Lesson video

Hello, Mr. Robson here.

Welcome to maths.

Great choice to join me today.

We're reading from graphs.

If reading from graphs is anywhere near as wonderful as reading from books, this lesson is going to be super.

Let's find out, shall we? A learning outcome is I'll be able to read and interpret points from a graph in order to solve problems. Some key words we're gonna hear today.

Linear.

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

Quadratic.

A quadratic is an equation, graph, or sequence whereby the highest exponent of the variable is two.

For example, X squared plus X is quadratic, but X cubed plus X is not.

A parabola is another word we'll come across.

A parabola is a curve where any point on the curve is an equal distance from a fixed point called the focus and a fixed straight line called the directrix.

The line of symmetry goes through the focus at right angles to the directrix.

Two parts of today's lesson.

Let's begin with reading from linear graphs.

In mathematics, we're don't just plot linear graphs.

We use them to solve problems. We do that by taking readings from them.

This is the graph of linear equation Y equals 2x minus one.

That should be familiar to you.

From it, we can read the coordinate three, five, but what's the significance of that? This means that the values X equals three and Y equals five must fit the equation.

If we substitute them in to the equation, Y equals 2x minus one, we get five being equal to two lots of three minus one.

It's a true equation.

The values X equals three Y equals five fit that equation.

Crucially, this also tells us the solution to the equation 2x minus one equals five.

What we're asking ourselves with that is when does the line Y equals 2x minus one have a Y coordinate of five? If we read across from the Y coordinate of five, we can read down to an X coordinate of three.

This tells us that X equals three is the solution to the equation 2x minus one equals five.

We solved a linear equation by reading from a linear graph.

Let's check you can do that.

Reading from this graph, that's the graph of Y equals nine minus 2x.

Reading from this graph, which of these is solution to the equation nine minus 2x equals one? Is it X equals one, X equals seven, or X equals four? Pause this video.

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

See you in a moment.

Welcome back.

How did we do? Hopefully, we said the solutions X equals four.

Why? Because what we're asking ourselves is when does the line Y equals nine minus 2x have a Y value of one and it only happens at this moment when X has a value of four.

There's other purposes for reading from graphs.

This one I particularly like.

This is the linear graph Y equals 8x minus 10.

We can use it to work out calculations such as eight multiplied by 1.

75 minus 10.

I can calculate them if you give me a pen and paper.

I'll have it done in a minute.

I can do it even quicker with this graph.

I need to substitute in X equals 1.

75, so I take that reading from the X axis up to the line and across to the Y axis where we can read a four.

Therefore, we know that eight multiplied by 1.

75 minus 10 equals four.

We used a linear graph to complete a calculation.

Let's check you can do that.

This is the linear graph Y equals 20x minus 0.

4.

I'd like to use that graph to work out 20 multiplied by 0.

06 minus 0.

4.

Pause this video, take a moment to do that.

Welcome back.

How did we do? Let's find out.

Hopefully, you read up from an X value of 0.

06 and then across the respective Y value, which was 0.

8.

Once you've identified that coordinate, we know 20 multiplied by 0.

06 minus 0.

4 equals 0.

8.

There's other purposes for reading from linear graphs too.

For example, estimation.

This is the linear graph Y equals 8.

4x plus 13.

65.

You can see from the scale on the axis, we haven't got a lot of detail here so we can only take an estimate, but estimates are really useful in mathematics.

For example, we might estimate four lots of 8.

4 plus 13.

65.

We can do that from our graph reading up from an X value of four and across to there, I'm estimating that vertical value to be 47, so I can estimate four multiplied by 8.

4 plus 13.

65 to be 47.

When I check that on my calculator, I found that it was accurate to the nearest whole number.

That's a pretty good piece of estimation from a linear graph.

Let's check that you can do that.

Which of these is most likely a good estimate for three multiplied by 8.

4 plus 13.

65? I've given you the coordinate there, so which do you think is the best estimate for the calculation? Is it 39, 40, or 41? Pause, have a conversation with a person next to you, or a good think to yourself.

Welcome back.

Let's see how we did.

Hopefully, you said 39.

Why would you say that? Well, we can estimate that coordinate to be three, 39.

