Lesson video

Hi there.

My name is Ms. Lambell.

You've made a really good choice deciding to join me today to do some maths.

Come on, let's get started.

Welcome to today's lesson.

The title of today's lesson is "Finding the percentage change," and that's in the unit Understanding multiplicative relationships: between percentages and proportion.

By the end of this lesson, you will be able to find percentage increase or decrease given both the start and the finishing values.

Let's get going with this.

A keyword that we'll be using in today's lesson is variables, and they are in proportion if they have a constant multiplicative relationship.

Today's lesson is split into two learning cycles.

In the first one, we will look at finding the percentage change, and we will look at this using our representations.

And in the second learning cycle, we will look at trying to find the percentage change using a multiplier.

Let's get going with that first learning cycle then, finding the percentage change.

Alex and Andeep are discussing who has made the biggest improvement in their times tables.

Alex says, "At the beginning of the year, my score was 80, and now it is 120." Andeep says, "At the beginning of the year, my score was 50, and now it is 80." Who has made the biggest improvement? Alex says, "I've made the biggest improvement as my score has gone up by 40." And if we look, Andeep's score has only gone up by 30.

Andeep says, "But we did not start on the same score." Is it fair to compare the increase in score alone if they both started with different values? So they both started on different scores.

What do you think? Well, let's take a look using a bar model.

Here are our two scores.

80 becomes 120, and 50 becomes 80.

Here is a bar model representing Alex.

So Alex's original score was 80.

And Andeep's original score was 50.

So we can see that Alex's original bar actually represented a higher score.

If we take our 80 and divide it into five equal parts, we end up with 16 in each part.

And if we do the same with Andeep, we end up with 10 in each part.

Let's now have a look at the increase.

If we take a look at Alex's increase, we can see that Alex's score has increased by 40.

What percentage of the bar model is 40? Well, it's 16 and 16 is 32, plus another 8, so I need half of that third bar.

Let's now take a look at Andeep's increase.

Andeep's score increased by 30.

How many parts of the bar model do I need to shade to represent 30? I need to shade three.

We can see then that although Alex's score increased by more than Andeep's, as a proportion of their original scores, Andeep's increased by more.

We can see that Alex's score increased by 50%, but Andeep's score increased by 60%.

So sometimes it is more appropriate to consider the percentage change rather than the exact change.

Like I said, here we can see that Alex's score has changed by 50%, and Andeep's score has changed by 60%.

Representing these problems on a double number line can often make it easier to see the percentage change.

Here's Alex's double number line.

100% is going to always be our original as we want to know what percentage of the original that increases.

So 100% is equal to the original score, and for Alex, that was 80.

We know Alex's new score is 120.

We need to work out, then, what percentage that is going to be on my double number line.

So I'm looking to find out what value I need to replace the question mark with.

I'm looking, then, for my multiplicative relationship between 80 and 120.

Remember, you could also look for the multiplicative relationship between 80 and 100.

And here, I've not bothered with simplifying fractions or even changing to decimals.

I've written it as 120/80.

We're going to do the same, then, to the percentage.

The question mark is 150.

We want to know the percentage change.

We're calculating the percentage change, and that word "change" is really, really important here.

How has the percentage changed? It's changed by 50%.

So we can see that using the double number line, we can clearly see that 50% change.

Now let's take a look at Andeep's.

Andeep's score, 100% to start with.

That's the original.

And his score originally was 50.

The new score for Andeep was 80, so we need to work out what the equivalent percentage is.

We're looking for that multiplicative relationship.

Now here, actually, it probably would have been easier to look for that multiplicative relationship between 50 and 100, but I've decided to stick with going horizontally.

So my multiplier is 80/50, giving me a score of 160.

Let's just check that then.

If I'd gone from 50 to 100 and found that multiplicative relationship, that would have been a multiplier of 2, and 80 multiplied by 2 is 160.

So a lesson to be learned here is sometimes it's easier to move vertically rather than horizontally.

Now we are calculating, remember, the percentage change.

By what percentage has Andeep's score changed? Well, he started off at 100%.

His score is now 160% of the original, so therefore the change is 60%.

