Lesson video

Hello there.

Mr. Robson here.

Superb choice to join me for maths, especially seeing as we're doing features of linear relationships.

I'm excited, so let's get started.

Our learning outcome is that we'll be able to recognise that linear relationships have particular algebraic and graphical features as a result of the constant rate of change.

Keywords for today, the word linear.

The relationship between two variables is linear when they change together at a constant rate.

When plotted on a graph, a linear relationship will form a straight line.

You'll be hearing that word a lot throughout today's lesson.

We're gonna start with looking at this constant rate of change.

If I said to you, "Calculate the first five terms of these sequences, 2n, n plus 3, n squared, and n cubed," I would hope that you'd draw me a table of values and generate those terms. Could you pause this video and do that now? You should have got these values.

The sequence 2n goes 2, 4, 6, 8, 10.

The sequence n plus 3, 4, 5, 6, 7, 8, et cetera.

What I'll ask you to think about now is what common feature do the sequences 2n and n plus 3 share that the sequences n squared and n cubed do not? Pause this video and see if you can answer that question with the person next to you.

Did you notice this? The sequence 2n changes by positive 2 from term to term.

The sequence n plus 3 changes by positive 1 from term to term, but the sequences n squared and n cubed didn't have the same change each time between their terms. As a result of this constant difference in the sequences 2n and n plus 3, we can call them linear sequences.

You might have heard these called arithmetic sequences.

They have this name because they have a constant difference between the terms. We can see the constant difference when we plot these sequences.

If I had a grid like so with the horizontal axis being n, the term number, and the vertical axis being T, the term value, I could plot a term number of 1 and a term value of 2, term number of 2 and a term value of 4, and I can see in that graph a constant change.

The same for n plus 3.

I can see in that graph a constant change.

The constant difference makes their points form a line, hence we call these sequences linear because they form a line.

By contrast, if I graph the sequences n squared and n cubed, the coordinates 1, 1, 2, 4, 3, 9, 4, 16, you can see that rate of change changing.

The same for the sequence n cubed.

That rate of change is changing.

Because of this graphical representation, we can see that these sequences are non-linear.

You can't draw a straight line through the points they plot.

Will this look any different if we plot algebraic relationships? By that, I mean a relationship between the x coordinate and the y coordinate on a Cartesian coordinate grid.

If we generate some coordinates for the line y equals 2x, y equals x plus 3 and y equals x squared, we'd get these y values for the respective x values, and we could plot these coordinates.

For the sequence, sorry, for the relationship y equals 2x, you see those y values changing by positive 2 each time.

For y equals x plus 3, you see them changing by positive 1 each time.

Y equals 2x, y equals x plus 3 have a constant difference in the y coordinates.

Y equals x squared, however, does not.

Not a constant difference.

When we graph this, you'll see how y equals x squared looks different.

Y equals 2x has those coordinates.

Y equals x plus 3, those coordinates.

Y equals x squared has those coordinates.

So y equals 2x, we can call it a linear relationship because we can draw a straight line through those points just as we can for y equals x plus 3, two linear relationships.

However, you can't draw a straight line through the coordinates we plotted for y equals x squared, so we call that non-linear.

If we look at the sequences and the relationships we've observed so far, we might be able to determine what it is that makes a sequence linear or non-linear.

The linear ones we've seen so far were the sequences 2n and n plus 3 and the relationships y equals 2x and y equals x plus 3.

The non-linear sequences and relationships we've seen so far, n squared, n cubed, and y equals x squared.

Spot the difference.

Can you see, without a graph, without the values in the sequence, without the coordinates, can you see from the algebraic notation what it is that determines whether something is linear or non-linear? Pause this video and make a suggestion to the person next to you.

I hope you said something along the lines of an exponent of 1.

You might have described it as a power of 1.

For example, in the sequence 2n, you could say that's 2n to the power of 1, or you could say the n has an exponent of 1.

By contrast, the non-linear sequences and relationships we've seen had an exponent that was not 1.

N squared, an exponent 2.

N cubed, an exponent 3.

Y equals x squared, the exponent was 2.

This is what determines whether a sequence or relationship is linear or non-linear, whether it has an exponent of 1 or not.

Quick check that you've got that now.

Which of these sequences will have a constant difference? I.

e.

