Lesson video

Hi, everyone, my name is Ms. Ku, and I'm really happy and excited to be learning with you today.

It's going to be a fun lesson full of some words maybe that you may or may not know, and we'll build on that previous knowledge too.

Super excited to be learning with you, so let's make a start.

In today's lesson from the unit Understanding Multiplicative Relationships, Fractions, and Ratio, we'll be checking understanding of multiplicative relationships.

By the end of the lesson, you'll be able to use proportionality as a comparison rather than just absolute values.

Today's lesson, we'll be looking at the keyword, starting with proportion.

And proportion is a part to whole, sometimes called part-to-part comparison.

And if two things are proportional, then the ratio of part to whole is maintained and the multiplicative relationship between part is also maintained.

We'll be looking at this during our lesson.

We'll be also looking at the word proportionality.

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

Same again, we'll be looking at this in our lesson.

Our lesson will be broken into two parts.

The first, we'll be comparing using proportion, and second, we'll be paying attention to that whole.

So let's make a start comparing using proportion.

Now, proportion is a part to whole, sometimes called part-to-part comparison, and you've seen many different types of proportions before.

For example, fractions, decimals, percentages, and ratios.

So what I'm gonna do, I'm going to show you this square.

What proportion of the whole square is shaded? I'd like you to tell me it as a fraction.

Well, hopefully you've spotted it's a quarter, and you can see this because there are four equal parts to our whole, and one part is shaded.

So that means the proportion which is shaded as a fraction is a quarter.

Now, let's have a look at the same square, and what we're going to do, I'm gonna shade this area in, and I'm gonna ask what proportion of the whole is shaded, but I'd like you to tell me it as a decimal.

Hopefully you've spotted, well, we have our whole here.

It's not very clear as a decimal.

So what I'm going to do, I'm going to draw lots of little squares.

I have a hundred of these squares because I've made my big square at 10 by 10.

That means one little tiny square is 0.

01.

Now, all I need to do is count how many of these shaded squares there are, and there are 25.

So that means there are 25 lots of 0.

01.

So the proportion that is shaded as a decimal is 0.

25, where we've represented the whole as one.

Now let's have a look at another type of proportion.

Here, I'm going to shade in this area and I want you to think what is this as a percentage? Well, hopefully you can spot, I've used this 10 by 10 grid again, so identifying what one tiny little square is is going to be 1%.

So if one tiny little square is 1%, what percentage is shaded? Well, 25% because if you count, there are 25 of those little tiny squares out of 100, which represents our whole.

So that means 25% is shaded.

Now, let's have a look at looking at the proportion that is shaded, but as a ratio.

Well, for every one shaded, there are three unshaded.

This was a nice little activity just to really show you that there are quite a few different ways in which we can show proportion, fraction, decimal, percentage, and ratio.

And all of these represent the same proportion.

Some people find fractions is an easier way to write and compare proportions, and we'll be looking at that.

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

Here, Jun and Alex want to find out what proportion is four pounds from 24 pounds.

Whose approach is correct? I want you to press pause and have a good look at their working out.

Well done.

So let's see how you got on.

Well, they're both correct.

What Jun did is he's written it as a fraction, four pounds over our 24 pounds, and then he's identified the highest common factor, which is four, thus changing the fraction to one multiplied by four over six multiplied by four, which then simplifies to one sixth.

Therefore, he says four pounds is one sixth of 24.

Now, Alex has used a bar model.

He's looked at that 24 pound representing the whole and thought, "Well, how many equal four pounds make that 24 pounds?" Well, from the bar model, you can see six lots of four pounds make that 24 pounds.

So therefore, because he sees four pounds fits exactly into 24 pounds six times, this means four pounds is one sixth of 24.

These are two great ways to identify what proportion is four pounds from 24 pounds.

Well done if you've got this one right.

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

Here, it says a full-Time Oak teacher receives 4,000 pounds a month and she gets a pay rise, so earns 4,500 pounds a month.

Now, a part-time Oak teacher receives 400 pounds a month, and he gets a pay rise and receives 800 pounds a month.

We need to have a look to see who has the best pay rise.

Think about the method that Jun and Alex did in the previous check.

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

Great work.

So let's see how you got on.

Well, first of all, let's have a look at this full-Time teacher.

I'm gonna put it as a bar model to begin with.

We know the full-time teacher received 4,000 pounds a month.

Then they receive a pay rise, which is now 4,500 pounds a month.

So we need to have a look at that 500 pounds and think, okay, what proportion is that 500 pounds from 4,000 pounds? Well, if you look at 500 pounds, how many 500 pounds fit exactly into 4,000 pounds? And hopefully, you can spot it's eight times.

So using our bar model, you can actually see 500 pounds represents one eighth of our 4,000 pounds.

So that means the full-time teacher got an eighth of a pay rise.

Another way to look at it is using fractions.

Well, we know it was a 500-pound pay rise over the original salary of our 4,000 pounds.

Simplifying our fraction, then we can spot a highest common factor of 500, thus giving us a fraction of one eighth.

Same again, we've identified the full-time teacher receives a pay rise of an eighth of her salary.

