Lesson video

Hello, thank you for joining us 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 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 where we are checking and securing your understanding of drawing distance-time graphs.

So we're gonna spend lots of time today drawing distance-time graphs and sketching distance-time graphs for different journeys.

With that in mind, you're definitely gonna want a ruler and a pencil so that you can do the tasks.

And you might want to have those handy so you can sketch things and think about things as we go through.

We're gonna talk a little bit today about displacement.

So displacement is the distance from a fixed starting point.

It's often called home, but just means a fixed starting point.

And the displacement is the distance when measured in a straight line.

So it's how far away from that start point are you at that moment in time.

So the first part of our lesson, we're gonna look at distance-time graphs.

And then the second part of our lesson, we're looking at displacement-time graphs, which is where that key word is coming in.

So if we have enough information, we can draw a distance-time graph for a journey.

So Alex says, I walk 12 minutes up the hill to the shops which are 500 metres away.

It takes me 10 minutes to buy what I need in the shop.

Then I walk the same distance home, but it only takes me 8 minutes.

So to start with, I've helped you out by drawing the axes for you.

So you'll see that we have time on the x-axis and I'm gonna do that in minutes.

And then I've got distance on the y-axis and I'm measuring that in metres.

So what we're gonna do now is we're gonna accurately plot this.

Even if you've done this before, I suggest you take this time to just remind yourself to make sure you don't make any mistakes.

So 12 minutes to walk 500 metres.

Now luckily, our x-axis is going up in 2, so that's nice and easy to plot.

And our y-axis is going up in hundreds, which is also nice and easy to plot.

At zero minutes, he has travelled zero metres, so this graph is gonna start at the origin.

It's important to remember that the fact that the journey is uphill does not affect the graph.

This is not a map of how the journey looks.

It is looking at the distance travelled over time.

It takes him 10 minutes to buy what he needs, so he's not gone any distance in that time.

So that's represented by a horizontal line.

And then the same distance home, but it only takes 8 minutes.

So it's 500 metres away from home, so we're doing 500 metres in 8 minutes.

Because this graph is total distance that needs to carry on increasing.

So it's another 500 metres.

So our graph looks like that.

So there's a few things we can check to see if we've got this correct.

So our overall distance should be 500 plus 500, which it is.

Our overall time should be 12 plus 10 plus 8, which it is.

We know we've got the horizontal line showing that you stopped for 10 minutes.

And we're expecting that final line segment to be steeper because he was walking quicker on the way home than he did on the way there.

So thinking about what we've drawn, what assumptions have we made about Alex's walking speed? What do you think? Well, I don't know if you said something like this.

So our model makes the assumption that Alex walked at a constant speed.

We have shown that by drawing straight lines.

Unless you're told otherwise, it is fair to assume that movement is at a constant speed when drawing distance-time graphs.

So we're gonna focus now on looking at scales and thinking about sensible scales.

So Jacob cycles to his friend's house 18 kilometres away.

He does 10 kilometres in 20 minutes, stops for 10 minutes, and then does the rest in 15 minutes.

So I'm gonna need to draw my axes.

And I need to think about what units I'm going to use.

So we're travelling in kilometres and minutes, so that's the easiest to plot.

Then if we look at our final distance and our final time, we'll know what our axes need to go up to and then we can think about a sensible scale.

So it's 18 kilometres and 45 minutes.

So Lucas says, the y-axis can increase in steps of 1.

Nora says, or in steps of 5.

I want you to look at those two suggestions.

What might be a problem with each of those ideas? What do you think? Right, so if we go up in steps of 1, you'll see that we only get to 9 kilometres.

We needed 18 kilometres.

If we go up in steps of 5, you'll see that we reach 18 kilometres before we're even halfway up the graph.

That means we'll end up with a really tiny graph, which will be difficult to plot, difficult to interpret, and we're not making use of the space we have.

So Laura says, let's think logically.

There are 10 major grid lines.

If you count them up, we've got 10.

