Lesson video

Hi.

Thanks for choosing to learn with me today.

In today's lesson, we're gonna be looking at how maths can help us with our everyday lives.

Let's get started.

Welcome to today's lesson on Checking a Claim.

By the end of today's lesson, you'll be able to analyse a claim using estimation to assess its validity.

Let's look at what that might mean.

We're gonna use a couple of keywords today in our lesson and they're probability and estimate.

Now, these words should be very familiar to you, but if they're not, feel free to pause now and read through their definitions on the screen.

Our lesson today is broken into two parts, and we're gonna begin by analysing headlines.

Articles and adverts often use statistics to influence opinion.

It's important that you verify these as they are not always correct.

A newspaper headline states, "Removing the 20 percent close tax will mean an average spend of 400 pounds a year will decrease to 320." Is this claim correct? No, it's not.

The author has incorrectly calculated the amount before a percentage increase.

Let's look at what they've done.

Now, if the value of the close without tax was 320 pounds, then with the 20 percent tax, they would in fact cost 384 pounds, not 400.

To calculate the cost of a 400 pound clothes shop without 20 percent tax, we could use a proportion table to help.

Now remember, the 400 pounds is after 20 percent was added.

So in other words, it's 120 percent of the original cost.

By dividing by 1.

2, we can get back to what 100 percent was, or 400 divided by 1.

2 is 333 pounds, 33 pence.

I've had to round there to the nearest penny.

In other words, it's not going to decrease as much as the article claims. Another article states, "It is 188 percent cheaper to live in Oakfield than Acorndale." Ah, look, there's that asterisk again.

Based on respective average house prices of 180 to 520,000.

Hmm.

Is this claim correct? Well, the phrasing of this percentage saving is definitely misleading.

It's not possible for something to be over 100 percent cheaper.

Did you spot that as well? So let's have a look at this.

If we were going from a house price of 180,000, then that's 189 percent increase.

But if we're going the other way, so we're looking at decreasing, this is only a 65 percent decrease.

See, the person who's reported this has got the percentages completely around the wrong way.

So moving to Oakfield is a saving of 65 percent, not 188 percent.

Another area where percentages can be misleading is when percentages are compared.

So look at this claim, "Spending on public services only expect to rise by five percent, because it's going from 18 percent to 23 percent." Now, is that claim correct? Well, the misinterpretation here comes from understanding percentage change.

The percentage points have increased by five percent.

That is true, but the amount spent has not.

Let's look at this.

If the budget was, for example, 100 pounds, then the spending would've increased from 18 pounds to 23 pounds, 18 percent of it to 23 percent of it.

Well, to go from 18 pounds to 23 pounds, what's that as a multiplier? Well, I multiply by 1.

28.

In actuality, it's percentage increase of about 28 percent.

So percentage of the budget has gone up by five percentage points, but the amount spent has increased by about 28 percent.

Bit of a difference there.

Let's do a quick check.

The price of an item of clothing rose from 52 pounds to 58 pounds.

What is the percentage increase in the cost of the item? And now, let's look at that the other way.

The price of an item of clothing fell from 58 pounds to 52 pounds.

What is the percentage decrease in the cost of this item? Pause the video while you work through this.

Welcome back.

Let's see how you got on.

Well, for the first one, I saw an 11.

5 percent increase, and for the second one, we noticed a 10.

3 percent decrease.

See how they're not the same? It's time for your first task.

Question one, a magazine article titled, "Look at the savings you would make if tax was abolished", states, the figures below.

They're not right, of course.

Could you please correct the prices of each item without VAT? In other words, what's in the purple bubble needs to be changed.

Pause and work on this now.

Welcome back.

In question two, there's a raffle, and there is a one in 2,180 chance of winning a prize.

In other words, there are 2,180 tickets and if you buy one ticket, then that's your chance of winning.

The headline reads, "Just buy two tickets and increase your chance of winning by 100 percent." I'd like you to have a go at answering each part.

Part A, what is the chance of winning with one ticket as a percentage? Part B, what is the chance of winning with two tickets as a percentage? Part C, what is the percentage point difference in the percentage chance of winning when you buy one ticket versus two tickets? And part D, explain why, although technically correct, the claim could be misleading.

