Lesson video

Hi everyone, my name is Ms. Ku 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 parts, 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 2 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 2 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 2 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 milliliters of puree, 4 garlic cloves, and 2 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 source it's 120 milliliters of puree to 4 garlic clothes to 2 pinches of basil.

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

So therefore, the multiplier has to be 120.

Pizza sauce times 120 is equal to puree.

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

Now what I want you to do is identify what's the multiplier from garlic clothes 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 clothes to gimme 30.

So multiplying the garlic clothes by 30 gives me the puree.

But what does this mean? It means for every one garlic clove, there are 30 milliliters 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, 4 divided by 2, the garlic divided by the pinches of basil, gives us 2.

So to multiply the basil by 2, we have our garlic cloves.

But what does this mean? Well, it means for every pinch of basil there are 2 garlic cloves.

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

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 4 equals dogs.

So that means dogs multiply by what equals cats.

So 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 8, divide by 4, which is 2.

So eggs multiply by 2 gives us the sugar.

Therefore, if you're trying to find out the multiplier from sugar to eggs, 4 divide by 8, which is a half.

So that means sugar multiplier 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 3 over 2 simplified.

But to work out the multiplier from phones to people, you do 50 divided by 75, which is 2/3.

So that means phones multiply by 2/3 equals people.

Now we know the multiplier from cats to dogs is 4, so that means there must be 5 cats, but what's the multiplier from cats to dogs? Well, that had to be 1/4.

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 4 eggs there are 8 grams of sugar.

So the formula is eggs times 2 equals sugar.

Sugar times a half equals eggs.

Now Izzy says, "For every 1 egg there are 2 grams of sugar." And Jun says, "For every 1 gram 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 1 egg there are 2 grams of sugar." Let's find out.

Using the ratio table, if we divide everything by 4, that confirms what Izzy said.

For every 1 egg there are 2 grams of sugar.

It also confirms that formula whereby you are multiplying the eggs by 2 to give the sugar.

You could also look at it as a bar model.

Referring back to what Izzy said, "For every 1 egg there are 2 grams of sugar." 1 egg, 2 grams of sugar.

1 egg, 2 grams of sugar.

1 egg, 2 grams of sugar.

And 1 egg, 2 grams 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 saying, "For every 1 gram of sugar, it's half an egg." Let's have a look at the ratio table.

Well, if we divide everything by 8, we have 1 gram of sugar is 0.

5 eggs, so it works.

You could also identify that multiply of eggs to sugar is still 2.

So Jun is still correct.

You could also look at it as a bar model.

Now Jun says, "For every 1 gram of sugar, it's half an egg." Here's half an egg and 1 gram of sugar.

Another half an egg, another gram 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.

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 5.

If you multiply the number of ticks by 5, you get the number of crosses.

So in other words, for every tick there are 5 crosses.

Kitten multiply by what give puppies? Multiply by 2.

If you look at the number of kittens in the bar model, multiply by 2, you get the number of puppies.

So that means for every kitten there are 2 puppies.

Let's have a look at C.

What's a multiplier for pounds to give dollars? Well, it's 2.

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 4 equals stars.

So for every star there's 1/4 of a circle.

Well done if you got those ones right.

C and D were quite tough.

Now it's time for your practice 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 2.

Question 2 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 3.

Question 3, Jacob spilled 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 and press pause one more time.

Great work.

Let's move on to question 4.

Question 4 says, Laura, Lucas and Alex are making papier mache.

Here are the instructions: For every 4 tablespoons of white flour use 240 milliliters of water.

Laura uses 8 tablespoons of flour and 480 milliliters of water.

Alex uses 6 tablespoons of flour and 242 milliliters of water, and Lucas uses 2 tablespoons of flour and 238 milliliters 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 grams of flour.

B, the amount of water times 8 gives us the amount of sugar.

So for every milliliter of water there are 8 grams of flour.

For C, the grams of sugar multiplied by 0.

5 gives us the flour.

So the ratio is for every 1 gram of sugar there's 0.

5 grams of flour.

Great work if you got this one right.

For question 2, using the bar models, this is what you should have got.

The shorts multiplied by 3 is equal to T-shirts.

That means for every pair of shorts there are 3 T-shirts.

Cats multiplied by 2 is equal to mice.

That means for every cat there are 2 mice.

And for C, every house multiplied by 2 equals a phone.

