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 where we're interpreting and drawing more complex real-life graphs.

We're looking at two key themes in today's lesson.

The first is interest.

There's a few key words that we need to be confident with.

So interest is the money added to savings or loans.

It can be simple interest, which is always calculated on the original amount, or compound interest, which is calculated on the original amount and the interest accumulated over the previous period.

And we'll explore that today.

We're also gonna look at exponential equations.

The general form for an exponential equation is y equals ab to the power of x, where a is the coefficient, b is the base, and x is the exponent.

The other theme we're gonna look at is displacement-time graphs.

So pause a video if you need to remind yourself on speed, acceleration, or deceleration.

So those are our two key themes today.

We're starting by looking at graphing interest rates.

So we can use time graphs to model lots of different situations where we are mapping how something changes over time.

You may have seen distance-time graphs, displacement-time graphs, depth-time graphs, speed-time graphs, all sorts of situations where we're modelling something changing over time.

Let's think about this example.

A hundred Pound is invested in a bank account, paying 1% simple interest per month.

Let's think about how we could graph this information.

So we've got time on the X-axis and we've got that in months and then what we'd be plotting is we'd be plotting how the value of the money changes over time.

So 'cause we're starting with 100 Pounds, we can plot the coordinate 0,100 and notice that I've put a broken scale in so that I can better see the trend of this graph.

Now after one month, I'm gonna have the hundred Pounds we invested, but also 1% simple interest.

1% of a hundred is one so after one month we'll have 101 Pounds.

Now because simple interest is always calculated on the original amount, each month we're gonna get another one Pound.

It doesn't matter how much money's now in my bank account, I'm getting 1% of the original investment.

If I keep adding a Pound into the account every month you'll see that we have what looks like a linear relationship.

Now what we could do is draw lines between the points, a little bit like a time series graph.

However, it's important to remember that there is no data between zero and one month or one and two months because we're getting all the interest in one go every month.

So we can draw the line and that's gonna help us see the trends and solve problems and we can find the equation of that line.

But just be aware that there isn't any data between the months.

What that means is we can't use the line to interpolate results between the points.

The reason for drawing the line is to help us see the key features of the scenario.

So we have drawn a linear graph.

Simple interest produces a linear graph because there is a constant rate of money being added each month.

Pause the video.

What is the equation of the line we have drawn? So, the gradient of the line is one, this is the amount that's being added on each month.

1% of a hundred is one.

And the y intercept is 0,100 as 100 Pounds is the amount invested.

Our equation then could be y equals 100 plus x or y equals x plus 100.

Remember there's a coefficient of one in front of the x.

We just don't tend to write that.

So y equals one x plus 100 would be fine as well.

What do you think would happen to the graph and the equation if 200 Pounds was invested instead? Make a prediction before we look at this together.

Right, I'm gonna need to change my scale this time.

So I've done a broken scale and I've started at 200, but I'm still increasing in steps of one just as before.

Now after one month, the bank account would have 200 Pounds and then 1% of 200 is 2.

So 202 Pounds.

It's simple interest so we're gonna add that on each month.

Now 1% of 200 is greater than 1% of 100 so the money will grow at a faster rate.

So we've still got this constant rate of change.

We've still got a linear relationship.

But if we were to draw the line between all the points, it would have a steeper gradient.

The equation would be y equals 2x plus 200.

Okay, time to make another prediction.

What do you think would happen if I invested 100 Pounds but with a simple interest rate of 2% per month? What would the graph look like? What would the equation look like? So I need my original scale back where I'm starting at 100.

2% of a 100 is 2 Pounds.

So we're adding 2 Pound per month.

So we've got y equals 2x plus 100.

What about if we invested 100 Pounds with interest at 0.

5% per month? What would that look like? Good, you should be starting to compare these and thinking about how changing the values changes the equation and the line.

So we still need 0.

5% of a hundred 'cause that's what we're starting with and that's 50 P.

So the gradient of our line is now smaller, but our y incept is still 100 'cause that's the amount we're investing.

So y equals 0.

5x plus 100 or y equals a half x plus 100.

Remember the line is just a model because the data is only at those set points.

We can't use the line to interpolate between the monthly values.

So just to be aware that we're only using the equation, the line as a model to best fit our scenario.

Quick check then, which would be the equation of the line if graphing 500 Pounds invested at a simple interest rate of 1% per month? What do you think? Well done if you said B, y equals 5x plus 500.

1% of 500 is five Pounds.

So our line is gonna have a gradient of five.

We've started with 500 Pounds, so that's gonna be our y intercept.

Now let's look at compound interest.

Compound interest is a more common form of interest when investing money.

With compound interest, the interest is calculated on the original amount and any interest accrued over the previous periods.

