Lesson video

Hi everyone! My name is Miss Coe and I'm really happy and excited to be learning with you today! It's going to be a fun lesson, full of some words maybe that you may or may not know, and we'll build on that previous knowledge too.

Super excited to be learning with you so let's make a start! Hi everyone.

Under the unit understanding multiplicative relationships, fractions and ratio, we'll be looking at multiplicative relationships.

By the end of the lesson you'll be able to calculate the multiplier for any two given numbers.

We'll be looking at that keyword reciprocal.

Now remember, a reciprocal is that multiplicative inverse of any non-zero number and any non-zero number multiplied by its reciprocal is always equal to 1.

For example, the reciprocal of 3 is 1/3, or you could say the reciprocal of 1/3 is 3, and this is because 3 multiplied by 1/3 is equal to 1.

Another example is 5/4.

Well, the reciprocal of 5/4 is 4/5, and this is because if you multiply 5/4 by 4/5, it will equal 1.

We'll look at this word reciprocal in our lesson again.

Today's lesson will be broken into two parts.

First, we'll be looking at connecting numbers multiplicatively, and then we'll be looking at efficiently calculating a single multiplier.

So let's make a start.

I want you to have a look at this diagram.

Any two numbers can be connected by an additive and a multiplicative relationship.

So looking at these two numbers, can you fill in that missing number, given the operation and find the relationship between 10 and 30? What is that multiplicative relationship? What's the additive relationship? See if you can give it a go and press pause if you need more time.

Let's see how you got on.

Well, if we multiply 10 by 3, it gives 30, so our multiplier is 3.

If we add 20 to our 10, it gives 30.

So that means there is an additive relationship there, we're adding 20.

Any two numbers can always be connected by an additive or a multiplicative relationship.

Great work if you got this one right.

What I want you to do is have a look at this check question.

Three pupils were given this, 6 multiplied by something is 2.

Now, Laura says there is no multiplicative relationship as they use division.

Is Laura correct? And I want you to explain.

Well, hopefully you can spot, she's incorrect.

There is a multiplicative relationship.

We're dividing 6 by 3 to give 2.

Now when you divide by 3, it's exactly the same as multiplying by 1/3.

So there is a multiplicative relationship.

It's important to remember when dividing by a number, it's the same as multiplying by its reciprocal.

Great work if you've got this one right.

Now let's have a look at a check question.

What I want you to do is work out the missing numbers and explain how you found them.

A connects the number 3 to 9, additively and multiplicatively.

B connects the numbers 12 to 24.

C connects the numbers 4 to 28 and d connects the numbers 6 to 3.

See if you can find that relationship and fill in the missing numbers and don't forget to explain why.

Well done.

Let's see how you got on.

Well, for a, let's find out what that multiplier is.

9 divided by 3 is 3, so that means our multiplier is 3 and to find out what that additive relationship is, we simply do 9 subtract 3, which gives us 6.

So you can see our relationship multiplicatively and we can see our additive relationship too.

Let's have a look at b.

The multiplier can be found out by 24 divided by 12, which is 2.

The additive can be found out by 24 subtract 12, which is 12.

For c that multiplier can be found by 28 divided by 4, which is 7, and the additive can be found by 28 subtract 4, which is 24.

For d, the multiplier can be found by 3 divided by 6, which is 1/2, and the additive can be found by 3 subtract 6, which is at -3.

Great work if you got this one right.

Now I need to do is only using addition and subtraction, how can you connect these two numbers? Have a little think and press pause if you need more time.

Well, you could add 2 as a single step.

What about in two steps? What do you think we could do here? Well, you could subtract 3 to get the number 3 and then add 5 to get the number 8.

But there are an infinite number of steps here and that's important for you to remember.

There are an infinite number of ways in which you can connect any two numbers.

You could have chose 6 subtract 7, which is -1, and then add 9, which is 8.

You could have chose 6 add 100 and then subtract 98.

All of these connect the numbers 6 to 8.

These will all be equivalent to a single additive step.

In other words, you just simply add 2 because if you sum -7 with +9, it gives you 2.

If you sum that 100 with -98, it gives +2.

So you can efficiently work out that single additive step.

Now only using multiplication and division, how can you connect these two numbers? And I also want you to think how could you connect them in two steps? Well, let's have a look at the two step first.

Well, you could do 6 divided by 3, which is 2, then multiply by 4, which gives us our 8.

And just like before, there is an infinite number of ways in which you can do this for any two numbers.

You could do 6 multiplied by 4, which will be 24 and then divide by 3, which is 8.

You could do 6 divide by 6, which is 1, and then multiply by 8 to give you 8.

