Lesson video

Hi everyone.

My name is Miss Coo 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 under the unit understanding multiplicative relationships, fractions and ratios, we'll be looking at ratio language and notation.

And by the end of the lesson you'll be able to 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.

We're looking at those keywords proportionality and ratio again.

Proportionality means when variables are in proportion if they have a constant multiplicative relationship and ratio shows the relative sizes of two or more values and allows you to compare a part with another part in a whole.

Today's lesson will be broken into two parts, ratio language and notation first and then comparing different representations ratio.

So let's make a start.

Notation in mathematics is really important.

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

They are 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.

This can be set with ratio notation too.

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

For every three circles there are four squares.

I'm gonna represent it as a graph first.

Hopefully you can spot for every three circles there's four squares.

So this means for six circles there are eight squares, so on and so forth.

Now I'm gonna represent it as a bar model.

Three circles to four squares.

Represent it as a ratio table, for three circles, there's four squares.

Now we're going to write it in a concise way, and a concise way to write a ratio is to use a colon and we say it as the ratio of circles to squares is three to four.

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

So what I want you to do is have a look at this table.

I've done the first one for you.

So you can see the bar model and we write it as triangles to squares is one to three, and in words the ratio of triangles to squares is one to three.

See if you can fill in our table identifying the ratio notation and how we say it in words.

Give it a go and press pause if you need more time.

Well done.

Let's see how you got on.

Well, cats to dogs is one to one.

We say it as ratio of cats to dogs is one to one.

Apples to bananas is two to three.

So the ratio of apples to bananas is two to three.

The ratio of stars to hearts to circles is one to one to two.

Ratio stars to hearts, to circles is one to one to two.

Great work if you got this one right.

Now, the order of the ratio must match between the words and notation.

So you might notice using the same bar model, I'm going to use the ratio squares to triangles.

So it's three to one.

The ratio is exactly the same, but the word and notation is slightly different.

So it's important that you understand the wording and the notation must always be aligned.

So now let's have a look at another check.

I want you to fill in the gaps for the ratio and the double number line.

We're using that concise notation to show the ratio of R to C is one to three.

How do you think we fill that in on the double number line? The ratio of R to C is unknown, but have a look at that double number line and the ratio R to C is unknown, but have a look at that double number line.

See what you can figure out.

Press pause if you need more time.

Great work.

Let's see how you got on.

Well, if we know the ratio of R to C is one to three using our double number line, we still have our equivalent ratios.

One to three is the same as two to six, which is the same as three to nine.

Now the ratio of R to C is unknown, but look at our double number line, it's two to three.

So using our double number line and that equivalent ratios we know two to three is the same as four to six, which is the same as six to nine.

Next we don't know the ratio, but you can spot an equivalent ratio using our double number line 10 to 14.

If you know we have three intervals there and the middle number is 10, it must be five, 10 to 15, seven, 14, 21 because of those equal intervals, that means the ratio of R to C must be five to seven.

Well done If you've got this.

Let's have a look at another check.

For every nine ticks there are three crosses.

I want you to identify who is correct and explain.

Alex says all these ratios are the same and Aisha says all these ratios are different because they've got different numbers.

Who do you think is correct and I want you to explain.

Well done.

Let's see how you got on.

Well, Alex is correct because when they're simplified all the ratios are the same.

So let's have a look.

The bar model, if you were to divide it by two, it gives you one cross is three ticks and if we divide for every nine ticks there are three crosses by three, it gives you for three ticks there's one cross.

Looking at our ratio table, if you divide by 10, we have three ticks is one cross.

Ticks to crosses is 15 to five divided by five gives us three to one.

They are all the same ratio.

Yes, they use different numbers, but remember there's an infinite number of equivalent ratios out there, but they all represent the same proportion.

Well done if you got this one right.

Now let's have a look at another check.

We have four to six is the same as two to three.

Is that right? Jun says these ratios are the same because four times 1.

5 is six, two times 1.

5 is three.

Lucas says these ratios are the same because four divided by two is two and six divided by two is three.

Aisha says these ratios are different because they've got different numbers.

Who do you think is correct and explain? Well, hopefully you've spotted Jun and Lucas are both correct.

Jun has identified the multiplier between parts is 1.

5.

So multiplying four by 1.

5 gives six and the same multiplier of 1.

5, two multiplied by 1.

5 is three.

This is the same multiplier, so they equivalent ratios, but Lucas has simplified the parts using the highest common factor of two.

So four to six can be simplified to give two to three.

Both are correct.

Well done if you spotted this.

Going back to what Aisha said, remember there's an infinite number of equivalent ratios, all using different numbers, but the multiplicative relationship is the same.

