Lesson video

Hello everyone, how are you today?

Hope you are doing really well.

My name is Ms. Afzal and I'll be your teacher for this lesson.

I'm feeling really pleased about that because I think we've got an interesting topic ahead.

Our lesson is called ratio language and notation.

I hope that sounds interesting to you.

Our lesson comes from the unit of work, understanding multiplicative relationships, fractions and ratio.

So if you're ready to get into our subject, if you have focus, energy and enthusiasm, we'll begin our lesson now.

The outcome for today's lesson is I can understand and use the language and notation of ratio and use a ratio table to represent a multiplicative relationship and connect this to other known representations.

I hope that sounds interesting to you.

Let's dive into it.

Have some keywords in our lesson, proportionality and ratio.

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

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

So these are our keywords, proportionality and ratio.

Today's lesson is called ratio, language and notation, and it has two learning cycles, ratio language and notation, and comparing different representations of ratio.

Let's begin by exploring ratio language and notation.

Notation in mathematics is very important.

We use symbols and notation because, well share and share with someone.

Why do we use symbols and notation?

Thanks for sharing.

Let's find out.

We use symbols and notation because they're easier to read and understand.

They're concise and take up less space.

They can be used to represent complex concepts.

It allows mathematical ideas to be communicated more effectively than words.

Perhaps you came up with some of these ideas for why we use symbols and notation, and this can be said with ratio notation too.

For example, here are the different representations of the ratio of circles to squares.

For every three circles there are four squares.

Pause here and share with someone which of these four representations of ratios of circles to squares are you familiar with?

Have you used before?

Thanks for sharing.

A concise way to write a ratio is to use a colon, circles to squares, 3 to 4.

We say the ratio of circles to squares is 3 to 4.

This is much more efficient, concise, and takes up less space and uses mathematical notation.

It's a great way to write a ratio using a colon.

Let's have a check for understanding.

Fill in the table.

The first has been done for you.

We have the bar model.

You can see an image of one triangle and three squares.

Then we have ratio notation, triangles to squares, 1 to 3, and then in words, ratio of triangles to squares is 1 to 3.

And now I'd let you to complete the table.

So CD stands for cats and dogs.

So you have the bar model, and then I'd let you to complete the ratio notation and in words, and then the same for the next example, which stands for apples and bananas and the SHC, stars hearts and circles.

So pause here while you complete the table.

So how did you get on with completing the table?

Let's take a look at what it could have looked like for you.

So CD, cats and dogs, ratio notation, cats to dogs, 1 to 1, in words, ratio of cats to dogs is 1 to 1.

Next example, apples to bananas, 2 to 3, in words, ratio of apples to bananas is 2 to 3 and a ratio notation for our final example, stars to hearts to circles, 1 to 1 to 2.

And in words, the ratio of stars to hearts to circles is 1 to 1 to 2.

Well done if you completed a table in this way and the order of the ratio must match between the words and notation.

Let's have another check for understanding.

Fill in the gaps for the ratio and the double number line.

Pause here while you have a go at this.

Let's take a look at how you got on.

So in the first example we've got ratio of R to C is 1 to 3.

So on our double number line, we should have 1, 2, 3 and 3, 6, 9.

In our next example, the ratio of R to C is 2 to 3.

And to complete our double number line, we have 2, 4, 6 and 3, 6, 9.

And in our final example, the ratio of R to C is 5 to 7 and our double number line is 5, 10, 15, 7, 14, 21.

Well done if you filled in the gaps to the ratio and a double number line in this way.

Let's have another check for understanding.

For every nine ticks there are three crosses and we have Alex and Aisha saying two different things.

Alex says "All these ratios are the same.

" And Aisha says "All these ratios are different as they have different numbers.

" Who is correct?

Can you explain?

Pause here while you do this.

So the answer is that Alex is correct because when simplified, all the ratio are the same.

If we divide the bar model by 2, we have one cross and three ticks.

If we divide the ratio written in words by 3, for every nine ticks there are three crosses, we get for every three ticks there's one cross looking at the ratio table, ticks to crosses 30 to 10.

If we divide this by 10, we get 3 to 1.

And if we divide the ratio notation a 15 to 5 by 5 we get 3 to 1.

So when simplified all the ratios are the same, they all come down to 3 to 1.

Let's have another check for understanding.

Is 4 to 6 the same as 2 to 3?

Jun says "These ratios are the same because 4 multiplied by 1.

5 equals 6, 2 multiplied by 1.

5 equals 3.

" Lucas says "These ratios are the same because 4 divided by 2 equals 2 and 6 divided by 2 equals 3.

" And Aisha says "These ratios are different because they have different numbers.

" Who is correct?

And can you explain?

Pause here while you do this.

Well done if you said that Jun and Lucas are correct, Jun has identified that the multiplier between the parts is 1.

