Lesson video

Hello, Mr. Robson here.

Welcome to Maths.

Lovely to see you.

Today, we're modelling with graphs.

This should be a cracking lesson.

Let's get going.

A learning outcome.

I'll be able to model real life situations graphically.

Keywords, the rate of change is how quickly one variable changes in relation to another.

The gradient is a measure of how steep a line is, calculated by finding the rate of change in the y-direction with respect to the positive x-direction.

Two parts to today's lesson, and we're going to begin by reading distance time graphs.

Distance time graphs can model many real-life situations and they tell us many useful things.

Sofia's been recording a data from PE lessons and plots this graph.

"That's what I did a few years ago," she says.

"This is what I can do now." What can you read from this distance time graph? Pause this video.

Have a conversation with a person next to you or a good think to yourself.

What can you spot there? Welcome back.

Lots of things you might have noticed.

Did you notice that Sofia run 100 metres on both occasions? I hope so.

Did you notice that crucially her time has changed? She used to run 100 metres in 20 seconds.

Now she can do it in a very impressive 14 seconds.

Did you comment upon the gradient? Both lines have a consistent gradient.

They're straight lines.

This reflects a constant rate of change.

And because this is a distance time graph, that reflects a constant speed.

Finally, you might have commented about the steepness of the gradient.

The steeper gradient reflects a greater speed.

If we look at what's happened on 14 seconds for those two lines, it's clear to see this point.

Sofia used to be able to cover 70 metres in 14 seconds.

Now she can cover 100 metres.

That's created a steeper gradient which reflects a faster speed.

Sofia's smart-watch tracks her walk to school one day.

"This moment looks unusual on the graph," Sofia says.

What do you think happened here on her journey? Pause.

Tell the person next to you.

Have a good think to yourself.

See you in a moment.

Welcome back.

I wonder what you thought.

Izzy knows.

Izzy says, "That's when you stopped to pick me up." "Of course, a gradient of zero means I stopped." This horizontal line represents no change in distance.

That is a stop.

Let's check you've got that.

On this day, how long does it take for Sofia to reach Izzy's house? Look at that distance time graph.

At what point does Sofia stop? Is it 250 minutes, six minutes, or 12 minutes? Pause.

Have a think about that one now.

Welcome back.

I hope you said it's B, six minutes.

That coordinate there, six minutes in, we're at 250 metres and Sofia stops.

Another quick check.

On this day, how many minutes is Sofia waiting at Izzy's house? She stops on six minutes, but for how many minutes? Is it zero minutes, six minutes or 12 minutes? Pause, and have a think about that one now.

Welcome back.

I hope you said it's B, six minutes.

If we read along that horizontal line, we get to the coordinate 12, 250.

That means Izzy went from minute six to minute 12, stop still on 250 metres from home.

That's a duration of six minutes.

Final check.

What does this change in gradient tell us? What change in gradient? This change in gradient.

What does that tell us? Does it tell us there's been a change of direction? The road got steeper, or does it tell us it's been a change of speed? Pause this video.

Have a think about that one.

Welcome back.

I hope you said it's option C, change of speed.

It absolutely does not represent a change of direction or that the road got steeper.

It represents a change of speed.

The steeper gradient shows a faster rate of change of distance with respect to time, therefore a greater speed.

Sofia's smart-watch gives her this data one day.

"That's unusual." notes Sofia.

Izzy says, "I think your watch is broken." What do you think? What's happening there? Pause this video.

Have a conversation with the person next to you or a good think to yourself.

Welcome back.

I hope you notice that Izzy was right.

Something is wrong with the data.

A horizontal line can exist on a distance time graph because it represents no change in distance as time continues to change, but a vertical line cannot exist on a distance time graph.

That would represent distance, your position changing without any change in time, and that is impossible.

So when Izzy said, "I think your watch is broken," she's correct.

You won't see vertical lines on a distance time graph.

When Izzy went on to say to Sofia, "Unless you invented teleportation and didn't tell me," she's being a little bit cheeky, but correct, you'd hope a friend would share that with you.

Alex draws a map of the route he had to take to his friend's house because of road closures.

If you look at that block of houses, Alex has taken something of a convoluted route to get to his friend's house.

Alex says, "I think my distance time graph will have the exact same shape." Let's have a look.

The same line on a distance time graph.

Why is this graph not a possible model for Alex's journey? Pause this video and see if you can spot why.

Welcome back.

I hope you commented on that moment there.

What's wrong with it? This moment would represent travelling back in time.

Distance from home has increased, but time has gone backwards.

That's not possible.

Time has to move forwards.

Quick check you've got those concepts now.

Spot the error in this distance time graph.

Pause.

See if you can notice what's gone wrong.

Welcome back.

I hope you pointed out that moment there.

There's our error.

We can have a horizontal line on a distance time graph, but we cannot have a vertical line.

Another check, spot the error in this distance time graph.

Pause.

Do that now.

Welcome back.

I hope you pointed out that moment there.

As the model progresses, it appears to go backwards in time, which is not possible, time has to move forwards.