So that means our interest is gonna keep on increasing as the value of our money increases.

So let's look at 100 Pounds invested in a bank account paying 10% compound interest per year.

Now 10% is very unrealistic.

You're not gonna see that as an interest rate for a bank account.

However, it's gonna help us see what is happening here.

So how much money would be in the account at the end of each month? Well, let's look at month zero.

We're gonna start with 100 Pounds and then we need to add on 10%.

Now the quickest way to do that is to use a multiplier.

So you multiply by 1.

1, it will increase our value by 10%.

If I multiply that by 1.

1 again, and that's how compound interest works.

You increase the new amount by 10% and then that new amount by 10% and so on.

And then multiply the new value by 1.

1 and the new value by 1.

1.

So those are our values for the first four months.

Let's have a look at what that looks like on the graph.

Now because the increase in the interest each month is quite small, it's hard to see what is happening here.

If I draw a line between the first two points and then carry it on, you'll see that this is not a linear relationship.

Graphing software can be quite helpful here.

So I've plotted this on a graph and now we can see the trend better.

Again, I've chosen to join the points up to see the trends even though there's no data between them.

What do you notice about this compound interest graph? How does it differ to the simple interest graphs we had before? Right, it forms a curve.

Now I could have joined the points up with straight line segments, but I chose a curve 'cause I could see how the values were increasing and saw that it was gonna have an exponential relationship.

We'll delve into that in a bit more detail in a moment.

Using the curve better describes the nature of the graph and then I can use the equation of the curve to solve problems and model this situation.

If I zoom out further, we can see the nature of the curve even better.

So that's over about 55 months there and it is an exponential curve.

The value of y at the y-intercept is still 100 'cause that's the amount invested.

Because the interest is compounded, the value of the account increases exponentially.

So you may have seen equations of exponential curves before.

Let's see if we can work out the equation of the exponential curve we just looked at.

So month zero was a hundred.

Month one we said we could do a hundred times 1.

1.

Month two is gonna be a hundred times 1.

1 times 1.

1 or a hundred times 1.

1 squared.

To get the third month we can do a hundred times 1.

1 cubed and so on.

And that is our formula for compound interest, which you may have explored in the past.

Now let's look at that first month.

A hundred can be written as a hundred times one and one can be written as anything to the power of zero.

So if I use the same multiplier that I use for the other months, I can actually write that as a hundred times 1.

1 to the power of zero.

Equally 1.

1 is 1.

1 to the power of one.

And now you might be able to see a pattern in how we generated those values.

Let's put them into a table of values.

So those are the values we had before and hopefully you can see that to get any month we just do a hundred times 1.

1 to the power of that month.

So that's just the equation of the curve.

So we've got y equals a 100 times 1.

1 to the power of x.

And what that means is you could now use graphing software, you could type in that equation and it would draw the exponential curve.

Right, how do you think the graph and the equation would change if we change the interest rate to 1%? Have a think and make a prediction.

Right, well our multiplier is gonna change.

We're still investing 100 Pounds, but now we are multiplying by 1.

01 to get the amount after each month.

So in order to find our general equation, we'll have y equals 100 times 1.

01 to the power of x.

Again, I can plot that on a graph, but we're not gonna see the general shape until I scroll out a little bit further.

So here are the graphs of 100 Pounds invested at 1% simple interest and 1% compound interest over 24 months.

The purple one's gonna be compound interest 'cause we're gonna start getting more and more interest each month as it's compounded.

Simple interest is a linear graph.

And there we go.

I've zoomed out to see a larger time period.

We can see those shapes better.

What do you think the graph and equation would look like if 200 Pounds was invested at 1% compound interest per month? Make a prediction and then we'll plot it.

So those are our values each month and our equation would be y equals 200 times 1.

01 to the power of x.

And there's our graph.

Quick check then.

Which of these would be the equation of the exponential curve if graphing 100 Pounds invested at a compound rate of 2% per month? Which of those do you think it is? So remember the general form of an exponential equation is y equals a times b to the x.

So it's C, 1.

02 is the multiplier to increase by 2% and we're starting with 100 Pounds.

So 100 multiplied by 1.

02 to the power of x.

So we can model depreciation in the same way.

So a car costs 15,000 Pounds new and the value depreciates at a rate of 10% per year.

So multiply to decrease by 10% is 0.

9.

So after one year the car will be worth 15,000 times 0.

9, which is 13,500.

Then we can keep going exactly the same way as interest, just this time we have a value which is gonna be decreasing each month.

The equation that we can use then is y equals 15,000 times 0.

9 to the power of x.

What do you think this will look like on a graph? Let's have a look.

So when our value for b is between zero and one, we get a decreasing curve.

