Lesson video

Hello, Mr. Robson here.

Great choice to join me for maths today, especially seeing as we're problem solving.

You love problem solving.

I love problem solving.

So let's get started.

Our learning outcome is I'll be able to use our knowledge of interpreting real-life graphs to solve problems. A couple of keywords you're gonna hear throughout our learning today: linear and gradient.

The relationship between two variables is linear if, when plotted on a pair of axes, a straight line is formed.

The gradient is a measure of how steep a line is.

It's calculated by finding the rate of change in the y-direction with respect to the positive x-direction.

Two parts to our learning today.

Let's begin with matching graphs.

Real-life graphs present a visual model of a situation.

For example, Aisha is cycling at a constant speed.

On a distance-time graph, this makes a linear graph, a straight line.

Why? It's because there's a constant rate of change between the two variables, distance and time.

In such models, it's important that we understand the link between distance-time graphs and speed-time graphs.

The distance-time graph shows that Aisha is cycling at a constant speed.

How would this look on a speed-time graph? Pause.

Have a conversation with a person next to you.

Maybe try and sketch that speed-time graph.

See you in a moment for the answer.

Welcome back.

Hopefully you described or drew a horizontal line.

Why? Well, speed is not changing, i.

e.

, speed is constant.

That's what a constant speed looks like on a speed-time graph.

We often see horizontal lines on distance-time graphs, too.

What would this look like on a speed-time graph? What is that horizontal line on the distance-time graph telling us? What's it gonna look like on a speed-time graph? Pause.

Have a conversation with a person next to you.

Maybe try and sketch this graph.

See you in a moment for the answer.

Welcome back.

Hopefully you drew or described something like that.

If Aisha's position does not change as reflected in that distance-time graph, then her speed is zero.

So we'll get horizontal line at speed equals zero.

We often see diagonal lines on speed-time graphs.

What's happening here? Well done.

We're going from slow to fast.

What do we call that? Of course, acceleration.

On this speed-time graph, Aisha has accelerated.

Now here's the big question.

What does acceleration look like on a distance-time graph.

Pause.

Have a conversation with a person next to you.

Perhaps try and sketch that distance-time graph representing acceleration for yourself.

See you in a moment for the answer.

Welcome back.

I wonder what you described.

I wonder what you drew.

A curve like that, perhaps? What's going on here? Well, a steeper gradient means a greater speed on a distance-time graph.

At the start, we're going slow.

At the end where the gradient is much steeper, that's how a fast speed, the curve shows our acceleration on a distance-time graph.

We may also see this shape on a distance-time graph.

Another curve, but a slightly different one.

A curve getting less steep.

What's this modelling? Well done.

We're going from a fast speed at the start with the gradients really steep to a slow speed at the end when we've got a really gentle gradient.

Aisha has decelerated.

A curve like that on a distance-time graph shows deceleration.

What will that look like on a speed-time graph? Pause.

Have a conversation with a person next to you.

Perhaps try and sketch the speed-time graph that models deceleration.

See you in a moment.

Welcome back.

Hopefully you drew a line like that, representing us going from having a fast speed to a slow speed, i.

e.

, decelerating.

Quick check.

You've got this so far.

I'd like you to match the labels to the graphs.

There's six graphs and there's six labels to match to them.

Now do beware, some of these are distance-time graphs and some of them are speed-time graphs.

Pay particular attention to that fact.

Pause and do some matching.

I'll see you in a moment.

Let's see how we did.

Hopefully the first graph, the distance-time with a straight line, you put the label constant speed on there.

The second graph was a speed-time graph.

We go from having a high speed to a low speed, that's deceleration.

Next up, speed-time graph with a diagonal line with a positive gradient.

Well, that's acceleration, going from a low speed to a high speed.

Next up, a horizontal line on a distance-time graph means zero speed.

Where that's a stop when we see that on a distance-time graph.

Next up a horizontal line on a speed-time graph, well, that's a constant speed.

What's not changing? Speed.

