Lesson video

Hi everyone, my name is Ms Koo, and I'm really happy to be learning with you today.

Today's lesson's going to be jam packed full of interesting and fun tasks for you to do.

It might be tricky or easy in part, but I will be here to help.

It's gonna be a great and fun lesson, so let's make a start.

In today's lesson from the unit, Understanding multiplicative relationships, fractions and ratio, we'll be expressing multiplicative relationships as ratios and as fractions.

And by the end of the lesson, you'll be able to express a multiplicative relationship as a ratio or as a fraction.

We'll be looking at proportionality, and remember, proportionality means when variables are in proportion if they have a constant multiplicative relationship.

And a ratio shows the relative sizes of two or more values and allows you to compare a part with another part in a whole.

And a fraction shows us how many equal parts are in a whole.

Today's lesson will consist of two parts.

We'll be looking at that multiplicative relationships as ratios first, and then we'll be looking at those multiplicative relationships as fractions.

So let's make a start.

Remember, proportion is a part to whole, sometimes part to part comparison.

And if two things are in proportion, then the ratio of part to whole is maintained, and that multiplicative relationship between parts is also maintained.

As a result, lots of multiplicative relationships can be formed and it's important to understand what each multiplicative relationship represents.

For example, a tomato sauce for one pizza is made using this recipe.

For one pizza sauce, it's 120 millilitres of puree, four garlic cloves, and two pinches of basil.

Now, what do you think the multiplier from the number of pizza sauces to puree is? I'm going to pop it into a ratio table, just because it makes it a little bit easier.

So you can see for one sauce, it's 120 millilitres of puree to four garlic cloves, to two pinches of basil.

So what do we multiply pizza sauces by to give puree? Well, to find out the multiplier, I'm going to do the puree, which is 120, divided by the pizza sauce, which is one.

So therefore, the multiplier has to be 120.

Pizza sauces times 120 is equal to puree.

This means for every one pizza sauce, there is 120 millilitres of puree.

Now what I want you to do is identify what's the multiplier from garlic cloves to puree.

See if you can use the ratio table to identify it.

Well, hopefully you spotted to find that multiplier, I'm going to divide the puree by the garlic cloves to give me 30.

So multiplying the garlic cloves by 30 gives me the puree, but what does this mean? It means for every one garlic clove, there are 30 millilitres of puree.

That's what our formula represents.

Next, what is the multiplier from pinches of basil to garlic cloves? Well, hopefully you're spotted here, four divided by two, the garlic divided by the pinches of basil gives us two, so to multiply the basil by two, we have our garlic cloves.

But what does this mean? Well, it means for every pinch of basil, there are two garlic cloves.

Identifying that multiplier is so important as well as understanding what that multiplier represents.

So let's have a look at a check.

Here's a ratio for orange juice.

For every five millilitres of orange juice concentrate, we use 100 millilitres of water.

And Izzy and Lucas are going to work out the multiplier from orange concentrate to water.

Izzy says, "100 divided by five is 20, so the multiplier is 20." And Lucas says, "Well, five divided by 100 is five over 100, which simplifies to one over 20." And Lucas says the multiplier is one 20th.

Who is correct and explain what the other pupil worked out? Let's see how you got on.

Well, Izzy is correct.

Multiplying the orange concentrate by 20 gives the amount of water needed.

But what did Lucas work out? Well, Lucas worked out the multiplier from water to concentrate.

Now let's have a look at a check.

I've given you some ratio tables and I'd like you to identify the formulas.

So for A, what do we multiply eggs by to give sugar? What do we multiply sugar by to give eggs? For B, what do we multiply people by to give the phones? And what do we multiply phones by to give the people? And for C, cats multiply by four equals dogs, so that means dogs multiply by what equals cats.

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

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

Well, to work out the multiplier for eggs to sugar, you do eight divide by four, which is two.

So eggs multiply by two gives us the sugar.

Therefore, if you're trying to find out the multiplier from sugar to eggs, four divided by eight, which is a half.

So that means sugar multiply by half gives you the eggs.

For B, to work out the multiplier from people to phones, you do 75 divided by 50, which is three over two simplified.

But to work out the multiplier from phones to people, you do 50 divided by 75, which is two thirds.

So that means phones multiply by two thirds equals people.

Now we know the multiplier from cats to dogs is four, so that means there must be five cats.

But what's the multiplier from cats to dogs? Well, that had to be a quarter.

Well done if you got this, and in particular, did you spot anything with the relationship between these multipliers? Let's have a look at another check question.

