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 real-world ratios, and it's under the unit, Ratio, and by the end of the lesson you'll be able to use your knowledge of ratios to real-world context.

Now we'll be looking at the word proportion, and proportion is a part to whole, sometimes part to part comparison, and if two things are proportional, then the ratio of part to whole is maintained and the multiplicative relationship between parts is also maintained.

Today's lesson will be broken into three parts.

We'll be looking at ratios and recipes first, then ratios and maps second, and then other real-world ratio problems. So let's make a start looking at ratios and recipes.

Now, ratio tables are efficient ways to show the multiplicative relationship between parts or the whole, and recipes are excellent examples of how we use ratios, and when a recipe is followed correctly, it can make some delicious food.

Alternatively, if the recipe is not followed correctly, it can have some untasty outcomes.

So let's have a look at an example.

Here's a recipe for a sponge cake.

Now the recipe states vanilla cake consists of 120 grammes of butter, 120 grammes of flour, and 120 grammes of sugar, and two eggs.

Laura says, "I'm just gonna add 40 grammes of each ingredient and one more egg to make the cake a little bigger." And Izzy says, "Well, I'm going to add 60 grammes of each ingredient and one more egg to make the cake a little bigger." Who is correctly following the proportions of the recipe? And I want you to explain why.

Have a little think.

Well, Izzy is keeping to the proportions of the recipe.

Basically, she's adding a half of each ingredient.

You can see here.

So that means 180 grammes of butter would be used, 180 grammes of flour would be used, 180 grammes of sugar would be used, and three eggs would be used.

I'm gonna show you why these are all in proportion.

Well, multiplying the original quantities by a half gives you 60 grammes of butter, 60 grammes of flour, 60 grammes of sugar, and one egg.

We can also spot that these ratios are in proportion by multiplying that ratio by three, giving us the 180 grammes of butter, 180 grammes of flour, 180 grammes of sugar, and all three eggs.

Alternatively, you could even multiply the original recipe by 1.

5, but clearly you can see that multiplicative relationship using our ratio table, and you can see all these ratios are in proportion.

Now let's have a look at what Laura did.

When Laura add 40 grammes of each ingredient, one egg.

It's not in proportion.

Yes, if you divide each of our ingredients by three or multiply by one third, we get 40 grammes of butter, we get 40 grammes of flour, and we get 40 grammes of sugar, but we do not get one egg.

Laura's recipe is not in proportion, so I won't be eating anything made by Laura.

Now it's time for a quick check.

Which of the following ingredients are in proportion to this batter mix? 160 grammes of flour, 200 millilitres of water, and eight grammes of salt.

So you can give it a go.

Press pause for more time.

Well done.

Let's see how you got on.

Well, hopefully you spotted A is in proportion.

If you multiply each of the recipe quantities by two, it gives these proportions.

Well done.

B is not in proportion.

The flour and water have been multiplied by 1.

5, but this would mean that the proportion of salt should be 12 grammes, not 14 grammes, and for C, yes, it is in proportion because if you multiply each of the recipe quantities by a quarter, it gives those proportions.

Really well done.

Ratio tables allow you to use multipliers for any part of the ratio or the whole, so you can amend a recipe.

For example, this recipe is for chocolate brownies for eight people, and let's see if we can work out the quantities of each ingredient for the chocolate brownies for 20 people.

Now to do this, it's so much easier to represent it in a ratio table, so I'm putting this wonderful recipe into a ratio table like this.

Now, there are lots of different ways to convert the recipe to serve 20 people.

I'm just gonna show you a couple of different methods here.

Because we're serving 20 people and our original recipe states it serves eight people, let's see if we can identify a common factor of both eight and 20.

I'm going to use four.

So I'm going to adjust our recipe so it serves four people.

To do this, I simply multiply everything by one half.

That means I have all the ingredients to serve four people.

Now I have all my ingredients for four people, I simply multiply this by five, so I have my ingredients for 20 people, thus giving me all of these ingredients here.

This is one method to convert our recipe, which serves eight people into a recipe for 20 people.

An alternative method, and I think it's more efficient, would be to find the single multiplier that connects eight to 20, and that single multiplier would be 2.

5, so I'm going to multiply all my ingredients by 2.

5, giving me exactly the same answer as before.

Both methods are absolutely fine to use.

For me, finding that single multiplier is a little bit more efficient.

Well done.

So let's move on to a check.

Here's a recipe for a banana smoothie.

What I want you to do is work out the quantities for each ingredient for 16 people and for 14 people.