5.

This means they're equivalent ratios.

And Lucas has simplified the parts using the highest common factor of 2.

Aisha says that "These ratios are different because they have different numbers.

" However, there are an infinite number of equivalent ratios.

All use different numbers, but the multiplicative relationship is the same.

This is why Jun and Lucas were correct.

And now it's time For your first task.

I'd like you to fill in the table.

The first has been done for you.

So we have the bar model of BBBH, the ratio notation, B to H, 3 to 1, and in words, ratio of B to H is 3 to 1.

I would like you to fill in the table now for the ratio notation and ratio in words of squares to circles, T-shirts to shorts and carrots to apples to oranges.

Enjoy your task and I'll see you when you're finished.

And now for the next part of your task, all these represent the same ratio, fill in the blanks, pause here while you do this.

And for the next part of your task, I'd like you to match the ratio table with the ratio notation.

Pause here while you do this.

For the next part of your task, I would like you to work out the equivalent ratios equivalent to 4 to 5.

Pause here while you do this.

And now for the final part of your task, I would like you to work out the equivalent ratios to the ratio of 2 to 3 to 1.

Pause here while you do this.

It's good to be back with you.

So how did you get on with that multi-stage task?

Let's have a look at how you filled in the table.

Squares to circles, 1 to 2, and in words, ratio of squares to circle is 1 to 2.

T-shirts to shorts, 2 to 3.

Ratio of T-shirts to shorts is 2 to 3 and carrots to apples to oranges.

The ratio notation is 1 to 2 to 2.

And in words, ratio of carrots to apples to oranges is 1 to 1 to 2.

How did you get on with filling in the blanks for all of these different representations of ratio?

So we have pounds to dollars is 2 to 3.

The double number line 2, 4, 6, 3, 6 9.

In a bar model, we have two pound signs, three dollar signs, and we can seat £12 to $18.

How did you get on with matching the ratio table with the ratio rotation.

We have 3:1, 2:3, 3:5, 5:1.

And for 24 to 80 we divide by 8 which gives us 3 to 10, 15 to 25 we divide by 5, 3 to 5, 25 to 5 we divide by 5, 5 to 1, and did you match the ratio tables with the ratio notation in these ways?

Well done if you did.

How did you get on with working out the equivalent ratios equivalent to 4 to 5?

If we multiply by 2 it gives us 8 to 10.

We multiply by 5 we have 20 to 25.

If we multiply by 8 of 32 to 80, if we divide by 4, 1.

25, if we divide by 10, we have 0.

4 to 0.

5.

And if we divide by 2 we have 2 to 2.

5.

Well done if you worked out the equivalent ratios in this way.

And working out the equivalent ratios to 2 to 3 to 1.

Multiplying by 4 gives us 8:12:4.

Multiplying by 11 gives us 22:33:11.

Dividing by 2 gives us 1:1.

5.

:0.

5.

Multiplying by 10 gives us 20:30:10 divided by 3 gives us two thirds to one to one third and multiplying by 5 gives us 10:15:5.

Well done if you worked out the equivalent ratios in this way.

Well done for having a go at this task, and now we're onto our next learning cycle, comparing different representations of ratio.

Knowing the different forms of writing ratios is important as it deepens understanding of how the multiplicative relationship can be seen.

Regardless of how the ratio is represented, the multiplicative relationship is the same for all equivalent ratios.

They represent the same proportion.

Can you think of some advantages and disadvantages with using each type of ratio Representation?

So we have bar model, graph, ratio table, double number line and ratio notation.

Pause here and share with someone what are the advantages and disadvantages with using each type of representation.

Perhaps you said something like this.

Bar model, advantages, good visual representation.

Disadvantages, takes a little time to draw.

Graph, advantages, shows lots of equivalent ratios.

Disadvantages, accuracy depends on axes.

Ratio table advantages, multipliers can be easily found.

Disadvantages, can lose the visual relationships.

Double number line shows the relationships well.

Disadvantages, difficult to get the scale correct, and ratio notation, advantages, efficient and concise.

Disadvantages, harder to see relationships.

Well done if you completed the table in this way.

A chocolate bar is made up of three parts raisins and four parts chocolate.

How many grams of raisins will be in a 35 gram chocolate bar?

Pause here and share with someone.

Using a ratio table, 3 raisins, 4 chocolate, total, 7.

And if our chocolate bar is 35 grams, that means we've multiplied the 7 by 5 to get to 35.

So we multiply the raisins by 5 which will give us 15 and chocolate, 4 multiplied by 5 will give us 20.

So 15 grams will be raisins.

And using a ratio notation, 3 to 4, total, 7, total of a chocolate bar is 35 grams, we've multiplied by 5 so we multiplied with 3 by 5 giving us 15 and a 4 by 5 giving us 20.

So that's 15 to 20.

