# Lesson video

In progress...

Hello, my name is Mrs. Behan.

And for this lesson, I'm going to be your teacher.

Today we're going to learn to derive facts from other facts that we know.

How many number facts do you think you know? Because I bet today we'll be able to double it if not even triple it.

So without further ado, let's make a start.

So we're going to begin by checking out our lesson agenda.

So we're going to recap on how to derive addition facts.

This is something that you will have looked at previously.

We're then going to look at strategies for deriving more facts, but based around multiplication and division.

There will then be a practise activity followed by an independent task for you to have go at by yourself.

And because I know that you want to know how you've got on, I will make sure I go through the answers with you.

There are just two things you're going to need, something to write with, so a pencil or a pen, and something to write on, so paper would be ideal.

If you haven't got those things to hand, just pause the video now whilst you go and get them.

Try to work in a quiet place where you're not going to be disturbed for the rest of the lesson.

All of the facts hidden under the grey boxes are associated with five plus two equals seven.

What facts do you think are hidden underneath? Pause the video here and write down six possible facts.

It should take you about 30 seconds if that.

When you're ready, come back to me and we'll see if you and I both got the same facts.

How did you get on? Easy peasy? Okay, let's have a look.

The first fact I came up with was two plus five equals seven.

So I made one new fact from five plus two equals seven using the law of commutativity.

I know that if I change the order of the addends, so the five and the two, then the sum will still be the same, I still have seven.

I rearranged my equation and used the inverse operation.

So I now know that seven subtract five is equal to two.

I now know that seven subtract two is equal to five.

Again, rearrange my calculation.

I used this to derive another new fact.

I've also worked out that 70 subtract 20 is equal to 50.

Going back to addition now, and I know that 20 plus 50 is equal to 70.

I used this calculation over here, and I made every part 10 times greater.

I also worked out that 10 plus four is equal to 14.

Do you know what I did to get this new fact? I actually doubled each part, so it's now twice the size.

We have already seen how we can derive facts from an addition calculation.

We saw the remainder connection to subtraction, we use the inverse operation.

In this lesson, we are going to learn to derive facts by starting with a multiplication or a division fact.

Derive means to get more from, just like we got more calculations from five plus two equals seven.

So what do we already know about multiplication and division? I'd like you to just pause the video for a moment whilst you have a think, or maybe you could talk to somebody else in your house to see what they know about multiplication and division.

Don't worry if you can only think of one or two things, because when you come back, I'm going to share some more ideas with you.

Welcome back.

So did you find some ideas? Did the people in your house have some ideas for you? Let's take a look at mine.

Multiplication is commutative.

That means that we can change the order of the factors and the product will still remain the same.

Division is the inverse of multiplication.

It has a very special relationship.

We can rearrange our calculations.

It's basically the opposite operation.

Repeated addition is a way of multiplying.

So I could take five plus five plus five which would equal 15.

I could also use five multiplied by three which gives us the product of 15.

I can divide by grouping or sharing.

So the divisor in an equation will tell me either the group size or how many groups to share my whole between.

I can use place value to multiply or divide.

So we can place extra zeros at the end, or we can remove zeros from the end of a number, depending on whether we're multiplying or dividing.

And doubling and halving, if we double, that means that we are multiplying by which number? That's right, two.

Doubling means multiply by two and halving means divide by two.

So to help us derive new facts from multiplication and division facts, we're going to take four multiplied by three equals 12.

So down the left hand side of the screen, I'm just putting up the ideas that I came up with earlier.

You will also have some more ideas of your own.

So I'll just remind you of them.

Multiplication is commutative.

Division is the inverse of multiplication.

Repeated addition is a way of multiplying.

Division by sharing or grouping.

Use place value to multiply or divide.

And doubling and halving.

Okay, so here is an array that shows four multiplied by three.

So we've got a four, three, oh, cross get off, we've got four, three times, or we can see we've got three, four times, okay, or three groups of four, whichever way you want to look at it because it's commutative.

So we want to make new facts now so we could change the order.

This is using our multiplication is commutative.

Because we know that, we can change the order of the factors, so we have three multiplied by four is equal to 12.

We know that division is the inverse of multiplication.

So if I do a bit of reshuffling of the numbers, I know that 12 divided by four is equal to three.

If I remember this fact that repeated addition is a way of multiplying, I can see that four plus four plus four is also equal to 12.