, which one will be linear? 4n plus 6, n squared minus 7, 7 minus 4n, or 5n to the power of 6? Which of those will have a constant difference? Pause this video and have a think about that.

I hope you said 4n plus 6 will have a constant difference.

Why? Because the exponent of n is 1.

This will be an arithmetic linear sequence, therefore it'll have a constant difference.

N squared minus 7 will not.

The exponent of n is not 1.

That will be non-linear.

Option c, 7 minus 4n.

Just because the coefficient of n is negative, it doesn't matter.

It's all about the exponent.

The n exponent is still 1, therefore it will be an arithmetic or linear sequence with a constant difference.

Option d, what do you think? Well done.

The exponent is not 1.

That will not be a linear sequence.

What about algebraic relationships between the x coordinate and the y coordinate? Which of these relationships will be linear? Y equals 3x plus 8, y equals 3x cubed plus 8, y squared equals 3x plus 8, 1 minus 3x equals y.

Which of those will be linear? Pause and have a think.

The first one, option a, is linear, y equals 3x plus 8.

The x exponent is 1.

The y exponent is 1.

That will be a linear relationship.

By contrast, b will not, y equals 3x cubed plus 8.

The exponent of x is not 1.

It's 3 on this occasion.

That will not be linear.

C was a tricky one because y equals 3x plus 8 would be linear, but y squared equals 3x plus 8 is not.

Both the x and y exponents have to be 1 in order for it to be linear.

That will not because it's y squared, an exponent of 2.

The final one was again a negative coefficient for the x term.

However, that does not affect the fact the exponents of 1 will have a linear relationship.

Looking at this in a slightly different context now, Alex starts with two sweets.

Every day, he's given three more.

This will form an arithmetic sequence, a practical example of an arithmetic sequence.

He starts with two sweets, and we add three every day.

Provided Alex doesn't eat any of those sweets, he'll have 2 sweets, then 5 sweets, then 8 sweets, then 11 sweets, then 14 sweets.

You'll see that constant difference of positive 3.

That's the 3 extra sweets he's given each day.

The arithmetic or linear sequence 3n minus 1 has a constant difference of positive 3.

That's what's represented in these numbers, 2, 5, 8, 11, 14.

We could visually represent this sequence to have a closer look at what linear means, what this constant difference means.

With a visual representation, we can see the constant difference of positive 3.

Remember, Alex started with 2 sweets, and then he was given 3 more and 3 more again and 3 more again and 3 more again.

Can you see the sequence in that visual representation? Can you see why we call this a linear sequence? I hope so.

Will we see something similar when we plot the algebraic relationship y equals 3x minus 1? Not the arithmetic sequence 3n minus 1, the algebraic relationship y equals 3x minus 1.

We'd start with a table of values to find our coordinates, and the y coordinates would be 2, 5, 8, 11, 14.

Those numbers look familiar, a constant difference of positive 3, and when we graph them, does it look the same as that sequence? Does it not? I think it's the exact same thing, more or less.

Every time we move forward 1 on the x axis, we're going up by 3 on the y axis.

You can almost see those three extra sweets each time but in that algebraic relationship.

Again, can you see the fact that this relationship is linear? Y equals 3x minus 1 has that table of values, those coordinates.

The relationship between two variables is linear when they change together at a constant rate.

That's the definition I gave you at the start of the lesson.

The relationship between two variables is linear when they change together at a constant rate.

For every change of positive 1 in x, there is a change of positive 3 in y.

You can see it in the table and on the graph.

From the x coordinate 2 to the x coordinate 3, that's a change of positive 1.

The y coordinate changed from a 5 to 8, a change of positive 3.

We can see that again on the graph, and this pattern continues in our table of values and on the graph, a constant rate of change, both variables changing together at a constant rate.

That's what generates our linear relationship.

Quick check now.

Does this set of coordinates have a linear relationship? Pause this video.

Have a good look at those coordinates.

Linear relationship? You tell me.

Pause now.

At first, it looks like that is a linear relationship.

On closer inspection, the answer is no.

The y coordinates go 1, 3, 5, 7, 9, 11, 13.

It's a constant difference of positive 2.

That's linear, isn't it? It's not, because the x coordinates don't have a constant rate of change.

We don't have both variables changing together at a constant rate.

The x coordinates begin with an increase of positive 1 and then change to an increase of positive 2.