Jun and Izzy look at the ratio table and the multipliers and identify this ratio.

For every four eggs, there are eight grammes of sugar.

So the formula is, eggs times two equals sugar, sugar times a half equals eggs.

Now Izzy says, "For every one egg, there are two grammes of sugar." And Jun says, "For every one gramme of sugar, there is half, 0.

5, of an egg," who's correct? See if you can give it a go, and press pause if you need more time.

Well, let's see who's correct.

Both are correct, they've just written the ratio in a different way, so let's have a look.

There are lots of multiplicative relationships which can be formed, and it means that there are different approaches to how a question can be answered.

So let's have a look at what Izzy did.

Well, Izzy said, "For every one egg, there are two grammes of sugar." Let's find out, using the ratio table, if we divide everything by four, that confirms what Izzy said, for every one egg, there are two grammes of sugar.

It also confirms that formula, whereby you are multiplying the eggs by two to give the sugar.

You could also look at it as a bar model.

Referring back to what Izzy said, for every one egg, there are two grammes of sugar, one egg, two grammes of sugar, one egg, two grammes of sugar, one egg, two grammes of sugar, and one egg, two grammes of sugar.

Both the bar model and the ratio table all are equivalent to what Izzy is saying.

Now let's have a look at Jun.

Jun's saying, "For every one gramme of sugar, it's half an egg." Let's have a look at the ratio table.

Well, if we divide everything by eight, we have one gramme of sugar is 0.

5 eggs, so it works.

You could also identify that multiply of eggs to sugar is still two.

So Jun is still correct.

You could also look at it as a bar model.

Now Jun says, "For every one gramme of sugar, it's half an egg." Here's half an egg and one gramme of sugar, another half an egg, another gramme of sugar, so on and so forth.

This is equivalent to the ratio, it's just simply written in a different way.

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

Here, I want you to use the following bar models to fill in those gaps.

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, for A, ticks multiply by what gives crosses? Well, it's five.

If you multiply the number of ticks by five, you get the number of crosses.

So in other words, for every tick, there are five crosses.

Kitten multiply by what give puppies? Multiply by two, if you look at the number of kittens in the bar model, multiply by two, you get the number of puppies.

So that means for every kitten, there are two puppies.

Let's have a look at C.

What's a multiplier for pounds to give dollars? Well, it's two, but be careful of the second part, it says, for every US dollar, how many pounds are there? Well, for every US dollar, it's half a pound, you can see it here and here, that was a tricky one.

Well done if you got that one right.

For D, circles times four equals stars.

So for every star, there's a quarter of a circle.

Well done if you've got those ones right, C and D were quite tough.

Now it's time for your practise task.

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

Well done, let's move on to question two.

Question two shows the following bar models, I want you to fill in those gaps.

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

Great work, let's move on to question three.

Question three, Jacob spilt ink all over his work again, can you work out what is under those ink splats? We have some ratio information given below, read everything carefully before you fill in the bar model, and that ratio table, and those gaps.

See if you can give it a go, press pause one more time.

Great work, let's move on to question four.

Question four says, Laura, Lucas, and Alex are making papier mache.

Here are the instructions, for every four tablespoons of white flour, use 240 millilitres of water.

Laura uses eight tablespoons of flour and 480 millilitres of water.

Alex uses six tablespoons of flour and 242 millilitres of water.

And Lucas uses two tablespoons of flour and 238 millilitres of water.

Who used the correct ratio for making the papier mache? And explain the error of the other pupils.

And B, whose mixture will be too watery and explain? See if you can give this a go, press pause if you need more time.

Well done, let's have a look at these answers.

Well, for question one, we had to use our ratio table to fill in those missing amounts.

Let's see how you got on.

The number of eggs times 100 gives us the flour.

For every egg, there's 100 grammes of flour.

B, the amount of water times eight gives us the amount of sugar.

So for every millilitre of water, there are eight grammes of flour.

For C, the grammes of sugar multiplied by 0.

5 gives us the flour.

So the ratio is, for every one gramme of sugar, there's 0.

5 grammes of flour.

Great work if you got this one right.

For question two, using the bar models, this is what you should have got.

The shorts multiplied by three is equal to T-shirts.

That means for every pair of shorts, there are three T-shirts.

Cats multiply by two is equal to mice.

That means for every cat, there are two mice.

And for C, every house multiplied by two equals a phone.

So that means every house, there are two phones.

Really well done if you got this one right.

Now, let's find out what's under those ink splats.

