Lesson video

Welcome.

It's really nice to see you for today's algebra lesson.

My name is Ms. Davis and I'm gonna be helping you as you work your way through this video.

There's lots of exciting things that we are gonna talk about today.

I hope you are really, really looking forward to it.

Make sure you've got everything you need and let's get started.

Welcome to this lesson where we're interpreting and drawing real life graphs.

So the few key words that we're gonna look at today, you may have come across these before.

You might feel like you just need to remind yourself.

I'll talk you through those now.

So we're gonna look at time series graphs.

Now time series graphs show data over a time period.

The time period is always shown on the x-axis and the data points are joined chronologically by line segments.

There's an example of a sketch of a time series graph on the right hand side.

We're also gonna talk about extrapolation and interpolation.

Extrapolation is the process of estimating unknown values that are outside the range of existing data.

So sometimes when you use existing data to make a prediction about what's gonna happen in the future or to make a prediction about what happened in the past before your data values, that will be called extrapolation.

Interpolation is the process of estimating unknown values that are inside the range of existing data.

So let's get started Looking at time series graphs.

We're gonna start by recapping some of the real life graphs you may have seen already.

So distance time graphs show us how distance travelled changes over time.

Speed time graphs show us how speed changes over time.

Because we're looking at how the values change as time progresses, we plot time on the x-axis, and you'll be really used to seeing this by now.

So there's a distance time graph and a speed time graph.

There are other types of graphs which look at how values change over time.

Pause the video.

Can you think of any examples where we look at how something would change over time on a graph? Right, so you may have thought of something like depth time graphs or volume time graphs, or you might have thought about height time graphs if you're plotting something that is changing height over time, or you may have thought about those keywords that we started this lesson with.

You may have thought about time series graphs.

So time series graphs are line segment graphs with time on the x-axis.

And the y-axis contains a measure, such as frequency or percentage.

And we're gonna look at some today that have temperature plotted on the y-axis as well.

So all sorts of things can be plotted on the y-axis and we're seeing how they change over time.

The points are joined by line segments.

We're gonna go into a little bit of why and some of the problems with that in this part of the lesson.

Time series graphs can be really useful for businesses, for governments or local authorities, anyone who wants to analyse a trend over time.

So when plotting time series graphs, we're generally plotting set pieces of data which have been collected.

So this table shows the mean of the maximum daily temperatures in July in Whitby.

So you might wanna pause the video and just read through that table and see what it's showing you.

So this data could be presented on a time series graph.

Right, so here I've got my set of axis ready.

2015 is the first year we have data for, so that could be thought of as year zero.

2016 would then be one year later.

2017 would be two years later, and that would be our scale on our x-axis.

So what we could label our x-axis with is zero, one, two, three years, and they will be consistent time periods that we could plot.

However, because we're gonna analyse this in context, it's useful to keep the x-axis labelled with the years so it's easy to see which year we're talking about.

It's important that they are increasing in equal steps.

We also need to pick a sensible scale for the y-axis.

We could start at zero and increase in steps of five.

And then all you need to do is plot those values as accurately as you can.

And then to finish off our time series graphs, we would join them with line segments.

So you can see that I've started at zero and I've increased with steps of five.

What you might be thinking is there's a lot of wasted space on that graph.

So instead you could use an axis break and start at a different value.

You don't have to start at zero as long as you use an axis break to show you're not starting at zero.

So because my lowest value is 19.

4, I've decided to start at 19 instead and then plot my time series graph.

You can see that does stretch all the values, which can be a good thing.

It means it can be easier to see the values.

But using a broken scale can have negatives as well 'cause our graph you can see now shows a massive fluctuation between values, whereas actually in reality, there's only one or two degrees difference between most of the values.

So just be aware of those issues when changing your scales around.

Izzy says, "Halfway between July 2017 and July 2018 is January 2018." So she's used our graph and said, "So I can predict that the mean monthly temperature for January 2018 was 21 degrees." What is the problem with this statement? Pause the video and have a think.

