Lesson video

Hi everyone.

My name is Ms. 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 part, but I will be here to help.

It's gonna be a great and fun lesson, so let's make a start.

Hey everyone.

Under the unit "understanding multiplicative relationships, fractions, and ratio" we'll be looking at equivalent multiplicative relationships, and by the end of the lesson, you'll be able to appreciate that there are an infinite number of pairs of numbers for any given multiplicative relationship.

Now, the key words we'll be looking at is proportionality.

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

Now, two fractions are equivalent if they have the same value.

We'll look at these words during our lesson.

Our lesson today will consist of two parts.

The first will be looking at equivalent ratios, and the second will be using equivalent ratios.

So let's make a start on equivalent ratios.

I want you to have a look at these bar models.

What is the same, and what is different with the following ratios? So you can give it a go, and press pause for more time.

Well, hopefully you spotted, if I were to write it in a ratio table, the first bar model shows, for every three T-shirts, there are nine shorts.

The second bar model shows, for every two T-shirts, there are six shorts.

And the last bar model shows, for every one T-shirt, there are three shorts.

All these proportions are the same.

In other words, for every one T-shirt, there are three shorts.

You can see it here, and here.

For every one T-shirt, there's three shorts.

For every one T-shirt, there's three shorts.

As well as here.

One T-shirt, we have three shorts, one T-shirt, we have three shorts, and for one T-shirt, we have three shorts.

So all the proportions are exactly the same.

The total clothing, however, is different.

Well done if you got that one right.

Now what I want you to do is have a think, are there any more equivalent ratios out there to T-shirts and shorts? Have a little think.

You could have chosen 10 T-shirts to 30 shorts, 12 T-shirts to 36 shorts, 50 T-shirts to 150 shorts.

There's an infinite number of equivalent ratios, but they all show the same proportion.

In other words, for every one T-shirt, there are three shorts.

However, what I want to do, is I want you to have a look and see, how do we know if the ratios are equivalent? For example, here is a bar model and a ratio table.

How can you identify if these are equivalent? Well, hopefully you can spot, there are a few different ways to identify if they're equivalent.

You can use the multiplier.

In other words, the multiplier from apples to bananas will always be 1.

5.

If that multiplier is the same or constant, then we know the ratio's the same.

You could also look at multiplying or dividing apples and bananas by a constant, and it gives you the same proportion.

What is really important to remember is that all the ratios simplify.

In this case, for every two apples, there are three bananas.

So simplifying ratios will really help you identify other equivalent ratios.

Now what I want you to do is have a look at a quick check question.

Here, which of the following are equivalent to, for every three stars, there are four moons? See if you can give it a go.

Well, for every nine stars, there are 12 moons.

This is because if you multiply for every three stars, there are four moons by three, you get for every nine stars there's 12 moons.

Next, for every 15 stars, there are 20 moons.

They're in the same ratio, because if you multiply by five, you get an equivalent ratio.

Next, for every one star there are three moons.

Well, if you divide three stars by three, you get one star.

Four divided by three does not give you three moons, so that is not equivalent.

D, for every 1.

5 stars, there are two moons.

Yes, because if you divide by two, or multiply by one half, it gives you an equivalent ratio.

Well done if you got this one right.

Now what I want to do is have a look at two pairs of numbers, and the multiplicative relationship between them.

So I'm going to look at six to 18, and 24 and 72.

Now, hopefully you can spot, the multiplier is the same.

72 ÷ 24 is three, and the multiplier between 18 and six is also three.

Now let's have a look at the ratios.

Let's say A is six and B is 18, and A is 24 and B is 72.

We have equivalent ratios here, because we have equivalent multipliers.

Now let's have a look at another check.

Here are two pairs of numbers.

Do you think the ratios are the same? See if you can give it a go.

Press pause if you need more time.

Well done.

Yes, they are the same, because six to nine has exactly the same multiplier as 12 to 18.

Well done if you got that one right.

You can find equivalent ratios using the same multiplier within the ratio, or using the same multiplier on each part between the equivalent ratios.

For example, 10 to four.

Well, we know the multiplier would be two fifths.

If all of these have the same multiplicative relationship, you can just use the multiplier within the ratio to find the missing values.

So that means 20 multiplied by two fifths, two divided by two fifths, 15 multiplied by two fifths, would all enable us to calculate those missing values.

So we can use that multiplicative relationship between the ratios to find them.

If all of these have the same multiplicative relationship, you could just use the multiplier within the ratio to find the missing numbers.