We need to get up to 18 kilometres, So if you do 18 divided by 10, each one would be 1.

8.

Now that's not a particularly sensible value to use because it's gonna make everything really tricky to plot.

So instead of using 1.

8, maybe we could use 2.

So if we count up in twos and check that it fits on.

Perfect, we will be able to plot up to 18.

So we used a little bit of logic there and a little bit of our division skills to work out what would fit.

Lucas can do the same with time.

So we also have 10 major grid lines, 45 divided by 10 is 4.

5.

Well again, 4.

5 minutes is not gonna be a particularly sensible scale to use.

So let's try 5.

And again that goes up to 50.

We only need it to go up to 45, so that would work as well.

And now we can plot that.

Pause video if you want to double check my drawing.

Let's have a look at another journey.

So Jun cycles 20 kilometres an hour, stops for an hour, then cycles 4 kilometres in the next half an hour, breaks for 10 minutes, then cycles 6 kilometres in the next 20 minutes.

So quite a lot going on in this journey.

Laura says, I've done the calculations and steps of 3 will work on the y-axis.

Can you see a potential problem though with her idea? What do you think? All right, well our first two values are not multiples of 3, so they're gonna be harder to plot.

But going up in threes, plotting 20 kilometres and 4 kilometres is not gonna be easy to do accurately.

Also, there are five minor grid lines between each major grid line.

So if each major grid line is 3, 3 divided by 5 is 0.

6.

So that means each minor grid line is 0.

6, which again does not make these values easy to plot.

It's gonna be hard to find where 20 kilometres is and 4 kilometres is with any accuracy.

So what do you think might be a better step to use? Right, even though it doesn't use the whole grid.

I think a step of 5 would be easier.

Steps of 1 and steps of 2 would not fit on our graph.

So steps of 5 is the next sensible idea because there are five minor grid lines, each one's then gonna represent 1 kilometre.

And that's gonna make all our integer values easy to plot.

What about for time? What would be a sensible scale? I think 20 minute intervals makes the best use of the space.

10 minutes steps do not fit.

And because our intervals are often 10 minutes or 20 minutes or half an hour, 20 minute intervals are gonna be sensible.

It's not essential to write every number on the axis.

You'll see that I've chosen just to write 60, 120, and 180 to make it easy to see.

And then we can draw our graph.

There we go.

Pause the video if you want to check my graph one more time.

Okay, your turn.

Izzy is drawing a distance-time graph for this journey.

Take a second to read through her journey.

Which of these is the most sensible step for the major grid lines on the y-axis? See if you can think of a justification for your answer.

Give it a go.

I think 20 kilometres is going to be best.

The total distance travelled is 160 kilometres, so steps of 10 kilometres will not fit.

Steps of 40 would mean each minor grid line is 10 kilometres, which is quite helpful, but it only uses up half the space.

So we'll have a little graph that'll be tricky to read.

So 20 kilometres is gonna be best.

Okay, we're gonna do the same this time, but for time.

I think there's a couple of choices this time.

I think either 20 minutes or 30 minutes.

The total time is 180, so all of these options would fit.

So you just want to think about what's gonna be the easiest to plot.

Now 20 minute steps, we'll use all the space available.

However, if you look at our time stages, they're either 30 or 60.

So actually going up in steps of 30 minutes might make it really easy to plot.

You won't need to go halfway between any of the grid lines then.

There's many moments where a sketch will suffice.

We're not thinking about axes and scales anymore, we're just having a sketch.

So this is Alex's scenario this time and we'll work through it together.

So total distance is 400 plus 600, which is 1,000.

And the total time, we've got 5 minutes, another 5 minutes, and 10 minutes, so 20 minutes in total.

So we need to mark our axes.

I'm gonna go up to 1,000 metres on the y-axis and 20 metres on the x-axis 'cause that's our final time.

And then I'm gonna sketch this.

So if I'm doing 400 metres in 5 minutes, it should be less than halfway to 1,000 and approximately halfway to 10 minutes.