Speed is constant.

Finally, a curve on a distance-time graph.

A curve whereby the gradient is getting less steep, that represents deceleration.

Most models reflect the fact that journeys are made up of multiple moments.

So if we look at a journey of multiple moments that Aisha's making, Aisha starts to cycle and accelerates up to a constant speed.

This is a distance-time graph.

That is acceleration up to a constant speed.

Can you see the curve at the beginning representing the acceleration? Going into a straight line, representing the constant speed.

Next, Aisha decelerates to a stop and takes a break.

That is a deceleration to a stop on a distance-time graph.

Finally, she starts again accelerating to a greater constant speed.

What do you think this is gonna look like? Well done.

A curve to represent that acceleration and then a line of a steeper gradient because it said greater constant speed.

How might these three stages look on a speed-time graph? Notice this is not a distance-time graph.

It's a speed-time graph, but we're gonna look at the same three moments of Aisha's journey.

Aisha starts to cycle and accelerates up to a constant speed.

That'll look like that on a speed-time graph.

The diagonal representing our acceleration and the horizontal representing her being at a constant speed.

The next moment was decelerating to a stop and taking a break.

We decelerate back to speed equals zero, and then a horizontal line representing stopping at that speed of zero.

Finally, accelerating to a greater constant speed.

Diagonal line to represent the acceleration and notice now we're at a higher point on the vertical axis because it's at a greater constant speed.

Quick check.

You've got this.

I'd like you to describe the three moments in this model.

Come up with your own words.

This is a distance-time graph.

What's happening at A, what's happening at B, and what's happening at C? Write three sentences, one for each.

I'll see you in a moment.

Welcome back.

You might have said, for A, a constant high speed, it's a straight line on a distance-time graph, that's a constant speed.

Really steep gradient, that's high speed.

For B, deceleration to a lower constant speed, the curve taking us to a less steep gradient.

For C, you should have said something along the lines of deceleration to a stop.

The horizontal finish on this distance-time graph means Aisha has stopped.

Next, I'd like you to match that same model, that distance-time graph, to the correct speed-time graph.

There's three speed-time graphs on the right hand side to choose from.

Which one represents that distance-time journey? Pause, take your pick.

Welcome back.

Hopefully you said B.

Why B of all three options? Well, B started with a constant speed, A didn't, but it had two moments of deceleration, ah, C did not have two moments of deceleration, and then finally came to a stop.

B, at the end there, represents coming to a stop.

We can also model with displacement-time graphs.

Aisha starts to cycle and accelerates up to a constant speed.

It's the same shape, the curve representing the acceleration, a straight line representing the constant speed.

She decelerates to a stop and takes a break.

Well, it's looking pretty similar so far.

She accelerates again to a constant speed and returns home.

This is a displacement-time graph.

The graph is showing us how far displaced Aisha is from her home, so when she accelerates to a constant speed and returns home, it looks like that.

The last part of this journey, we return to where displacement equals zero, i.

e.

, Aisha is back home again.

This journey can be shown on a distance-time graph.

We've got acceleration to a constant speed at the start, deceleration to a stop, and at the end of our displacement-time graph, remember Aisha accelerated and then returned home.

On a distance-time graph, it looks like that.

Why the difference? Well, the distance travelled continues to increase.

This graph tells us nothing of direction.

Remember, displacement-time graphs tell us a bit about direction, distance-time graphs don't.

Here's Jacob with a common misconception.

Jacob says, "Surely this moment shows you decelerating? The graph is going down." Do you agree with Jacob? Pause, tell the person next to you or maybe make a suggestion to me aloud at the screen.

Welcome back.

Well done.

We're not agreeing with Jacob.

This is a very common error when we see displacement-time graphs.

I should correct Jacob by saying, "No, I accelerate toward constant speed; I'm just travelling in a different direction." That's the stop on the displacement-time graph.

That's a moment of acceleration.

It's just acceleration in a different direction, and that's a moment of constant speed, but again, it's a constant speed in a different direction.

