Lesson video

Hi, welcome to our proportion problems lesson where we will be solving problems involving the relative size of two quantities.

All you'll need for today's lesson is a pencil and piece of paper.

So pause the video now and grab your things if you haven't got them already.

Here's our agenda for today.

Then we'll go on at looking at enlargement, you'll do some independent learning and then a final quiz.

So straight onto enlargement.

Here Zaara is making bracelets, using the guidelines below.

So she has got a sort of recipe, if you like, for making a bracelet and these are the ingredients that she will need.

So to make two bracelets, she will need 50 centimetres of elastic, 20 blue beads, 10 red beads, 10 purple beads and six star beads.

So can you think about what you can say about one bracelet, using the information you've been given and how you know? Pause the video now and write down your thoughts.

So we know that in order to make one bracelet, she will need to take the instructions for two bracelets and she will need to divide it by two.

And in this case, that means that she is enlarging the recipe by a scale factor of 1/2.

So she's dividing by two, she's halving everything in the recipe in order to get one bracelet.

Now, you might be thinking that makes absolutely no sense because enlarging, it has the word large in it, means to make something bigger.

But actually, in the mathematical definition of enlargement is just to change the size of an object.

So if you're enlarging something mathematically, you can be making it bigger or smaller.

And in this example, the enlargement makes the quantity smaller by 1/2.

So let's have a look at what that looks like for each different part of the bracelet recipe.

So going from making two bracelets to one, we are halving every part of it.

So 50 centimetres of elastic halved will mean that she needs now 25 centimetres of elastic.

Rather than 20 blue beads, she'll only need half that amount, so she'll need 10 blue beads.

10 red beads becomes five and the same for purple.

And where she needed six star beads to make two bracelets, she just needs three star beads to make one bracelet.

So each part of this recipe for the bracelet has been enlarged by a scale factor of 1/2, which means that it has been halved and divided by two.

Let's have a look at another example.

Now, what if Zaara wanted to make 12 bracelets this time? How many of each item would she need? You might be feeling confident to have a go at this by yourself.

So if you are, pause the video and figure it out.

How are you going to scale this recipe in order to make 12 bracelets? If you still need a little bit more practise with me, then keep the video going and we'll do it together.

So if we're going from making two bracelets to 12 bracelets, we need to think about by what scale factor would we be enlarging? So in order to go from two to 12, we're going to be enlarging by a scale factor of six, which means that we're multiplying everything by six.

So this time, the enlargement means that the quantities are getting bigger, whereas in the last one, they were getting smaller.

We were enlarging by a scale factor of 1/2 and dividing by two.

This time, we actually are making the quantities bigger.

So you can see the relationship here between two bracelets and 12 bracelets.

12 is six times greater than two.

Therefore it's an enlargement of scale factor six.

So again, we just work through and we multiply each by six.

So now we start with the 50 centimetres of elastic.

Think about using our related facts.

So if we know what five times six is, which is 30, then we know what 50 times six is.

It's just going to be 10 times greater because 50 is 10 times greater than five.

So 50 times six is 300.

Again, we'll use our related facts.

Two times six is 12.

So 20 times six must be 120, which is 10 times greater than 12.

10 red beads becomes 60 because that's 10 multiplied by six.

And then the six star beads, enlarged by a scale factor six, means that we will need 36 star beads.

So now it's your turn to have a go at one independently.

I'd like you to pause the video and work out the quantities of each ingredient for the different amounts of slime.

So you'll start with the recipe in the middle where we have two bowls of slime.

And then you're going to enlarge it to eight bowls of slime.

And then also, you're going to enlarge it to one bowl of slime.

Pause the video now and work out the quantities.

So to go from two bowls of slime to eight bowls of slime, we need to think about the relationship between two and eight.

I know that getting from two to eight, I'm going to be enlarging by a scale factor of four, which means multiplying by four.

So for each of the ingredients, I'm going to multiply that by four and again, I'm using my known facts.

So if I know five times four, then I know 50 times four.

So we'll have 200 mills of PVA glue.

480 mills of jelly solution, four drops of food colouring and 440 mills of water.

Now I'm looking at going from two bowls to one bowl of slime.

And I know that in order to get from two to one, I'm going to be enlarging by a scale factor of 1/2, which means dividing everything by two.

So 50 divided two means 25 mills of PBA glue.

60 millilitres of jelly solution.

Half a drop of food colouring.

And 55 millilitres of water.

So let's have a look at another one together.

Here we have a recipe for 12 cakes and you can see the quantities of each ingredient needed in order to make 12 of these cupcakes.

And just an aside, if you are ever looking at a recipe and you see this tsp, that means teaspoon, which is five mills.

So one teaspoon of vanilla is needed for this recipe.

So Liman only wants to make three cakes but the recipe is for 12.

Liman says, "I need to enlarge by a sale factor of four to find out how much of each ingredient I need to make three cakes." Is he correct? To get from 12 cakes to three cakes, does Liman need to enlarge by a scale factor of four.

So Liman is not correct because if he enlarges by a scale factor of four, he will end up with 48 cakes because 12 times four is equal to 48.

What he actually needs to do is enlarge by a scale factor of 1/4, which means that he'll be dividing everything by four.

12 divided by four is equal to three.

So now we know that he's enlarging by a scale factor of 1/4, have a think about how many grammes of margarine would Liman need? So what we're looking to do here is to divide the quantity of margarine by four because we're enlarging by a scale factor of 1/4.

So 140 divided by four is equal to 35 grammes.

Another one for you to do independently.

So you have a recipe that makes three smoothies.