So just be aware that if your multiplier is greater than one, we have that increasing exponential curve.

And when that value for b is between zero and one, then we have this decreasing curve.

Using this model, will the car ever have a value of zero Pounds? What do you think? No, there's going to be an asymptote.

Well done if you remembered that word, at y equals zero.

So it'll get closer and closer and closer, but it's never going to cross the X-axis.

Which of these could be the equation for the exponential curve shown? Well done if you said C, the y intercept should be 200 and we're decreasing, which means the value for b needs to be between zero and one.

So we've got 200 times 0.

75 to the power of x.

Right, time for you to have a go.

I'd like you to choose the correct sketch for the graphical model of each scenario.

Off you go.

Now have a go at question two.

Read all the information carefully and answer those three questions.

For question three, I'd like you to match each of these scenarios to one of the graphs drawn.

When you've done that, can you write the equation for each of the curves? Give that one a go.

Now try question five.

Make sure you read the information carefully and see if you can answer those five questions.

Give it a go.

And finally we've got a computer depreciating at a rate of 20% a year.

Can you sketch the graph and write the equation? Give it a go.

Let's have a look then.

So A matches with E and we should have a linear graph with positive gradient.

B matches with G.

We should have a curve where the rate that the value is increasing over time is increasing.

For I, we've got another curve, but this time our values are decreasing over time, but the rate in which they're decreasing over time is slowing down.

So for two, check your graph looks like mine.

The equation will be y equals 0.

1x plus 20 or y equals a 10th x plus 20.

And then Aisha is incorrect.

Money is only added every month.

She can't make statements about the amount between months.

That doesn't make sense.

Pause the video if you need to check your graph.

For 3A, pause and just make sure you've got those matched up and then we'll go through the equations.

Very good.

So A, you should have y equals 10 times 1.

01 to the power of x.

B, y equals 10 times 1.

03 to the power of x.

C, y equals 20 times 1.

05 to the power of x.

D, y equals 50 times 1.

01 to the power of x.

And E, y is 50 times 1.

03 to the power of x.

You might wanna just take some time to have a look at how those different equations look different on the graphs.

For question five, you should have 51 Pounds 25 after one month.

After 12 months we can use an exponent of 12.

So we've got 67 Pounds and 24.

And then it's gonna be easiest if we plot these every 12 months.

It will just be a sketch because our values will be hard to plot accurately.

So have a look at my values for each year and then see if your graph looks similar to mine.

The equation is then y equals 50 times 1.

025 to the power of x.

And then we can estimate when we got to 150 Pounds and it's about 45 months.

But remember interest is only added each month, so it does need to be a whole number of months.

Because we've only plotted them every 12 months we can look between the points and make an estimate as to when that value is 150.

And our computer.

So again, pause the video and check that your sketch looks like mine.

And your equation should be y equals 900 times 0.

8 to the power of x.

Now we're gonna have a look at displacement-time graphs.

So we know that a straight line on a distance-time graph represents constant speed.

And a curve represents acceleration.

What is the difference between the second and the third graphs? So the second graph is showing that speed is increasing over time and the third graph is showing speed decreasing or deceleration.

So we're now gonna look at acceleration and deceleration on displacement-time graphs and compare it to the distance-time graphs.

Sofia runs at a constant speed to the end of her road.

She stops to rest for a short time and turns around and runs back at a constant speed.

On a displacement-time graph, this would look roughly like this.

Sofia said, "In reality I accelerated to start running and decelerated before I stopped." And then she did the same on the way back.

What do you think this is gonna look like on the graph? Okay, well there'll be a curve showing acceleration, then the straight line segment to show her constant speed, and then a different curve showing deceleration.

Of course we have a horizontal line when Sofia is stationary.

Sofia says, "I did exactly the same on the way back so the graph will look like this." What mistake has Sofia made? Right, well she's a drawn distance-time graph, but the displacement-time graph should have a downward sloping line as she returns home.

So here Sofia is accelerating from stationary to a constant speed.

So the curve is gonna look like this.

You can see she's gone from a horizontal line to a line with negative gradient so we need that sort of connecting curve.

And then the decelerating curve is gonna look like this and eventually that will become horizontal to show that she's back home.

Sofia says, "Because the way back was identical to the way there, my graph has a line of symmetry." So we could sketch Sofia's journey on a speed-time graph too.

Accelerating at the beginning means speed is increasing so a diagonal line.

Horizontal line showing constant speed on a speed-time graph and because she becomes stationary at the end, that line with negative gradient is gonna reach the X-axis again, which is a speed of zero.

So horizontal line on the X-axis to show speed is zero.

And then this part on displacement-time graph represents acceleration, remember? So that's accelerating back towards home.

