Lesson video

Hello, thank you for joining us for today's algebra lesson.

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

There's lots of ideas today that you're gonna want to think about and give yourself plenty of time to think for yourself before I give you some hints and help you out.

With that in mind, make sure you pause the video to give yourself that time to think.

You might also want to make sure you've got a pen and piece of paper and maybe a pencil and a ruler for any of the graph work that you're going to need to do.

Let's get started then.

Welcome to today's lesson on checking and securing an understanding of reading from context-based graphs.

So what we're gonna look at today is different types of graphs in context and whether we can interpret what they are telling us.

So we're gonna talk a little bit today about rate of change.

So the rate of change is how one variable changes with respect to another.

If the change is constant, then there is a linear relationship between the variables.

So we might look at how if distance increases at a constant rate over time, that means we have a constant speed.

So we're looking at the rate of change of distance with respect to the time that that takes.

So we're gonna start then at looking at distance-time graphs.

So distance-time graphs are a type of graph which model the distance an object travels with respect to time.

You may well have come across these before.

Because we are modelling the distance moved as time progresses, we plot time on the x-axis.

Now, it's always useful to be aware of the units of time that you're working with, 'cause this could be seconds, minutes, hours, so just make sure you're taking note of that when you're working with these graphs.

Then we plot distance on the y-axis.

And again, these lots of different units that we could use here, so make sure you know what you're working with.

So we can use these to model scenarios.

The idea with modelling is it might not be a perfect model, but it gives us a general idea of what is happening in a real-life situation.

Real-life situations don't always fit exactly with pure mathematics, but we can model them closely so that we can use our math skills to solve problems. So here Aisha has recorded data from a recent jog and she has modelled it as a straight line on a distance-time graph.

So what would this point represent on her graph? What do you think? That's it, it tells us that she ran a total of 2000 metres in 10 minutes.

You could write that as two kilometres in 10 minutes.

Okay, so what does that point there represent on her graph? We just need to check our scale, make sure that we're reading correctly.

So halfway between four and six is obviously five.

So after five minutes, she has run 1000 metres or a kilometre.

It's important to note that after five minutes, she's gone 1000 metres.

In double that time, so in 10 minutes, she's gone double the distance, 2000 metres.

So why is the graph a diagonal line? See if you can explain this.

Why is it a diagonal line rather than a horizontal line or rather than curve? What's that telling us? So what this is saying is that as time progresses, the distance she has run increases.

With this example as well, it's important to note that it starts on zero zero, so this is a direct proportion graph.

So it starts at zero zero and there is a consistent rate of change.

Like we said before, if she ran one kilometre in five minutes, then she can do double than in double the time.

So that's telling us that she's running at a consistent speed.

Now, this is a really important thing to note.

So you might want to repeat this after me or write this down.

So on a distance-time graph, a diagonal line represents moving at a constant speed.

On a distance-time graph, diagonal lines are constant speed.

Let's look at another one.

So Aisha decides to try a slightly longer route.

And we can see here that after eight minutes, she has run 1600 metres.

You might wanna pause and just check that scale.

There are five steps between 1000 and 2000, therefor each one must be 200.

Can you explain then what happens after this point on the graph? Right, well the shape of the graph changes.

So there's another line segment which is less steep than the previous one.

What do you think that means then about Aisha's run? Good, we're really starting to interpret these graphs now.

So what this means is, she started to run at a slower speed, but it's still a straight line, so it's still a consistent speed, just slower than before.

So on a distance-time graph, the steeper the line, the faster the movement.

So that first line segment has a steeper gradient, therefor it's representing moving faster, because you can go further in less time.

Okay, let's have a look at this one.

What happens to the graph after 14 minutes? See if you can explain what happens on the graph and what do you think that means about her run? So we've got a horizontal line segment now.

And what that means is there's no increase in distance in that time.

So six minutes has progressed on the x-axis and we haven't gone any further.

So that means she was stationary for six minutes.

And again, this is another important point, so you might wanna repeat it after me or jot it down.

So on a distance-time graph, a horizontal line represents no movement.

Quick check then.

Aisha runs a 10k.

She start off at a fast speed and we're gonna assume that that's a constant speed.

Halfway through she takes a short break for water and then she finishes the run at a slower but still constant speed.

Which of these sketches could represent Aisha's run? What do you think? Well done if you said that last one.

So the key bit with the first one is that there's a steeper line segment after the break.

So that means she'd be running faster after the break and the information says she finished at a slower speed.

The second one doesn't have a horizontal line to represent stopping.

