Lesson video

Welcome.

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

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

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

I hope you're really, really looking forward to it.

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

Welcome to this lesson on distance time graphs.

By the end of this lesson, you'll be able to calculate time intervals and speed for sections of a distance time graph.

Couple of keywords that we're going to use today.

Just pause the video and make sure you're happy with those.

So we're gonna start by looking at how to calculate distance from distance-time graphs and displacement time graphs.

So here is a distance time graph for a journey.

How far was travelled over the whole journey? Aisha's given this a go.

She says, "They did 50 kilometres in one hour, 100 kilometres in 1.

5 hours, stopped, then did 225 kilometres in two hours, stops again and the final 25 kilometres in an hour." She's then added those up and got a total of 400 kilometres.

What do you think about Aisha's answer? I wonder if you said something similar to Andeep.

Andeep says, "I think you've over complicated this Aisha.

There's a really quick way to see total distance." Wonder if you can spot it.

Of course, this is a distance time graph.

400 kilometres is the y-coordinate of the end of the final line segment.

So in order to find total distance, we just have to find the end of the final line segment, and it'll tell us how far was travelled in total.

So some coordinates can be easier to read than others.

So the difficulty with this can be checking the scales and making sure that you're reading them correctly.

So pause the video.

What was the total distance travelled for this journey? So first, we need to work out what each line represents.

So between 20 and 30, there are five steps.

So 10 divided by five is two.

So that means each grid line is worth two.

And you can always count through and check that that works, 22, 4, 6, 8, 30.

That does work.

So that means that point there is going to be 24 metres, and it can sometimes be helpful to use your ruler either to draw on a line or just to put up against your page.

If you're trying to show this to somebody else, then you're definitely gonna want to draw the line on your page so they can see where you've got that number from.

So this time, we've got a displacement-time graph.

These can sometimes be called distance from home graphs.

And what that means is we're moving away from a start point.

We might also be moving towards the start point again later.

So in this graph for instance, you can see that we've moved away from a fixed start point.

We've been stationary for an amount of time, and then we've moved back towards the fixed start point, and that's why our displacement is decreasing.

Aisha says, "The journey finishes at zero metres, but I know the journey was longer than that." Well that's because this point shows that the journey finished at the same place that it started.

That doesn't have to happen with displacement-time graphs.

So to work out the overall distance, we can't just look at the end of the final line segment.

We have to consider the parts of the journey moving away from home, and the parts of the journey moving back towards home.

So it's got 1000 metres walking away from our start point, then 600, then 400 moving back towards our start point.

Total distance is 2000 metres.

Why do we not need to include the horizontal line segment? What do you think? Well, if we assume that all movement is in a straight line from home, which we're always doing with our displacement-time graphs, then a horizontal line represents no movement because your displacement hasn't changed over that time.

That means no distance has travelled and therefore we don't need to add anything onto our distance travelled.

Andeep says, "We didn't need to add the two return sections separately, we can see that the journey was one kilometre out and one kilometre back." He's absolutely correct.

We can use that as a bit of a shortcut so we don't have to keep adding up the separate line segments.

Just make sure that you're considering all the times that you're moving away from home and all the times that you're moving towards home.

Right, this is a displacement-time graph.

You can see the y-axis has been labelled distance from home.

Andeep says, "It's 20 kilometres there and 20 kilometres back.

So the total distance is 40 kilometres." Is he correct? I gave you a little bit of a clue for this one on the previous slide.

He hasn't taken into account the start part of the journey.

He needs to think about all the times that we're moving away from home, and all the times that we're coming back towards home.

So we've got four kilometres away and then we've had to come back.

Then 20 kilometres away from home, and we can consider that all in one go, that's fine because all the line segments have positive or zero gradient, so they're all moving away from home.

And then we can consider that last section when we're coming back to home.

So in total, there's 48 kilometres.

Alright, quick check then.

True or false? On a distance-time graph, the overall distance is the y-coordinate of the final point of the journey.