So you can give it a go.

Press pause if you need more time.

Well done.

Let's see how you got on.

Well, here's my recipe in a ratio table.

It serves four people, so to find out what my ingredients would be for 16 people, I'm gonna simply multiply by four, thus giving me the following quantities of each ingredient.

Now, to find out the quantities of each ingredient for 14 people, I'm gonna simply multiply my recipe for four people by 3.

5, and this gives me the recipe for 14 people.

Well done if you got this.

Now, ratio tables also allow you to use multipliers for any part of the recipe.

For example, here's a recipe for vegetable soup for 12 people, but Jun only has six carrots.

How many of each ingredient does he need when he only has six carrots and how many people will this soup serve? Well, let's put this in a ratio table first.

Putting in a ratio table, I have this.

So focusing on our carrots, there are a few different ways to calculate how many of each ingredient we need.

One method would be to identify a common factor of nine and six, which is three, so what's the multiplier that connects nine to three? Well, it's multiplying by one third, so multiplying all the other ingredients and the serving number by a third, gives us this.

Now we need to work out the quantities for six carrots, so I need to multiply it by two.

Multiplying by two gives me the following quantities and also tells me it will serve eight people, but a more efficient approach would be to find the single multiplier that connects nine to six.

That would be two thirds, so multiplying each part of our recipe by two thirds gives us exactly the same answer, and states it will still serve eight people.

Really well done if you spotted this.

Great work, everybody.

So let's have a look at your questions.

Carefully read the recipe and the question.

Press pause if you need more time.

Well done.

For part B, it continues, and Jacob only has 420 grammes of flour, so how much of each ingredient is needed to make the biggest carrot cake he can, and how many people will it serve? Press pause for more time.

Well done.

Let's move on to question two.

Question two, calculate the ingredients needed for the following number of cookies, and D says, you've got plenty of sugar, flour, and chocolate chips, but only one kilogramme of butter.

What's the largest number of cookies that you can make? Press pause for more time to answer these questions.

Well done.

Let's move on to question three.

Sam has created her own recipe and called it a monster cake.

Write down the proportions of each ingredient for 12 monsters and 30 monsters.

See if you can give it a go.

Press pause for more time.

And then question four wants you to create your own recipe and your own questions and answers whereby you're changing the serving.

Great question.

Gives you an opportunity to be creative here.

Press pause for more time.

Really well done.

Let's move on to these answers.

For question one, you should have these quantities.

Press pause if you need.

Question one B, you should have these quantities.

Press pause if you need.

For question two, we should have these quantities.

Press pause if you need.

For B, you should have these quantities, and for C, you should have these quantities, and for D, hopefully you figured out, it would be 200 cookies is the biggest amount of cookies you can make.

Press pause if you need more time.

Question three, here are our answers to question three A and B.

Press pause if you need more time to look at the working out, and for question four, I hope you enjoyed creating your own question and answers.

Really well done.

Great work everybody.

So let's move on to ratios and maps.

Now here's a map and a map scale.

What do you think the scale of one to 800,000 means? Have a little think.

Well, a map scale refers to the ratio between distance on a map and the corresponding distance in real life.

For example, one to 800,000 on a scale map means that one centimetre on the map equals 800,000 centimetres in real life.

It can also mean one millimetre on the map is the same as 800,000 millimetres in real life.

So converting to a more appropriate units, it basically means one centimetre is eight kilometres in real life.

It's really important to know length conversions in this part of our lesson, so let's quickly convert the following.

I want you to spend a few seconds converting 2.

4 kilometres into centimetres, 8.

9 metres into centimetres, and 2 million centimetres into kilometres, and here's the little box just showing you the conversions if you need.

Well done.

Let's see how you got on.

Well, hopefully you've spotted 2.

4 kilometres is the same as 2,400 metres, which is the same as 240,000 centimetres, 8.

9 metres in centimetres is 890 centimetres, and 2 million centimetres is 20,000 metres, which converts to 20 kilometres.

Really well done if you've got this.

Ratio tables are very good, efficient methods to show map scales so we can calculate map and real-life lengths.

For example, the distance on a map between two houses is six centimetres, and the distance in real life is eight kilometres, so let's write the map scale.

Well, we know that the map is six centimetres and the real life's eight kilometres, so when writing ratios correctly, they've got to be in the same unit, so let's convert.

Converting both to centimetres as it'll be easier, means our map's six centimetres and our real life's 800,000 centimetres.

