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Oh, hello there.

Mr. Robson here.

Welcome to maths.

Lovely of you to join me today.

Today we're estimating journeys from nonlinear graphs.

You might already know how to calculate the distance travelled on a linear speed-time graph, but do you know what we do when those speed-time graphs are nonlinear? Let's find out, shall we? Our learning outcome is that we'll be able to estimate the distance travelled for nonlinear speed-time graphs.

A keyword that you're gonna hear a lot throughout this lesson is estimate or estimation.

A quick estimate for a calculation is obtained from using approximate values, often rounded to one significant figure.

We've got two parts to our learning today and we're gonna begin by estimating distance from a nonlinear graph.

On a speed-time graph, we can calculate the area bound by the graph and the horizontal axis to find the distance travelled.

So we just need to find the area of this shape.

What shape have we got? I can see one, two, three, four, five sides.

I don't happen to know the formula for area of a pentagon like this so we could break the area up into three different polygons and sum their areas.

What have we got? Oh, a triangle, a rectangle and a trapezium.

I know the formula for working out the area of a triangle, a rectangle and a trapezium, so let's work through these.

The area of that triangle, formula for area of a triangle is half of base times height, so my base is four, my height is 16.

I want to do a half of four multiplied by 16.

That's going to be 32.

The triangle's got an area of 32.

The rectangle, formula of a rectangle is base multiplied by height.

I see a base of four, a height of 16, four lots of 16 is 64.

The rectangle's got an area of 64.

The rectangle's the interesting one.

I use the rectangle to justify why it's the area bound by the graph and the horizontal axis that tells us the distance travelled.

If you look at that moment from four seconds to eight seconds, what's happening? Well done.

We're at a constant speed of 16 metres per second.

If you travel for one second at 16 metres per second, you've travelled 16 metres.

Travel like that for two seconds, you've travelled two lots of 16 metres.

Three seconds, three lots of 16 metres, four seconds, four lots of 16 metres, you've travelled 64 metres.

The area of that rectangle is 64 and it tells us in the context of a speed-time graph, we've travelled 64 metres in that section.

Onto the third polygon, trapezium.

Do you remember the formula for area of a trapezium? Well done.

It's half of brackets, the upper base, plus the lower base, close brackets, and multiply by the height.

So for that trapezium, and you might want to rotate your head 90 degrees in order to see the trapezium clearly, we'll calculate the area by doing a half of 16 plus 24 and multiplying it by a height of two.

That would give us 40.

The trapezium's got an area of 40.

Now that we know we've got an area of 32, 64 and 40, we can sum those together and we can say with a total layer of 136 that the distance travelled in these 10 seconds is 136 metres.

That's an exact calculation provided.

That's an accurate model of the speed and time of this particle.

The reality is that an object's movement is rarely modelled by straight lines.

You're more likely to see a graph like this with a few curves.

For this graph, we could map the same polygons from the example we just saw onto this graph and that will give us an estimate for the distance travelled.

Can you see the exact same three polygons? I could sum those areas, and this time, I'll say the distance is approximately 136 metres.

It's important that we say approximately and use the approximate sign there because these gaps mean our result is not entirely accurate, but it is a very good estimation.

Quick check you've got this so far.

In order to estimate the distance in this speed-time graph, we could, is it A, draw one right triangle with the hypotenuse going from the start to the end of the graph? Is it B, map multiple polygons onto the graph and sum their areas? Or C, count the squares under the graph.

Pause and make your choice.

Welcome back.

Hopefully you didn't say option A.

If you did that, your right triangle would've looked like.

So you'll notice this is not an accurate estimation.

There's huge gaps between the curve and hypotenuse of our triangle.

B was the right choice.

Mapping multiple polygons onto this graph gives us a really good estimate.

C would not have worked.

That would not have been very accurate at all.

Now that we've mapped multiple polygons onto this curve, what I'd like you to do now is sum the areas of those polygons to estimate the distance travelled in this speed-time graph.

Pause and work this one out now.

Welcome back.