That breaks the linear relationship.

Our definition of linear, remember, the relationship between two variables is linear when they change together at a constant rate.

If you were to graph these coordinates, and perhaps some of you did, you could have seen the fact there was not a linear relationship.

Our line goes a little crooked in the middle, and that's when the rate of change in the x coordinates changed.

It broke our linear relationship.

Practise time now.

I'd like you to classify these relationships into linear and non-linear.

Which relationships? These relationships scattered all around the screen.

I'd like you to categorise them into the box linear and the box non-linear.

Hmm.

To which does each belong? Pause this video and ponder that now.

Question 2, plot the coordinates for these relationships for the given range of x values and show graphically their constant rate of change.

I'd like you to do that for part a, the relationship y equals 2x minus 5.

Complete the table of values, plot the coordinates, show the rate of change, and once you've done that for that relationship, part b is the relationship y equals 3 minus x.

Pause and give this a go now.

Question 3, this set of coordinates has a linear relationship with a constant rate of change.

Find the missing values, plot the graph, and show that constant rate of change.

Pause the video.

Give this a go.

Feedback time, classifying our relationships into linear and non-linear.

Linear relationships, we're looking for the ones that have an exponent of 1.

Y equals 4x, y exponent of 1, x exponent of 1.

Y equals x plus 4, y exponent 1, x exponent 1.

4y equals x.

That might look a little bit unusual to you.

You might not have seen them written that way before with a y coefficient that is not 1.

Does it matter? It's still linear because the y exponent is still 1.

The x exponent is still 1.

X plus y equals 4, still got exponents of 1 on both those variables.

Y equals 1/4x, exponents of 1.

Y equals 0.

4x, exponents of 1.

They were all of our linear ones.

Non-linear, y equals 4x squared.

X squared.

That's not an exponent of 1.

That will not be linear.

Y to the power of 4 equals x plus 4.

That y exponent is no longer 1.

It's 4.

That's non-linear.

Y equals 2x to the power of 4.

That's an exponent of 4, not 1.

That's non-linear.

Y equals x to the power of 4 plus 4, exponent not 1, non-linear, and y equals x the power of 0.

4.

That's a beautiful graph if you happen to have the time to go and plot it, but not linear.

That is not an x exponent of 1, hence it's in the non-linear category.

For question 2 part a, your table of values should have come out with the y coordinates negative 3, negative 1, 1, 3, 5, and when plotted, they should have looked like that, and then crucially, I said show on the graph their constant rate of change.

So in your table of values, you see every time x changes by positive 1, y changes by positive 2.

You can show that on the graph like so, and that relationship continues right the way through each coordinate.

You can show that constant rate of change.

If you haven't done that now, pause this video and copy this.

Then you draw a straight line through all the points.

Because of their constant rats of change, this makes a linear relationship.

We'll do the same thing for part b, fill in the table of values.

You should've had the y coordinates 2, 1, 0, negative 1, negative 2.

When plotted, it would look like that.

The constant rate of change looks slightly different here, but we can still show it.

As the x coordinate changes by positive 1, the y coordinate changes by negative 1, and that is reflected throughout each of those coordinates.

Once again, we get a linear relationship because of the constant rate of change.

Question 3, we were given a set of coordinates with some y coordinates missing, but we were told it was a linear relationship with a constant rate of change.

You should have spotted that the y coordinates have a step of positive 3 between them.

Find the missing values, plot the graph, and show the constant rate of change.

This one's slightly different than the ones you've seen already because the coordinate plotting is no different, but instead of going up in consecutive integer values and having a constant change of positive 1, the x coordinates here had a change of positive 2 but a constant change of positive 2.

The y coordinates changed by positive 3.

Linear? Non-linear? Well, of course it's linear.

Look at those coordinates.

They're forming a straight line.

We can show this constant rate of change of both variables, positive 2 in x, positive 3 in y, on our graph between each of our coordinates.

Can you see how the constant rate of change in the table of values matches the constant rate of change in the graph? This is what gives us that linear relationship.

On to the second half of our lesson now where we're gonna look at graphical and algebraic representations of linear relationships.

All of these coordinates fit the relationship y equals 15 minus 3x except for one of them.

Have a look at those coordinates.

Substitute them into that relationship and see if you can spot the one that doesn't fit.

