Lesson video

Hi everyone, my name is Ms. Ku, and I'm really happy that you're joining me today because we'll be looking at the wonderful unit ratio.

I really hope you enjoy the lesson because I know I will.

So let's make a start.

Hi, everyone, and welcome to this lesson on checking and securing understanding of equivalent ratios, and it's under the unit Ratio.

By the end of the lesson, you'll be able to appreciate that any two numbers can be connected via a multiplicative relationship.

Our lesson will be looking at the keyword reciprocal.

And remember, a reciprocal is the multiplicative inverse of any non-zero number.

Any non-zero number multiplied by its reciprocal is always equal to 1.

For example, the reciprocal of 3 is 1/3.

This is because 3 multiplied by 1/3 is 1.

Another example would be 5/4.

The reciprocal of 5/4 is 4/5 because if you multiply 5/4 by 4/5, the answer is 1.

Today's lesson will be broken into two parts.

We'll be looking at connecting numbers multiplicatively first and then moving on to equivalent ratio second.

So let's look at connecting numbers multiplicatively first.

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

So can you find the missing number given the operation and find the relationships between these numbers? See if you can give it a go.

Press pause if you need more time.

Well, hopefully, you spotted you're multiplying the 10 by 3 and that gives us 30, or you could add 20 to our 10, thus giving us our 30.

So we're just gonna focus on multiplication now.

So focusing on this multiplicative relationship.

When it's been found, how do you think this helps us find the relationship between 30 and 10? Well, hopefully, you've spotted if you multiply 30 by 1/3, it gives us 10.

In other words, we use the multiplication of the reciprocal.

You might also think of it as the inverse operation.

So what I want you to do is have a quick check.

See if you can use your knowledge on that multiplicative relationship and those reciprocals to find out those missing numbers and those missing multipliers.

See if you can give it a go.

Press pause if you need more time.

Well done.

Let's see how you got on.

Well, for a, hopefully you spotted 8 multiplied by 3 gives us our 24, so therefore, 20 multiplied by the reciprocal of 3 is 1/3 gives us our 8.

For b, 15 times 2/3 is 10.

So what do we multiply 10 by to give us 15? Well, it would be the reciprocal is 3/2.

For c, hopefully, you've spotted that we have to use the reciprocal of 1/4, which is 4.

That gives us the answer of 23.

And for d, let's say we can identify that reciprocal again, which is 5/3.

So that means we have these missing numbers.

Really well done if you've got this.

So knowing that any two numbers can be connected by an additive or multiplicative relationship allows a middle number to be used to help make the connection more easily seen.

For example, can you identify the multipliers connecting these numbers? We have 6, then we're gonna make it 24, and then we're going to make it 8.

What do you think those missing multipliers are? Well, hopefully, you spotted we have 6 times 4 gives us our 24, and then multiplying this by 1/3 gives us our 8.

So these multipliers can be combined to give us a single multiplier.

If you multiply 4 and 1/3, that makes 4/3.

In other words, the single multiplier that connects 6 to 8 is 4/3.

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

For example, let's have a look at 2/5.

We're going to make 2/5 one, and then make 1 into 7.

So what do we multiply 2/5 by to make 1? Well, we multiply 2/5 by the reciprocal, which is 5/2.

Then, what do we multiply our 1 by to give 7.

Well, we simply multiply by the 7.

So combining these multipliers, we have 5/2 multiplied by our 7 gives us 35/2.

That's our single multiplier.

So in summary, using 1 like this is sometimes called the unitary method.

So now let's have a look at a quick check question.

I'll do the first part, and then I'd like you to do the second part.

We're going to work out the single multiplier between 8 and 15.

So I like to draw this diagram to illustrate that multiplicative relationship.

I'm using 1 as that stepping stone.

So what do I multiply 8 by to give 1, and then what do I multiply the 1 by to give 15? Well, remember, I'm multiplying the 8 by its reciprocal, which is 1/8.

That gives me the 1.

And then from there, I simply multiply the 1 by the 15 to give me my 15.

Combining these multipliers, 1/8 multiplied by 15, gives me the single multiplier of 15/8.

Now, I'd like you to work out the single multiplier between 6 and 14.

If you want to draw a diagram to help, please do.

Press pause for more time.

Well done.

Let's see how you got on.

Well, I'm going to use this diagram using one as that stepping stone again.

What do I multiply 6 by to give 1? Well, it's 1/6.

What do I multiply 1 by to give 14? Well, it's 14.

So as a single multiplier, multiplying that 1/6 by 14 gives me 14/6, which I've then simplified to give 7/3.