Reading from the vertical axis, you can see it's not quite reached 40.

It's certainly not above 40, so it wouldn't be options B or C.

Therefore, we can estimate three lots of 8.

4 plus 13.

65 to be 39.

Reading from graphs can also be used in a practical context.

For example, the eight pupils measure their height and shoe size.

They plot the results on a graph and they find it makes a linear relationship.

The linear relationship looks like that.

Modelling a linear relationship like this enables us to make predictions.

For example, from a height of 170 centimetres, we can read a shoe size of 10, therefore, we'd conclude or we'd expect a pupil of height 170 centimetres to have a shoe size of 10.

Let's check you can do that.

I'd like you to use this model to predict the height of a pupil with a size seven shoe.

Pause this video, take that reading, and see what you can predict.

Welcome back.

I hope you read from a shoe size of seven, a height of 155 centimetres and made the conclusion I would expect a pupil of shoe size seven to have a height of 155 centimetres.

Well done.

Practise time now.

To question one.

This is the graph of Y equals three minus a half of X.

I'd like to use the graph to solve these three equations.

My hint for you is that if I were doing this, I'd be drawing all over that graph to solve those equations.

Pause, have a go at these three problems now.

Question two.

Use this graph of Y equals 1.

75x minus 0.

3 to calculate 1.

75 times 0.

8 minus 0.

3, 1.

75 times 0.

4 minus 0.

3, and 1.

75 multiplied by negative 0.

4 minus 0.

3.

I guarantee you will find that last one easier reading from this linear graph than you would to calculate it.

Pause, try those three problems now.

For question three, the Oak pupils plot their age against how many books they've read and they find a linear relationship.

For part A, I'd like you to use the graph to estimate the age of a pupil who has read 400 books.

For part B, I'd like to use the graph to predict how many books a pupil will have read by the time they've reach age 18.

Pause and do those two problems now.

Welcome back.

Feedback time.

The graph of Y equals three minus a half of X, utilising that to solve these equations.

For A, we were solving three minus a half of X equals one.

When does that line have a Y value of one? Well, that occurs when X equals four.

You should have taken that reading and found a solution X equals four for part A.

For part B, three minus half of X equals zero.

When does that line have a Y coordinate of zero? It happens there when X equals six.

For part C, three minus half of X equals five.

When does our line have a Y coordinate of five when X equals negative four? I don't know what you think, but I found those three easier to solve by reading from my linear graph.

Question two feedback.

I said use this graph to make three calculations.

For part A, we should have taken a reading from X equals 0.

8.

When we go up to our line and across to the Y axis we find a Y value of 1.

1.

For part B, we need the reading from the moment when X equals 0.

4, go up to our line across the Y axis, and we find that also to be 0.

4.

For part C, we're reading from the moment when X equals negative 0.

4 and we find a y value of negative one.

For question three, we found a linear relationship of age against books read.

We're using the graph to estimate the age of a pupil who's read 400 books.

We'll read across from 400 books and read down to the horizontal axis where we find the age of 14.

For part B, using the graph to predict how many books a pupil will have read by the time they reach age 18.

Let's read up from age 18 and we'll find that to be about 575.

An approximation in that region is a good one.

Onto the second half of the lesson now.

Reading from quadratic graphs.

It's not just linear graphs that we can read from.

This is the graph of the quadratic equation Y equals X squared.

We can read the coordinate four, 16.

This means that the values X equals four and Y equals 16 satisfy the equation.

When we substitute them in to Y equals X squared, we do indeed find that 16 is equal to four squared.

That's a true equation.

Those values satisfy our equation.

Jun and Jacob are discussing this graph and Jun says "This also means that X equals four is the solution to the equation X squared equals 16." That's awesome, Jun.

Well done.

But Jacob says "You're right, but I don't think that's the only solution." How interesting.

Can you see what Jacob means? Pause this video, have a conversation with a person next to you, or a good think to yourself.

See you in a moment.

Welcome back.

Did you see what Jacob meant? Let's have a closer look.

Jun says "You're right, Jacob.

There are two moments when Y equals X squared has a Y value of 16." We won't just read from the first quadrant.

We can read a moment in the second quadrant because that's a parabola and it has symmetry.

