Lesson video

Hello, Mr. Robson here.

Great choice to join me for maths again today.

We're plotting coordinates generated from a rule.

Let's get learning.

Our learning outcome for today is how we're going to represent, algebraically and graphically, a set of coordinates constructed according to a mathematical rule.

Keywords that I'll be using throughout this lesson.

Arithmetic sequence, also known as a linear sequence.

An arithmetic, or linear sequence, is a sequence with a difference between successive terms is a constant.

For example, 6, 11, 16, 21 has a constant difference of +5, therefore, it's arithmetic, or linear.

A non-example would be 1, 2, 4, 8, which does not have a constant difference.

Therefore, it's not arithmetic.

We'd call that non-linear.

Two parts to today's lesson.

First, we'll start by plotting arithmetic sequences.

Secondly, we'll be plotting coordinates that follow a rule.

Let's get going with some arithmetic sequences.

Which arithmetic sequence is bigger? 2n or n + 2? Sofia says, "I think 2n, because multiplication is more powerful than addition." Aisha says, "I think both can be the bigger one." Hmm.

Who do you agree with? Pause this video and tell the person next to you.

Aisha is correct.

2n could be the bigger one, and n + 2 could be the bigger one.

It depends on the value of n.

What do we mean by that? It's easiest to show you if we plot the first five terms of the arithmetic sequences.

We'll need a table of values for 2n.

We'll need a table of values for n + 2.

To find the first five terms of the sequence 2n when n = 1, n is 2 lots of 1, that's 2.

When n = 2, 2 lots of 2, that's 4.

For the sequence n + 2 when n = 1, 1 + 2 makes 3.

We can generate the first five terms of both of those sequences and we'll get these values, 2, 4, 6, 8, 10 for the sequence 2n.

3, 4, 5, 6, 7 for the sequence n + 2.

If I lay those sequences next to each other, you can see what Aisha meant.

2 and 3 are the first terms of each sequence.

For the first term, n + 2 is bigger.

For the second term, they're equal.

For the third term, and any term beyond it, 2n will be the bigger of the two sequences.

So Sofia had a point, but it only kicked in on the third term and beyond.

If we plot these sequences graphically, the relationship will be far easier to see.

We can plot the sequence 2n by taking a grid and labelling it with term number on horizontal axis and term value on the vertical axis.

We'll plot a term number of 1 and a term value of 2 there.

A term number of 2, a term value of 4 there.

And so on, and so forth, for the rest of those terms. Once plotted, we can see why arithmetic sequences are also known as linear sequences.

You see that those coordinates are on a straight line.

If we plot n + 2, I'll do this in a different colour, so you can compare the two sequences, we plot the coordinate (1, 3), (2, 4), (3, 5), like so.

If I label those points 2n and n + 2, you can see what Aisha meant.

Initially, the sequence n + 2 is greater.

At the second term, they are equal.

Then for all terms beyond 2n is greater.

Can you see how this visual representation of those two sequences helped us to answer that problem? The graph of the sequence, 2n, Jun spots the linear pattern, draws a line, and says, "The 2.

5th term is 5." That makes sense.

I look down to the term number, that reading will be 2.

5.

I'll look across the term value.

The value is 5.

So, is that right? Or, is there something wrong with this statement? Pause this video and have a conversation with the person next to you.

5 is not in this sequence.

The sequence goes 2, 4, 6.

It bypasses 5.

There is no 2.

5th term in this sequence.

So for that reason, we don't draw a line when we're plotting arithmetic sequences.

For this sequence, n, the term number, has to be a positive integer.

So, you leave your arithmetic sequence plotted like so.

We don't draw a line.

Let's just check you've got that.

What are the missing term values for the sequence 3n + 1? I'll give you the first term value.

It's 4.

But what about the second, third, fourth, and fifth term values for that sequence, 3n + 1? Would they be 5, 6, 7, 8? 6, 9, 12, 15? Or 7, 10, 13, 16? Pause this video and have a think about that problem.

It's C.

7, 10, 13, 16.