Now I want you to do another check.

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

I want you to work out that single multiplier that connects each number.

See if you can give it a go.

Press pause if you need more time.

Well done, let's see how you got on.

Well, hopefully, you've spotted that single multiplier connecting 3 to 8 is simply 8/3.

And for the second part, that single multiplier would be 50/9 because it's the product of 10/9 multiplied by 5.

Well done if you got this.

So using the unitary method or 1 as that middle step allows the single multiplier to be found between all types of numbers.

For example, what is the single multiplier that connects 0.

4 to 2/3? Well, to do this, drawing that connective diagram will help.

Then I'm gonna convert my 0.

4 into a fraction.

0.

4 is the equivalent of 4/10, which is equivalent to 2/5.

Using fractions just makes things a little easier sometimes.

Now from here, let's identify that middle step to be 1.

While multiplying that 2/5 by its reciprocal gives us that middle step of 1.

Then, multiplying 1 by 2/3 gives me my 2/3.

So finding the product of 5/2 and 2/3, we have our single multiplier, which is 10/6 or simplified down to 5/3.

So therefore, we know the single multiplier that connects 0.

4 to 2/3 is 5/3.

And you can see it in our calculation.

0.

4 times 5/3 is equal to 2/3.

Well done if you got this.

Now it's time for a quick check.

Knowing that the single multiplier that connects 0.

4 to 2/3 is 5/3, what do you think is the single multiplier that connects 2/3 to 0.

4? Well, hopefully, you spotted it's the reciprocal.

It's 5/3.

Well done.

Let's have a look at another check.

Alex and Sofia are both given the same question.

They're asked to identify the single multiplier which connects 1.

2 to 5.

You can see Alex's work on the left, and you can see Sofia's work on the right.

Here are their answers and the working out, but can you the differences in their working? See if you can give it a go.

Press pause if you need more time.

Well done.

Well, hopefully you've spotted Alex has not simplified the fraction throughout the calculation but Sofia has.

Both have correctly identified the single multiplier, but Sofia has simplified the fraction.

That's the only difference.

Great work, everybody.

So now it's time for your task.

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

See if you can work out that single multiplier that connects each number.

Press pause if you need more time.

Well done.

Let's move on to question 2.

I want you to identify that single multiplier as a single fraction where appropriate, which connects these numbers.

If you want to draw that diagram to help, please do.

Press pause for more time.

Great work.

Let's move on to question 3.

Same again, I want you to identify the single multiplier as a simplified fraction where appropriate which connects the following numbers.

Once again, if you want to draw that connective diagram, please do.

Press pause for more time.

Great work.

Let's move on to question 4.

Love this question.

Simply fill in the blanks.

Press pause as you'll need more time.

Great work.

Let's go through these answers.

Mark them.

Press pause if you need.

Fantastic.

Question 2, same again.

Mark them.

Press pause if you need more time.

Well done.

For question 3, mark these.

And press pause if you need.

Great work.

And for question 4, really tough one.

Massive well done if you got this one right.

Great work, everybody.

So now it's time to look at equivalent ratios.

Now here's some bar models showing the relationship between suns and stars.

What does each bar model have in common? Have a little think.

Well, hopefully, you can spot the proportion of suns to stars is the same.

In other words, for every three suns there are two stars.

You can see it here and you can see it in the next bar model.

3 suns to 2 stars.

3 suns to 2 stars.

So if I were to write them as a ratio, the first bar model shows sons to stars to be 64, and the second bar model shows the ratios to be 3 to 2.

As you can see from our bar model, these proportions are the same, so that means these ratios are equivalent.

But if we were to remove the bar models, how can we see that these ratios are equivalent? So let's pop them in into a ratio table and have a look.

Well, first of all, once the ratios are in our ratio table, we can see the multiplicative relationship more easily.

You can see if we multiply 3 to 2 by 2, we get our 6 to 4.

Alternatively, you could multiply 6 to 4 by 1/2, which gives us 3 to 2.

Or you could find the multiplier between the suns and the stars.

Well, to find that multiplier, we simply use those connective diagrams. You can use one as that middle step and work out the single multiplier which connects the suns to the stars is 2 to 3.

And this would be the same as working out that multiplier of 3 to 2.

Well, working out the multiplier, it gives us the same, 2/3.

So using our knowledge and finding that single multiplier really does help us identify those equivalent ratios.

As a result of these multipliers, we know that these ratios are equivalent, and this makes identifying equivalent ratios much easier without the need for bar models.

So let's have a look at a quick check.