And then in part E, use trial and improvement to find how many tickets you would need to buy to increase the probability of winning to one percent.

Pause while you work through this.

Question three, explain the mistake in each headline.

So I've told you now they are wrong.

You need to find that mistake and explain it.

Part A, local employment rates fell by 10 percent last year and 15 percent the previous year.

In just two years, employment rates have fallen by 25 percent.

In part B, 28 percent of men and 22 percent of women say they only clean their teeth once a day.

A whopping 50 percent of adults are not cleaning their teeth twice a day.

And then in part C, decrease your food expenditure by 300 percent a week by taking your own lunch to work, based on a takeaway lunch cost of 28 pounds a week and a homemade lunch cost of seven pounds a week.

Pause while you work out, and then explain the mistake in each headline.

Time to go through our answers now.

In question one, without our 20 percent VAT, our television, which is costing 450 pounds, would actually have cost 375, and the car seat, which with VAT is 180 pounds, actually costs us 171 pounds and 43 pence without the VAT.

In question two, what's the chance of winning with one ticket as a percentage? Well, I've written this to two significant figures.

It's absolutely fine if yours is more because I didn't tell you what you had to round to.

So if you round to this, well done.

Part A, I've got 0.

046 percent, in part B, 0.

092 percent.

So what is the percentage point difference? Well, there's an approximate percentage point increase of 0.

046, which means that in part D, it's mathematically accurate.

Doubling the number of tickets does indeed double your chance of winning because 0.

092 is double 0.

046.

This is the same as an increase of 100 percent.

However, that is a bit misleading because the chance of winning is so small that doubling that chance is still only a very small percentage.

The chance has only increased by 0.

046 percentage points.

People who don't understand probability may well find this quite misleading.

So in E, I said you could use trial and improvement to find how many tickets you need to make the probability of winning one percent.

Well, when I bought 21 tickets, I had 0.

009633.

I hadn't quite made it to one percent.

It's 22 tickets where I finally reach that one percent.

Just in case you're interested, it is perhaps worth noting that most national lotteries have a probability of winning the jackpot of at least one in 14 million with some having a much smaller probability than that.

Question three, explain the mistake in each headline.

Well, when calculating two successive percentage changes, you can't just add the percentages together.

So I've done an example here, and you might have done an example with numbers two to illustrate your point.

So if there were 100 people employed one year and that number decreases by 15 percent, we're left with 85 people.

Decreasing that by 10 percent means I'm left with, well, 76.

5, so I have to round that obviously, this is a decrease of 23.

5 percent, not 25 percent.

In part B, this is not how percentages work.

The percentage are of two different populations.

So if we assume that the same amount of men and women were together in the population, then the total percentage of people will be the mean average of the two.

In other words, 25 percent.

Also, this was the number of people who said they brush their teeth once a day.

There may be people who don't brush their teeth every day or brush their teeth more than twice.

So the claim that 50 percent of people are not cleaning their teeth twice a day is definitely wrong.

Then in part C, a value cannot decrease by more than 100 percent.

So well done If you spot that straight away.

It is true that increasing seven pounds by 300 percent is the same as multiplying by four, which would give us 28 pounds, but this doesn't work in reverse.

The calculation should be seven divided by 28, which is a 75 percent decrease because it equals 0.

25.

So what they could have said was it cost four times as much to buy takeaway lunch, that would be true, or decrease your food expenditure by 75 percent.

The each week part may also be misleading because the timescale doesn't affect the percentage at all.

It's now time for the second part of our lesson where we're gonna use estimation to check.

So we can check if certain claims are realistic using estimation.

Having a rough idea of certain statistics can be really helpful for this.

So what I'd like you to do is pause the video and have a go at estimating each of these.

In other words, do you think the population of the UK is? What about the population of the world? Pause and have a go now.

Welcome back.

Now, you may have your figures.

I'm gonna go through what I've got for mine.

So the answer to all these remember are rough answers, and most of these will change over time.

So you may wish to use the internet to find the most recent figures.

I'm going with the figures from 2024.

At that point, the population of the UK was approximately 70 million.

The population of the world was roughly 8 billion.

The average life expectancy of the UK, well, that does depend on gender and when you were born, but at the moment it's approximately 80 years.

The average number of people per household in the UK is 2.