Now it's time for your task.

I want you to fill in the table and the first one has been done for you.

So you can give it a go.

Press pause for more time.

Well done.

Let's move on to question two.

Question two shows five different ways of showing the same ratio.

Can you fill in that bar model, that ratio table, that double number line, ratio using the notation, using the completed graph? See if you can give it a go.

Press pause if you need more time.

Well done.

Let's move on to question three.

Match the ratio table with the ratio notation.

I'll give you a hint.

You might need to simplify some of these ratios down a little bit more to find the equivalent ratio.

Give it a go, and press pause if you need more time.

Well done.

Moving on to question four.

We have the ratio four to five.

What are the other equivalent ratios? Give it a go and press pause for more time.

For question five, we have another ratio two to three to one.

Work out those other equivalent ratios.

Press pause for more time.

Well done.

Let's go through these answers.

Well for question one you should say three to one.

So the ratio books to eyes is three to one.

For the second one, one to two.

So the ratio of squares to circles is one to two.

T-shirts to shorts, we say two to three.

So the ratio of T-shirts to shorts is two to three.

Next, the ratio of carrots to apple to orange is one to two to two.

Great work if you've got that one right.

Now, let's identify the equivalent ratio using these five different representations.

Using the graph, hopefully you can spot, we have a ratio of two pounds to $3.

So I can fill in starting my double number line first.

Two to three is the same as four to six, which is the same as six to nine.

Now let's represent this as a bar model.

Every two pounds is $3, so that means 12 pounds is $18.

Incredible work if you've got this one right.

Question three, my advice was to simplify.

I'm going to start simplifying the second one first.

This simplifies to three to one.

The third one simplifies to two to three.

The fourth one simplifies to three to five, and the fifth one simplifies to five to one.

Now look for me at the other ratios beneath.

I can divide this by eight to give me three to 10.

The next one, divide by five gives me three to five.

And the last one divide by five gives me five to one.

Simplifying our ratios enable us to find the equivalent ratio.

So that means we have this pairing.

Massive well done if you've got this one right.

For question four we have a ratio four to five.

So what are the other equivalent ratios? Hopefully you spotted we times this by two, we times this by five, we times this ratio by eight, we divide the ratio by four or times by a quarter, we divide this one by 10 or times by one 10th, we divide by two or times by a half to give us 2.

5 here.

Well done if you've got this one right.

And for question five we have the ratio of two to three to one.

Let's find our equivalent ratios.

Multiplying by four gives us this.

Multiplying by 11 gives us this.

Multiplying by a half or divide by two gives us this.

Multiplying by 10 gives us this.

Dividing by three or times it by a third gives us this.

And multiplying by five gives us this.

Well done.

Great work everybody.

So let's move on to the last part of our lesson comparing different representations of ratio.

Now know the different forms of writing ratio is important as it deepens our understanding of how that multiplicative relationship can be seen.

But regardless of how the ratio represented, the multiplicative relationship is the same for all equivalent ratios and they represent the same proportion.

I want you to have a look at this table and I want you to think about some advantages and disadvantages with using each type of representation.

So you can give it a go and press pause for more time.

Well, let's have a look at some advantages first.

You might have said bar models are great visualisations.

Graphs show lots of equivalent ratios and ratio tables, well multipliers can be found easily.

The double number line is great 'cause it shows those relationships very well and ratio notation is efficient and concise, but some disadvantages.

Well, the disadvantage with a bar model is it does take a little bit of time to draw.

The disadvantage with the graph is the accuracy does depend on the axis.

The disadvantage with the ratio table is you can lose that visual relationship.

Double number lines, well it's difficult to get the scale correct and ratio notation, it is difficult to see those relationships.

So all these different representations have their own advantages and disadvantages.

Let's have a look at a quick check.

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

How many grammes of raisins will be in a 35 gramme chocolate bar? See if you can give it a go and press pause for more time.

Well done.

Let's see how you got on.

Well, I'm gonna use a ratio table and I'm gonna show three parts raisins to four parts chocolate.

So that means I have a total of seven parts.

Well I don't want a total of seven parts.

I want a total of 35 parts.

So what would my multiplier be? Well, I have to multiply each part by five, thus giving me 15 grammes of raisins to 20 grammes of chocolates.

So 15 grammes of raisins will be used.

Now I'm gonna use the ratio notation.

I'm going to show that raisins to chocolate is three to four, still identifying a total of seven.

I want a total of 35, so I need to multiply by five.

Thus giving me the ratio of 15 to 20.

I'm still getting 15 grammes will be raisins.

I really wanted to emphasise how you can work out a question using the ratio table or the ratio notation.

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

I'm going to do one on the left and I'd like you to do the one on the right.