Quick check.

You've got this.

I'd like you to match the model.

Our model is a displacement-time graph to the correct distance-time graph.

There's three distance-time graphs on the right hand side.

Which one reflects that displacement-time graph? Pause and take your pick.

Welcome back.

Hopefully you said C.

Why would you say C? Well, C shows us at a stop to begin with.

Then a moment of acceleration, then a moment of constant speed, then a moment of deceleration, and finally a stop.

Practise time now.

Question one, we're gonna match the distance-time graphs to the speed-time graphs.

There's four distance-time graphs, four speed-time graphs.

You should come outta this question with four correctly matched pairs.

Pause and have a think about this one now.

Question two, we're gonna match the distance-time graphs to the displacement-time graphs, four distance-time graphs, four displacement-time graphs.

Therefore, you're looking for four pairs again.

Question three, these six graphs model two journeys.

Group the graphs accordingly.

For each journey, you're going to want a displacement-time graph, a distance-time graph, and a speed-time graph.

Welcome back.

Feedback time.

Question one, we were matching distance-time graphs to speed-time graphs.

So there are four distance-time graphs.

We should have matched distance-time graph A to speed-time graph F.

We should have matched distance-time graph B to speed-time graph G.

We should have matched C to H and D to E.

Question two, we were matching again.

There's the four distance-time graphs.

We should have matched distance-time graph A to displacement-time graph G, distance-time graph B to displacement-time graph E.

We should have matched C to H and D to F.

Question three, we have six graphs that were modelling two journeys, and I asked you to group them.

If we start with the two displacement-time graphs, keep those separate and then let's match up the distance-time graphs.

Displacement-time graph A will match to distance-time graph F.

Displacement-time graph E will match to distance-time graph D, and then A and F we match to speed-time graph C, leaving speed-time graph B to be matched with E and D.

Onto the second half of our learning now where we're going to be sketching graphs.

A sketch can be just as useful as a carefully drawn graph.

We were sketching this story.

Aisha starts running and accelerates up to a slow, constant speed.

We'd start by sketching that on a distance-time graph.

After some time she accelerates to a higher constant speed and our sketch would continue like so, and then upon tiring she decelerates and stops.

We would draw a curve into a horizontal line.

We've sketched a distance-time graph of this journey.

A sketch does not need a scale on the axes; it just needs to show key features, such as constant speed, acceleration, stopping.

On this graph, the curve at the start shows acceleration into a constant speed, further acceleration to a higher constant speed, and then a curve showing deceleration to a stop.

If I can pick those features out of your distance-time graphs, they're good sketches.

We can sketch the same journey on a speed-time graph.

Aisha starts running and accelerates up to a slow, constant speed.

After some time, she accelerates to a higher constant speed.

Upon tiring she decelerates and stops.

Again, our sketch doesn't have any labels on the axes, but we can pick out the key moments like acceleration, a constant speed, further acceleration, a faster constant speed, deceleration and a stop.

If I can pick those features out of your sketches as speed-time graphs, they're good graphs.

Quick check.

You've got this so far.

I'd like you to sketch this journey on this distance-time graph.

Three moments, what will this journey look like? Pause and give this one a go.

Welcome back.

Hopefully Aisha's walking at a constant speed, you sketched a straight, diagonal line like so.

She decelerates to a slower speed, a curve to show deceleration, a lesser gradient, and then accelerating into a run at a higher constant speed would look like so.

Next little check, I'd like you to sketch the same journey that Aisha took, but this time on a speed-time graph.

Sketch me an axes reading speed and axes reading time and then try and sketch this journey.

Pause and do this now.

Welcome back.

Aisha's walking at a constant speed.

You should have started with that.

She decelerates to a slower speed.

That's how your sketch follows.

Then accelerates into a run at a higher constant speed.

Your sketch should have looked like so.

Another quick check.

You now have both a distance-time, and speed-time graph of this journey.

What I'd like you to do is apply these labels.

As eight labels on the right hand side, where will you put them across those two sketches? Pause.