Now, Aleeyah has 200 millilitres of pineapple juice, so she wants to adapt this recipe.

So in order to make three smoothies, you would need 600 millilitres of pineapple juice but if Aleeyah only has 200 millilitres, how many smoothies can she make? And in this case, how much yoghourt and ice cubes would she use? Pause the video now and make your notes.

So what we're looking at here is the relationship between 600 and 200.

I know that 200 is 1/3 of 600.

That means that I'm enlarging by a scale factor of 1/3.

600 divided by three is equal to 200.

So therefore, I can see that this makes three smoothies but I have to divide it by three, which means she'll only be able to make one smoothie.

So for her recipe, how much yoghourt does she use? I'm going to divide the quantity of yoghourt by three.

So 300 divided by three means that she'll need 100 millilitres of yoghourt and the same with the ice cubes.

Nine ice cubes divided by three means that she will need three ice cubes.

Now it's time for you to complete some independent learning.

So pause the video and complete the task and then click restart once you're finished so that we can go through the solutions together.

So in your first question, you had a recipe for 20 cookies and we're asked to scale the ingredients to make 10 cookies, 60 cookies and one cookie.

So we'll start with the 10 cookies.

To go from 20 cookies to 10, we're going to enlarge by a scale factor of 1/2, which means that we're dividing the quantities by two.

So this recipe goes from making 20 to 10 cookies and you can check all of your ingredients that each of them have been successfully divided by two.

I'm not sure how you put in 1/2 an egg, so this is not a very accurate recipe.

Now, we'll go on to 60 cookies.

I'm thinking about the relationship between 10 and 60.

I know that 10 multiplied by six is equal to 60, therefore I'm enlarging by a scale factor of six.

I'm multiplying everything by six.

And here are my new quantities of each of the ingredients.

Now, for the last one, I'm looking to make one cookie.

Now, I could have looked at the relationship between 60 and one, which means that I would be dividing everything by 60.

I would be enlarging by a scale factor of 1/60.

But I can think of a better way of doing it.

I can see a clearer relationship between 10 and one or an easier-to-manage relationship.

So I know going from 10 to one cookie, I'd be enlarging by a scale factor of 1/10 and dividing everything by 10, so these would be my quantities of each ingredient.

So you can see from this recipe that you would not be able to make one cookie very accurately because how you make 1/20 of an egg, I cannot see that you would be able to do that very accurately.

Now, in the second part of the question, you're asked how many cookies are made with 625 grammes of castor sugar? So I'm looking at this original quantity of 125 and thinking about the relationship between that and 625.

I know that 125 times five is equal to 625 grammes, which means that I've enlarged by a scale factor of five.

So I've enlarged from 20 cookies to 100 cookies.

In question two, you had a soup recipe to make six bowls of soup.

And you were asked to scale it to make 30 bowls and then 120 bowls.

So in order to get from six to 30, I enlarged by a scale factor of five.

So multiply every quantity by five.

And here is my new recipe.

And then to get from 30 to 120, I'm enlarging by a scale factor of four.

So all of my quantities there have been enlarged by scale factor four, multiplied by four.

The second part of the question asks how many bowls are made from 2.

7 kilogrammes of tomatoes? So I looked at my original recipe where 900 grammes of tomatoes were needed and I thought in order to do some conversions, I'm going to need this to be in grammes.

So that's equal to 2,700 grammes.

And in order to get from 900 grammes to 2,700 grammes, I multiply by three, which means I enlarged the recipe by a scale factor of three.

So six enlarged by scale factor three is equal to 18.

So that would make 18 bowls.

For question three, we're back to our bracelet formula.

And part a asks you how many bracelets can you make if you have 42 star beads? So for two bracelets, you need six star beads.

If you have 42, how many bracelets would that be equal to? So the relationship between six and 42 is that six multiplied by seven is 42.

So we're enlarging by scale factor seven.

So then we multiply each quantity by seven.

So we're going from making two bracelets to 14 bracelets.

50 centimetres of elastic to 350.

140 blue beads and 70 of each of the red and purple beads.

So for question four, we have a recipe for 12 cakes.

And Fatma wants to make 120 cakes.

She's looked in her cupboard and found a bag of flour, which is half full and it originally contained three kilogrammes of flour.

So we want to know does she have enough flour to make 120 cakes? And if not, how many cakes could she make? So the first thing to think of, this is a multi-step problem is about the flour.

So she's found a bag of flour, which is half full and it did have three kilogrammes of flour.

So we now know that it has 1.

5 kilogrammes or 1,500 grammes of flour, which is half of the original three kilogrammes.

Now, in order to make 120 cakes, she needs to enlarge her recipe by scale factor 10.

Relationship between 12 and 120 is that 120 is 10 times greater than 12.

So how much flour would she need if she were to enlarge the original recipe by scale factor 10? So the original says 200 grammes.

It means that she would need 2,000 grammes of flour but she doesn't have that.

She has 1,500 grammes of flour.

So how many cakes could she make? So she could make up to 90 cakes.

So we know that if we multiply 200 grammes by seven, that gives us 1,400 grammes of flour.

And then if we multiply it by 7.

5, that gives us 1,500 grammes of flour.

But if we multiplied it by eight, that would give 1,600 grammes of flour, which would be too much.

So we've scaled it up by a scale factor of 7 or 7.

5 to give us the right quantity of flour.

Great work today.

I hope you really enjoyed that lesson.

I really like scale factor because it's something you can really use in real life and it will be great if you could try it with doing some recipes and cooking in the kitchen.

Before you go, don't forget to complete your final quiz.