But these will all give you an equivalent to a single multiplicative step.

So let's say we can find out what that multiplicative step is.

We can write the divisions as a multiplication.

So if we change that, the division of 3 or the division of 6 into a multiplication of the reciprocal.

This means we'll have the multiplication of 1/3 and the multiplication of 1/6.

Then let's see if we can combine these to make a single multiplier.

4 multiplied by 1/3 is 4/3, or you could do 1/6 multiply by 8, which is also 4/3.

So the single multiplicative step to connect 6 to 8 is 4/3, and using two or more multiplicative steps can be easier mentally as well.

So let's have a look at a check.

I want you to complete each of these in two steps only using multiplication and division.

See if you can give it a go.

Press pause if you need.

Well done.

Let's see what you could do.

Well, one example could be 12 divide by 6 is 2, then multiply by 10, which is 20.

That connects 12 to 20.

For the second in question, you could do 18 divide by 3, which is 6 and multiply by 2.

There are lots of different ways, but let's check how they're equivalent to a single multiplier.

Well, the division of 6 is the same as the multiplication of 1/6, and then we multiply by 10, that means we have a single multiplier 5/3.

For the second example, the division of 3 is the same as the multiplication of 1/3.

Multiply by 2 gives a single multiplier of 2/3.

Great work if you got this one right.

So when working with proportional relationships, finding the multiplier in one direction makes it so much easier to work out the multiplier and to go in the opposite direction.

For example, 5 to 10, we know the multiplier is 2, but if you were going 10 to 5, that means the multiplier, that means we're dividing by 2.

But remember, let's replace the division with a multiplication of its reciprocal, which is multiplication of 1/2.

So multiplying by the reciprocal is the inverse of our multiplier.

Let's have a look at another one, 3 to get 12.

Well, we know we're multiplying by 4, so we're going from 12 to 3.

What do you think the multiplier would be? Well, we know we're dividing by 4, but the multiplier would be 1/4, which is the reciprocal of our 4.

Now what I want you to do is have a look at this quick check question and I'd like you to identify the missing numbers.

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

Great work! Let's see how you got on.

Well, if you multiply 8 by 4 gives 32.

So what do you multiply 32 by to give 8? Well, it's a 1/4.

51 multiplied by 1/3 gives 17, so that means 17 multiplied by the reciprocal of 1/3, is 3 which gives us 51.

25 multiplied by 2/5 is 10.

So that means 10 multiplied by 5/2 is 25.

Fantastic work everybody! So let's move on to your task.

I'd like you to work with the multipliers of the following pairs of numbers.

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

Well done! Let's move on to question two.

Question two wants you to work out the multiplier from A to B, and then you need to work out the multiplier from B to A.

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

Excellent! Let's move on to question three.

Question three wants you to fill in these missing numbers.

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

Well done.

Let's go through these answers.

Well, for question one, the multiplier from 8 to 16 is 2.

For B, the multiplier for 3 to give 15 is 5.

For C, the multiplier for 1 to give 7 is 7.

For D, the multiplier from 100 to give 20 is 1/5.

The multiplier for 1.

3 to give 3.

9 is 3, and the multiplier for 12 to give 9 is 3/4.

Did you manage to do that in one step or two steps? Either way, the single multiplier would be the same.

Great work if you got this one right! For question two, well, to work out the multiplier from A to B, you simply do 4 divided by 2, which is 2.

So therefore the reciprocal would be 1/2, which is the multiplier of B to A.

The multiplier of A to B is 4.

So the reciprocal would be 1/4.

The multiplier from A to B is 3/2, so the inverse multiplier would be 2/3.

The multiplier from A to B is 1/5, so from B to A it would be 5.

And the multiplier from A to B is 6, so that means the multiplier from B to A is 1/6.

Great work if you got this one right! For question three, let's see how you got on.

Well, for A, we know we multiply by 8 by 3 to give 24.

So 24 multiplied by 1/3 is 8.

For B, we have 15 multiplied by 2/3 is 10, so that means 10 multiplied by 3/2 is 15.

For C, well, we know from 92 to 23 it'd be multiplied by a 1/4.

So that means 23 to 92, you multiply by 4.

And for D, great question if you figured this one out, you have the missing value is 30 and the multiplier is 5/3.

Well done.

Great work everybody! So let's have a look at efficiently calculating a single multiplier.

Now you've looked at how the multiplier between two numbers can be found using multiple steps in combining them.

So for example, 6 multiplied by 4 is 24 and then multiply by 1/3.

From here, we know the single multipliers found by multiplying those multipliers to give us a single multiplier, 4/3.