Now the reason we join the points together is to make it easier to see a trend.

So you've got lots and lots of data points.

By joining them up, we can track whether they're increasing, decreasing.

It's a little bit easier to see.

However, we don't actually know what happens between these data points.

We've collected data in July of every year.

We don't know what's happening with the data between the Julys.

And if we think logically, temperature changes seasonally, so there will be massive fluctuations in temperatures between successive Julys.

So it's not actually going to follow line segments.

What the temperature will do if we collected temperature monthly, the temperature will go up and down and fluctuate between those Julys.

So Andeep says, "I knew extrapolation could give inaccurate results, I didn't know interpolation could too." Yeah, he's got a really good point here.

If you extrapolate, if you look outside the range of data, it can give you inaccurate results because what you are doing is you're predicting what might happen in the future when you don't know whether that trend is going to continue.

The problem with interpolation is you don't know what happens between the data points.

So if there's a nice linear relationship between the data points, it's gonna be easy to interpolate and give it accurate results.

If you have something like temperature, which is gonna fluctuate massively between the years, then interpolation is not gonna give you accurate results.

So Izzy says, "I guess it depends on the type of data.

This graph shows the population of my town between 2001 and 2021." Could Izzy use this to estimate the number of people in her town in 2015? What do you think? So this is interpolation again.

So she could look at 2015 and make a prediction.

There are three data points.

Just like the previous graph, there's no way to know what happens between these points.

They might have built a new development in the town in 2014 or 2015 and suddenly the amount of people there jumps rapidly, or there could be a reason why lots of people left the town one year.

So we don't know what's going to happen between those points.

However, this time we can see an increase in trend and it's not as obvious that it's gonna fluctuate like weather, that we know is gonna fluctuate throughout the year.

So she can make this prediction that there might have been 7,200 people in her town in 2015, but it would only be an estimate.

There'd be no way of knowing if that is accurate or not without having data from that time.

Okay, now we're gonna have a look at varying the time intervals.

So different time intervals can be useful for different situations.

So this time series graph shows data about the mean number of hours worked per week by full-time workers in the UK.

You can see that the data points are taken yearly.

I've labelled every five years.

If you look at each minor grid line between that, you can see we've plotted a point every year.

Pause the video.

What does this graph show? Okay, some of the things you might have thought about, you could see that the weekly working hours were decreasing between 2000 and 2008.

Then there's a sharp drop in 2009.

The UK suffered an economic crash in 2008 that may have impacted the hours worked in 2009.

You could then see the hours worked increased up to 2014 and then they were slowly starting to decrease again before we hit 2020.

And there was a national lockdown in 2020, which is why we've got that massive decrease in the hours worked.

You might have also said that by 2022, which is the last data point we have, that value had not returned to the pre-lockdown rates yet.

So a store manager wants to know what time of year he might need to hire extra cover staff because his staff aren't working.

Why is this graph not useful to answer this question for the store manager? Right, well the data is plotted yearly so it doesn't show which months people work more or less.

So what do you think a more useful scale would be if the store manager wanted to answer this question? Yeah, we'd need time to be plotted monthly so he can see which months are more popular for holidays.

Okay, have a look at the new graph that's been drawn.

What does this graph now show? What decisions might the store manager make from this graph? So we've got two different years plotted by two different colours.

Both years less hours were worked per week in August and September.

You can see that there is a drop between August and September.

This might mean that the manager needs to hire more staff or offer incentives for staff to work during this time.

A few things to be aware of with this graph though, this is across the UK.

A better understanding of working hours might come from the store collecting its own data from its own staff.

Equally, connecting each month does not account for changes within the months.

So for example, people might work less at the end of December because of the holidays.

We can't see that on this graph.

It's also useful to look at the scale.

You'll see that we've not started at zero so we should have a broken scale.

What that means is the values are actually only changing between 36.

2 hours and 36.

7 hours.

That's only actually half an hour per week fluctuation.

So that might not actually be a problem for the store manager.

So it is worth just checking that scale and seeing what that variation actually is.