Jacob says, "Exactly.

There are two solutions to X squared equals 16.

They are X equals positive four and X equals negative four." That's awesome, you two.

Well done.

So you need to be aware and because quadratic graphs form the shape of a parabola, we can sometimes take two readings.

Make sure you spot both.

Quick check you've got that now.

I'd like you to find the two moments when X squared equals nine.

And as a reminder, there are two moments when that happens.

Pause this video, see if you can spot them both now.

Welcome back.

I hope you took this reading three, nine and also this reading negative three, nine.

There are two moments when X squared equals nine.

They are when X equals positive three and when X equals negative three.

Parabolas are really useful for modelling real life situations.

Aisha's playing basketball and notices the path of the ball follows that of a parabola.

There's a diagram.

You might have noticed this if you play basketball.

We could put that onto an axis and start to take some readings.

For example, when is the ball five metres above the ground? We'd need to read across from a height five metres and we'd find that moment when the ball is 2.

25 metres in distance from Aisha and also that moment when the ball is 4.

8 metres from Aisha.

So we know the ball is five metres above the ground after 2.

25 and 4.

8 metres in distance from Aisha's position.

Crucially, our graph enables us to clearly see that there are two moments when the ball is five metres above the ground.

Let's check you've got that.

This parabola models the flight of a toy glider.

What height is the glider at after three seconds? Is it 225 metres, 150 metres, or 300 metres? Pause this video, have a conversation with a person next to you or a good think to yourself.

See you in a moment for the answer.

Welcome back.

I hope you said option B, 150 metres.

We'd read up from three seconds and across to a height of 150 metres.

There's only one value when reading up from the horizontal axis.

Another check.

This parabola models a flight of a toy glider.

At what time is the glider at 225 metres? Slightly different question that one.

At what time is the glider at 225 metres? Four options there.

Pause this video, see if you could pick the right one.

Welcome back.

I hope you said option A, five seconds.

I hope you also said option D, 15 seconds.

When we read across from 225 metres, we can clearly see there are two moments when the glider is at that height.

Five seconds and 15 seconds.

So beware.

We had two values when we read across from the vertical axis.

Practise time now.

Question one.

This is the graph of the quadratic equation Y equals half of X squared plus 4x and isn't it beautiful? It might look complicated, but you will be able to use it to solve these three equations.

My hint if I was doing this, I'd be drawing all over that graph to find the solutions to those equations.

Pause, see if you can do this now.

Question two.

This parabola models the flight of a cricket ball being thrown.

Use the graph to read A, the height of the ball at a distance of 18 metres, B, the distance when the ball is at height of nine metres, and C, the distance when the ball is at a height of 12 metres.

Pause and solve those three now.

Feedback time.

It looked like a tricky problem but we're just reading from a graph.

Solving the quadratic equation half X squared plus four X equals 10 for part A.

Well, we need to read when is that graph giving us a Y coordinate of 10 and it's these two moments when X equals negative 10 and when X equals positive two.

For part B, when does that graph give us a Y coordinate of zero? Well, it's these two moments when X equals negative eight and when X equals zero.

How about part C? Was this different somehow? I hope you spotted that.

When does the graph have a Y coordinate of negative eight? All of a sudden, there's only one moment where that happens and that's when X equals negative four.

In part A and B, we had two solutions.

In part C, we had only one solution.

Question two feedback.

We were using this graph to read a few things.

Part A, the height of the ball at a distance of 18 metres, so we look for a distance of 18 metres and we read the height of the ball and we find it's got a height of five metres at that moment.

Just one reading there.

For part B, the distance when the ball is at a height of nine metres, we read from a height of nine metres and we find these two distances, six metres and 14 metres.

Part C, the ball never reaches the height of 12 metres.

The graph shows us that, and this happens in mathematics.

Sometimes we have one solution.

Sometimes we have two solutions.

There'll be times when there's no solution.

There was no solution to Part C.

Sadly, that's the end of the lesson now, but in summary, we can read and interpret points from a graph to solve problems that could be to solve an equation or to solve a real world problem in context.

The graph we read from may be linear, but not always.

For example, we might read from quadratic graphs to solve problems. I 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.