And division by sharing or grouping, so I've used the numbers in a different order now.

So 12 divided by three is equal to four.

So I could say 12 shared into the group size of three, each group has three in it, there would be four equal groups of three.

So 12 divided by three is equal to four.

So there we are, we've got four facts there for the price of one.

I'd like you to pause here and see how many facts you can come up with if I show you that 10 divided by two is equal to five.

Here is an array which will give you some help.

And on the left hand side of the screen are all the different ways that will help us to derive facts.

When you're ready, come back to me and we'll have a look together at different facts that we can derive.

I'm sure you've done a cracking job there.

Let's go through some different facts.

So using our understanding that multiplication is commutative, I have put that we've got two times five equals 10, and five times two is equal to 10 also.

So I've actually used the inverse of multiplication there as well.

I understand that we can use repeated addition, so instead of five multiplied by two, I've written five add five equals 10.

Okay, division and multiplication have that special relationships.

So even though we started off with a division calculation, we're now using addition and multiplication as well.

I know that 10 divided by five is equal to two, so I've rearranged these numbers here.

Okay, and here the dividing by two is showing me I've got 10 divided into groups of two, which means I have five groups of two.

But in this calculation 10, how many groups of five go into 10? There are two groups of five, one group of five there and one group of five there.

I can use a repeated addition, so two plus two plus two plus two plus two is equal to 10.

So that adding these groups, instead of these groups, which I had over here for five plus five is equal to 10.

Now I've changed the sizes of the numbers, I have made them 10 times greater.

So if each one of these spots was worth 10, I had a value of 10 instead of one, this would be 100 divided by five.

So 100 divided into five equal groups would give me a value of 20, because this would be a group here, this would be a group here, and so on.

Or 100 divided by two would give me two groups, the value of each would be 50.

I can also use this as well making it 10 times greater, 20 multiplied by five equals 100, and 50 multiplied by two equals 100.

So I've actually got there one, two, three, four, five, six, seven, eight, nine, nine facts for the price of one.

I'm sure that most of those facts you would have been able to work out.

We did a lot of work using multiplication is commutative, a lot of work about the inverse, but we didn't really look at doubling and halving.

Now, doubling and halving can create a whole load more facts.

So let's have a look at how we can create more facts using doubling and halving.

Let's play around for the very easy calculation.

Okay, we've got two multiplied by four is equal to eight.

You'll notice that I've colour coded each of the numbers.

So one of the factors is pink, one factor is green, and the product is blue.

And I've done that so that you can watch how each number is changed when we double or halve.

Here is our array showing two equal groups of four or four equal groups of two.

Now then, I have doubled our first factor, I have doubled two, so now we have four multiplied by four.

The product changes to 16.

Can you tell me what the relationship is between eight and 16? That's right, 16 is double eight.

So if we've doubled one of the factors, our product also doubles.

This time I've kept the first factor as two, but I've doubled the second factor.

I've doubled four, so now we have eight.

The product is 16.

And we've just seen that the relationship between eight and 16 is that 16 is double eight.

So if we double the second factor, the product also doubles.

What did I do this time? I've changed our first factor.

That's right, I've halved it.

So instead of two multiplied by four, we're just going to look at one multiplied by four.

And again, the product has changed.

What is the relationship between four and eight? That's right, four is half of eight.

So one multiplied by four is equal to four.

This time we have halved the second factor, so two multiplied by two is equal to four.

We've got loads of facts already and we've not even looked at division.

Here we go, eight divided by four equals two.

Can you remember what we call it when we use the opposite operation? That's right, it's the inverse.

So we've rearranged our equation, we've started with the whole, we've got four in each group, we have two equal groups of four.

What happened here? We've actually used place value to multiply and divide here.

We have multiplied our eight by 10 to make it 10 times greater, we've kept the divisor the same, but our quotient is 10 times greater.

So if we've used placed value here, 80 divided by four is equal to 20.

And I'll just remind you that the first factor we were given was two multiplied by four, we've changed our questions loads here.

So we know that we can halve or double one or both of the values.

You can do that at the same time, or you can do it at different times.

So like we said over here, we doubled the first factor, but not the second.

And on here, we doubled the second factor but not the first.

I wonder what it would look like if we doubled both factors at the same time.

Maybe you could work that one out.

You have worked so hard this lesson.

Well done.

Just think we started with around three calculations, and now that has grown to around 25 or more calculations.