So that means every house there are 2 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 2 parts water, 6 parts glue, and 3 parts paper.

Really well done if you got this one right.

And for question 4, using ratio tables really helps you out.

So let's have a look at Laura first.

Well we know the original ratio is 4 parts flour to 240 part milliliters water.

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

If you have a look at what Laura did, is she had 8 tablespoons of flour, multiplying this by 60 means she's following the correct ratio.

Now for Alex, if you look at the 6 tablespoons of flour, multiply by 60 should have used 360 millimeters of water.

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

And Lucas for 2 tablespoons of flour, that means 2 multiply by 60, he should have used 120 milliliters of water.

So he hasn't followed the correct ratio either.

But where did they make their mistakes? Well, Alex added 2 tablespoons of flour and then added 2 milliliters of water.

And Lucas subtracted 2 tablespoons of flour and then subtracted 2 milliliters of water.

Well done if you 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.

Let's have a look at our multipliers.

Well we know from Laura, the multiplier was 60, so that means for every tablespoon of flour there are 60 milliliters 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 milliliters of water.

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

So clearly Lucas' 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 recognizing 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? So 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 2 cats there are 3 dogs.

What's the fraction of cats? Well, you can see there are 2 cats out of a total of 5 animals.

So our fraction is 2/5.

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 parts represents the denominator of the fraction.

So if you have 2 cats and 3 dogs, that means we have a total of 5 animals.

So the denominator represents our total.

Well done if you spotted this.

So given fractions, decimals, percentages and ratio are common form 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 2 cups of flour for every 3 cups of water.

What do you think the multiplier would be from flour to water? Well, it's 3 divided by 2, which is 3/2.

So that's a multiplier.

So what would the multiplier be for water to flour? Well, hopefully you've spotted it'd be 2 divide by 3, which is 2/3.

So water multiplied by 2/3 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 2/5, 2 out of the total parts of 5.

And the fraction of batter which is water, is 3/5, 3 out of the total 5 is 3/5.

Hopefully you can really see that interchangeable relationship between a ratio table, multiplies, 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.

While the ratio table should have been completed to show one part water for every 4 parts orange.

So water multiplied by 4 is equal to orange.

Orange multiplied by 1/4 equals water.

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

Next, let's see the fraction of juice, which is water.

Well it's 1/5.

The fraction of juice which is orange? Well, it's 4/5.

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

Well done if you got this one right.

Now let's move on to another check.

There are 3 shops advertising 3 different chocolate bars.

Shop A says, "For every 1 gram of chocolate, there's 2 grams 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 1/4 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 or hopefully you can spot the chocolate multiplied by 1/4 gives the raisins.

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

Well, for shop A, we know a third of raisins.

So 2/3 is chocolate.

For shop B, a half of raisins, and that means we know a half is chocolate.

And for C, now we have our lovely ratio table.

We can see 1/5 of raisins and 4/5 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 2.

Question 2 says, there are 4 types of smoothies for sale, all containing strawberries, apples and bananas.

Jacob like 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.

So 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 3/4.

The multiplier from cat to dog is 3.

So this means for every cat there are 3 dogs.

For B, the fraction of hearts is 5/7.

The multiplier for presents to give hearts is 5/2.

So that means for every present there are 5/2 hearts.

And for C, the fraction of oranges is 4/6.

You can cancel down to give you 2/3.

So that means if you multiply the apple by 2, it gives you the orange.

So for every apple there are 2 oranges.

Great work if you got that one right.

For question 2, we have these 4 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 2 parts, and banana is 5 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 as one part, apples is 2/3 and bananas is 2.

I'm going to make each of the parts integers so I'm gonna multiply everything by 3 to give me strawberries is 3 parts, apples is 2 parts, and bananas is 6 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 represents banana.

I'm gonna make them all integers by simply multiplying by 2.

So I have one part is strawberry, 2 parts is apple and 2 parts of banana.

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

So for Jacob, he likes strawberries.

For A, what fraction of strawberries? It's 1/8.

For B, what fraction of strawberries? It's 1/3.

For C, what fraction of strawberries? It's 3/11.

And for D, what fraction of strawberries? It's 1/5.

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, 1/4 of it is an apple.

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

For C, 2/11 of the smoothie is an apple.

And for D, 2/5 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 proportions.

So that means it has to be shop B.

Shop B has the smallest proportion which is banana.

Great work, everybody, if you've 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.