What do you think? See if you can justify your answer.

That is true.

A distance-time graph shows the total distance travelled at any given time.

So to get the overall distance, we just need to find the furthest time and read off what the distance was.

Okay, another idea.

On a displacement-time graph, the total distance is always double the furthest distance from home.

Is that true or false? And what's your justification? That one is false.

You need to take the distance travelled after each change of direction into account.

On top of that, journeys do not need to start and end at the same point.

It's possible to travel away from home and not come all the way back home again, or even on a displacement-time graph.

So what you need to do is consider the separate sections when you're moving away from home and then coming back home again.

So after each change of direction.

Time to put this into practise then.

To start with, I'd like you to work out the total distance travelled for these journeys.

Think carefully about your scales.

When you're happy with those, we'll look at the next step.

Well done.

So this time we've got some displacement-time graphs.

I'd like you to work out the total distance travelled in these journeys.

And finally, "These displacement-time graphs show the journeys that Jun and Andeep took delivering letters to pupils in five adjacent classrooms." So what you'll see is that they started at a fixed start point and then they moved away and then back towards, or maybe away in the other direction and back towards, okay? As they're visiting different classrooms all next to each other, probably on a corridor.

Get those a go and then we'll come back and look at them together.

Alright, so the first one, we just need to look at our y-coordinate.

It doesn't matter that the x-coordinate is a little bit tricky to read, we just need to know the total distance.

So that's 200 metres.

Make sure you've got your units.

For B, so between 60 and 80 there are four steps.

20 divided by four is five.

So each minor grid line is five.

So that represents 65 kilometres.

And for this one we're gonna need to consider the sections moving away from home.

So for A, you can see that 700 metres is the furthest away from home, and then they did the same on the way back.

So 1,400 metres or 1.

4 kilometres.

For B, we're gonna have to consider the separate sections.

So moving 10 kilometres away from our start point, then 10 kilometres back.

Then we're going 70 kilometres away from our start point, and then 40 kilometres back, and then we're going another 15 kilometres away again.

So just need to add all those separate sections up, and we get 145 kilometres.

So then we have these displacement-time graphs.

So if you look at all the individual sections, Andeep moves 20 metres away from his start point.

And then to get back to his start point, it's another 20 metres.

But then he goes 15 metres in the other direction and then 15 metres back.

So that's 70 metres in total.

Jun does a lot more toing and froing.

So he goes 15 metres in the opposite direction and 15 metres back.

Then 10 metres in that same direction and 10 metres back.

And then he goes 20 metres in the other direction, 20 metres back, and then 15 metres in that same direction and 15 metres back.

So in total that's 120 metres.

And then make sure you read the question, that's 50 metres more than Andeep.

Right, how far had they travelled after five minutes? We need to find five minutes on the x-axis, we need to check our scale.

Each grid line is one minute, and then again we need to make sure we're adding up all the line segments.

So we've got 20 metres away and then 10 metres back for Andeep.

So that's 30 metres in total.

For Jun, 15 metres away and 15 metres back, 10 metres away, 10 back, and an extra five metres before we get to five minutes.

So that's 55 metres in total.

And finally, 6.

5, 0 is the point that the graph first crosses the x-axis on Andeep's graph.

That could represent Andeep returning to his start point before delivering the letters to the classrooms in the other direction.

If you imagine, he started off in the middle of the classrooms, he went one way first and then came back the other way.

When he's gone back past his start point, he'll then have negative displacement.

If you're still a little bit unsure about the negative displacement, do not panic, it's a slightly more challenging idea.

But it works in just the same way as positive displacement just in the other direction.

So now we're gonna have a look at calculating speed.

So we can use distance-time graphs to calculate speed.

Have a look at this graph.

How far was travelled in one hour? So we can see, looking at this graph, that we have travelled 100 kilometres in one hour.

If we put that into a ratio table, we can see that for the first stage of the journey, the speed was 100 kilometres per hour, and that ratio table is gonna be more useful when we're looking at time periods that are not one hour.