We could have got even more calculations and facts from that.

I'd like you now to pause the video and see how many facts you can derive if I give you the calculation six multiplied by four equals 24.

Don't forget everything that we have learned so far.

This is how I got on.

Have a look and see if you got the same facts as me.

I used commutativity to rearrange the equation.

I love playing around with number facts, it's why maths is one of my favourite subjects.

So I've taken four multiplied by six equals 24 and created all of these different facts.

How many did you come up with? Did you get more than I did? I'm sure that you did.

I wonder how many possibilities there are if we carried on using the inverse or doubling and halving the new facts that were created.

I bet there would be thousands.

Just pause on the screen for the moment and just see if you can understand the relationship between the main fact and the new ones that I derived.

When you ready, come back to me and we'll have a look together.

Okay, let's take a few examples and we'll see how they relate to the main fact and where they came from.

So four multiplied by six is equal to 24.

So I used commutativity over here, and we can see that six multiplied by four is equal to 24.

I doubled the first factor to make 12 multiplied by four which meant that my product doubled and is now 48.

Let's have a look down here.

So I've used the inverse on these two calculations.

So 24 divided by six equals four, that's because I understand the relationship with multiplication.

I've then rearranged these numbers, so 24 divided into four equal groups is equal to six in each group.

I made the 24 10 times greater.

So 240 divided by six is equal to 40.

So I've used my place value understanding there too.

Let's have a look for something different.

Oh, on this one I've doubled the second factor.

Four multiplied by 12 meant that my product doubled as well, which is equal to 48.

What did I use up here? 40 multiplied by six and 60 multiplied by four.

That's right, I used my place value understanding over here, and I used a lot of commutativity as well.

I used place value understanding for this one too, 400 multiplied by six equals 2,400.

Remember that anything multiplied by 100, you can place a zero for the tens and an extra zero for the ones and the digits move over two places to the left.

I'm sure you are ready and raring to go to derive as many number facts as you can for each of these number facts in your independent task.

Don't forget you've got commutativity, using the inverse operation, place value and doubling halving which will get you tonnes more facts.

Once you've had a go at that, there's a second activity for you to do here.

So in each of these tasks, you've been given a number fact.

And you're being asked which calculation could you solve by deriving a new fact? So on your paper, you should copy out the sentence at the bottom filling in the blank.

So I can solve blank by using blank.

Now after that "by using" phrase, that's where you will say I used commutativity, or my understanding of place value, or by using doubling and halving to derive new facts.

Good luck and have fun and I will see you once you've had a go at your task.

Let's see how many facts you managed to come up with.

Here are a few that I came up with.

40 multiplied by nine equals 360.

90 multiplied by four equals 360.

I actually doubled the nine and halved the four to get the next fact.

18 multiplied by two is equal to 36.

18 multiplied by four is equal to 72.

I've put plus division on the button there, because all of those facts are just including multiplication, there will have been tonnes more to do with division.

Let's take 27 divided by three equals nine.

So for this one, I used some division facts.

So I rearranged the nine and the three.

So 27 divided into nine equal groups which mean that I have three in each equal group.

I've used my understanding of place value, 270 divided by nine is equal to 30.

And now I've rearranged the equation, 270 divided by three is equal to 90.

I've used the inverse operation for nine multiplied by three, and then I doubled the second factor.

Nine multiplied by six is equal to 54.

My product double too.

I doubled the first factor here, 18 multiplied by three is equal to 54.

Let's take eight multiplied by three is equal to 24.

I started off playing around with place value.

So I made number eight 100 times greater, 800 multiplied by three equals 2,400.

16 multiplied by three equals 48.

What did I do there? That's right, I doubled the first factor.

And the effect it had on the product was that that was doubled too.

Eight multiplied by six is equal to 48 I doubled the second factor.

And now I've used the inverse operation.

24 divided by three equals 8, 48 divided by three equals 16, and 48 divided by eight is equal to six.

For our number fact of three multiplied by seven equals 21, the calculation we could solve by deriving a new fact was 210 divided by seven.

You can see there's a clear relationship there in the numbers.

So I can solve 210 divided by seven by using place value, because 21 has been made 10 times greater, and the inverse operation.

Instead of multiplication, we're now using division.

And in our second example, we can also derive a new fact of 28 divided by two.

So I can solve 28 divided by two by using halving and doubling.