Well, hopefully you've spotted our bar model should consist of two parts water, six parts glue, and three parts paper.

Really well done if you got this one right.

And for question four, using ratio tables really helps you out, so let's have a look at Laura first.

Well, we know the original ratio is four parts flour to 240 part millilitres water.

So that means tablespoons of flour multiply by 60 gives us our millilitres of water.

If you have a look at what Laura did, if she had eight tablespoons of flour, multiplying this by 60 means she's following the correct ratio.

Now for Alex, if you look at the six tablespoons of flour, multiply by 60, should have used 360 millimetres of water.

So that means he's not used the correct ratio as he's used 242 millimetres of water.

And Lucas, for two tablespoons of flour, that means two multiply by 60, he should have used 120 millimetres of water.

So he hasn't followed the correct ratio either.

But where did they make their mistakes? Well, Alex added two tablespoons of flour and then added two millilitres of water, and Lucas subtracted two tablespoons of flour and then subtracted two millilitres of water.

Well done if you've got this one right.

It's important to remember ratio uses a multiplicative relationship, not an additive one.

What we had to do next was identify, well, whose mixture is too watery and explain why? Well, let's have a look at our multipliers.

Well, we know for Laura the multiplier was 60.

So that means for every tablespoon of flour, there are 60 millilitres of water.

For Alex, the multiplier was 40 and a third.

In other words, for every tablespoon of flour, there's a 40 and a third millilitres of water.

And for Lucas, for every tablespoon of flour, he used 119 millilitres of water.

So clearly Lucas's ratio is far too watery.

Well done if you got this right.

Great work, everybody, so let's have a look at the second part of our lesson, which is multiplicative relationships as fractions.

Now, fractions, decimals, percentages, and ratio are common forms to show proportion, because proportion is a part to whole, sometimes part to part comparison.

And recognising how a ratio can be written, for example, a bar model, ratio tables, sentence, et cetera, can help interchange between these different forms of proportion.

So let's have a little look.

Can you fill in the ratio table and the fraction using this bar model? See if you can give it a go, and press pause if you need more time.

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

Well, using our bar model, the ratio table should look like this.

For every two cats, there are three dogs.

What's the fraction of cats? Well, you can see there are two cats outta a total of five animals.

So our fraction is two fifths.

If the bar model wasn't there, how else could you see the fraction maybe using the ratio table? Well, hopefully you can spot the total part represents the denominator of the fraction.

So if you have two cats and three dogs, that means we have a total of five animals.

So the denominator represents our total.

Well done if you spotted this.

Let's have a look at a quick check.

I want you to use this bar model and complete the ratio table, and what is the simplified fraction for the ticks? See if you can give it a go, then press pause if you need.

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

Well, our ratio table should look like this, four to six.

So what does it look like as a fraction? Well, we know the total is 10, so the proportion of ticks as a fraction is four over our 10, simplified gives us two over five.

Simplifying our fraction could have been done by simply looking at the four over 10, identifying our highest common factor and simplifying from there.

Or you could have simplified using our ratio table.

So using our ratio table, an equivalent ratio would be two ticks to three crosses, which gives a total of five, thus giving us exactly the same fraction, two fifths.

Well done if you got this one right.

So given fractions, decimals, percentages, and ratio are common forms to show proportion, we are able to interchange between them to show the same ratio.

For example, let's have a look at this ratio table, which refers to two cups of flour for every three cups of water.

What do you think the multiplier would be from flour to water? Well, it's three divided by two, which is three over two, so that's our multiplier.

So what would the multiplier be for water to flour? Well, hopefully you've spotted, it'd be two divide by three, which is two thirds.

So water multiply by two thirds equals flour.

Can you spot that relationship between those multipliers? Next, let's have a look at a fraction.

Well, what fraction of the batter is flour and what fraction of the batter is water? You can use the ratio table or those multipliers to help.

I think it's easier to look at the ratio table to spot the fraction of batter that is flour is two fifths, two outta the total parts of five.

And the fraction of butter which is water, is three fifths, three outta the total five is three fifths.

Hopefully you can really see that interchangeable relationship between a ratio table, multipliers, and the fraction.

Well done, let's have a look at a check.

Given the bar model showing the ratio of orange juice to water, I want you to fill in the ratio table, identify the multipliers and the fraction.

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 ratio table should have been completed to show one part water for every four parts orange.

So water multiplied by four is equal to orange.

Orange multiplied by one quarter equals water.

Did you spot that relationship between those multipliers? Well done if you spotted they are using the reciprocal.