Let's see how we did.

Hopefully you found that the triangle had an area of 12, the rectangle an area of 24, the trapezium an area of 28, that last tall rectangle, an area of 40.

Then we summed those together.

Yeah, total area of 104 metres.

So we would say the distance is approximately 104 metres and I do hope you used the approximately sign there, not an equal sign.

For some graphs, we can state whether our estimate is an underestimate or an overestimate.

If we map one polygon here, a triangle, and say that the area is 120, we say the distance travelled is approximately 120 metres, but this is an underestimate.

Can you see why that's an underestimate for the true distance travelled? Well done.

Because the area we calculated is less than that of the area bound by the graph and the horizontal axis, we know that the true distance travelled is greater than the 120 metres we calculated.

Quick check you've got that.

What I'd like you to do here is use the area of the triangle to estimate the distance travelled in this model and state whether your estimate is an underestimate or an overestimate.

What triangle? This triangle.

Pause, work out that area and state whether that, as an approximation for the distance travelled, is an overestimate or an underestimate.

Welcome back.

Hopefully you said the area of that triangle is 70.

Base of seven, height of 20, seven by 20's 140, half that is 70, so we'd say the distance travelled is approximately 70 metres.

Then we'd state this is an overestimate.

The area of our triangle is greater than the area bound by that graph and the horizontal axis.

Hence, this is an overestimate.

Practise time now.

For question one.

I'd like you to map polygons onto this graph and sum their areas to estimate the distance travelled.

Notice, I said polygons plural.

I'd like to see multiple polygons mapped onto this graph.

Pause and do this now.

Question two.

I'd like you to use a trapezium to estimate the distance travelled in this speed-time graph.

For part B, I'd like you to state where your estimation is an underestimate or an overestimate.

Pause and do this now.

Feedback time.

Let's see how we did.

For question one, I asked you to map polygons, plural, many polygons onto this graph and sum their areas to estimate the distance travelled.

Hopefully your polygons look something like so.

I've mapped a triangle and a rectangle and a trapezium and they're a pretty good estimation for the area underneath this curve.

So my triangle had an area of 16, my rectangle had an area of 16, my trapezium an area of 64.

Sum those together, get a total area of 96 metres, so I can say my distance travelled is approximately 96 metres.

Hopefully you got something in the same region as 96.

For question two, I asked you to use a trapezium to estimate the distance travelled in this speed-time graph.

My trapezium looks like this.

You notice my upper base goes in the coordinate 4, 10 to the coordinate pair 12, 10.

So the area of my trapezium is half of bracket, eight plus 15 multiplied by the height of 10.

That gave me an area of 115.

So I can say my distance travelled is approximately 115 metres, and I can say that it's an underestimate because the area of my trapezium is less than the area bound by my graph and the horizontal axis.

If you did a different trapezium and your trapezium maybe went to the coordinates 1, 10 and 14, 10, then you might have a larger area and you might be stating it's an overestimate and that would be absolutely fine as an answer.

Onto the second half of our learning now where we're going to be improving the accuracy of our estimates.

Well, I'm excited to find out how we do that.

We could use one polygon to estimate the distance travelled in this model.

One polygon giving us a trapezium with an area of 116 metres, but the large gap between the graph and the polygon means our estimation is not very accurate.

How could we improve the accuracy of our answer? What do you think we could change about the maths we've done here in order to be a little more accurate? Pause, have a conversation with the person next to you or a good think to yourself and I'll be back to talk about the solution in a moment.

Welcome back.

I wonder what ideas you came up with.

I wonder if you said we could solve this problem by mapping more than one polygon onto our graph.

If we do that, the gap between the graph and the polygons is less.

This improves the accuracy of our estimation.

We want our polygons to map that graph as closely as possible.

In fact, the more polygons we use, the more accurate our estimate becomes.

After mapping four polygons to this graph, the distance between the polygons and the actual graph is minimal, which means we'll have a very good estimate.

There is, of course, a playoff here.