Sofia and Izzy are looking at another graph.

Sofia says, "Is this bit and error? Can the lines go down?" What do you think? What's your mathematical intuition telling you? Izzy knows.

"Yes they can.

This part represents your return journey." On a distance time graph, the upward lines typically represent an outbound journey, and the downward lines typically represent the return journey.

Sofia says, "I see, this is me travelling 400 metres to my friend's house.

And this is me travelling 400 metres back to my house." Izzy says, "Yes, you travelled 800 metres in total." Quick check you've got that.

This graph represents Izzy's journey to a friend's house.

How far did Izzy travel in total? 10 metres, 300 metres, 20 metres, or 600 metres? Pause.

Have a think about that one.

Welcome back.

Maybe said it's option D, 600 metres.

You could have justified that by saying it's 300 metres out and 300 metres back.

That's 600 metres in total.

Practise time now.

Here is a distance time graph of a cycle ride that Jacob took.

For part A, I'd like to know how far away he was at 30 minutes.

For part B, what happened on 40 minutes? For part C, at what time did Jacob start to return home? For part D, how far did Jacob's cycle in total? And for part E, was Jacob's furthest point from home the high point, or low point, in altitude of this cycle? Which was it? And I'd like you to write a sentence to justify your answer.

Pause, and do those questions now.

For question 2, Alex shows you a distance time graph, which he drew before this lesson.

He knows there are moments in it which cannot be true.

For each labelled moment, declare if that moment is possible or impossible and write a sentence for each explaining why.

Pause this video, and do that now.

Feedback time now.

Jacob's cycle ride.

For part A, how far away was he at 30 minutes? We'd read from 30 minutes up and across to a distance of nine kilometres.

What happened on 40 minutes? Well, on 40 minutes, he stopped.

He might have gone on to say, there was a 10 minute stop after 40 minutes.

At what time did Jacob start to return home? From that moment there, 90 minutes in.

For part D, how far did Jacob's cycle in total? 30 kilometres.

15 kilometres on the way out and 15 kilometres back.

Part E was a little bit trickier.

Was Jacob's furthest point from home the high point, or low point in altitude of this cycle? That's a bit of interpretation to be done here.

I hope you said it's the high point and justified it by perhaps saying the gradient on the journey home is the steepest part of the graph which tells it's the highest speed.

That would therefore be Jacob cycling downhill.

You might have justified it by saying Jacob has a break on the way out there but not on the way back.

This would suggest the route back was easier, which would make it downhill.

What you were not allowed to write was the return journey is going downhill on the graph.

That would not be a satisfactory justification.

For question 2, labelling each of the moments A to E, declaring them possible or impossible with a sentence to justify.

You might have written A, is possible.

It represents a stop in the journey.

B and E were impossible.

They were vertical lines.

We should know we can't see on a distance time graph.

It will represent a change in position in zero time, not possible.

C, was possible.

This is a move towards home at a constant speed.

D, was impossible, because it would represent going back in time.

Onto the second half of the lesson now.

Drawing distance time graphs.

Sometimes we already have the information, and we have to draw the distance time graph.

Here, I've access, labelled my axes, D for distance and T for time.

With a grid, we can plot these points accurately.

Let's take Aisha's journey for example.

"I walk to the shops one kilometre away, it takes 20 minutes.

I'm 20 minutes in the shop, then I walk home at the same speed.

Half way home, I stop for 10 minutes to drop some items off with a relative.

Then I walk home at the same speed I walked earlier." We can plot this entire journey.

We'll start with the journey to the shops.

That is 20 minutes in reaching a kilometre away.

The units on that vertical axis being metres, 1,000 metres, one kilometre.

That's the start of Aisha's journey.

Next, we plot that coordinate.

"I'm 20 minutes in the shop." There we go, let's stop for 20 minutes in the shop.

"I then walk home at the same speed." It's important that we plot that coordinate, draw that line and acknowledge the same speed means the same change in distance over the same change in time, so both our lines there would have the same gradient, but one is positive and one is negative.

"Halfway home I stop for 10 minutes." That's a stop for 10 minutes.

"Then I walk home at the same speed I walked earlier." There we go, the conclusion to the journey.

Our model makes the assumption that Aisha walked at a constant speed when she was moving, hence the straight lines.

This is standard unless you're informed otherwise in a simple distance time graph model like this one.

Let's check you've got that.

What is the first coordinate that should be plotted for this journey? This is a story of a cycle ride that Jacob took.

I won't read the whole thing to you.

You're going to read it part by part, but what's the first coordinate that we should plot? Is it 20, 10, 10, 10, or 10, 20? Pause.

See if you can pick the right one out.

Welcome back.

I hope you said it's A, 20, 10.

That would be 20 minutes in and reaching 10 kilometres.

That's that coordinate there.

Next, what's the next coordinate that should be plotted? We can see that Jacob cycled 10 kilometres in 20 minutes.

Which one's next? Is it 20, 25, 35, 10, or 35, 25? Pause this video.

See if you can pick out the right one.