So the line continues straight until 2, 200.

What does that mean about our speed? Right, well a straight line means constant speed.

So that means that for the next hour, we're also travelling at 100 kilometres per hour.

In two hours, we can then do 200 kilometres.

Let's look at the speed for the second part of the journey.

So we can then see that we have travelled 200 kilometres.

But now it's taken four hours.

So if 200 kilometres is travelled in four hours, this is where our ratio table becomes helpful.

We can work out the speed by working out the kilometre per hour.

So speed is the unit ratio between distance and time.

So to get the distance travelled per one hour, if we divide by four, we can see that that's 50 kilometres per hour.

So what have we calculated by doing this? Can you spot the similarity between this and maybe something that you've done before? So we've calculated that the speed is 50 kilometres per hour, but what we've done is we've found the gradient of the line.

You may well have used ratio tables to find gradients of lines before.

So the speed of each individual section of the journey is the gradient of the distance-time graph.

You might wanna pop that in your notes or you might wanna repeat it after me.

So on a distance-time graph, speed is the gradient of the line.

Izzy says, "In science, we use this formula for calculating speed.

Is this the same thing?" And she's written speed equals distance divided by time.

Well, let's think about it.

That's what we're doing here with our gradient.

We found the distance, we found the time, and then we've done the change in distance divided by the change in time, or the change in Y divided by the change in X.

So that formula that you may have seen before is the same thing as finding the gradient of the line, which is calculating speed.

For both of these methods, we do need to be careful with our units.

So let's calculate the speed of this first part of the journey, and then we're gonna think about our units.

So you can see that in 12 minutes we've travelled 500 metres.

And then if we divide by 12 on our ratio table, we can see that can be written as 41.

7.

But what unit is our answer in? Right, well I don't know if you said metres per minute.

So if we can go 500 metres in 12 minutes, that's 41.

7 metres per minute.

Right, Sam says, "Metres per second is a much more common measurement." So what could we do to change our answer into metres per second? Pause the video.

What would you do? Right, if you look at this new graph, you'll see that I've changed the axes.

So now the scale is in seconds.

So instead of going up two, four, six, eight minutes, I've gone 120, 240, 360 seconds.

Now, all my units are gonna be metres per second.

So I've got 0.

694 metres per second.

Sam says, "We didn't need to change the whole axes, we could've just written 12 minutes in seconds.

We do that by multiplying by 60." So of course, you could do that as well, if you didn't wanna redraw your axes, it's perfectly sensible to do what Sam says and just change the time in minutes into time in seconds.

Okay, time for a check.

So Izzy has this graph.

She said, "The gradient of the line for the first part of the journey is a half, so the speed is 0.

5 kilometres per hour." Have a look at her working.

What mistake has she made? Well done if you said that time is being measured in minutes.

So she's actually calculated that it's 0.

5 kilometres per minute.

If she wanted it in kilometres per hour, she would've needed to do something a little bit different.

We're gonna have a look at that now then.

So often, we want the speed in kilometres per hour or miles per hour.

We've got our time in minutes.

So what we can do is we can change our time in minutes into time in hours.

So how could we write 20 minutes in hours? See if you can figure this one out, Right, so well done if you remembered you could write it as a fraction.

We know that there's 60 minutes in an hour.

So 20 minutes would be 20 out of 60.

You could simplify that just 2/6 or 1/3 if you wished.

So now we can write it like this.

So the time is actually 1/3 of an hour.

And the distance is 10.

And then to get one hour, we can divide by 1/3 or times by three, we know they're the same thing.

So in one hour it would be 30.

So that's then 30 kilometres per hour.

Alternatively, you could use our knowledge there are 60 minutes in an hour.

And if we had 10 kilometres in 20 minutes, then we can work up to 60 minutes.

So instead of dividing to get to one minute, we can work up until we get to 60 minutes, 'cause that's an hour.