Now, let's have a look at the part-time teacher.

Well, we know the part-time teacher received 400 pounds, and the pay rise was also 400 pounds.

Well, how many times does 400 pounds go into 400 pounds? Well, it's one time.

So that means the fraction the part-time teacher receives from 400 pounds is one whole, which is exactly the same as a 100% pay rise.

Writing this as a fraction, 400 over 400, well, 400 over 400 is the amount received over what the original salary was is one whole, which is 100% So that means the part-time teacher receives a 100% or one whole of a pay rise of his salary.

So therefore who receives the best pay rise? Well, it's the part-time teacher.

This was a great example to show you don't focus on the numerical increase, it's the proportion that you need to look at.

The full-time teacher received a 500-pound pay rise, which looks like a lot compared to a 400-pound pay rise, but actually, in proportion to their salary, the part-time teacher got 100% pay rise while the full-time teacher only got one eighth, which is 12.

5% pay rise.

Well done if you've got that one right.

Now, let's have a look at your task.

I want you to use the following bar models and subdivide to work out what is three pounds as a proportion of the whole amount.

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

Well done.

Let's move on to question two.

Question two wants to use your fractions.

What proportion of 12 pounds is out of 40 pounds? What proportion of 12 pounds is out of 15 pounds? And what proportion of 12 pounds is out of 12 pounds? See if you can give it a go, and press pause if you need more time.

Well done.

Let's look at question three.

Question three says test A is outta 20, and Laura gets 12 out of 20 correct.

Test B is outta 50, and Sofia gets 40 outta 50 correct.

Who got the highest proportion of answers correct? See if you can give it a go, and press pause if you need more time.

Well done.

Let's look at question four.

Question four shows a table.

Which pupil got the best allowance increase, and I want you to explain why.

We have Aisha, who used to get 15 pounds a month, but now it's increased by six pounds a month.

Andeep used to get 10 pounds a month, and now it's increased by 2.

50 pound.

Izzy used to get 12 pounds a month, now it's increased by four pounds.

And Jacob used to get 30 pounds a month, and now it's increased by six pounds.

So who got the best allowance increase? Okay, so let's have a look at these answers.

So question 1A, hopefully you can spot, we needed to identify what is three pound as a proportion of 30 pounds.

Well, 10 lots of three pounds fit into 30 pounds, so that means three pound represents one 10th of 30 pound.

For B, three pound as a proportion of 12 pound.

Well, you can see from our bar model, three pounds fits exactly into that 12 pounds four times, so that means three pounds represents a quarter.

And for C, three pounds fits into that 15 pounds exactly five times, so that means three pounds is one fifth of 15 pounds.

Well done if you got that one right.

Now, let's have a look at these fractions.

Simplifying, what proportion of 12 is outta 40? Well, simplifying our fractions, we write it as 12 over 40, giving us three outta 10.

For B, what proportion of 12 pounds is out of 15 pounds? Writing it as our fraction, 12 over 15, simplifies to four over five.

What proportion of 12 pounds is out of 12 pounds? 12 over 12 simplifies a simplified fraction, writes as one.

Well done if you've got this one right.

For question three, who got the higher proportion? Well, let's have a look at Laura first.

Laura got 12 outta 20.

This simplifies to three fifths.

This means that Laura got three fifths of the test correct.

Now, Sofia got 40 outta 50, so this represents the simplified fraction four fifths.

So that means this represents the proportion that Sofia got right.

So who got the highest proportion of answers right? Well, it's Sofia.

Sofia got the highest proportion of answers correct, as she got four fifths of the test right, whilst Laura got three fifths of the test right.

Well done if you got that one right.

Question four, we had to work out the fractional increase.

So Aisha got six over 15, which is an increase of two fifths of her allowance.

Andeep got 2.

5 over 10, which simplifies to five over 20, which is a quarter, so he got a quarter of his allowance increased.

Izzy got four over 12 or a third of her allowance increased.

And Jacob got six over 30, or one fifth of an allowance increase.

So who got the best allowance increased? Well, it's Aisha.

Aisha got two fifths of her allowance increased, which is a bigger proportion than a quarter, a third, and one fifth.

Well done if you've got that one right.

Great work, everybody.

So let's have a look at paying attention to the whole.

Now, when using proportionality as a comparison, attention has to be paid to the whole rather than assuming the measure of proportion is the absolute value.

For example, if Alex gets 50% in test A and Aisha gets 30% in test P, can we identify who's done better? Well, we can't really because there's not enough information to make a comparison, as we don't know the total marks in each test.

If I add the fact that test A had four questions and test B had 100 questions, is it easy to identify who's done better? Well, yes, it is.

Aisha got more answers correct, as she did more questions.

Granted it was 30% that she got correct, 30% of 100, which is 30 questions correct.

Alex got 50% correct, but it's after four questions, so that means he only got two questions right.

So it's really important to pay attention to the whole rather than focusing on the proportion.

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

Lucas and Jun both have a chocolate bar each.

Lucas has a 60-gram chocolate bar and Jun has a 30-gram chocolate bar.

Now, Lucas eats one third of his chocolate bar and June eats a half of his chocolate bar.