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

Laura only has 120 millilitres of milk.

How much orange juice does she need? Well firstly, if we know the ratio of orange to milk is two to three, that means if she only has 120 millilitres of milk, what's the multiplier? Well, it had to be times by 40, so that means multiplying the orange by 40 gives me 80.

So how much orange does she need? She needs 80 millilitres of orange.

Now your turn.

Here a question states, to make glue, it's five parts of PVA and two parts are water.

Andeep only has 20 millilitres of PVA.

How much water should he add to make the glue? See if you can give it a go.

Press pause if you need more time.

Let's see how you got on.

Well, we know the ratio of PVA to water is five to two.

If he only has 20 millilitres of PVA, that means we have a multiplier of four.

Multiplying by four gives me eight parts for water.

So that means, Andeep needs eight millilitres of water.

Looking at another check question, here we have a question involving Sam and Sofia.

In a bag there are only triangles and squares and the ratio triangles to squares as four to one and there are 40 shapes in total.

Sam says triangles to squares is four to one, so 40 to 10 there are 40 triangles and 10 squares.

Sofia says triangles to squares is four to one, giving a total of five.

So the ratio would be 32 to eight, giving a total 40.

There's 32 triangles and eight squares.

Who's correct and explain where the other pupil made their mistake.

Well hopefully you've spotted Sofia is correct.

Sam misunderstood as there's a total of 40 shapes, not 40 triangles.

Great spot if you've got that one.

Now it's time for your task.

I want to fill in the blanks to complete the work.

Press pause for more time.

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

Question two states, in a bag there are only stars and circles and the ratio of stars to circles is two to three.

Now if there's 20 stars, how many circles are there? And if there's 65 shapes in total, how many stars are there? Give it a go.

Press pause for more time.

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

Question three says the chocolates in this box either have red or blue wrappers.

The ratio of red to blue is two to one.

Question three A says if there are 10 red chocolates, how many blue chocolates are there? B says, if there are 10 blue chocolates, how many red chocolates are there? C says, is it possible for there to be 10 chocolates in total? Why or why not? And D says there are 10 more red chocolates than blue.

So how many are there in total? So you can give this a go.

Press pause one more time.

Well done.

Let's see how you got on.

Well for question one, the ratio of cheese to tomato is three to four, but then we have an equivalent ratio of 15 to 20.

It does state that the total of 35 grammes means there's 15 grammes of cheese and 20 grammes tomato.

So let's identify a multiplier.

It should have been five.

Yoghourt to strawberry.

It's in the ratio of five to nine.

The multiplier is four, so that means we should have the ratio of 20 to 36.

This means a total of 56 millilitres means there are 20 millilitres of yoghourt and 36 millilitres of strawberries.

So for part C we have the ratio of star to tick to squares.

Now we have a multiplier six.

So that means we can work backwards.

So that means we know our ratio here must be three to three to one because when we multiply by six we get 12 to 16 to six, thus giving us a total shapes of 34, which means there are 12 stars, 16 ticks and six squares.

Well done if you got this one right.

Question two, in a bag, there are only stars and circles in the ratio of two to three.

If there's 20 stars.

Well if there's 20 stars, the multiplier for stars must be ten.

Two multiply by 10 is 20 stars.

So that means three multiplied by 10 is 30 circles.

Well done if you've got this one right.

B, if there are 65 shapes in total.

Well, stars to circles is two to three, giving a total of five.

To make 65 we multiply by 13, thus giving us 26 stars to 39 circles to giving us a total of 65.

That means there are 26 stars.

Great work if you got this one right.

Question three A we know chocolates and the ratio of two to one.

If there's 10 red chocolates, well, we multiply two by five to make 10.

So we multiply the blue by five.

So that means there are five blue.

If there are 10 blue chocolates, that means we multiply the blue by 10 to make 10 blue.

So we multiply the red by 10 to make 20 red.

Is it possible for there to be 10 chocolates in total and explain why? Well, no, because the total parts for chocolates is three, two, add one is three and 10 is not a multiple of three.

So for every two red need to have one blue, meaning that chocolates are added in groups of three.

Now there are 10 more red chocolates than blue chocolates.

So how many are there in total? Well, if there are 10 more red than blue, the difference between red and blue is one.

If we multiply this by 10, that means the difference between red and blue is now 10, thus giving us 20 red and 10 blue.

This gives us a difference of 10 chocolates between red and blue.

Great work everybody.

In summary, we use symbols and notation because they're easier to read and understand, they're concise and take up less space and they can be used to represent complex concepts and it allows mathematical ideas to be communicated more efficiently than words.

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

Great work everybody.

It was wonderful learning with you.