So about there.

Then he waits for 5 minutes, I'm gonna have a horizontal line and two 5s is 10.

So that should take me halfway to 20.

And then the final line segment, we're going 600 metres in a further 10 minutes.

So we should get up to 1,00 and up to 20.

We need to make sure all the key information is marked.

You've got options with how you'd like to do this.

So you could mark it on the axis.

So I'd have to mark where my 5 and my 10 were and my 20.

I'd also have to mark where my 400 was or you can put them on as coordinates, which sometimes makes it easier to see.

And then we can check our key features.

We've definitely got the section where Alex is stationary and our sketch is good enough to show that the final section was slower than the first section.

That first section's got a steeper gradient.

Quick check then.

Which of these is the right estimate for the first coordinate for Izzy's scenario? Read the scenario again.

Which is the best coordinate? Well done if you said B, it needs to be approximately halfway between zero and 1 hour.

And it needs to be approximately halfway between zero and 80 kilometres.

Okay, I've plotted the first two stages for you now.

So we're then up to the section where she says then we stopped for half an hour, we did the final 20 kilometres in an hour.

So what is the right estimate for the next coordinate of this sketch? Off you go.

We should be happy now that stopping, being stationary, means that there is a horizontal line.

Time has passed but she's not moved any further.

Time for a practise then.

Please make sure you are using a ruler and a pencil for these.

And that you are thinking about being really accurate with where you're plotting your points.

Give this one a go.

Well done.

You're gonna have to think more about your scale this time.

So draw your axes on, making sure you give yourself lots of space.

Then think about what your scale is gonna be on both of your axes.

Don't forget to label your axes as well.

Come back when you're ready for the next one.

So for this journey, I'd like you to suggest the disadvantage of each of these suggested steps for the x-axis.

You're gonna need to read through the journey and I've said we could do the major grid lines in steps of an hour, or in steps of 30 minutes, or in steps of 10 minutes.

Could you think of a possible disadvantage for each? Off you go.

Perfect.

This is the same journey as before.

Now you're gonna have a go at plotting it.

So think about what we said about the previous time intervals.

Can you find a better time interval that will work? Do the same for distance and then plot your journey.

Give it a go.

And finally this time, all I want from you is a sketch.

What you do need to do is include all necessary information on your sketch.

So you could do that as coordinates or you could do that by marking all the relevant places on the axes.

When you're happy with your answer, we'll look through it together.

Well done.

You need to pause the video and just check that you've got each of these sections correct.

Notice that the gradients of the third and the fourth line segments look similar, but they're not.

So there's actually two separate line segments between 2 hours 40 minutes and 3 hours 50 minutes.

So do make sure you've got two separate line segments with two separate gradients in that middle part.

Okay, you might have gone with slightly different labels on the axis.

I went with the one that I thought was the most sensible.

So I've done time, I've done two major grid lines as an hour.

So each major grid line is 30 minutes.

And distance, I've gone up in steps of 5 miles.

Pause the video and check you've got that correct.

Make sure that your line is definitely finishing at 32 miles, whatever that looks like on your scale.

So you might have said something like total time is 3 hours 4 minutes, going up in steps of an hour would not make use of the space.

We'd end up with a really squished graph that would be hard to read and hard to plot but wouldn't make use of all that space.

For 30 minutes.

30 minutes isn't a bad suggestion.

The main problem with that is that each minor grid line would represent 6 minutes.

And that might be quite tricky to plot, especially when we've got to plot 20 minutes and 44 minutes.

10 was possibly a little bit easier to see because that's not gonna fit on our graph.

The total travel time is 184 minutes, so steps of 10 will not fit.

So I thought it'd be more sensible to go with steps of 20 minutes.

So each major grid line is 20 minutes and you'll notice that I only wrote on 60, 120, and 180.

And that was just to make it clearer so I didn't have lots of cramped numbers near each other.

For distance, I got up in steps of 25.