Izzy and Alex are given this ratio table, and they're comparing the ratios 3 to 5 and 33 to 55.

Show, using four different multipliers, that these ratios are equivalent.

And out of curiosity, which multiplier is the easiest to show the equivalence.

See if you can give it a go.

Press pause if you need more time.

Well done.

Let's see how you got on.

Well, you could have multiplied 3 to 5 by 11 giving you 33 to 55, or you could have multiplied 33 to 55 by 1/11 giving you 3 to 5, or perhaps you may have multiplied 3 by 5/3 to give you 5, and that would be the same as 33 multiplied by 5/3 also gives you 55.

Alternatively, you may have multiplied 55 by 3/5 to give you 33 or multiplying the 5 by 3/5 also gives you 3.

So we have four of these wonderful multipliers.

So there are four multipliers showing the equivalent between the ratio 3 to 5 and 33 to 55.

Usually, the easiest is the integer.

Now I'd like you to do a quick check.

Here are some ratio tables showing equivalent ratios.

I want you to fill in the missing ratios and try to use the easiest multiplier.

See if you can give it a go.

Press pause if you need more time.

Well done.

Let's see how you got on.

Well, for me, the easiest multiplier here would be multiplying 4 to 5 by 6, giving me 24 to 30.

For b, the easiest multiplier would be multiplying the 10 by 2.

5 giving me 25.

So that means 4 multiply by 2.

5 gives me the 10.

For c, the easiest multiplier would be dividing by 5.

In other words, multiplying by 1/5.

This gives me the missing number to be 12.

And for d, the easiest multiplier would be multiplying by 4, identifying the missing number to be 16.

Really well done if you've got this.

Now I want you to another check.

Which of the following is equivalent to 15 to 36? And I want you to show your working out.

Press pause if you need more time.

Great work.

Let's see how you got on.

Well, identifying the multiplicative relationship between 15 to 36, I know that single multiplier is 2.

4 or 12/5.

So let's see if you multiply 2.

4 or 12/5, what are the other equivalent ratios? Well, if it was 20, that means multiplying by 2.

4 would give us 48.

So our first ratio is equivalent and it's correct.

If it was 1, well, we multiply by 2.

4, and it gives me 1 to 2.

4.

So this ratio is correct.

If it was 5 multiplied by 2.

4 gives me 12.

So this ratio is correct.

And lastly, if it was 5/3, if I multiply by 2.

4, same again, it's correct.

In other words, all of these ratios are equivalent to 15 to 36.

Massive well done if you got this.

Great work, everybody.

So let's have a look at your task.

Which of the following ratios are equivalent to 2 to 5? And which of the following ratios are equivalent to 8 to 3? See if you can give it a go.

Press pause if you need more time.

Well done.

Let's move on to question 3.

Which of the following ratios are equivalent to 2 to 3? And for question 4, which of the following ratios are equivalent to 6 to 8 to 4? Press pause for more time.

Great work.

Let's have a look at question 5.

For question 5, here are some ratio tables showing pairs of equivalent ratios.

I want you to find the missing parts.

See if you can give it a go.

Press pause for more time.

Well done.

And lastly, question 6 shows a lovely spider diagram, and we're asked to fill in the diagram to show all the equivalent ratios to 12 to 15 to 21.

See if you can give it a go.

Press pause if you need more time.

Well done.

Let's move on to question 7.

Question 7 says, Andeep says the dimensions of UK bank notes are in proportion.

Is he correct? I want you to explain your working out.

See if you can give it a go.

Press pause for more time.

Well done.

Let's move on to these answers.

Well, hopefully, you've spotted these are the equivalent ratios.

Well done.

For question 3 and 4.

Here are our equivalent ratios.

Press pause if you need.

For question 5, these are our missing values.

Press pause if you need more time to mark.

For question 6, great work if you got any of these, this was tough.

And for question 7, well, the multiplier from the width to the length on our five-pound note is 1.

92 to 3 significant figures.

The multiplier from width to length of our 10-pound note is 1.

91 to 3 significant figures.

And the multiplier from width to length is 1.

90 to 3 significant figures.

So they're not exactly in proportion, but if you were to round to two significant figures, then we can say that they are in proportion.

Well done, everybody.

So in summary, any two numbers can be connected by an additive and a multiplicative relationship.

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

Remember, we can use multipliers to work out equivalent ratios.

And using ratio tables allows the multipliers to be seen more easily and can help efficiently calculate unknown ratios.

Massive well done, everybody.

It was tough today, but it was wonderful learning with you.