4.

And again, this is 2024.

The average wage in the UK, again, it depends on which average you are using here, but it's approximately 35,000 pounds a year.

And the capacity for coffee mug, about 300 millilitres, and the mass of the smartphone, again, this is a 2024 smartphone, it's approximately 170 grammes.

So for the rest of this lesson, I'm going to use these figures, but you are very welcome to use your own.

Izzy says, "I reckon there must be about 1,000 secondary schools in the UK." Do you think her claim sounds reasonable? Well, it's really hard to tell at the moment.

We only live in experience a very small area of the country at one time.

It's hard to try and imagine a figure for the whole country, but we can make this easier by considering the facts we know and applying them on a wider scale.

So how can we begin to estimate the number of schools in the UK? What do you think? Pause the video and have a bit of a brainstorm now.

Welcome back.

Let's see if you came up with an approach similar to mine.

Sam says, "I live in a medium sized town and there are five secondary schools that I know of." Hmm, is that a good starting point? Well, there may be other secondary schools that Sam's not aware of.

It's hard to estimate the number of towns in the country, and big cities will have many more schools than towns do.

So although this seems like a reasonable start, there's just too many factors which are too hard to estimate accurately.

Instead, it might be helpful to start with a statistic we do know, and that's reasonably accurate.

So to work out what we think the number of schools in the UK might be, well, we know the UK population is approximately 70 million.

We know the average life expectancy is roughly 80 years.

So what percentage of the population do you think are secondary school age? Hmm, what might that be? Well, thinking about 10 percent is probably reasonable.

Now, of course this does depend if you're including post 16 students as well in the secondary school.

Let's go with it being 10 percent.

Well, that's roughly 7 million pupils therefore.

What can I do now? Well, I can estimate the number of pupils per school.

Think about your own school.

Is it a small school or a large school? Now, depending of course on what size your school is, we'll give you an estimate for the number of pupils.

But an average school size, a sensible estimate would be around 1,000 pupils.

So if there are approximately 1,000 pupils per school, that would mean 7 million divided by 1,000, 7,000 schools.

So Izzy's this prediction of 1,000 seems a little low compared to this.

Remember, of course this uses population, so it doesn't consider the type of school such as secondary, middle, high, school, independent.

There are lots of different variables.

Now, our area of volume and capacity skills can also help us to check claims. An article in magazine reads, "Many young people are drinking enough fizzy drinks in a month to fill a bath!" Well, what things do we need to know to estimate whether this claim seems realistic? Pause and make a list now.

Well, we need to know the size of a bath, the size of a bottle of fizzy drink, the number of bottles of fizzy drink young people drink.

There are lots of ways to start this.

That was just three.

Let's begin with how big do you think the average bath is? Well, you could use your own height to help you.

The interior length might be around 150 centimetres.

The interior width might be roughly 60 centimetres.

If you sat in a bath, think about how it might feel.

Could you lay down in it? How far across can you stretch? The claim said, fill a bath.

Well, how deep does that have to be to be considered full? Well, let's go with roughly 30 centimetres in depth.

If you want to know what that looks like, you can always get out one of the longer rulers and see where 30 centimetres is on it to get an idea of how deep that bath is going to be.

Now, you can use my figures or your own.

Let's work out the volume of the interior of the bath while using these figures.

We're gonna say it's 150 times 60 times 30.

So again, we're gonna estimate that our bath has a cuboid shape.

That would give us 270,000 cube centimetres.

So if one cube centimetre is equivalent to one millilitre, how many litres would it take to fill this bath? Using my figures, that would be 270,000 millilitres.

Dividing that by 1,000 means it's 270 litres.

So does drinking 270 litres of fizzy drink a month sound reasonable? Don't know about you, but that doesn't really seem like a value that's easy to interpret at the moment.

So what could we do to that? Well, we could say there are 30 days in a month, roughly.

If I divide the 270 by 30, that means I've got to drink nine litres of fizzy drink a day to get that in a month.

Again, that is on average.

A big bottle of fizzy drink is normally two litres.

So to drink nine litres a day, I've got to drink four and a half large bottles of fizzy drink.

That doesn't sound very reasonable to me.

The person who wrote this article is likely to have underestimated the size of a bath or interpreted the word full very differently.