And again I've labelled every other one, so I've labelled 50, 100, 150, and 200.

Again, you need to pause the video and just check that your graph looks the same as mine.

If you've used a different scale, you need to check your coordinates are nice and accurate.

So you'll notice that a lot of my coordinates were not on major grid lines.

You need to check that you've plotted those accurately.

Well done.

And finally, our two sketches.

Pause the video and check your answers.

Make sure that you've got those key pieces of information either as coordinates or written on your axes.

Right, well done.

There's lots of key bits of information there, particularly about scale and about axes and units that we are thinking about.

And all those skills we're gonna need to continue thinking about with the rest of this topic.

So now we're gonna look at displacement-time graphs.

So displacement is the distance from a starting point when measured in a straight line.

So this is Alex's scenario from earlier and that was the distance-time graph we drew.

How will a displacement-time graph look different? Well, let's look at it together.

So if Alex walks 12 minutes up the hill to the shops, which are 500 metres away.

And we're assuming he's walking a straight line away from home.

That's gonna mean that the first part of his journey is gonna be the same as on a distance-time graph.

Because the distance he is away from home is the same as the distance he has walked 'cause he is walking away from home.

We're still gonna need a horizontal line to show that he's stationary.

But this time, as he returns from home, his displacement is going to decrease.

'Cause he is getting closer to home again, his distance from home is getting smaller and smaller and smaller.

Because he's travelling the same route, so he is walking in a straight line at a constant speed like we assumed earlier, we can draw the final section with a negative gradient.

So that's him returning home and he does that quicker, so you can see that final line segment is steeper.

So those are our two different graphs next to each other.

After how many minutes is Alex 250 metres from his house? What do you think? So it's the displacement-time graph that's gonna be the easiest to use here.

We can see that at 6 minutes, he's 250 metres away from home.

But we can also see that at 26 minutes, he's 250 metres away from home.

Because he's coming back towards the house again.

Without the scenario, we can't get that information from the distance-time graph.

So displacement-time graphs can sometimes give us more information about the nature of a journey.

So here's a graph of Izzy's cycle ride.

What does this graph tell you about what happened after 40 minutes? Right, well she returned home again, before then setting off again.

So if we assume Izzy is travelling in a straight line, the steeper the gradient, the faster her speed, just like a distance-time graph.

What's useful with the displacement-time graph is we can see where she turned around and came back home again.

If we were to draw this on a distance-time graph, we would need a less accurate scale 'cause we need to go all the way up to 40 kilometres rather than just 20 out and 20 back.

And we'd lose that information about the direction of travel.

We haven't got those coordinates anymore where she turned around and had to come home again.

However, what information is easier to see from a distance-time graph? What do you think? The total distance travelled is a lot easier to see from a distance-time graph.

So if you're not interested in direction and you just want to see how far someone has travelled in total, it's a distance-time graph you want.

Right, time to have a little think then.

Which of these is an advantage of a distance-time graph? What do you think? Right, the best answer is C.

From a distance-time graph, you can easily see how far you've travelled after any period of time.

B and D are also true.

So if you went with B and D, I can understand why.

However, these things are true for displacement-time graphs as well.

They're not an advantage of a distance-time graph over a displacement-time graph.

Okay, so this time, which of these is an advantage of a displacement-time graph? What do you think? I think there were two good answers this time.

It tells you when you're travelling away from a fixed start point and when you're travelling towards the start point.

And you can see at which points of the journey you were the same distance from the start.

So which points of the journey were you essentially in the same place? One on the way out, one on the way back.

And you can't see that from a distance-time graph.

You can from a displacement-time graph.

So Andeep is looking at this displacement-time graph.

So toy car being pushed along the ground.

He says, I didn't think we could have negative values on these graphs.

I wonder what you thought.

So although it does not make sense for distance to be negative, displacement can be negative.

Could you think about what each stage of this graph might represent? He's pushing a toy car along the ground.

Have a little think.