Now using the middle value suggested, fill in the missing multipliers for each of the following.

So 15, what do we multiply by to give 3? And then what do we multiply again by to give 21? And then identify what that single multiplier is.

For B, 8, what do we multiply by to give 1? Then what do we multiply that 1 by to give 3? And I want you to think about what that single multiplier is.

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

Well done.

Let's see how you got on.

Well, hopefully you spotted 15 times 1/5 is 3, multiplied by 7 is 21, so while single multiplier is 7/5.

For 8, well, if you multiply by 1/8, then you get 1, multiplied by 3 gives you 3, so our single multiplier is 3/8.

Using the knowledge that a number multiplied by its reciprocal is equal to 1, we can easily work out that multiplicative relationship between all numbers using 1 as the middle step.

For example, 2/5 multiply by something is 1, and then we can multiply 1 by a multiplier to give 7.

So what do we multiply 2/5 by to give number 1? Well hopefully remember, we can multiply any number by its reciprocal.

So if you multiply 2/5 by its reciprocal, we get 1 and then we can simply multiply that 1 by our 7 to give us our 7.

Identifying our common multiplier, we simply multiply 5/2 by 7 so single multiplier is 35/2.

When we use 1 like this, it's sometimes called the unitary method.

So let's have a look at a check.

Here are some numbers connected multiplicatively using 1 as the middle step.

Or you to work out that single multiplier that connects each number.

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

Well done! Let's see how you got on.

Well, the single multiplier that connects 3 to 8 is 8/3.

3 multiply by 1/3 gives us our 1, then multiply by 8 gives us our 8.

So combining those two multipliers, we have 8/3.

9/10 multiplied by its reciprocal of 10/9 gives us our 1, multiplying by 5 gives us our 5.

So that means combining those two multipliers, 10/9 multiplied by 5 gives us 50/9.

Well done if you spotted this.

I'm going to do the first question, I'd like you to do the second question.

We need to work out the single multiplier between 8 and 15.

Well, let's use that middle step of 1.

What do I multiply 8 by to give 1 and then multiply 1 by to give 15? Well, 8 multiply by 1/8 gives me 1, and then 1 times 15 gives me 15.

Combining those two multipliers gives me a single multiplier of 15/8.

Now I'd like you to try and do a question on your own.

I want you to work out the single multiplier that connects 6 to 14.

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

Well done.

Let's see how you got on.

Well, I'd use that middle step of 1.

What do I multiply 6 by to get 1? Well, it's 1/6, and then I'm going to multiply by 14, thus giving me a single multiplier of 14/6, which gives me a simplified fraction of 7/3.

Great work if you got this one right.

Fantastic work everybody! So let's move on to your task.

Here are some numbers connected multiplicatively and using 1 as that middle step, workout the single multiplier that connects each number.

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

Well done.

Let's see how you get on with question two.

Question two wants you to write the number in the box to make the calculation correct.

So if you want to use those diagrams to help and use 1 as a middle step, please do.

Press pause for more time.

Great work.

Let's move on to question three.

Question three has our diagrams partially filled.

Can you fill in those missing numbers? See if you can give it a go and press pause one more time.

Well done.

Let's see how you get on with these answers.

Well, for question 1a, hopefully you spotted we have the multiplier of 1/3 and 5 giving us a single multiplier of 5/3.

For b, we have the multiplier of 7/4 and then multiply by 6 to give me 42/4, which simplifies to 21/2.

For c, we have a multiplier of 1/5 and then multiply by 2 to give us a single multiplier of 2/5.

For question two, let's find out what that single multiplier is.

Well, for a, using that middle step as 1 we know we multiply by 1/8, then multiply by 16.

So that means combining them together to make a single multiplier of 2.

For b, 8 to give 1, we multiply by 1/8 and then multiply by 12 to make a single multiplier of 3/2.

For c, well, we multiply by 1/8 and then multiply by 10, which gives us a single multiplier of 5/4.

And for d, we multiply by 1/8 and then multiply by 9 to give us a single multiplier of 9/8.

Great work if you got this one right.

For three, did you fill in these missing gaps? You should have had 4 as it's the reciprocal of 1/4, so that means 4 to 9 has a single multiplier of 9/5.

For c, you should start off with 10 to give 8, and then we have these missing values.

Great work if you've got this one right.

So in summary, there always exists a multiplier for any pair of values.

And multiplying a number by its reciprocal will always have the product of 1, which then can be used with a multiplier to connect one to another number.

And the multiplicative relationship can be shown in multiple steps, but where the single multiplier is the product of those multipliers.

Great work everybody, well done.