Lesson video

Hello, Mr. Robson here.

Super choice to join me for maths again, defining features of linear relationships today, this should be wonderful.

Let's get learning.

Our learning outcome.

We'll be able to appreciate that there are key elements to any linear relationship.

The rate of change and the intercept point.

They're going to be keywords today, rate of change and intercept.

The rate of change is how quickly one variable changes with respect to another.

If the change is constant, there is a linear relationship between the variables.

Intercept, an intercept is the coordinate where a line or curve meets a given axis.

By the end of this lesson, you'll be very familiar with both of those terms. Two parts to today's lesson.

We're gonna begin with looking at the numerical features of linear relationships.

I'm going to start you with something with which you'll be very familiar.

The first five terms of the arithmetic sequence, 3n + 2.

I'd like you to find those and then I'd like you to describe the sequence in words.

Pause this video, work out the first five terms, and gimme a sentence to describe that sequence.

So you would've got the terms five, eight, 11, 14, 17, and then your sentence might have been something along the lines of the sequence starts on five and has a constant difference of positive three every time.

Lots of ways you might have phrased that, there was no escaping the starting point of five and the fact that it's increasing by three every time or a constant difference of positive three.

Those words are unique to that sequence and that sequence is unique to those words.

Here's the first five terms of a difference sequence, the sequence, 3n squared.

Those terms will be three, 12, 27, 48, and 75, but I can't call that an arithmetic sequence, a linear sequence.

What is it about that sequence that means it's not linear? It's the difference between the terms, it's not constant.

We have an increase of positive nine, Then positive 15, positive 21.

It's not a constant difference between the terms. What if we stop looking at arithmetic sequences and we look at linear relationships? Instead of relating the term number N to the term value, T, we relate an X coordinate to a Y coordinate in what we call a linear relationship.

For linear relationship Y = 3x + 2 we'd get those Y coordinates for those respective X coordinates.

Five, eight, 11, 14, 17.

That's the coordinates for the linear relationship Y = 3x + 2 If I compare that to what I showed you earlier, the linear sequence, 3n + 2.

what similarities do you see? Pause this video and say some similarities to the person next to you.

Lots of things that you might have pointed out.

Five, eight, 11, 14, 17, they're the same numbers.

It's the same common difference or constant difference, positive three.

You might have said if we plotted that linear relationship and we plotted that linear sequence, we plot the same coordinates, one, five, two, eight, three, 11, et cetera.

If you looked at it like this, it tells you something very useful about these relationships sequences.

We can call this a constant rate of change.

In the table on the left, you can see that for every change of positive one in X, there is a change of positive three in Y.

In the table on the right, the arithmetic sequence, for every change of positive one in N, there was a change of positive three in T, we call this a constant rate of change.

When plotted, we can see this rate of change.

If I plot the arithmetic sequence 3n + two, I get those coordinates at one, five, two, eight, three, 11.

That's my rate of change numerically in the table of values.

Then you see this reflected on the graph for every change of positive one in N, there is a change of positive three in T, that change is reflected between every coordinate we plotted.

If I look at the linear relationship, Y = 3x + 2, we've already observed we plot the same coordinates, the table reflected the same constant rate of change For every change of positive one in X, there's a change of positive three in Y and we can see that reflected on the graph.

For every change of positive one in X, there's a positive change of three in Y.

That is a constant rate of change.

The difference between when we plot arithmetic sequences and when we plot linear relationships is that when we're using an X input, X can be any value.

Integer values, non integer values, we're not limited to X being one, two, three, four, five so we could complete this table of values and plot slightly differently the linear relationship Y = 3x + 2.

We already know the outputs or the Y coordinates when X is one, two, three, four.

What about the rest of the values in that table? Pause this video and see if you can work them out.

You should have got that.

Now, you might have gone through and plugged negative one into the linear relationship three lots of negative one plus two gives me the Y output of negative one, or you might have spotted, well, it's two steps to get from five to eight, so one step must be 1.

5 and completed the table that way.

Whichever way you did it, we'd get the same values.

When plotted, our graph now looks like so.

We've plotted coordinates in three quadrants.

We've plotted negative X coordinates.

We could keep going and plot more and more and more X- Y coordinates, we could go on infinitely, in fact, and if we did so we'd form a straight line.