We can now see that as a percentage of their original scores, Andeep's improvement was greater.

We can also calculate percentage change using a ratio table.

So we've just looked at a bar model, and we've looked at a double number line.

And we know that a double number line and a ratio table are practically the same thing, but it's just the way we draw them is slightly different.

Alex's cat has been to the vet for her annual checkup.

And if you've visited me with other lessons, then you may have seen Alex's cat before.

She features quite a lot.

Last year, her mass was 5.

5 kilogrammes.

This year, it is 4.

4 kilogrammes.

So we can see that Alex's cat has lost some weight.

We want to know what the percentage change is in her mass.

So not the fact that she's lost 1.

1 kilogrammes.

What percentage of her original weight has she lost? 100% will be the original as we want to know what percentage of her original mass she has lost.

And that's really important.

100% is the original.

Here's my ratio table.

100% is equal to the original weight, and Alex's cat originally weighed 5.

5 kilogrammes.

We want to work out what percentage 4.

4 is.

We're looking for that multiplicative relationship between the mass because this is where we have both values.

And I've decided here to do that, because there's decimals in there, I decided to write that as a decimal.

So the multiplicative relationship between 5.

5 and 4.

4 is multiplied by 0.

8.

Over here, then, for our percentage, we also need to multiply by 0.

8, giving us 80.

Remember, we are calculating the percentage change.

So our answer here is not going to be 80.

It needs to be the change in percentage.

The change in percentage is 20%.

Alex's cat has lost 20% of her original mass.

She's lost 1/5 of what she originally weighed.

I'd like you now to have a go at this question.

Spot the mistake.

The cost of a house increases from 260,000 pounds to 291,200 pounds.

What is the percentage increase in price? We've got a ratio table.

I'd like you to check carefully the ratio table and then also check the percentage increase is 112%.

Pause the video, and when you're ready, come back and we'll check your answer.

Okay, where did you find the mistake? And hopefully, you spotted that the mistake was only in the very final step.

My ratio table was spot on.

My ratio table was brilliant.

But what we hadn't done was consider the difference in percentages by how much has it changed? So it's not the final percentage I end up with in my ratio table.

It's the difference between that and 100.

And we can see that actually, the difference in percentages was 12%.

That's really important and a really common mistake I see lots of people write here because we're used to writing the final thing in the ratio table as our answer.

But remember here, we're working out a percentage change.

It didn't mention change in the question, but it's asking what the percentage increase was, okay? And it had increased in price, so with a change in percentage.

Here's your task, your first task, to have a go at.

Calculate the percentage change from the old to the new values, okay? Really important that old is your original.

And then you're gonna shade in the answers in the grid to reveal an image.

Increases are identified by I, and no surprises, decreases are identified by a D.

So if it is a percentage decrease, and imagine it was a 15% decrease, then you would shade in anything that says 15% and a D.

And then if it was, for example, a 2% increase, you would shade in anything that says 2% and an I.

So you're going to shade them in, and it will reveal an image.

Good luck.

Use your calculators for this, no problems. Use one of the methods that we've used, so either bar model, double number line, or ratio table.

Good luck with it, and I'll be here waiting when you get back.

Great work.

Question number two.

Below are some items that were bought and sold by a market stall holder.

You need to rank the items based on percentage change.

So I've given you how much the market stall holder purchased the item for and how much they sold it for.

What I'd like you to do is calculate the percentage change and then rank them based on the percentage change.

So you've got a mobile phone, some sunglasses, a tablet, shoes, and a cap.

Again, yes, of course, use your calculators, you'll need to with this type of question, but make sure you're showing me all steps of your working.

Good luck.

Okay, so here are our answers.

Hopefully, you've got the same as me, so it should have revealed a percentage symbol.

So check yours looks exactly like mine does.

And then question number two.

Order would have been mobile phone, shoes, tablet, sunglasses, or cap, or you may have those in the other order.

So in reverse, cap, sunglasses, tablet, shoes, and mobile phone.

Great work on that.

Now we can move on to our final learning cycle.

We're going to be using multipliers as a way to find a percentage change.

So we're going to try and leave behind the representations, but you don't have to.