So if we multiply both sides by three, since this is exactly the same as what we did before, just written slightly differently, then we've got 30 kilometres in an hour.

Just be aware that the gradient of the line is still a half.

We are writing our speed as 30 kilometres per hour, 'cause that's the units we want.

But if we were just asking what the gradient was, the gradient remains as a half.

So this graph shows Izzy running a 300 metre race.

What was her speed? Give this one a go.

Right.

Well to get the speed, we need to calculate the gradient.

So you need to pick a coordinate that is definitely on the line and is on the corner of two grid squares.

So you could've gone with 20, 100.

40, 200 or 60, 300.

I just went with 60, 300 in this case.

Looking at our units, that is 300 metres in 60 seconds.

So per second, that is five metres per second.

Now I didn't tell you what units to work in, so you could've written that as 300 metres per minute, but metres per second is a lot more common.

And if we're finding the gradient of our line, our distance is in metres, our time is in seconds.

So our gradient is in metres per second.

So here we've got a distance-time and a displacement-time graph for the same journey.

We're gonna look at that last section on the distance-time graph.

So we've travelled 500 metres in eight minutes.

So that's 62.

5 metres per minute.

I'm happy with my speed being in metres per minute for the moment.

Now if we look at the gradient at this line, because as X increases, Y decreases, we're gonna have a negative gradient.

So we've got -62.

5.

"Well hang on," says Sam, "on the displacement-time graph, the gradient is -62.

5.

Is the speed different to on the distance-time graph?" What do you think? Well, we know they represent the same journey, so no, we know the speed should be the same.

So what has happened? Well, a negative gradient on a displacement-time graph represents moving in the opposite direction.

So if we assume that all movement is in a straight line, we can find the gradient as normal, but then write the speed as the positive value.

So like Sam says, "Even though the gradient is -62.

5, we could write the speed as 62.

5 metres per minute." Right, time for you to have a go.

What is the speed for the second stage of this journey? So in four hours, we've travelled 500 kilometres.

The negative gradient, 'cause we're moving back towards our start point.

Our speed, if we divide both by four, we get -125 kilometres in one hour.

But the speed would be 125 kilometres per hour.

So when a journey has multiple sections, we can find the average speed by looking at the total distance travelled and the total time taken.

So here you can see that we've got a two stage journey with two different speeds.

We're gonna have a look at Aisha's average speed.

Well, let's look at the journey in total.

She has travelled 2,200 metres in 14 minutes.

So we use our ratio table.

I use my calculator to help me.

So I've got approximately 157 metres per minute.

Because that's taken into account her entire distance and her entire time, that's an average speed.

Right, Izzy says, "Her speed for the first part of the journey is 200 metres per minute and the second part is a hundred metres per minute.

Can I just find the average of those values to get the average speed?" What do you think? Well, let's have a look.

That's not gonna work, 'cause 150 is the average of 100 and 200.

And the answer was 157, wasn't it? So let's think about why that's happened.

Well, she maintained the fastest speed for longer, so that's gonna bring her average speed up.

So we can't just find the average of the two speeds 'cause the amount of time that that speed was maintained is gonna affect the overall average.

So to get our average, we need the total time and the total distance.

So time for a check, what's the total distance travelled in this journey, and therefore what is the average speed? See if you can get this one.

Well done if you said that the total distance is gonna be 1000 kilometres.

But 500 moving away from home, and 500 moving back towards home.

So we've got 1000 kilometres in six hours.

So our average speed is approximately 167 kilometres per hour.

Time for practise then.

So this time, I'd like you to spot the mistake in each of these attempts to calculate the speed of the journey.

So you need to read the question, read the attempt, can you spot the mistake? Can you maybe suggest what they could do to improve on their answer and not make that mistake again? Once you're happy with those one's, come back for the next bit.

Well done.

So this time, we've got three journeys.

Can you find the average speed of those journeys? Off you go.

So here's a distance-time graph of a bike ride, which June completes.