So it must be that last one.

We've got a steep line segment to start with, then we've got a horizontal line segment to represent stopping and then a slightly less steep line segment for her slower speed.

Okay, so when we interpret distance-time graphs, it's important to understand what is meant by distance.

We can see slightly different graphs.

So here we've got two graphs.

We've got a distance-time graph and a distance from home time graph.

These are gonna represent the same journey, I've just plotted them on two slightly different graphs.

Pause the video and think about what is same in these two graphs and what is different.

Off you go.

Right, hopefully you got some good chance to think about those, maybe you discussed those with somebody else.

So this is what I came up with.

So to start with, we've got a line segment showing 500 metres was travelled in six minutes, and that's the same on both graphs.

Even though it looks longer on the second graph, that's to do with the scale, and we'll come to that in a moment.

But if you read off the y-axis, you can see that we've travelled 500 metres, if you read off the x-axis, that's taken us six minutes.

The other similarity is that we have a horizontal line showing no movement for four minutes.

Now the differences.

So one of the differences is that the scales are different.

You'll see that we go all the way up to 1000 on the distance graph, whereas we only need to go on to 500 on a distance from home graph, and we'll look at why in a moment.

The axes are labelled differently.

So the first one is showing us distance and by that we mean total distance travelled and the second one is showing us a distance from home or a distance from like a fixed start point, and we'll get into that a moment.

What that means is, the second graph has a downwards sloping line, a line segment with negative gradient at the end.

So what is the actual difference between these? Well, in a distance-time graph, you're measuring total distance travelled.

So that means that the line segments will always have a positive gradient or a zero gradient.

Because we are measuring total distance, it's not possible for that to ever decrease.

It can stay the same and it can increase if we travel a bit further, but it won't ever decrease.

Whereas in distance from home graph, our distance from home can decrease.

If you have a negative gradient on this sort of graph, it represents a return journey.

And again, by home we just mean a fixed start point.

So if you walk away from a fixed start point and then walk back to that start point, the distance from home is telling you how far away you are from that start point, either on the way there or on the way back.

So Andeep is going to visit the Eiffel Tower.

If we walk from the station, then go up in the lift, I think the distance-time graph will look like this.

What is the problem with Andeep's graph? Well done if you said that a vertical line is impossible.

It would represent travelling a distance in zero time.

This is impossible, time has to move forward.

We can't travel a distance and it not take any time at all.

It might take a really short amount of time, but it will take some time.

So that line should have some kind of positive gradient.

Right, Andeep's done it properly now.

I have drawn our journey, including stopping at the top and coming back down.

Sofia says, "If we're coming down in the lift, shouldn't the line be slowing downwards?" What do you think? Now, Andeep was right the first.

Because the y-axis is total distance, it's not a distance from home graph, it's just total distance, the distance is always going to increase.

The thing about a distance-time graph is it doesn't tell us about direction of movement, so it doesn't tell us if we're coming back the same way that we've gone out.

Andeep says, "If time is always moving forwards, does this mean that this graph is impossible too?" Yes, it is, because the horizontal axis represents time and time has to move forward, we can't have line segments going backwards in time.

So we can't have vertical line segments or line segments showing a move backwards in time.

And that's true for a distance from home graph as well.

We can negative gradients on a distance from home graph, but they have to moving forwards in time from a previous point.

They can't be moving backwards in time from a previous point.

Right, time to check your understanding.

So this is a distance from home graph, which sections of this graph are impossible? Come back when you think you've got your answers.

Right, well done if you said C, because distance is travelled as time decreases and time has to move forward.

And then E is a problem, because that's showing distance travelled over zero time.

Equally, G is showing distance travelled over zero time, even though it's return journey, it's still impossible to have a vertical line.

In this case F is absolutely fine, because we've got distance from home, a negative gradient is gonna represent the return journey as long as it's moving forwards in time from the previous point.

Right, time to have a practise then.

I'd like you read this distance-time graph.

Think carefully about the units and have a go at answering these questions.

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

Well done.

This should be fun.

So this time, I'd like you to describe Andeep's journey from his graph.

So you can use any context that you like.

He's on a cycle ride.

And I want you to think about what is happening at each stage of the journey and see if you can tell a little bit of a story about what is happening.

It's important that you reference each part.

Get as much key information in as you can.

Off you go.

Well done, so I have sketched this time three distance-time graphs.

So you'll notice there's no units.

I'd like you to match the distance-time graphs with the three scenarios.

You're thinking about key parts of the graph.

Give it a go.