Next, let's see the fraction of juice which is water, well, it's one fifth, the fraction of juice which is orange, well, its four fifths.

Did you spot that relationship between the fractions? Remember, the sum of the fractions has to be a whole.

Well done if you got this one right.

Now, let's move on to another check.

There are three shops advertising three different chocolate bars.

Shop A says, for every one gramme of chocolate, there's two grammes of raisins.

Shop B says, a half of the chocolate bar has raisins and a half of the chocolate bar is chocolate.

And C says, chocolate multiply by a quarter gives you a number of raisins.

Now Aisha loves raisins, so which shop should she buy the chocolate bar from? B, Andeep loves chocolate, so which shop should he buy the chocolate bar from? See if you can give it a go, and press pause if you need more time.

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

Let's identify them as fractions first.

Well, for shop C, I'm going to write a ratio table.

Well, hopefully you can spot the chocolate multiply by a quarter gives the raisins.

So now I'm gonna identify fractions for them all.

Well, for shop A, we know a third of raisins, so two thirds is chocolate, for shop B, a half are raisins, and that means we know a half is chocolate, and for C, now we have our lovely ratio table, we can see a fifth of raisins and four fifths is chocolate.

So Aisha should go for shop B, because she really likes raisins, and Andeep should go for shop C, because he really likes the chocolate.

Great work, this is a nice example of how we can use fractions to show the greatest proportion.

Now it's time for your task.

For question one, you have a bar model, for each question, you need to identify the fraction, the ratio, and that formula.

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 says, there are four types of smoothies for sale, all containing strawberries, apples, and bananas.

Jacob likes strawberries, Sofia likes apples, so we need to find out which smoothie should Jacob and Sofia pick.

We also need to find out which smoothie has the least banana.

And I'd love you to create your own smoothie, so the ratio of bananas is greater than the ratio of apples, which is greater than the ratio of strawberries.

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

Great work, everybody, so let's go through these answers.

Well, for question one, how did you get on? Hopefully you spotted the fraction of dogs is three quarters.

The multiplier from cat to dog is three.

So this means for every cat, there are three dogs.

For B, the fraction of hearts is five sevens.

The multiplier for presents to give hearts is five over two.

So that means for every present, there are five over two hearts.

And for C, the fraction of oranges is four sixths, you can cancel down to give you two thirds.

So that means if you multiply the apple by two, it gives you the orange, so for every apple, there are two oranges.

Great work if you got that one right.

For question two, we have these four shops, and we know Jacob likes strawberries, so which smoothie should he pick? So let's identify what we have, I'm going to identify the ratio table for shop A, B, C, and D, and identify the proportion from here.

Well, the parts for shop A is strawberries is one part, apples is two parts, and banana is five parts.

For shop B, all the fruit is split equally, so each one, each ingredient has one part.

For shop C, using that formula or the ratio, you should have, strawberry is one part, apples is two thirds, and bananas is two.

I'm going to make each of the parts integers, so I'm gonna multiply everything by three to give me strawberries is three parts, apples is two parts, and bananas is six parts.

Next, let's have a look at shop D.

Shop D has a half parts of strawberry, apple represents one part, and one part it represent banana.

I'm gonna make them all integers by simply multiplying by two.

So I have one part is strawberry, two parts is apple, and two parts is banana.

Now from here, let's identify a proportion so I can compare.

So for Jacob, he likes strawberries, for A, what fraction is strawberries? It's one eighth, for B, what fraction is strawberries? It's a third, for C, what fraction are strawberries? It's three elevens, and for D, what fraction are strawberries? It's one fifth.

So that means Jacob should pick shop B, as it's got the highest proportion of strawberries in their smoothie.

Let's have a look at Sofia, Sofia likes apples.

So let's have a look at the proportion of apple in each smoothie.

Well, for A, a quarter of it is an apple.

For B, a third of the smoothie is an apple.

For C, two elevenths of the smoothie is an apple.

And for D, two fifths of a smoothie is apple.

So that means she should pick shop D, as this has the highest proportion of apples.

Next, which smoothie has the least banana? Looking at bananas, here are our proportions, so that means it has to be shop B.

Shop B has the smallest proportion, which is banana.

Great work, everybody, if you got this one right.

So proportionality means when variables are in proportion if they have a constant multiplicative relationship.

There are lots of multiplicative relationships which can be formed, and it's important to understand what each multiplicative relationship means.

Given fractions, decimals, percentages, and ratio are common forms to show proportion, we are able to interchange between these to show the same ratio.

Great work, everybody, well done.