Now, considering that both of the units are the same now, we do not have to include the unit, so now we have a ratio of six to 800,000.

From here, we can simplify, as this can help us with calculations later on.

This means the ratio is three to 400,000, therefore the map to real-life ratio is three to 400,000.

Now let's have a look at a quick check, where I want you to write the map scale of the following as a simplified ratio in the form map to real life.

For part A, the map length is 12 centimetres and the real life length is eight kilometres, and for B, the map length is 2.

4 centimetres and the real-life distance is 60 metres.

See if you can write these as a simplified ratio.

Press pause for more time.

Well done.

So, hopefully you've got a ratio table here showing that the map to real-life ratio is three to 200,000, and for B, here's my ratio table here, and notice how I've got some decimals, so all I'm gonna simply do is convert into integers by multiplying each parts by 10 and then simplify, so that means I have the map to real-life ratio is one to 2,500.

Lots of different ways you could do that, but you will still get the same simplified ratio.

Well done.

So once a correct map scale has been calculated, we can use this to help calculate map length or real-life lengths.

For example, on a scale drawing, a building has a length of 12.

4 centimetres and a width of four centimetres.

The real-life length of the building is 62 metres, and what we're going to do is work out the real width of the building in metres, so let's use our ratio table again.

We know the map length is 12.

4 centimetres and we know the real length is 62 metres.

Now converting to the same units, allows us to write a ratio correctly, removing those units given the fact that both of them are given in centimetres, so now we have 12.

4 to 6,200.

Now, one method to work out what the width would be is to use the unitary method, and all I'm going to do is find out what one centimetre is on the map.

So using our knowledge on multiplicative relationships, I'm going to divide everything by 12.

4, telling me that the map length of one centimetre is 500 centimetres in real life, so that means, given the question wants us to find the map length of four centimetres, I'm going to multiply by four, telling me that in real life, the building is 2,000 centimetres, which is exactly the same as 20 metres.

So you can see how we've continued to use our ratio table as well as that unitary method to work out what one centimetre in map length would be, and then we can calculate four centimetres.

Now it's time for a check.

A map has a scale of one centimetre to 20 kilometres, but the distance between Edinburgh and Bristol is 500 kilometres.

What is the simplified map scale in the form map to real life, and what is the distance on the map between these two cities? I want you to give your answer in centimetres.

Press pause if you need more time.

Well done.

Well, hopefully you've drawn a ratio table.

Well, there are lots of different ways in which you could have illustrated this.

We know one centimetre is 20 kilometres, converting them to the same unit means I have the ratio of one to 200,000.

From here, I need to work out what 500 kilometres would be, so I need to multiply my 200,000 by 25, as this would be my 500 kilometres in centimetres.

In the map length it would be 25 centimetres, so my answer would be 500 kilometres would be 25 centimetres on the map.

An alternative method would be to keep the units labelled in the table and simply multiply by 25.

That's absolutely fine as well.

Great work everybody.

So now it's time for your task.

Read the questions carefully and I want you to write the map scale of the following as a simplified ratio in the form map to real life.

Give it a go.

Press pause for more time.

Well done.

Let's move on to question two.

A map has a scale of three centimetres to 12 kilometres, and the distance between the two towns in real life is 42 kilometres.

What is the distance on the map between these two towns? Give your answer in centimetres.

Press pause if you need more time.

Well done.

Let's move on to question three.

A map has a scale of 14 to 50,000.

This distance between the two towns on the map is 3.

5 centimetres, what is the distance in real life between these two towns? And I want you to give your answer in kilometres.

Give it a go.

Press pause if you need.

Well done.

And for question four, Aisha and Laura place their rulers on a map, and the map scale is one to 625,000.

What is the approximate real-life distance in kilometres between Cheltenham and Swindon, and Newbury and Reading? See if you can give it a go.

Press pause for more time.

Well done.

Let's go through these answers.

You should have had these.

Press pause if you need.

Great work.

Let's move on to question two.

One option is to correctly write the ratios and then convert it from centimetres into kilometres.

Another option is simply labelling the units and the ratio table and then multiply.

Same again.

You still get 10.

5 centimetres.

For question three, we should've had a real life distance of 0.

125 kilometres.

Well done.

And for question four, well, the approximate distance between Cheltenham and Swindon is 50 kilometres, and for question four B, the approximate distance between Newbury and Reading is 31.

25 kilometres.

Well done.

Great work everybody.

So let's move on to other real-world ratio problems. So many ratios you've probably seen in real life.