What is his speed for each section in kilometres per minute? Then I'd like you to convert your answers into kilometres per hour.

Give it a go.

So this is the same bike ride.

I want to know what Jun's fastest speed is, and what's his average speed is? Off you go.

And finally, here's a displacement-time graph for a cycle ride Izzy completes.

I'd like to know what her fastest speed was, and what her average speed was? When you're happy with your answers, come back and we'll look through it together.

Well done.

So the first one, you might've said something like the values have been divided the wrong way.

Speed is distance divided by time.

Your suggestion might have been to use a ratio table.

So it's four kilometres in six hours.

So it's 0.

67 kilometres per hour.

We need to do the distance divided by the time, the change in Y divided by the change in X.

For the second one, why don't you spotted that they didn't take the scale on the x-axis into account.

So we can't just count the squares.

We need to see there is an increase of eight hours, and an increase of five kilometres.

So it's five kilometres in eight hours or 0.

625 kilometres per hour.

And this last one is just incorrect units.

So the answer that we've been given, 0.

4, would be in kilometres per minute.

If we wanted the answer in kilometres per hour, we need to think about what 30 minutes is as an hour.

Well, that's not too bad, 30 minutes is half an hour.

So we've gone 12 kilometres in half an hour, so that'd be 24 kilometres per hour.

Let's have a look at our average speed.

So for two, the total distance is 50, the time is five, so the average speed is 10 metres per second.

For B, the total distance is 50.

The total time this time is four.

So our average speed is 12.

5 metres per second.

Now I just want to draw your attention to the fact that this time we could've worked out the two speeds and found the average.

Because the two time periods are the same.

The first line segment spans two seconds, and the second line segment spans two seconds.

So you could find the two speeds and then find the average.

However, because that only works when the time periods are the same, it's easier to stick to the method of looking at the overall distance and the overall time.

For C, the overall distance is 70 metres.

We need to add up the two different parts of the journey.

The overall time is five seconds, so we've got 14 metres per second.

It doesn't matter, that part of that journey is coming back in the other direction, because we're just working out the speed, and the speed is 14 metres per second.

And then with Jun, in kilometres per minute, you've got 20 divided by 60, which is a third or 0.

33 kilometres per minute.

Then you've got 0.

5 kilometres per minute and 0.

125 kilometres per minute.

For B, you've got choices of how you want to convert it into kilometres per hour.

I just went back to the original question and looked at the scale for the first one.

I went, hang on a minute, That's 20 kilometres in 60 minutes.

Well, 60 minutes is an hour, so that's 20 kilometres per hour.

I could do the same for the second one.

So that from 60 to 80 is 20 minutes, that's a third of an hour.

So 10 divided by third or 10 x 3 is 30 kilometres per hour.

You might have also noticed that to get from kilometres per minute to kilometres per hour, you can multiply by 60.

The last one then is 7.

5 kilometres per hour.

So Jun's fastest speed was 0.

5 kilometres per minute or 30 kilometres per hour.

We've done the hard work for that one.

And the average speed is easier, 'cause total distance is 35, total time is two hours, so 17.

5 kilometres per hour.

And finally, to get Izzy's fastest speed, we want the steepest gradient.

I've worked out all four gradients.

Well done if you manage to save yourself time by identifying the steepest gradient.

So it's that final stage of the journey, which is 0.

35 kilometres per minute.

It doesn't matter that it's a negative gradient.

If you change that to kilometres per hour, it's 21 kilometres per hour.

Her average speeds, we need all the distances added up.

So the total distance is 40 kilometres.

Total time is 160 minutes, which is 0.

25 kilometres per minute or 15 kilometres per hour.

Well done for all your hard work on that one.

Thank you for joining me today.

We've done lots of work on distance-time graphs.

Main thing we've looked at is how to calculate speed from a linear distance-time graph, and that'd be a really, really useful skill.

If you want to just read through what we've covered today, feel free to pause the video and check through that now.

Really glad you decided to join us, and I look forward to seeing you again.