15 grams will be raisins.

Let's have a check for understanding.

An orange smoothie is made up of two parts orange juice and three parts milk.

Laura only has 120 milliliters of milk.

How much orange juice does she need?

So we've got orange to milk, 2 to 3, milk, we have 120 mils.

3 has been multiplied by 40 to get to 120.

So we need to do the same to the 2 which gives us 80.

Laura needs 80 milliliters of orange.

And now it's your turn.

To make a glue, five parts are PVA and two parts are water.

Andeep only has 20 milliliters of PVA.

How much water should he add to make the glue?

Pause here while you work this out.

Perhaps you worked out in this way.

PVA to water, 5 to 2, Andeep has 20 milliliters of PVA, to get to that, we've multiplied the 5 by 4.

So we need to do the same to the 2, 2 multiplied by 4 gives us 8, Andeep needs 8 milliliters of water.

Well done if you came to this answer.

Let's have another check for understanding.

In a bag there are only triangles and squares.

The ratio of triangles to squares is 4 to 1.

There are 40 shapes in total.

Here's Sam, "Triangles to squares, 4 to 1, 40 to 10.

There are 40 triangles and 10 squares.

" And Sofia, "Triangles to squares, 4 to 1 which gives us a total of 5, triangles to squares, 32 to 8, the total being 40, there are 32 triangles and 8 squares.

" Who is correct?

Explain where the other people made their mistake.

Pause here while you do this.

Well done if you said Sofia is correct, Sam misunderstood, as there a total of 40 shapes, not 40 triangles.

And now it's time for your task.

I would like you to fill in the blanks to complete the work.

Pause here while you do this.

Next, in a bag, there are only stars and circles.

The ratio of stars to circles is 2 to 3.

Here are your questions.

A, if there are 20 stars, how many circles are there?

And B, if there are 65 shapes in total, how many stars are there?

Pause here while you answer these questions.

Three, the chocolates in this box either have red or blue wrappers.

The ratio of red to blue is 2 to 1.

Answer these questions.

A, if there are 10 red chocolates, how many blue chocolates are there?

B, if there are 10 blue chocolates, how many red chocolates are there?

C, is it possible for there to be 10 chocolates in total?

Why or why not?

And D, there are 10 more red chocolates than blue.

How many are there in total?

Pause here while you answer these questions.

So how did you get on with your task, filling in the blanks to complete the work.

Cheese and tomato, to get from 3 to 4 to 15 to 20, we multiplied by 5, yogurt to strawberry, 5 to 9, we multiplied by 4 which gives us 20 to 36.

A total of 56 milliliters means there is 20 milliliters of yogurt and 36 milliliters of strawberries.

And final example stars to ticks to squares, 2 to 3 to 1 multiplied by 6 gives us 12 to 18 to 6.

A total of 36 shapes means there are 12 stars, 18 ticks and 6 squares.

Well done if you filled in blanks in this way.

How did you get on with answering these questions?

Our ratio of stars to circles is 2 to 3.

If there are 20 stars how many circles are there?

Stars to circles, 2 to 3, you need to multiply by 10 to give us 20 to 30.

So there are 30 circles.

And stars to circles, 2 to 3 is a total of 5, multiply by 13 to get us to the total of 65, which gives us a ratio of 26 to 39.

So there are 26 stars in total.

And chocolates in the box with a ratio of red to blue, 2 to 1.

If there are 10 red chocolates, how many blue chocolates are there?

Red to blue is 2 to 1, to get us to 10 red we've multiplied by 5, so if we multiply the one blue by 5, we know that there are 5 blue chocolates.

If there are 10 blue chocolates, how many red chocolates are there?

We know the ratios 2 to 1, so with 10 blue chocolates we've multiplied by 10, so if we multiply the 2 by 10, that gives us 20, 20 red chocolates.

Is it possible for there to be 10 chocolates in total?

Why or why not?

No, as the total parts for chocolates is 3, and 10 is not a multiple of 3.

Every two red needs one blue, meaning that chocolates are added in groups of three.

And finally there are 10 more red chocolates in blue.

How many are there in total?

So we've got 2 to 1.

If we multiply by 10, that gives us 20 to 10.

There are 20 red and 10 blue.

This makes a difference of 10 chocolates between red and blue.

And that gives us 30 chocolates in total.

Well done if you answered in this way.

Well done for having a go at this task.

In our lesson, ratio language and notation, we've covered the following.

We use symbols and notation because they're easier to read and understand.

They're concise and take up less space.

They can be used to represent complex concepts and it allows mathematical ideas to be communicated more effectively than words.

Ratio notation allows us to represent bar models, ratio tables, graphs, and double number lines in a concise and efficient way.

Well done everyone for having a go at this lesson.

It was great to explore ratio and language and notation together.

I look forward to seeing you at another lesson soon.

Bye for now.