Pause this video and have a go at that now.

Did you spot that it's negative 5, 0 which doesn't fit? When we substitute in an x value of negative 5, we don't get a y value of 0.

You can see that more clearly if you graph the relationship.

Five of those coordinates are on the line y equals 15 minus 3x.

The coordinate negative 5, 0 is not.

The coordinate that does not fit the linear relationship is not part of that linear pattern.

Some pupils are discussing a coordinate, the coordinate 2, 5.

Jacob says, "I think there's a linear relationship, a linear relationship between the x and y coordinates." Jacob suggests that's y equals x plus 3.

Okay, 2 plus 3 makes 5.

Makes sense.

Aisha says, "I think there's a linear relationship y equals 7 minus x." 7 minus 2 makes 5.

Okay.

Sofia says, "I think there's a linear relationship y equals 2x plus 1." 2 lots of 2 plus 1 makes 5.

Hmm, who is right? Pause this video.

Make a suggestion to the person next to you.

Well, with the limited information we've got, they might all be right.

If we draw those relationships, the lines of those relationships, y equals x plus 3, y equals 7 minus x, y equals 2x plus 1, they all go through the coordinate 2, 5 or the coordinate 2, 5 belongs to all of those relationships, but what if we knew a second coordinate? If we knew there was another coordinate, negative 5, 12, does that change our answer? It does.

We can see that the line y equals 7 minus x is the only one which contains that coordinate.

So it's now only Aisha's relationship that works for both of those coordinates.

Quick check now.

The coordinate 6, 13 belongs to which linear relationship? Y equals 3x minus 5, y equals x plus 7, y equals 2x plus 1.

Pause this video and have a think about that now.

It fits the relationship y equals 3x minus 5.

3 lots of 6 minus 5 is 13.

It also fits the relationship y equals x plus 7.

6 plus 7 makes 13, and it also fits that third relationship, y equals 2x plus 1.

2 lots of 6 plus 1 is 13.

The coordinate 6, 13 belongs to all of those linear relationships.

Which coordinates fit the linear relationship y equals 5x minus 14? Four coordinates.

Can you test them each and see which ones fit that linear relationship? Pause this video and try that now.

Coordinate a, 2, negative 4 fits.

5 lots of 2 minus 14 is negative 4.

Coordinate b, negative 4, negative 34 fits.

5 lots of negative 4 minus 14 is negative 34.

Coordinate c fits as well.

5 Lots of 3.

4 minus 14 is 3.

It's the last one does not fit, negative 14, 0.

If you substitute in an x value of negative 14, you would get a y coordinate of negative 84, not 0.

You might have been fooled by this one because 0, negative 14 would have fitted the linear relationship y equals 5x minus 14.

Practise time now.

I'd like you to match the coordinate pairs, their respective relationships, and graphs.

A, B and C are pairs of coordinates.

D, E and F are relationships, and G, H and I are the respective graphs, but which coordinates match to which relationship match to which graph? Pause this video and have a think about that now.

Right, let's go through those answers.

If I start with the graphs, a useful thing to do would be to match the coordinates to the quadrants in which they're plotted.

It's a useful start.

The coordinates negative 4, negative 1 and positive 8, positive 5 would be in the third quadrant and the first quadrant.

That's the only graph with two coordinates, one of which is in the third quadrant, one of which is in the first quadrant.

So it must match those two coordinates.

The graph H has a coordinate in the second quadrant and a coordinate in the fourth quadrant.

So that could be coordinates negative 4, 5 and 8, negative 1.

Graph I has a coordinate in the second quadrant and a coordinate in the first quadrant.

So that could be negative 4, 1 and 8, 7.

We can now check by substitution which relationship those coordinates fit.

The coordinates negative 4, negative 1 and 8, 5 fit the relationship y equals x plus 2 over 2.

The coordinates negative 4, 5 and 8, negative 1 fit the relationship y equals 3 minus 1/2x, and the coordinates negative 4, 1 and 8, 7 fit the relationship y equals 1/2x plus 3.

That's the end of the lesson now.

In summary, linear relationships when plotted on a pair of axes form a straight line.

For example, y equals 7x plus 3 is linear.

We also saw examples of non-linear.

An example of non-linear would be y equals 7x cubed.

That is not linear.

I hope to see you again soon for more maths.

Bye for now.