Right, quick check then.

This graph shows hours work per week in the UK between 2005 and 2010.

A news article uses this to claim that in summer 2006, people worked an average of 37.

2 hours a week.

Why might this be inaccurate? Pause the video.

What would you say? All right, there's a few ideas you may have thought about.

We don't actually know what the data points represent.

They're plotted yearly, but we don't know what time of year.

So this article seems to be suggesting that we've plotted it January each year, but we don't know that that's the case.

Those values could actually be a mean from the whole year or they could be from July of that year.

We don't know.

So we can't make an assumption that halfway between 2006 and 2007 is summer of 2006.

Equally, we don't know what happens between the data points.

So even if those points were January of every year, there's no way of knowing how the values would fluctuate throughout the year.

I wonder if you said something similar to either of those points.

Okay, a different article claims financial difficulties caused people to work significantly less in 2009.

Does the graph support this statement? What would you say? So there is a clear drop in weekly hours in 2009.

We can see that on our graph.

However, this is only actually a drop of 0.

3 hours from the previous year, which is 18 minutes.

So you'll have to decide whether you think the news article saying people work significantly less fits with the idea that on average, they're working 18 minutes a week less.

Time for you to have a go at drawing some yourself.

I'd like you to use the axes and the scale given to plot a time series graph for this data.

When you're happy with that, we'll look at the next bit together.

So for question two, you've got exactly the same data, but this time I've changed the axes scales.

You'll see that I've put in a broken scale and I've started at eight degrees C.

I'd like you to plot the points on this new scale being as accurate as you can and then answer question B.

What is the advantage of using a broken scale here? You might also want to think of the disadvantages.

Okay, for question three, these two graphs, A and B, show the total amount spent by citizens of the UK on visits abroad.

Which graph will be most useful in each scenario? Read through those scenarios and explain your choice.

Off you go.

Let's have a look then.

So your graph is gonna look something like this.

Try and plot your points as accurately as you can.

It was quite tricky in this case 'cause a lot of the points were quite similar and the scale was quite small.

When we changed the scale, you should have this graph.

This one was easier to be accurate for.

So please make sure that you've plotted these in the right place.

For question B, what's the advantage? Well, because all the values are between eight and 10, using the broken scale allowed us to be more accurate.

It was also easier to see the variation in temperature between the years.

The disadvantage though, is that it can over exaggerate the small variations.

Okay, so for A, I think graph B is gonna be the better one 'cause it has the monthly breakdown.

So if they want to decide when to put on more flights, they want to know when more people are going on holiday.

So more people spend money in August.

So that's implying that more people are going on holiday, higher demand, so more flights needed.

You can't see that from the yearly graph.

For B, graph A is gonna track the trend over time.

So if we're deciding whether the company should invest more money in more planes, then we need to see if more people are going on holiday over time, which means A is gonna be easier to use.

So in 2018, for example, you might be looking at that and saying more people are going on holiday, more people are spending more money on holiday, it's gonna be a good business investment to get more planes.

In 2020/2021, it might not have been a good investment to increase the amount of planes you had.

And finally, I think this one depends a little bit on what you want to see.

So if you're trying to decide how to price your flights, it might be useful to see the fluctuations in months 'cause you might be able to say then that, right, we need to make our flights more expensive in August 'cause more people are going on holiday and we're gonna make the most money.

We might need to lower prices in December because less people want to go on holiday.

So if we lower the prices, that might encourage more people to go on holiday and increase our profits.

Or you might use graph A if you want to see whether profits are increasing year on year.

If more people are flying each year, that might mean we wanna put our prices up to increase profits, or it might mean that we need to lower our prices to compete with other companies.

You may well have come up with other ideas as well as to which graph would be good for this one.

Right, so now we're gonna have a look at some other real life graphs.

We're gonna look at volume time and depth time graphs.

So Jun is filling his paddling pool with water.

His paddling pool has a rectangular base and straight sides.

So it can be modelled as a cuboid.

"I want to track the volume in the pool over time," says Jun.