And finally, a climber drives to a local rock formation, walks down to the bottom, climbs to the top, then walks back down to their car.

Have a look at this distance-time graph and can you explain all the mistakes? Give it a go.

Let's have a look then.

So after on hour, we have travelled 50 kilometres.

After two and a half hours, the car has stopped.

Well done if you added that it stopped for 30 minutes or half an hour.

The overall journey, so we just need to look at the furthest distance travelled, and that was 400 kilometres.

Now, I wonder if you came up with this idea.

So if the journey clock said five and a half hours and we can see it actually took six and a half hours, there's an hour somewhere that the car hasn't recorded.

Now, the car was stopped for 30 minutes at two separate points.

So that means it was stopped for an hour.

So maybe when the car was stopped, the journey clock was turned off.

So maybe if you stopped for a break, you might turn the car engine off, it might stop recording the journey time.

If you came up with any other sensible ideas, that's great as well.

And finally, the highest speed, we're looking for the steepest line segment, and that's between three and five hours.

So I hope you came up with some fun stories about what Andeep was doing on his cycle ride.

Some key points that you may have included.

So he travelled four kilometres in 10 minutes, but then returned home at the same speed.

So you might've said that he'd forgotten something and had to go back home.

He then travelled 16 kilometres in 40 minutes and then he had to slow down for the next 20 minutes.

So maybe he was cycling up a hill, maybe he was getting tired.

And then he took a 10 minute break, so it's up to you what you've talked about during that 10 minute break.

Then the next two kilometres he did really quickly.

So I was thinking maybe he going back down the hill again.

And then he turned around and went back home again.

So then going back home, he then did at a consistent speed.

Okay, so let's have a look at these graphs.

So the first one, you're looking at that bottom left graph.

So we've got the steep line in the middle segment representing the train journey, which is gonna be faster than the escalator or climbing the stairs.

For B, that should be the top graph, because we're gonna be slower running up the hill than running down again.

And C could be the driving on the motorway, because when we're going through the roadworks, we're gonna have to go slower.

And then finally, our climber.

So this distance-time graph doesn't make sense at all.

Maybe the person that drew this thought they were drawing a map or something of their journey.

It's not a distance-time graph.

So this is gonna be a problem, because this says we're going backwards in time.

It doesn't mean that we're walking in a different direction down to the bottom of the rocks.

That doesn't make sense.

This means we're going backwards in time.

That's impossible.

And this definitely does not mean climbing up a rock face, 'cause that's definitely gonna take time.

So we can't have a vertical line.

And then I wonder if you got this bit.

This is a distance-time graph, so it shouldn't have line segments with negative gradient.

It's not a distance from home, it's a total distance.

Also, this is a steeper line segment than the first line segment, so that would suggest they were walking back to their car quicker than they drove, which doesn't really make sense either.

Right, well done if you spotted all those mistakes.

Now we're gonna have a look at some other real life graphs.

So we can use graphs to model all sorts of scenarios.

So here we've got a graph, we still got time on the x-axis and this time we're gonna plot the depth of water of time rather than the distance travelled over time.

So water is run from a bath tap into a bath at a constant rate.

So the inside of the bath can be modelled as a cuboid and therefor it's got a rectangular cross section.

So let's see what a graph would look like of the depth of water over time.

Because the cross section of the bath is the same at all depths, the depth is gonna change at a constant rate therefore we have a linear graph.

So you have a constant rate of change, so a linear graph.

Okay, what about if I have a slightly shorter bath.

Still has the same width.

Well, this time it's gonna fill up quicker, so it will take less time to get to the same depth.

We still have a constant rate of change, so we still have a linear graph, but it's steeper this time.

Ooh, so this time we have a bath with a ledge for a seat.

We've still gotta fill up to the top.

What do think that will do to our depth-time graph? Make a prediction before we have a look.

Okay, well it's still gonna increase at a constant rate until it reaches the seat, but then because the seat adds extra space to our bath, it's gonna take longer to fill up.

It's still gonna be a linear line segment, because it's still increasing at a consistent rate, but slower than before.

So it's gonna be less steep.

It's gonna have a smaller gradient than the previous line segment.

Okay, well so far we haven't been very realistic, because most baths have ends which are at an angle for comfort.

So what will be different when we fill this bath? Pause the video.

What do you think this will look like? Right, so there's not gonna be a sudden change, like there was with the one with the seat and this time, the bath is getting continuously wider.

So that means the rate in which the depth is increasing will be continually slowing.

It's gonna get slower and slower for the bath to get deeper as the cross section gets wider.