Because we form a straight line, we call it a linear relationship.

Our rate of change hasn't changed, however, just because we've added more X coordinates.

Rhe rate of change is still true that for every change positive one in X, there is a change of positive three in Y.

Let's check you've got that now, which of these is true for the linear relationship Y = 8x - 1.

You have a table of values to reflect that linear relationship.

Which of those three statements is true? Pause this video and tell the person next to you.

I hope you said A, for every change of positive one in X, there is a change of positive eight in Y, but was that the only answer? You can see that answer there.

You can't see that answer at all.

There's no change of negative one in Y, but we should have been able to see this one too.

For every change of positive two in X, that's a change of positive 16 in Y.

You can see that in the table now.

Practise time now, question one.

These are tables of values for linear relationships.

Fill in the missing values and complete the statements about the rate of change.

I'd like you to fill in the missing values in that table A and then complete the sentence for every change of positive one in X, there is a change of what in Y.

Once you've done that for part A, you can repeat for part B, pause this video and give this a go now.

Part C and D of question one are very similar.

Table of values with missing values, which I'd like you to complete, and then there's a gap in each of the sentences about the rate of change.

Can you pause this video, complete the tables, and complete those sentences.

Question two, use the graph to complete the table and write a sentence about the rate of change.

You need to read the coordinates, fill in a table of values, and then gimme a sentence a bit like the ones you just completed for question one about the rate of change you observe here.

Pause this video and do that now.

Part B for question two, very similar, but you'll notice a tiny difference.

I'd like you to complete those coordinates, find the missing y coordinates, and write me another sentence about the rate of change here.

Pause and try this now.

Feedback.

In table A you can see the difference between two and seven is five, so we should have been able to fill in the rest of those values.

The Y coordinates would read negative eight, negative three, two, seven, 12, 17, 22, and you observe from that table for every change of positive one in X, there is a change of positive five in Y.

Part B was slightly trickier because we had a negative value, negative four, and we had two steps to get from there to 10.

We should have observed that their steps of seven each time giving us Y coordinates of negative 18, negative 11, negative four, three, 10, 17, and 24.

A statement about the rate of change should have read, for every change of positive one in X, there was a change of positive seven in Y.

For part C, we had one, two, three, four steps to get from two to eight.

Halfway between those two would've been five and halfway between two and five would've been 3.

5, so these must have been steps of 1.

5.

We can say for every change of positive one in X, there is a change of positive 1.

5 in Y.

The rate of change doesn't always have to be a whole number or integer value.

For part D, something different again.

We should have noticed that this was a decreasing set of Y coordinates 22, 17, 12, seven, two, negative three, negative eight.

For every change of positive one in X, there was a change of negative five in Y.

Our rate of change was negative in Y.

For question two, using the graph to complete the table, we should have read the Y coordinates, negative seven, negative five, negative three, negative one, one, three, five, and a sentence about the rate of change should have read something like for every change of positive one in X, there was a change of positive two in Y.

You could of course have said for every change of positive two in X, there was a change of positive four in Y and a statement like that would also have been true.

For part B, slight difference now, these coordinates were a little trickier to read.

The Y coordinates were 1.

4, 1.

1, 0.

8, 0.

5, 0.

2, and negative 0.

1.

And our rates of change were slightly different.

Again, for every change of positive point in X, that's a change of negative 0.

3 in Y, a rate of change of Y, which is both negative and non integer.

Onto the second half of the lesson, let's look at the graphical features of linear relationships.

Again, I'd like to start you with something very familiar, arithmetic sequences 2n + 5.

If I said generate the first five terms, you'd gimme a table of values completed like so and you'd get the first five terms of the sequence, seven, nine, 11, 13, 15.

You could then say this sequence starts at seven and changes by positive two.

That description is unique to this sequence.

What if I changed the words around a little bit? What if I said start at two and add positive seven each time? How does that differ? We'd get the sequence two, nine, 16, 23, 30, but crucially we would only ever get that sequence.

No other sequence can come from that worded rule.

A given starting point and a given constant difference generates a unique arithmetic sequence.

So if such a thing is true for arithmetic sequences, is such a thing true for linear relationships? The linear relationships Y = 2x + 5.

I could generate a table of coordinates like so.