But on a speed-time graph, we don't have any information about direction.

So we just have a diagonal line representing the acceleration.

If her speed was exactly the same, then we're gonna have another horizontal line segment at the same height as our previous one.

And then decelerating again.

Pause the video if you want to check any of those parts.

Okay, which of these journey segments show speed increasing? These are displacement-time graphs.

What do you think? Well done if you said B and D.

The first one is showing constant speed.

Then we've got speed increasing, moving away from home.

Speed decreasing, moving towards home or deceleration.

And speed increasing, moving towards home.

Think about what the gradient of a straight line segment would be.

If we added it to the end of that curve we'd have a very steep straight line segment.

So that would be fast speed, but moving towards home.

Time for you to have a practise.

Here is a sketch of a displacement-time graph for a cycle ride.

Can you describe what is happening at each point? Think about direction, think about speed.

What does a curve mean? What does a straight line mean? Give it a go.

So this is the same graph as you had before.

I want to know at which point of the journey was the cyclist travelling fastest? True or false, on the journey out it took the cyclist longer to accelerate to a constant speed than to decelerate to a stop? So just looking at the journey away from home.

And for D, I'd like you to draw a speed-time graph for that journey.

Have a think about your answer to B.

When the cyclist was travelling fastest, that'll be our highest speed.

So see if you can be as accurate as you can with where you're drawing your line segments.

Give it a go.

Question two.

Andeep has drawn this displacement-time graph for a cycle ride.

I'd like you to think about which parts of the journey are unrealistic and which are impossible, and see if you can explain your answers.

Once you're happy with that, come back for the next bit.

And finally, this is an activity you could do with a partner.

I'd like you to invent your own scenario.

So it could be a walk or a cycle ride or a skiing trip, anything you like and sketch a displacement-time graph.

Obviously for a displacement-time graph to have any negative gradients or downward sloping line segments, you need to be moving away from home and then back towards home.

See if you can describe your journey to a partner and see if they can draw the correct graph.

So you're gonna need to think about speed, you're gonna need to think about direction, you're gonna need to think about accelerations and decelerations.

If you want a different challenge, you could describe a displacement-time graph to your partner and see if they can draw the correct speed-time graph.

Once you've played around with that, come back for the answers.

Well done.

So that first section of curve shows us accelerating but moving away from the start point.

That first straight line segment is maintaining a constant speed away from home.

Then decelerating to a stop.

The horizontal line at the top is showing that the cyclist is stationary.

Then they are accelerating but moving back towards the start.

Then we have a straight line segment showing travelling at a constant speed.

We have another straight line segment further down, which has a steeper gradient, so that means they're travelling faster.

In between those two, we've got acceleration and that makes sense.

If we're travelling at one speed and then another speed, we're gonna have acceleration between those two speeds.

Then at the end we've got decelerating to a stop.

Pause the video if you need to check any of my wording.

So for B, the fastest speed is that final section of constant speed on the way home.

So before that last curve to show deceleration is our fastest speed, it's our steepest line segment.

For C, that is true.

The accelerating curve takes more time than the decelerating curve.

It has a further horizontal distance on the graph.

Right, and those two elements are gonna help us draw our speed-time graph.

You might wanna pause the video and check yours against mine.

There's a few key features I want to draw your attention to.

We said that that final part of the journey was the fastest, so that should be higher on our speed-time graph.

So you'll see that second peak is higher than the first.

You could have used a ruler to measure how much time was taken out with each stage of the journey and used the same gaps in time on the speed-time graph.

So you'll see the first acceleration took more time than the later decelerations and the later accelerations.

Feel free to study that graph in a bit more detail now and then we'll look at the next bit.

So here's some ideas you might have said.

So it's unrealistic that he would've started at a constant speed.

If he started stationary, he'd need to accelerate to get up to a constant speed.

B, this is actually impossible.

Time has to move forward.

We can't have a curve curving back on itself.

C is a problem.

If we're stationary at the top, then it's impossible to decelerate and that curve is showing decelerating, so that's not gonna work.

You might have also said that coming to a stop without deceleration is unrealistic.

So we should really have a curve at the end if we're being realistic.

And if you had a go at the paired activity, make sure you check that your partner drew the correct graph.

If they didn't, were you clear about your moments of acceleration and deceleration? Did you explain when they were travelling the fastest? Did you think about changing direction and explaining that? The same for B.

Make sure the steepest gradients on the displacement-time graph are shown as the highest speeds on the speed-time graph.

Well done.

Thank you for joining us today.

We've talked about lots of different types of real-life graphs.

You may want to spend some more time exploring those in more detail.

If you want to read through what we've learned today, pause the video and do that now.

Thank you very much for joining me and I look forward to seeing you again.