Don't worry if you've not seen this before.

I'll run you through it once you've had a little think yourself.

Okay, so the first section is car moving in a straight line away from a start point.

So 'cause it's a displacement-time graph, we need a fixed start point.

So B would then mean the car moving back to that start point.

And because it's symmetrical, those two journeys are gonna be the same speeds on the way out and on the way back.

That third section would be the car moving past the start point but in the other direction.

Because we've got a fixed start point and we've got a fixed direction an negative displacement would actually mean in the other direction.

If it's something you're finding a little bit tricky at the moment, don't worry, it is a bit of an extension concept.

Then that last section is the car turning round again and moving back to the start from the other direction.

All of these have the same gradient, either positive or negative, so all of these will actually be the same speed.

Right, your turn.

Which sections of this displacement-time graph are impossible? If you're finding the idea of negative displacement a bit tricky, don't worry.

Think about what you know about impossible graphs.

What things will be impossible? Give it a go and we'll talk through it together.

Okay, A and B are fine.

The first problem is with C, distance is travelled as time decreases, Time always has to move forwards, so we can't have a line going backwards in time.

E is also problematic.

We can't have vertical lines 'cause that means that your displacement has changed and it's not taken any time.

So you've moved a distance in no time.

Now F is absolutely fine, it's moving towards the start point.

G is fine.

That means moving past the start point in the opposite direction.

H is fine, it's stationary.

It's just being stationary in the opposite direction to where you started.

And then I and J is just moving towards the start point and then past the start point again.

Right, time for you to have a go.

So I'd like you to read Izzy's journey and plot a displacement-time graph.

You're gonna need to think of your own scale and make sure you label your axes.

Give it a go.

Well done.

For question two, I'd like you to draw a distance-time graph for the information shown on this displacement-time graph.

If you are not sure about that negative displacement below the x-axis, think about how much distance has been moved and how much time that has taken, and what's that gonna look like on a distance-time graph.

Give it a go.

And finally, Lucas has drawn a distance-time graph for his journey to his friend's house and his journey back.

If he stays at his friend's house for 20 minutes, can you draw this on a displacement-time graph? Think about your scale before you start drawing this.

Off you go.

Lovely, I'd like you to pause the video and check your graph looks like this one.

So let's have a look at this second one together.

The first two sections are gonna be exactly the same, just on a slightly different scale.

And then it doesn't matter if we're walking back towards home, it's still 10 kilometres in 10 minutes.

So we can draw that on our distance-time graph.

What that means here is that you end up with only one line segment, which was two line segments on the displacement-time graph.

Between 30 and 60 minutes, the speed was the same, just going in different directions.

So as soon as you put that on a distance-time graph, you lose the fact that they are different directions and it looks like one continuous movement.

Don't forget that your axes need to be labelled.

So we've got time in minutes and distance in kilometres.

Once you're happy with your graph, we'll have a look at the last one together.

And finally, we needed time to go up to 40 minutes.

So I've gone with each grid line as 2.

But to make it clearer, I've only marked on each multiple of 10.

We need to go up to 600 metres, so going up in 100 metres is gonna be the easiest on the y-axis.

And then just pause the video and check you've got the same graph as me.

Well done.

We've done lots of work today on being accurate when drawing graphs.

Drawing your own scales and your own axes can sometimes be a trickier skill.

It's really important that you practise that because it'll be useful in other subjects and real life situations as well.

So today we've looked at how distance-time and displacement-time graphs can be drawn.

Displacement-time graphs remember can be called distance from home graphs as well.

We've looked at how distance-time graphs are helpful 'cause they show the total distance over time, whereas displacement-time graphs can be helpful 'cause they show the distance from a start point at any given time.

And then we spend lots of time thinking about sensible scales, how we can work that out if we're not sure what scale to use, how we can check that it's going to fit, and then using those to accurately plot our journeys.

Right, thank you for joining me today.

I hope you found that really useful and something that you can use going forward.

I look forward seeing you again.