So where does that start? The linear relationship, Y = 2x + 5 with that table of coordinate values, where does it start? Pause this video and make a suggestion to the person next to you.

Well, it looks like it starts at seven, but we can't say that this starts at seven because we could have an X input of x equals zero.

If I add that to the table, my table of values, the Y coordinates now read five, seven, nine, et cetera, so do I say it starts at five? I can't say that because we could have x inputs of X equals negative one, x equals negative two, x equals negative three, x equals negative four.

We could have any number of X inputs.

If we plotted the linear relationship, Y = 2x + 5 with all these coordinates, we'd get a line like that.

Where does that line start? We can't say it starts anywhere.

Once plotted, we can see it has no start and no end.

In fact, it goes on infinitely in both directions.

However, the line does have some unique features which help us to define it.

One of those features is that moment there, which we call the y intercept.

The Y intercept is where our line meets the y axis.

In the case of this line, it's at the coordinate zero five.

That's our y intercept for the linear relationship, Y = 2x + 5.

We could say something else about it as well.

The line Y = 2x + 5 intercepts the Y axis at zero, five and has a constant rate of change of positive two.

When I describe the linear relationship, Y = 2x + 5 in this way, I will get that line and only that line.

A Y interceptive negative five, and a constant rate of change of positive two.

Let's check you've got that now, what is the Y intercept of this line? Is it six, (0, 6) or (-2, 0)? Pause and tell the person next to you.

I hope you said (0, 6).

The line intercepts the Y axis at the coordinate (0, 6).

If you said answer A, six, the Y value when we intercept the axis is six, but we don't describe the Y intercept as so, we need to use the full coordinate.

So please describe your Y intercepts as (0, 6).

What's the Y intercept of this line? Is it (0, -3), (0, 3), or (-3, 0)? Pause, tell the person next to you.

I hope you said A, (0, -3).

We would say the line intercepts the Y axis at the coordinate (0, -3).

No graph this time, just a table of values, but what is the y intercept of the line plotted from this table of values? Three options there.

Pause and take your pick.

I hope you said option C, the coordinate (0, 5).

The line would intercept the Y axis when the X coordinate is zero.

That's why the Y intercept is (0, 5).

Practise time now, question one.

Izzy has completed a table of values and plotted coordinates for linear relationship Y = 4x - 3.

You can see her table of values, you can see the coordinates.

You can see the line that she's drawn.

She says the line Y = 4x - 3 starts at one and has a rate of change of positive four each time.

Could you write a sentence to put in Izzy's book explaining what she needs to change? Pause and do that now.

Question two, a matching exercise.

I'd like you to match the graphs to their unique descriptions.

For example, which graph of those three matches the description intercepts the Y axis at (0, 5) and has a rate of change negative two.

See if you can match those three statements to those three graphs.

Pause and do that now.

Question three.

I'm going to give you two graphs and ask you to write unique descriptions for them.

Be sure to include the words y intercept and rate of change in your descriptions.

Please pause and get writing.

Question one.

This was Izzy's graph and statement about the line Y = 4x - 3.

I hope you said something along the lines of the line Y = 4x - 3 has no start and no end.

You need to continue the line in both directions, and plot zero and negative X coordinates.

A table of values could have had a few more values added and then the graph would've reflected the fact that line goes on infinitely in both directions.

Question two, we were matching descriptions to these three graphs.

The first graph has an intercept, or intercepts the Y axis at (0, -5) and a rate of change of positive two.

The second graph intercepts the Y axis at (0, 5) and a rate of change of negative two.

The third graph intercepts the Y axis at (0, 2) and has a rate of change of positive five.

For question three, we were writing descriptions for these two graphs and I asked you to include the keywords Y intercept and rate of change.

So for graph A, I hope you wrote something along the lines of intercept the Y axis at (0, 4) and has a rate of change of negative one.

And for graph B, I hope you wrote something along the lines of intercepts the Y axis at (0, 1.

5), and has a rate of change positive 0.

5.

We've reached the end of the lesson now.

We've learned to appreciate that there are key elements to any linear relationship.

The key elements being the rate of change and the intercept point with the Y axis.

Those descriptions are unique to any given linear relationship.

I hope you've enjoyed this lesson and I hope I see you very soon for some more mathematics.

Goodbye for now.