I could map 10 trapezia onto this graph and I'd have a really good estimate, but I don't want to work out the area of 10 trapezia, so I think I'll stick at four.

This'll be a pretty good estimate.

Quick check you've got all that.

Which of these models will give the best estimate for the distance travelled in this speed-time graph and why? When I say why, I'd like you to write a sentence to explain that.

Pause, make your choice and write a sentence now.

Welcome back.

Hopefully you said C.

I wonder how you chose to explain and justify why that one will give us the best estimate.

You might have said the more trapezia, the better the estimate because the gap between the graph and the polygons is less.

That would've been a lovely explanation.

Practise time now.

Question one, I'd like to find the sum of the area of the polygons in each case and write a sentence comparing the accuracy of the three estimates for the distance.

Pause and do that now.

For question two, I'd like you to find an accurate estimate of the distance travelled in this speed-time graph.

When I say I'd like an accurate estimate, I obviously mean don't map one polygon onto that speed-time graph.

I'd like you to map multiple polygons but not too many because I don't want you working out the area of eight polygons.

Anyway, you have a play with this graph.

Let's see how accurate you can be while still maintaining a good level of efficiency.

Pause and try this problem now.

Welcome back.

Feedback time.

For question one, I asked you to find the sum of the areas of the polygons in each case and write a sentence comparing the accuracy of the three estimates for distance.

For A, we had one polygon, a trapezium.

Your working out would've read 1/2 of, brackets two plus 20 multiplied by a height of 12.

That would've given you 132.

So with an area of 132, that polygon, we can say that the distance travelled is approximately 132 metres.

Do be sure to use the approximately equal to sign there, not the exactly equal to sign.

For B, you had two trapeziums. The first one's 1/2 multiplied by brackets, two plus six multiplied by the height of six.

The second one would've been 1/2 of bracket six plus 20 multiplied by a height of six.

The two area, therefore 24 and 78.

We sum those together to get 102.

If the area of those two polygons is 102, we can say the distance travelled is approximately 102 metres.

For part C, we've got a little bit more work to do here, but we can do it.

That's three trapezia.

The first one being 1/2 of brackets, two plus four multiplied by height of four.

The second one, 1/2 of brackets, four plus nine multiplied by a height of four.

And the third one being 1/2 of brackets, nine plus 20 multiplied by a height of four.

You should have had 12, 26 and 58 for those three trapezia respectively.

Sum that together and you get an area of 96.

So we can say distance is approximately 96 metres.

I then asked you to write a sentence comparing the accuracy of the three estimates for distance.

Look at the range of estimates we've got here.

Our first one, you knew it was an overestimate.

You could see that from the graph, but look at that number, approximately 132 metres.

Contrast that with the answers in B and C, 102 metres and 96 metres.

We're getting far closer to the truth, but wow, A was something of an overestimate.

So hopefully you said something along the lines of A is the least accurate estimate because the large gap between the one polygon and the actual graph.

And then followed it up with B and C are better estimates because more polygons were used.

Estimate C is the most accurate one.

If you said something along those lines, well done.

For question two, I asked you to find an accurate estimate of the distance travelled in this speed-time graph.

Now you know if we're being accurate, you weren't just going to map one polygon onto that graph.

You'd have had a gross overestimate then.

Hopefully your solution looked something like this.

I've got four trapezia with area of 5, 8, 14, and 23 respectively.

When I sum those together, I get an area of 50.

So we can say the distance travelled is approximately 50 metres.

Now, I hope you mapped at least three polygons to get an accurate estimate somewhere in the region of 50 metres.

Your estimate might not be exactly 50 like mine, but it's okay.

We're estimating.

If you're somewhere in the region of 50, then your answer is a pretty good piece of approximation.

Well done.

Sadly, we're at the end of the lesson now, but we have learned, we've learned that polygons can be used to estimate the distance travelled for nonlinear speed-time graphs.

Increasing the number of polygons used increases the accuracy of the estimated distance.

Hope you enjoyed this lesson as much as I did and I look forward to seeing you again soon for more mathematics.

Goodbye for now.