You can stick with those representations if you prefer.

Let's go.

Aisha and Jacob have both bought a T-shirt in a sale.

Oh, I like a good sale.

They know the original cost and how much they paid, but what they want to do is work out who got the best percentage discount.

So they're being a bit competitive here.

They want to know who saved the biggest percentage.

Aisha said hers was 20 pounds originally, and she only paid 15 pounds for it.

And Jacob says his was 30 pounds, and he paid 24 pounds for it.

We want to work out which person got the biggest percentage discount.

We are trying to work out a percentage of the original amount.

We don't know what that percentage is yet, so I've just called it p.

So we are finding a percentage of the original amount, so that's important.

How much was the T-shirt originally? It was 20 pounds.

And we know that that percentage of 20 pounds is equivalent to the amount of money that Aisha paid for her T-shirt, which was 15 pounds.

We know that "of" and multiplication can be exchanged in maths, so we end up with p% multiplied by 20 equals 15 pounds.

I'm gonna rearrange now to find p.

What is the inverse of multiply by 20 pounds? It's divide by 20.

Remember, we can write a division as a fraction.

So p is equal to 15/20.

We need to now convert this to a percentage because remember, we were working out percentage change.

p is 0.

75.

So I know that 15/20 is 3/4, which is 0.

75, but that is not the percentage, is it? That's the decimal equivalent.

So we need to convert that back into a percentage by multiplying by 100, meaning 75%.

But what was the percentage change? 15 pounds is 75% of the original price.

Therefore, Aisha had a 25% discount on her T-shirt.

We were looking at by what percentage had the price changed, which is 25%.

So Aisha saved 25%.

Now let's take a look at Jacob.

So we're working out what percentage of the original, the 30 pounds, is equal to the price that Jacob paid.

We're gonna switch that "of" and we're gonna exchange it for a multiply.

We're then going to rearrange to find p.

So p is 24/30.

We're gonna convert that to a percentage.

So firstly, let's convert it to a decimal and then to a percentage by multiplying by 100, giving us p is 80%.

24 pounds is 80% of the original price.

Therefore, Jacob had a 20% discount on his T-shirt.

So we now know that Aisha actually saved the biggest percentage.

Now we'll take a look at this problem.

Aisha and Jacob's teacher has written a percentage on a card.

Aisha chooses a number, and the teacher increases this by the percentage on the card.

What percentage is on the card? So we've gotta solve a little bit of a mystery here.

Aisha tells us that she chose the number 62, and Jacob says, "Our teacher says the new number is 76.

88." Just as we did before, we are finding a percentage of the original amount, which in this case is our original number, the number we were thinking of, which was 62.

And we know that once we found that percentage of 62, the answer is 76.

88.

Let's follow through our steps then.

We're going to change the "of" for a multiply.

We are going to rearrange to find p.

So p is 76.

88/62.

We're gonna convert to a percentage.

So firstly, convert to a decimal, and then to a percentage.

We get 124%.

76.

88 is 124% of the original number.

Therefore, their teacher chose 24%.

Remember, it's the change in percentage.

We started with 100%, it's now 124%, meaning we must have been increasing by 24%.

Now I'd like you to have a look at this question, and you are going to spot the mistakes.

There is more than one.

Aisha and Jacob's teacher has written a percentage on a card.

Aisha chooses a number, and the teacher changes the number by a percentage on the card.

What percentage is on the card? So exactly the same problem as previously.

But this time, we are going to have a different percentage written on the card, and Aisha has chosen a different number.

And we can see here that Aisha has chosen 60, and Jacob says that the teacher says the new number is 36.

I'd like you to pause the video, look through those workings out really carefully, and decide what mistakes you think have been made.

Now, often with questions where I'm trying to spot mistakes, I cover up the solution that's been given to me and I work it out myself and then compare them.

So you may prefer to do that.

Pause the video, and then we'll check and make sure you found all of those mistakes on your return.

What did you come up with? Now, this was a tricky one to spot, and I've put this in purposely because I know that this is one of the places where people make the most errors.

And that is that the 60 and the 36 are the wrong way around.