Izzy says, "Jun's eating more chocolate than Lucas because a half is greater than a third." I want you to have a think and explain why's Izzy wrong? Well, hopefully you can spot the proportion of Jun's chocolate bar is greater than the proportion of Lucas's chocolate bar, but because Lucas's chocolate bar is twice the size of Jun's chocolate bar, that means Lucas has eaten more.

If you think about it, Lucas has eaten a third of a 60-gram chocolate bar, so that means he's eaten 20 grammes.

Jun has eaten half of his chocolate bar, so that means he's eaten 15 grammes.

We know 20 grammes is bigger than 15 grammes, so Lucas has eaten more.

Granted the proportion that Jun has eaten is bigger, but if the question wants to know who's eaten more, then it's important to recognise what the whole represents.

Let's have a look at another check question.

Jun is asked which of these is greater? Do you agree? And I want you to explain.

Well, Jun is given 2%, 100%, a half, 300%, and Jun says, "I choose 300% because it's the highest proportion.

Do you agree? Well, hopefully you can spot, well, without knowing the whole, you can't really make an informed decision.

For example, it could be 300% of nothing.

Now, let's add some values to it.

Let's look at the whole.

What is the greatest amount of money, 2% of a million pounds, 100% of 11,000 pounds, a half of 20,000 pounds, or 300% of 2,000 pounds? See if you can give it a go.

Press pause if you need more time.

Well done.

So let's see how you got on.

Well, 2% of a million pounds is 20,000 pounds, 100% of 11,000 pounds is 11,000 pounds, a half of 20,000 pounds is 10,000 pounds, and 300% of 2,000 pounds is 6,000 pounds, so what's the greatest amount of money? 2% of a million pounds.

It's the lowest proportion, but you have to look at the whole, and the whole here represents a million.

Well done if you got this one right.

Now, let's have a look at your task.

For question one, I want you to insert the correct inequality, the equals, less than, or greater than, for the following.

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

Well done.

Let's move on to question two.

Question two shows the Oak Academy pupils have a cashless account.

Now, all Oak pupils have been excellent and the Oak teacher can award them either 20% of what is in their account, or two pounds more of what is in their account.

Which offer should each pupil receive to get the best deal? So you can give it a go.

Press pause if you need more time.

Well done.

Let's have a look at question three.

Question three wants you to tick which is the best offer, or if there's not enough information, put a tick in that box too.

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

Well done.

Let's look at question four.

Question four says, "Jacob is looking for a new bike.

Two shops sell the same bike." Shop A has a 20% off.

Shop A also says you get 20% off, plus an extra 10% off the sale price.

Shop B says you get 30% off.

Jacob says both have 30% off in total, so the bikes cost the same in both shops.

I want you to have a little think and see if you can do this question to identify is Jacob correct or not? Great work, everybody.

Let's go through these answers.

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

50% of 900 is greater than 10% of 4,000.

50% of 900 is 450, 10% of 4,000 is 400, so 450 is greater than 400.

For B, a half is greater than a third.

And for C, a half of 24 is less than a third of 60.

Half of 24 is 12 and a third of 60 is 20.

For D, two thirds is less than four fifths.

For E, two thirds of 30 is less than four fifths of 50.

Two thirds of 30 is 20, four fifths of 50 is 40.

F, 1% of 900 is 900% of one.

Well, it's equal.

Well done if you got this one right.

For question two, this was a great question, and you could have worked out 20% of what's in each account and compare it to that two pound and identify what would be the best option.

Alternatively, you can identify anything over 10 pounds should always go for the 20% offer, but Jun has 10 pound, so he could get either.

If you've got that one right, massive well done.

For three, let's see where you put that tick.

Well, if A is offered 10% of 100 pounds and B is offered 5% of 200 pounds, that means both are the same, so it doesn't matter what you choose.

If A is offered 79% and B is offered 100%, there're really not enough information to decide which is best.

100% of nothing is still nothing.

For the third one, three eighths of 24 pounds, and B is offered a half of 19 pounds.

Well, B is better.

Three eighths of 24 pound is nine pound, and a half of 19 pounds, 9.

50 pounds, so that means B is better.

And lastly, A is offered 20% more money than B.

Although we don't have any quantities here, it doesn't matter what's being offered because A will always get 20% more.

Now, let's have a look at that question four.

So for shop B, we get 30% off.

So 30% off 120 pounds is 84 pounds.

For shop A, we get 20% off, which is 24 pounds, so thus giving us 96 pounds, then we get another 10% off.

So 10% of 96 pounds is 9.

60 pound.

Then we take that off that 96 pound again, thus giving us 86.

40 pound.

So are both prices the same? No, they're not, so Jacob is incorrect.

Taking off 20% and then taking off 10% is not the same as taking off 30%.

Great work if you got this one right.

So in summary, proportion is a part to whole, sometimes part-to-part comparison.

If two things are proportional, then the ratio of part to whole is maintained and the multiplicative relationship between parts is also maintained.

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

And remember when using proportionality as a comparison, you must pay attention to the whole rather than assuming the measure of proportion is the absolute value.

Well done, everybody.

It was great learning with you.