Knowing that they have the same multiplier, we're able to work out the other missing value, because we have equivalent ratios, as they all have the same multiplier.

So in other words, we can multiply 10 by two to give 20, so we multiply four by two to give us eight.

You could multiply the four by a half to give two.

So multiplying the 10 by a half gives you five.

Let's have a look at the last one.

Well, for here, because these are equivalent ratios, you can pair up any pair to help you find the missing parts in the equivalent ratio.

For example, you could use the five to two to help you find out the equivalent ratio, by simply multiplying by three, multiplying by three we have six, thus giving us these answers.

So when calculating mentally, you may prefer to work either way.

Either multiplying by two fifths, or by multiplying equivalent ratios by a constant multiplier.

These are still equivalent, as they are using the multiplier within the ratio.

Now let's have a look at a check.

These pairs and numbers are all in the same ratio, and I want you to find the missing numbers.

See if you can give a go, and press pause if you need more time.

Well done.

Let's see how you got on.

Well, the missing numbers here are 15, 30, and 7.

5.

Lots of different ways you could do this.

You could multiply the three by three to give you nine, and the five by three to give 15.

You could multiply the five by 10 to give you 50, and the three by 10 to give you 30.

You could multiply the three by 1.

5 to give you 4.

5, and the five by 1.

5 to give you 7.

5.

Lots of different ways.

Great work if you got this one right.

Well done.

So let's have a look at a check question.

Jun and Alex both draw the following on squared paper.

Is the ratio of shaded to unshaded the same for both pictures? Have a little think, and press pause if you need more time.

Well done.

Well, hopefully you spotted, yes, they are the same.

For Jun, for every eight shaded, there are 12 unshaded, and for Alex, for every two shaded, there are three unshaded.

If you divide Jun's ratio by four, you get Alex's ratio.

Thus, the ratios are the same.

Great work if you got this one right.

Well done.

Now it's time for your task.

What I want you to do is tick which ratio table is equivalent to, for every three ticks, there are four crosses.

See if you can give it a go, and press pause if you need more time.

Well done.

Let's move on to question two.

For question two, it wants you to pair up the equivalent function machines left to right, and identify the multiplier.

See if you can give it a go.

Press pause if you need more time.

Well done.

Moving on to question three, shade in to make the letter L, using the ratio three shaded to one unshaded.

For B, shade in to make the letter H using the ratio 12 shaded to 13 unshaded.

And for C, shade in to make the letter C, using the ratio one shaded to one unshaded.

See if you can give it a go, and press pause for more time.

Well done.

Let's move on to question four.

Question four wants you to identify the multipliers for the following function machines.

See if you can give it a go, and press pause for more time.

Great work, everybody.

Let's go through these answers.

For question one, your first ratio table should be three ticks to four crosses.

That's directly the same as for every three ticks, there are four crosses.

The next equivalent ratio is 1.

5 ticks to two crosses, because we multiplied our ratio by a half.

The next equivalent ratio is 15 to 20, because we multiplied our ratio by five.

The last equivalent ratio is nine ticks to 12 ticks, because we multiplied our ratio by three.

Great work if you got this one right.

For question two, let's identify those multipliers.

Well, we have a multiplier of two for four to give eight, and two to give four.

So that means these are equivalent.

Next, we have a multiplier of three.

Nine to give 27, and three to give nine.

So we have another equivalent ratio.

Next we have 10 to 40, and two to eight, as they have the multiplier of times by four.

Lastly, 30 times a half gives 15, is equivalent to 36 multiplied by half, which is 18.

Well done if you got that one right.

For question three, let's see how you got on.

You should have got this L shape.

You should have got this H shape.

And could have got this C shape.

Well done, as long as the ratio is the same.

Lastly, for question four, you should have had these missing values and multipliers.

Great work if you got this one right.

Excellent work everybody.

So let's move on to equivalent ratios.

Now, knowing that there are an infinite number of equivalent ratios allows us to compare to the whole.

For example, the only pets in a street are cats and dogs.

For every one cat, there are two dogs, and if there are 33 pets in total, how many cats are there? Well, let's put this in a ratio table.

In a ratio table, we have one cat for every two dogs.

We know there's a total of three animals.

Now, we're interested in a total of 33 pets.

At the moment our total is three.

So what's our multiplier? It has to be 11.

So multiplying each part by 11 tells us how many cats we have, 11, and how many dogs we have, 22.

So that means we have 11 cats out of 33 pets.

It's clear the equivalent ratios from our ratio table states that for every cat, there are two dogs, and this is exactly the same as, for every 11 cats, there are 22 dogs.