Have a think about this now.

Welcome back.

Let's see where we might put those eight labels.

Your slowest constant speed should be there on the distance-time graph and there on the speed-time graph.

Your fastest constant speed should be labelled there.

The moments of deceleration are there, moments of acceleration there on the respective graphs.

Some journeys are best modelled on a displacement-time graph.

This journey's broken down into these four parts.

Let's see what it looks like when we sketch a displacement-time graph.

Let's start with Aisha leaving home and accelerating to a walk at a constant speed.

Next, she decelerates to a stop to collect a friend.

Then together they accelerate to a walk and stop upon reaching the gym.

Finally, after a long stay at the gym, Aisha jogs home.

The key features we can see on the graph, there's a short stop, a longer stop, and then the fastest speed.

It's a negative gradient, but it's a steeper gradient.

So crucially, on a time-distance graph, it represents our fastest speed.

We can take this displacement-time graph and sketch a speed-time graph for the same model.

At the beginning, we've got our short stop.

Then we go again before a longer stop.

Finally, we go our fastest speed.

Can you see how this displacement-time graph and this speed-time graph show the same journey? Quick check.

You've got this.

I'd like to sketch a displacement-time graph for this journey.

Four parts to Aisha's journey this time.

I wonder what your displacement-time graphs are going to look like.

Pause and get sketching.

Welcome back.

Hopefully you drew Aisha leaving home and accelerating to a fast jog.

The keyword being fast should have a steep gradient there.

She decelerates and stops at the swimming pool.

After a long stay at the pool, Aisha accelerates to a jog home.

Halfway home, Aisha slows her jog down.

You should have that moment of deceleration into a slightly less steep gradient on the journey home.

I'd like you to sketch next the speed-time graph for this journey.

There's a displacement-time graph we just sketched.

What would a sketch of the speed-time graph look like to match the same journey? Pause, get sketching.

Welcome back.

Hopefully you do acceleration to a high speed before a long stop at the pool, before acceleration to a high speed and then deceleration to a slower speed.

Well done.

Practise time now.

Question one, I'd like you to sketch the distance-time and speed-time graphs of this journey.

Alex is away cycling three parts to this journey.

What will that distance-time graph and speed-time graph look like for this journey? Pause.

Sketch those two graphs.

Question two, part A, I'd like you to sketch a distance-time graph to match this speed-time graph.

For part B, I'd like you to get creative.

I'd like you to write a short bullet-pointed description to describe this activity.

Pause and do that now.

Question three, I'd like you to sketch a speed-time graph to match this displacement-time graph.

Pause and do that now.

Feedback time.

Question one, we were sketching the distance-time, and speed-time graphs of this journey.

We'll begin with cycling at a constant speed, then accelerating upon hitting a long, downhill section, then at the foot of the hill decelerating to a stop.

That's what your two sketches should have looked like.

I should be able to pick out the key features from your graphs.

Question two, part A, we were sketching a distance-time graph to match the speed-time graph.

Your distance-time graph should look like that.

Acceleration to a high speed into a stop, acceleration to a high speed into a stop before accelerating again to a lower speed.

For part B, I asked for a short bullet-pointed description to describe the activity.

You might have used this example.

Alex is interval training; he accelerates to a sprint, then decelerates to a stop.

Secondly, he accelerates into a second sprint and then decelerates to a stop.

Then he accelerates into a slow jog to finish his session.

Question three asked you to sketch a speed-time graph to match this displacement-time graph.

You should start with constant speed into a deceleration, a lesser speed, and then accelerating to a high speed before a long stop and then acceleration to a higher speed again.

We're at the end of the lesson now.

We have learned that knowledge of interpreting real-life graphs can be used to solve problems. Key features of a distance-time, or displacement-time graph, can be shown on a speed-time graph.

And useful sketches can effectively model journeys.

Hope you've enjoyed this lesson as much as I have, and I look forward to seeing you again soon for more mathematics.

Goodbye for now.