplication facts.

So, here we've got a number, number 35.

What would the factors of 35 be? How could we work these out, I wonder? Take a moment to have a think for yourself.

Andeep's suggesting we can draw an array, right? We've just been looking at those arrays, and that helped us to find factors, didn't it? Izzy's saying, hmm, we could use our times table facts.

Drawing arrays might take a bit too long.

Let's have a look at multiplication grid and see if we can use this to help us.

Let's find 35 on our multiplication grid.

We can see that 35 is in the 5 times table, so 5 is a factor of 35.

We can also see that 35 is in the 7 times table, so 7 would be a factor of 35.

And here, you can see the rectangular arrangement that we've created with the tiles next to it.

So we could say that 5 and 7 are factors of 35.

Let's look at another example.

Let's find 27.

27 is in the 3 times table, so we can say that 3 is a factor of 27.

And 27 is in the 9 times table, so we can say that 9 is a factor of 27 as well.

And again, you can see our rectangular array down the bottom demonstrating the rows and the columns.

We could have 3 rows and 9 columns, so we can say that 3 and 9 are factors of 27.

So let's have a look at this in a little bit more detail now.

We can see our 3 rows and our 9 columns, and we could represent this as an equation.

3 multiplied by 9 is equal to 27.

We could also remove the squares in the middle and say that that whole area now represents 27.

So, linking our language to our multiplication equation, we can say that a multiplication equation has the arrangement of a factor multiplied by a factor is equal to a product.

Or we could say that the product is equal to the factor multiplied by the factor.

So we could say that 3 is a factor, 9 is a factor, and therefore 27 is the product in this example here.

So 3 is a factor of 27, 9 is a factor of 27, and 27 is the product of 3 and 9.

Now let's link this to a division equation.

Where would these factors be represented in our division equation? Have a look.

Can you see where they're represented in the equation? Now, we don't use the term factors in a division equation.

The language that we use is here.

We can say that the dividend divided by the divisor is equal to the quotient.

So let's see what these numbers represent in this equation.

Well, the 27 represents the dividend.

However, the 27 in our multiplication equation was the product, wasn't it? And then the 3 could represent our divisor here.

Again, that was one of our factors in our multiplication equation.

And then the 9 could represent the quotient here.

So again, the 9 was one of our factors in a multiplication equation.

However, this is now representing the quotient in our division equation.

So we could say that the factors in the multiplication equation represent either the divisor or the quotient in a division equation.

Okay, time for you to check your understanding now.

True or false? 2 is a factor of 28.

That's right.

It's true, isn't it? And which of these justifications helps to reason that for you? That's right.

It would be B, wouldn't it? 28 divided by 2 is equal to 14, so 2 and 14 are factors of 28.

And we know that because in the division equation, the 28 is representing the dividend, the 2 is representing the divisor, and the quotient is represented by the 14.

And therefore, we know that the divisor and the quotient represent the factors, so we can say that 2 and 14 are the factors of 28.

Okay, and another quick check.

Can you use the multiplication grid to find the factors of 21? Take a moment to have a think again.

Okay, well, let's find 21 on our grid.

We can see that 21 is in the 3 times table and in the 7 times table, so 3 and 7 are factors of 21 and 21 is the product of 3 and 7.

You may have noticed that 21 was placed twice on the grid.

And again, you can see it's aligned in both the 7 and the 3 times table.

So we don't need to repeat those factors each time.

We can just say that 3 and 7 are factors of 21.

Time for you to practise now.

What I'd like you to do is to complete each of the equations and find the factors.

I'd like you to work from left to right in each row and then go down to the second row and then down to the third row.

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

Okay, let's see how you got on.

So we can say that 3 multiplied by 4 is equal to 12, so 3 and 4 are factors of 12.

If we know our multiplication facts of 3 multiplied by 4 is equal to 12, then we can say that 12 divided by 4 would be equal to 3.

So again, still, using that division equation, we know that the divisor and the quotient represent the factors, so we can say that 3 and 4 were factors of 12.

Next row.

Something multiplied by 6 is equal to 24.

So using our times table facts, we know that 4 multiplied by 6 is 24, so 4 and 6 are factors of 24.

And then looking at our division equation, we know that 24 divided by something is equal to 6.

Well, that must be 4.

So again, divisor here is 4 and the quotient is 6.

We know they represent the factors in a multiplication equation, so we can say that 4 and 6 are factors of 24.

And then the last example.

7 multiplied by something is equal to 42.

Well, 7 6s are 42.

Therefore, 7 and 6 are factors of 42.

And then again with a division equation, 42 divided by 7 is equal to something.

Well, using our multiplication fact, we know that 42 divided by 7 would be 6.

Therefore, the divisor this time is 7, and the quotient this time is 6.

We know they represent the factors, so 7 and 6 are factors of 42.

Well done if you managed to get all of those.

Okay, that's the end of our learning for today.

Thanks for joining me, and hopefully you're feeling a lot more confident about understanding what a factor is and how you can find these either through arrays or multiplication facts.

So, to summarise what we've learned today, when two factors multiply together, they form a product.

Factors can be represented in arrays by the number of rows and columns.

Every number has 1 and itself as a factor.

And in a division equation, the factors are represented by the divisor and the quotient.

Thanks for joining me today.

I've really enjoyed that lesson, and hopefully you have too.

Take care, and I'll see you again for the next one.