Welcome back.

I hope you said it's option B.

That's that moment there.

We're resting for 15 minutes.

No change in distance, but 15 more minutes in time will take us to the coordinate, 35, 10.

But what is the next coordinate after this rest? Is it 15, 15, 50, 25, or 50 15? Pause.

See if you can pick the right one out.

Welcome back.

I hope you said C, 50, 15.

That's us cycling again and reaching our destination 15 kilometres away from home and 15 minutes later in time.

That takes us to that coordinate there.

What's next? Is it 25, 15, 75, 15, or 50, 40? Read on in the story and pick the right coordinate.

Pause now, and do that.

Welcome back.

I hope you said B, 75, 15.

That's this moment here.

We hang out there for 25 minutes.

A stop for 25 minutes.

No change in distance, but a change in time.

Taking us to that coordinate there.

Final one.

What's next? Is it 100, 30, 75, 40, or 100, 0? Pause.

See if you can pick out the right one.

Welcome back.

I hope you said C, 100, 0.

That's us cycling straight to home in 25 minutes.

25 more minutes in time, reaching a distance of zero kilometres from home.

There are many moments in maths where a sketch of a model will suffice.

Here's Aisha's journey one day.

"I walk to my friend's house 400 metres away.

That takes five minutes.

I wait five minutes when picking them up.

Then we walk together to the cinema 600 metres further away, but we're chatting so we walk a little slower.

It takes us 10 minutes." It's useful to start by totaling the distance and time.

I should walk 400 metres and 600 metres.

That's 1,000 metres, a kilometre.

As we look at the story, there's five minutes, 10 minutes, and five minutes to account for.

That's 20 minutes in total.

We can label those on our axes.

We're not gonna label the entire axes.

This is just a sketch.

Let's start with this moment.

"I walk to my friend's house 400 metres away and it takes five minutes." We'll plot that moment there.

It's just shy of halfway to 1,000 metres and it's about a quarter of 20 minutes.

Next, "I wait five minutes when picking them up." We know we've reached 10 minutes, so let's plot that point about halfway to 20 minutes.

Finally, "We walk together a further 600 metres and that takes us 10 minutes." That takes us to our final destination, 1,000 metres away after 20 minutes.

Our sketch is good enough to show this slower, less steep, gradient.

To further benefit readers, we can mark the coordinate as we plotted.

We started after five minutes, reaching 400 metres.

After 10 minutes, we were still at 400 metres, and ultimately reached 20 minutes and 1,000 metres.

Quick check you've got that.

Which is the right estimate for the first coordinate of this sketch? Read Aisha's story and then tell me, is it coordinate A, B, or C? Pause, and have a think about this one now.

Welcome back.

I hope you said it's option B.

Why option B? Because it was any point that was less than halfway to eight minutes and less than halfway to 1,200 metres.

We've started the journey now.

Which is the right estimate for the next coordinate of this sketch? Pause this video, continue Aisha's journey and tell me which you arrive at coordinate A, B, or C.

Pause now.

Welcome back.

I hope you said it's option B.

Why option B? One minute stop takes us to four minutes in total and that's halfway to eight minutes.

There's the journey so far.

My last little check, what are you gonna plot in order to complete the time distance graph of this journey? Pause now, see if you can tell me that coordinate.

Welcome back.

I hope you said we plot that point there, which is eight minutes in reaching 1,200 metres.

Well done.

Practise time now.

For question 1, I'd like to use this grid to plot Jacob's long distance training ride.

I won't read that story to you.

It's down to you to read and pick out those key moments and plot those key coordinates.

Pause, and do that now.

For question 2, sketch the graph of Izzy's car journey to visit relatives.

To reiterate that keyword there, sketch doesn't have to be entirely accurate.

You don't have to label your entire axes.

I'd just like to see some good approximations of coordinates in there.

Pause, and try this now.

Feedback time now.

Question 1, we were using this grid to plot Jacob's long distance training ride.

Your distance time graph should have looked like that.

I'll ask you to pause this video now and just check that your coordinates match mine, your graph matches mine.

In question 2, we were sketching the graph of Izzy's car journey.

It's useful to note the journey is 160 kilometres in total and 180 minutes in total.

The first moment that we should have plotted was a half hour in and 40 kilometres, approximately 30, 40.

The next coordinate one hour on the motorway to travel 100 kilometres.

That would take us to a total of 90 minutes and a distance of 140, which should have been plotted about there.

Then the stop, that should have been a horizontal line adding another 30 minutes, taking us to that coordinate 120 minutes in, 140 kilometres in.

And finally, we reach 180 minutes in with 160 kilometres travelled.

I hope your graph looked a little bit like mine, and I do hope you labelled those key moments with the right coordinates.

That's the end of our lesson now.

In summary, we can model real life situations graphically, an example being a distance time graph.

Horizontal lines on distance time graphs represent moments where there is no change in distance.

Vertical lines are not possible as this would represent a change in distance in zero time.

Thank you for participating in this lesson today.

I look forward to seeing you again soon.