So at no point is there gonna be a constant rate anymore, it's gonna be continuously slowing down.

So it's gonna look something like this.

Pause the video if you need to check through that again.

Right, Andeep has drawn this depth-time graph.

What do you think this could be representing? Think about the different stages.

Right, so this could be the depth increasing at a constant rate, so we can have any container with vertical sides being filled at a constant rate.

In fact, we just need a container where the sides are parallel so that the cross section is the same at every depth.

So they don't necessarily have to be vertical and parallel, but they just need to be parallel so that the cross section is the same at all depths.

Then the depth is constant.

So maybe the water is turned off, 'cause the idea here is the depth isn't increasing as time increases.

And then for this one, we can see that the depth is decreasing at a constant rate.

So you might've said something like water is being drained from the container if that was done at a constant rate.

Andeep says, "It will be impossible to have vertical line on these graphs." He's absolutely correct.

Just like before, we're plotting time on the x-axis and time always has to be increasing.

Even if you did something like throw a bucket of water into the bath, it's still gonna take some time, even if it's a really tiny bit of time.

So your line segment might look like it's almost vertical, but it won't be perfectly vertical, 'cause some time has to have passed.

Time for a check then.

Assuming they were filled up at a constant rate, which container could this depth-time graph represent? What do you think? Well done if you picked C.

It's the only one that has vertical sides for all the sections, so it's the only one which will be three constant rates of change.

Time to have a practise.

I'd like you to match each container with its depth-time graph.

I'd like you to assume that they are all being filled at a constant rate.

Give that one a go.

And your turn now, so some Oak pupils are filling their different shaped water bottles from a water fountain.

Again, we're assuming that they're being filled at a constant rate.

I'd like you to have a go at sketching a depth-time graph for each.

So think about the different sections of the water bottles.

Will it be a constant of change? Or is it gonna be slowing down? Or is it gonna be speeding up? Once you're happy with your sketches we'll have a look at the answers.

Let's have a look then.

So A, you're gonna have a constant rate of change and then it's going to get quicker and quicker and quicker for that depth to increase, because our container is narrowing.

So that's gonna be the fourth one along.

For B, each section is gonna be a straight line.

It's got three constant rates of change.

We've got slower, quicker, then slower again.

So that's going to be the third one along.

C, I wonder if this caught any of you out.

C is just gonna be a linear graph.

It doesn't matter that the sides are on a slant, because they are parallel to each other, the cross section is the same all the way through that container.

So that's gonna be the first graph.

So D, because the container is narrowing to start with, the depth is gonna fill up quicker and quicker and quicker and then it's gonna be constant when we get to that tall thin part of the container.

So that's gonna be the final graph.

And then E, because our container is getting wider, that means the rate of change is going to be decreasing constantly.

So that's gonna be that second diagram from the left which has that curve.

And then let's see if your sketches look similar to mine.

So for A, we need a straight line segment to start with.

Then we can see that the container is widening, so we need a curve showing that the depth-time is slowing down.

And then we need another straight line and that's gonna less steep than we started with, because our container is now wider.

For B, we should just have a constant rate of change, so a straight diagonal line.

For C, so we need a curve to represent the depth-time graph slowing down.

Then we need a straight line segment for the majority of the water bottle and then the rate of change is going to increase because the container is narrowing and then we have a short straight line segment at the top where we've got that very top of the water bottle.

And D, this one looked quite cool.

So we should have a curve showing that the depth-time is increasing over time and then we need to curve 'round the other way to show that it is slowing down as the container is getting wider and then we need a curve back the same way as the start as the container narrows again.

So you get this kind of wiggle where the speed is increasing then decreasing, then increasing again.

Right, well done, I hope you had a bit of fun sketching out those graphs and looking at those different shaped water bottles.

If you have a water bottle of your own, you might wanna think about what your depth-time graph would look like.

Would it be similar to any of these? So today we've looked at distance-time graphs and what they can tell us about a journey.

We've looked at how diagonal lines represent constant speed, but horizontal lines represent no movement.

We've talked about the difference between distance time and distance from home time graphs and that we can have a downward sloping line on a distance from home graph, 'cause that represents the journey back towards the start point.

And then we played around with depth-time graphs as well.

So we can use graphs to model different situations.

There's different time graphs we can play around with.

The important thing with those is the fact that if the rate of change is constant, the graph will be linear, but there's lots of occasions that produce graphs that weren't linear.

So we just need to remember that not all graphs are going to be linear.

Right, thanks for joining me today.

It was really good to see you and I hope you'll chose to join us again.