It has the same multiplier.

So if you're thinking about it as a formula, cats multiplied by two is equal to dogs.

Now, given fractions, decimals, percentages, and ratio, all represent proportion, we can interchange between them to help us solve problems. So let's have a look at this again.

Using the ratio table, what fraction are cats? Well, we can see it's one third, because one cat out of a total of three.

So to find the total number of cats when there are 33 pets in total, what do you think we need to change? Well, if we write an equivalent fraction with a denominator of 33, we're simply multiplying one third by 11 over 11, thus giving us 11 cats out of the total pets.

So we still have 11 cats out of the total pets.

So you can see how powerful fractions, decimals, percentage, and ratios, can all represent exactly the same proportion.

Now let's have a look at a quick check.

Aisha and Sofia need to shade a grid containing 25 small squares.

According to the ratio, for every one shaded, there are four unshaded.

You can see what Aisha did, and you can see what Sofia did.

Who's correct? Have a little think, and press pause for more time.

Well, let's see how you got on.

Well, hopefully you spotted they're both correct.

Aisha is stating how many squares she needs to shade, and Sofia is stating the proportion of squares that need shading.

Aisha used a ratio table to show a total of 25 squares requires five shaded and 20 unshaded, and Sophia is showing, well, if the fraction of shaded is one fifth, to make the total number of squares 25, in other words, the denominator of 25, we multiply by five over five, thus giving us five over 25 is the proportion of squares that need to be shaded.

Really good work if you got this one right.

Now it's time for your task.

Using the ratio table, fill in the blanks.

Press pause, because you'll need more time.

Great work.

Let's see how you get on with question two.

Question two says, to make screed, the ratio of sand to cement is for every one kilogramme of sand, use four kilogrammes of cement.

I want you to show this in a ratio table.

I also want you to have a look at, how many kilogrammes of sand will Andeep need if he has 15 kilogrammes in total? And also, what fraction of the screed is sand? Give this a go.

Press pause if you need more time.

Well done.

Question three says, Jacob spilled ink all over the maths question, and his working out, but he's correctly shaded in the grid.

I want you to work out the maths question, and his working out.

See if you can give it a go, and press pause for more time.

Great work.

Let's see how you get on with question four.

Question four states that the ratio of the three different panels of fencing are, panel A multiplied by 1.

5 equals panel B, and panel A length multiplied by two equals panel C length.

The total length of fencing is 36 centimetres.

Can you work out the length of each panel? 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, question one.

Hopefully you've spotted.

You can fill in your ratio table by identifying the total is five, and the equivalent ratio is 20 stars to 30 hearts is 50 in total.

Therefore, for every two stars, there are three hearts, and the fraction which are stars is two thirds.

And when there are 50 shapes, there are 20 stars, and 30 hearts.

For B, filling in our ratio table, we've got a total of 11, and an equivalent ratio of 48 boys and 40 girls, giving us a total of 88 pupils.

This means, for every six boys, there are five girls.

The fraction, which is simplified, is five elevenths is the fraction of girls, and when there are 88 pupils, 48 are boys and 40 are girls.

Really well done if you got that one right.

For question two, in a ratio table, screed looks like this.

Sand is one part, cement is four parts, giving a total of five parts.

Now, Andeep has a total of 15 kilogrammes, so that means writing our equivalent ratio, we have three parts sand, 12 parts cement.

That means he needs three kilogrammes of sand.

But what fraction of the screed is sand? You can use any equivalent fraction given here, and you'll still find out the simplified equivalent fraction is one fifth.

For question three, Jacob spilled ink all over the maths question and the working out, but the shading was correct.

Hopefully, you spotted this is our working out and the original question.

Great work if you got this one right.

Lastly, we have to identify, what is the length of each panel? Well, drawing a ratio table, I have panel A as two, panel B as three, and panel C as four.

You can identify any equivalent ratio.

I've just chosen two, three, and four.

This means I have a perimeter of nine, but I want a total fencing of 36, so my multiplier is four.

Multiplying each length by four gives me eight centimetres is length A, 12 centimetres is length B, and 16 centimetres is length C.

Really well done if you got this one right.

So in summary, proportionality means, when variables are in proportion, they have a constant multiplicative relationship.

Given fractions, decimals, percentages, and ratio, all represent proportion, we can interchange between them to help us solve problems. Knowing that there are an infinite number of equivalent ratios allows us to compare to the whole, and problems can be solved more easily.

Great work, everybody.

Well done.