Remember, we need to find the percentage of the original number, and the original number was the number Aisha chose, which was 60.

So the 60 and the 36 are the wrong way around.

Also, the percentage change has not been found.

So here, we can see, this person says the answer is 166.

6 recurring percent, when actually, if we had set up the problem correctly to start with, our answer would have been 66.

6%.

Now let's correct those mistakes then.

So you corrected them.

I hope you did anyway 'cause I asked you to.

So we should have done p% of 60 equals 36.

So we're finding a percentage of the original number, and the original number Aisha chose was 60.

We're going to switch the "of" or exchange the "of" for a multiply.

We're going to rearrange to find p.

We're going to change to a decimal and then to a percentage.

So we get 60%.

But remember, it was a change.

And here we can see that it's gone down from 100% to 60, so that's why I've used the word decrease, and the change was 40%.

Let's have a go now, then, at these questions.

Some of the Oak pupils have decided that they want to work out who made the biggest improvement on their times table scores when compared to their starting scores.

So that's important.

They're not just looking at how many they've improved by.

They want to compare it to their original scores.

You're going to work out the percentage change for each of them, and then you're going to rank them in order from the smallest to the biggest improvement.

Pause the video, and then come back when you're ready.

Great work.

Question number two.

Three different brands are claiming they have reduced the sugar in their cereal by at least 30%.

And we see those bold claims, don't we, sometimes.

Part A is which brand cannot claim this? Which brand, actually if they say that, is telling a porky pie? They haven't actually reduced the sugar in their cereal by at least 30%.

And then part B, you're going to say which brand has reduced the sugar in their cereal by the most, by the biggest percentage.

So firstly, you'll need to work out the percentage change for each of the different brands, and then you'll be able to answer part A and part B.

Okay, good luck with this.

Make sure you show me all your workings.

Good luck, and I'll be here when you get back.

Question number three.

Below are some items that were bought and sold by a market stall holder.

Rank the items based on the percentage change.

So we had a question like this in the first learning cycle.

Slightly different.

Same items, but different purchase and selling prices.

And I'd like you, please, if you can to try and use that multiplier method, but it's absolutely fine to go back and use your bar model, double number line, or ratio table.

Okay, come back when you've got your answers to these, and then you can rank them in order based on percentage change.

Let's take a look at our answers then.

So number one, Laura's score had gone up by 15%, Sam's by 20%, Jun's by 10%, and Sofia's by 18%.

So in order from who made the biggest percentage, sorry, smallest percentage improvement to the biggest, it was Jun, Laura, Sofia, and Sam.

Question number two.

If we look at the percentage changes, we can see that Brand A reduced theirs by 30.

8% to one decimal place, Brand B by 30%, and Brand C by 9, sorry, 29.

8% to one decimal place.

Let's just go back to the questions then.

So it says which brand cannot claim this? And we can see that that's Brand C.

Yes, if they rounded to the nearest integer, then they would be able to make that claim, but actually to one decimal place, they have not reduced the sugar in their cereal by at least 30%.

Which brand has reduced the sugar in their cereal by the most? And we can see that that is Brand A.

Brand B, absolutely spot on.

It says least 30%, so remember, that will include 30%.

And then finally question three.

So the mobile phone was a 3% increase, sunglasses was a 2% decrease, tablet was a 10% increase, the shoes were a 3% decrease, and the cap was an 8% increase.

So the order were shoes, sunglasses, mobile phone, cap, and then tablet.

Or you may have those in reverse, tablet, cap, mobile phone, sunglasses, shoes.

Now we're ready to summarise our learning.

You've done really, really well today.

So key things to remember is remember we're looking for that percentage change.

When comparing changes, it is often appropriate to use the percentage change as this compares the exact change to the original.

And we saw various ways where actually, the person who made the biggest percentage change was not necessarily the person who had made the biggest exact change.

To calculate percentage change, we can use our bar model, double number line, ratio table, or multiplier.

It is essential to ensure that the original value is assigned to 100% or the amount you are finding a percentage of.

Great work today.

I'm really pleased that you've joined me, and I look forward to meeting up with you again very, very soon.

Bye!.