How could we sketch this on a graph? What do you think? Right, we can use a volume time graph.

"The volume is the amount of water in the pool.

So I think it will look like this." Do you agree with Jun? In order to answer this question, we possibly need some information about how the water is being run into the paddling pool.

So if we assume that the water is being run at a constant rate, then Jun would be correct.

As time increases, the volume will be increasing at a constant rate.

If we measure volume in centimetre cubed and time in seconds, what would be the units for the rate at which the volume of water in the pool is increasing? Can you work this one out? All right, it would be centimetres cubed per second.

It would be a compound measure.

In order to measure this rate of change, we could find the gradient of the line, and the gradient of the line would be the change in volume divided by the time.

So centimetres cubed divided by seconds, or centimetres cube per second.

Jun says, "I also have a smaller pool with the same shape for my dog." That is quicker to fill so I think the graph will look like this." Do you agree with Jun? Right, Jun is incorrect this time.

If he runs the water into the dog pool at the same rate as the main pool, then the gradients will be the same.

So the change in volume over time.

The volume of water in each pool will be the same at the same time, but the dog pool will fill up quicker.

That means it'll take less time to fill to the highest volume, and the highest volume will be less than the bigger pool.

So you'll have a line with the same gradient, but it won't be as long.

Laura says, "I have a paddling pool with a seat in each corner.

The pool fills up slower once the water level gets past the seats." What do you think her volume time graph would look like? Right, well there's Laura's paddling pool with the seats.

But again, if Laura is filling up her paddling pool in the same way as Jun, so the water is running at a constant rate, then the volume time graph will look the same.

The seats are not gonna affect that.

Now, if the pool has a smaller capacity, so if it can hold less water, then the line segment will be shorter than Jun's, but the gradient will be the same 'cause the rate that the water's being run into the pool is the same.

Can you think of a scenario where the volume time graph will not be one linear line segment? What do you think? Well, we need the rate of flow of water to change or be constantly changing.

So Jun is filling up his pool.

When it is roughly half full, Izzy helps him with another hose pipe.

Let's have a look at the volume time graph this time.

So there's Jun filling the pool.

Once Izzy joins, the pool is being filled at a faster rate, but it's still gonna be constant.

So you get a second line with a steeper gradient.

Jun says, "Actually it takes a few seconds for the water coming out the hose to get to maximum rate of flow." How might that affect the graph? What do you think? So there may be a curve at the beginning of the graph as the rate of flow of the water constantly increases until it gets to the maximum rate.

Then you'd have your linear line segment.

And if the same is true for Izzy's hose pipe, there may also be a curve section when she joins until her hose pipe gets up to maximum rate of flow.

And then we'd have a linear line segment again.

So we've looked at how volume can change over time.

Let's remind ourselves how to plot the depth over time.

Laura says, "The seats did not affect the volume time graph for my pool, but would they affect the depth time graph?" Well, yes.

If the water was run at a constant rate, then the depth of the water would increase at a faster rate whilst the seats were taking up space.

Izzy says, "If I helped, I reckon we could fill the pool in a way that the depth time graph is linear." So how could Izzy help to make this a single line segment? Right, well, if Laura filled up the pool on her own until the water got past the seats, we'd have that linear line segment as we have before.

Now as soon as the water gets up to the top of the seats and the pool gets bigger, Izzy could help by adding additional water at a constant rate to compensate for the larger cross section.

Now in reality, this is gonna be very difficult to get exactly right, but it would be possible.

Jun says, "I have the base of a basketball post which you fill with water.

The cross section looks like this." What do you think the volume time and the depth time graphs would look like if Jun filled them at a constant rate? We'll draw them together.

Well, it's been filled at a constant rate so the volume is gonna increase at a constant rate over time.

However, because the cross section is constantly narrowing, the rate at which the depth increases is constantly increasing.

So you'll get this curve.

Jun says, "How could I fill this so the depth time graph is linear?" Right, well, Jun would have to be constantly decreasing the flow rate at the same rate as the cross-section is narrowing.