For example, this is a common road sign.

I wonder if you've ever seen it, but do you know what it means? Have a little think.

Well, this ratio indicates the steepness of the road.

The one to eight means for every eight units travelled, the height decreases by one unit.

For example, for every eight metres horizontally, you've travelled one metre vertically.

That's what the ratio of one to eight means on this road sign.

And there are lots of different ratios that you might not even recognise.

For example, can you see what do all these images have in common? Have a little think.

Well, all these images are natural and man-made examples of the Golden Ratio, and the Golden Ratio is a mathematical concept that people have known about since the time of the ancient Greeks, and this is represented by this symbol, and this symbol is a Greek letter phi, and is a special number approximately equal to 1.

618 to three decimal places, and it's an irrational number like pi.

The Golden Ratio is thought to represent perfect beauty and is uniquely found through nature, but how does the Golden Ratio explain this spiral? So let's have a little look.

This spiral is seen so much in real life.

It's sometimes called the Golden Spiral.

Now, the Fibonacci sequence is formed by starting with the numbers one and one, and then summing the previous two terms. That makes the next term.

For example, starting with our numbers one and one, we sum them together, we get two.

Summing the two previous terms means we sum the one and two, which makes three.

Summing those two previous terms means we're summing two and three, which makes five.

Summing those previous terms means we sum the three and five, which makes eight, and what I want you to do, is just spend a little bit of time seeing if you can continue this sequence to find the next four terms. See if you can give it a go.

Well done.

Well, hopefully you spotted the next four terms would be 13, 21, 34, and 55, but how does this sequence link to the Golden Ratio and the Golden Spiral? So let's see if we can find the ratio between each term.

Well, if we divide one by one, we get one.

If we divide two by one, we get two.

Divide three by two, we get 1.

5.

Five divide by three gives us 1.

6 recurring, eight divide by five is 1.

6, so on and so forth, so what I want you to do is work out the ratio between the next terms in the Fibonacci sequence.

Well done.

Well, hopefully you spotted these are the next ratios, but what did you spot? Well, hopefully you spotted that these numbers, the ratio between the terms, are converging to phi, which is our 1.

618 to three decimal places.

Well done if you spotted this.

But how does this create the Golden Spiral? Well first of all, we start drawing a one by one, as you can see here, then I'm just gonna stick the next one by one, which is here.

Now, the next number in our Fibonacci sequence is two, so that means I draw a two by two next to that previous term.

The next term in our Fibonacci is three, so I draw a three by three next door term.

The next term is a five by five.

Then I draw my eight by eight, so on and so forth.

So now what I want you to do, is using squared paper, can you copy and complete the next two terms from our Fibonacci sequence? See if you can give it a go.

Well done.

Well hopefully, you've drawn an eight by eight here, and then a 13 by 13 here, and from this set of squares, we can connect each diagonal of each square with an arc, so let's have a look at our one by one first, and you can see this arc here.

Then our next one by one, so on and so forth, and we have created our Golden Spiral from our Fibonacci sequence, and the Golden Spiral is said to be seen in so much architecture and nature.

You've probably seen it yourself and not even realised.

Well done.

So let's have a look at the final part of our lesson.

Here, what I want you to do is measure the following, look at your own hand and measure the distance A, B, and C, and then I want you to measure the length of your hand, and then I want you to measure the distance from your wrist to your elbow.

Then I want you to calculate the following ratios, the distance C divided by distance B, the distance B divided by distance A, and then the distance from your wrist to your elbow dividing by the length of your hand, and then I want you to see if you can spot anything.

See if you can give it a go.

Press pause for more time.

Well done.

Well, let's see what you notice.

Well, I'm really hoping that the ratio of these lengths will be close to the Golden Ratio, 1.

618.

Don't worry if you don't get the exact value.

We're all not perfect.

You should get something really close to the Golden Ratio, but if you did get the exact value, very good.

That's regarded as perfection.

Excellent work everybody.

So in summary, ratio tables are efficient ways to show the multiplicative relationship between parts or the whole.

Recipes are excellent examples of how we use ratios, and when the proportions of a recipe are followed correctly, it can make some delicious food.

We can also use ratios when using map scales as this helps us calculate map lengths and real-life lengths when a scale is given, and finally, we also see so much ratio in real life.

For example, the Golden Ratio.

The Golden Ratio is represented as the symbol phi and is 1.

618 to three decimal places.

I really do hope you enjoyed the lesson.

It's been great learning with you.