How deep would a 140 centimetre by 50 centimetre bath be if 60 litres of drink was poured in? Well, 140 times 50 is 7,000.

Well, the area of the base of the bath is therefore 7,000 square centimetres.

60 litres is 60,000 millilitres.

Remember, we can convert that straight into cube centimetres.

It's still 60,000.

Well, 60,000 divided by 7,000 would be 8.

57.

It goes off a little bit, but that's roughly 8.

6 centimetres deep.

There's no way that's a full bath.

So even assuming people drink two litres of fizzy drink a day, i.

e.

one large bottle, this claim is not true.

Let's check you understand that.

An article claims you'll spend 10 percent of your life in a school classroom.

Answer the questions below and then use those answers to state whether you think this claim is realistic.

Pause the video while you work this out now.

Welcome back.

Let's see how you got on.

Well, how many hours a day are you in the classroom? I said approximately five.

And there are roughly 13 weeks of school holiday, if I include reception and post 16, it's 14 years that you are in school for.

And the average lifespan in the UK, remember, we said roughly 80.

So for my figures, that puts you at 25 hours a week in school, and that's 975 hours for the year.

14 years, that's 13,650 hours of a lifetime.

Well, if I divide that by 24, that tells me how many days of your life.

And then if I divide that, I can work at how many years.

Well, if we then look at that in terms of I'm in school for 1.

56 years out of the 80 years I'm alive for an average, I'm actually only in school for 1.

95 percent of my life.

So using these assumptions, even if I round, it's roughly two percent of my life is spent in school.

So that claim seems way too high.

It's now time for the final task.

In question one, an article reads, "The average person will spend 100,000 hours on their phone in their lifetime." Part A, estimate how long you think the average person will spend on their phone in lifetime.

So do write down any estimations you make.

And in part B, does the article's claim sound reasonable to you? Pause and do this now.

Question two, "A newspaper states that five percent of the UK has a car parked on it." Part A, estimate how many households there are in the UK.

B, estimate how many cars there are in the UK.

And then C, estimate how much space is taken up by those cars.

Pause and do this now.

And then Part D, assume the area of the UK can be modelled by the two rectangles shown on the screen.

What is the approximate area of the UK? And then part E, does the newspaper's claim seem valid to you? Pause and do this now.

Welcome back.

Let's go through the answers.

Now, again, I've used my approximate figures.

You may have different figures, and it's absolutely fine if you do.

So, I've said the average person spends five hours a day on their phone.

And if we do that over the course of the year, that's 1,825 hours a year.

Based on the average life expectancy, that's 146,000 hours over the course of your life.

So does the claim sound reasonable? Well, in that case, it is fairly reasonable.

It might even be a little low because I've said I think five hours a day is how long people spend on their phone.

But you may have said, actually it's about spot on.

It all depends how long you thought people spend on their phones.

Question two, we said five percent of the UK has a car parked on it.

Let's find out if that newspaper is right or not.

Well, based on the fact that the population in the UK is roughly 70 million and there's an average of 2.

4 people per household, that makes roughly 30 million households.

Well, if each household has two cars, this is now part B, that would be 60 million cars in the UK.

Now, that is quite a high estimate, but I didn't include car impounds, showrooms or other places that cars could be.

And part C, estimate how much space is taken up by the car.

Well, if a car measures five metres by one metre, then it takes up five square metres.

So multiply by number of cars.

I'm looking at roughly 300 million square metres of space taken up by cars.

Now, working at the area of the UK, I've gone for an estimate of 235,000 square kilometres.

Now, this is the bit where it's important.

I need to change that into square metres, or I need to change the square metres into square kilometres.

I went for that option.

So cars take up approximately 300 square kilometres.

What I've then done is go, well, what percentage of the UK is that? It's roughly 0.

13 percent.

So the article's claims seems to be a massive over-exaggeration.

Remember, I overestimated quite a bit and I didn't get anywhere near five percent.

Well done if you've got something similar.

Let's just check what we've done today.

Percentages can be misused in articles and statistics.

It's important to check statistics yourself to assess their validity.

Estimation can be used to check the validity of certain claims. Well done.

I hope you enjoyed doing this lesson today.

I look forward to seeing you for more maths in the future.