So again, that's gonna be almost impossible to do in reality.

What would the volume time graph look like in this case? Right, well, because he's turning down the rate of flow gradually, we should get this curve.

Right, quick check then.

Which of these could show a volume time graph for filling this container at a constant rate? Right, the first one.

The shape of the container does not affect the volume time graph.

What affects the volume time graph is the flow rate of the water into the container.

Right, water is poured into a container at a continuously increasing rate and this graph is produced.

Which container could it be? Right, any of them 'cause the shape of the container does not affect the volume over time.

It is how the water is being run into the container.

So it could be any of these.

Time for a practise then.

I'd like you to match the volume time graphs with the scenarios.

Once you're happy with those, come back for the next bit.

So Laura has a water bottle with this cross section.

I'd like you to sketch a volume time graph if it's filled at a constant rate and a depth time graph if it is filled at a constant rate.

Give those a go.

Right, Jun is filling this jug with water.

He fills it at a constant rate until the water reaches the dash line, and then he slowly turns the tap off so the rate of the flow slowly decreases.

Can you sketch a volume time graph for that scenario? And then the depth time graph actually looks like this.

I'd like you to describe what is happening at those four points in order to make the graph look like that.

Give those a go.

And finally, this is an activity that you could do with a partner.

So I'd like you to draw a container or draw a cross section of a container and then draw the depth time graph for your container, but don't show your partner.

Then describe the depth time graph to your partner and see if they can sketch your container.

Think about the different parts of the graph.

Think about whether you'll get curves or linear line segments.

Where will the line be steeper? Once you've played around with that, come back and we'll look at the answers.

The scenario A is gonna match with graph C.

You have a steep gradient from the middle line segment where someone's thrown in a bucket of water and the volume increases very quickly.

For B, you are going to have a horizontal line segment in the middle where someone has stepped on the hose and it's stopped pouring water into the pool.

For C, we need a curve as the rate of flow is increasing as the tap is slowly turned on.

And for D, it does not matter that it's a pirate ship shaped paddling pool.

It's still being filled at a constant rate.

So you get a linear graph.

For question two, because Laura's filling the water bottle at a constant rate, the volume time graph is going to be a single linear line segment.

The depth time graph, however, is going to look like this.

As the container widens, the rate at which the depth is increasing will slowly decrease.

Then it'll be constant when the sides are parallel and then the rate will increase as it narrows again.

Then Jun, so you should have a linear line segment when he's filling it at a constant rate and then a curve showing the rate of volume over time decreasing as he is turning the tapper.

Now let's see if we can interpret this graph.

So to start with, the width of the cross section is constant and is being filled at a constant rate, and that's why we get a linear line segment.

But at B, the cross section is narrowing, so we would expect to get a curve.

However, because he is turning the tap off slowly, he must be compensating for that shape decreasing.

For C, the water flow is still decreasing, but we have a constant width again, which means the rate at which the depth is increasing will be slowly decreasing.

And then at D, we have the horizontal line segment, so the depth is constant.

So that could be the tap turned off or the jug now completely full so the water depth can't get any higher.

Right, that was quite tricky to wrap your head around that last one.

So well done.

If you played around with a paired activity, here's some things you want to think about.

Vertical sided containers will have linear line segments.

If the container has different widths at different points, then the gradient will change.

The narrower it is, the steeper your depth time.

Sections that produce curves will be anything where the two sides are not parallel.

So they are widening or narrowing.

Check whether your partner has drawn the right container, and if not, have a discussion and see where there is that discrepancy.

So why have they not drawn what you drew? Was it something that you didn't explain using the right language? Do you need to go into a bit more detail? Did they misunderstand what you were describing? Hopefully you'll be able to describe it in a way that your partner drew the correct container.

If not, you might want to draw a different container and try again.

Well done.

We've looked at two very different types of real life graphs today.

If you want to just remind yourself of those key points, pause the video now.

Thank you very much for joining us.

You've worked really, really hard and I look forward to seeing you again.