3n + 1 would have those term values, 4, 7, 10, 13, 16.

A 3n sequence has to have a constant difference of +3, so that rules out option A.

And the reason it's option C instead of option B is because we can test that fifth term value.

If we're saying the fifth term value is 16, then when we substitute n = 5 into the expression 3n + 1, we should get 16, and that's only correct for option C.

Another check.

What term is plotted incorrectly for the sequence 3n + 1? Look at the graph.

Which one is incorrect? Is it the fifth term, the third term, or the first term? Pause this video.

Tell the person next to you.

It's the fifth term, and I've highlighted it for you.

And I hope you noticed it's not in line with the others.

In an arithmetic linear sequence, the points should be in a straight line.

True or false? Once plotted, we draw a line through the coordinates that you see on this graph.

Once you've answered whether it's true or false, choose one of those two statements to justify your answer.

Pause this video.

Have a think about that now.

You should have said false.

You should have justified it with we only know specific terms of a sequence.

We wouldn't have that line.

We'll just have those specific term values plotted.

If I said complete the term values for these sequences, 8 + 3n and 8 - 3n, you would substitute n, n = 1 into 8 + 3, 8 + 3 lots of 1, and then 8 + 3 lots of 2, 8 +3 lots of 3.

You'd get the term values 11, 14, 17, 20 and 23.

We did it for 8 - 3n, that's 8 - 3 lots of 1, and then 8 - 3 lots of 2, 8 - 3 lots of 3, and you'd get these values, 5, 2, -1, -4.

-7 What's different about the sequence 8 -3n versus the sequence 8 + 3n? Pause this video and tell the person next to you.

The common difference is -3.

In 8 + 3n, a common difference of +3.

In 8 - 3n, we've got a negative common difference.

That makes it a decreasing sequence.

Because it's a decreasing sequence, we end up with negative terms, -1, -4, -7.

This will affect the way we graphically represent these sequences.

8 + 3n will look like that.

It's an increasing sequence with a constant difference of +3 You can see that in those points.

An increase of 1 in the value of n.

An increase of 3 in the term value.

An increase of 1 in n, an increase of 3 in the term value.

However, the sequence 8 - 3n, we'd need to plot like so.

Because it has negative term values, we need two quadrants to be able to plot this sequence.

You can see there from that visual, it's a decreasing sequence with a constant difference of -3.

Quick check that you've got that.

Which of these could be the graphical representation of the sequence 13 - 5n? Three options to choose from.

Which one of them do you think represents 13 - 5n? Pause this video and have a think about that.

It's option A.

Why is that? Because option A and option B are both decreasing sequences, and they could be decreasing by 5, or changing by -5 each time.

<v ->5n will be a decreasing sequence,</v> but the key thing about 13 - 5n is the first term is positive.

The second term is positive as well.

Option B had no positive terms, which rules that one out.

Option C was ruled out because that's an increasing sequence, 13 - 5n.

The n coefficient of -5 means it must be a decreasing sequence.

So, option A it was.

Practise time now.

For question one, I'd like you to find and plot the first five terms of these arithmetic sequences.

Fill in that table of values for 4n - 10 and plot that sequence.

Fill in the table of values for 10 - n and plot that sequence as well.

Pause this video and give that a go.

Question two, matching exercise.

I'd like to match the sequences to their respective graph.

Three expressions for sequences.

Three graphs.

Match them up.

Pause this video and have a go at that now.

Question 1.

For 4n - 10, 4 lots of 1 - 10, 4 lots of 2 - 10, 4 lots of 3 - 10 should have given us the term values -6, -2, 2, 6, and 10.

In those term values, you can see a constant difference of +4.

When you plot those coordinates, you can see that constant difference of +4 again.

We start with negative terms, and we have a constant difference of +4.

For the sequence 10 - n, 10 -1, 10 - 2, 10 - 3, you should have got the term values 9, 8, 7, 6, and 5.

When you come to plot those, it would look like that.

You see a decreasing sequence with a constant difference of -1.

Matching the sequences to their respective graphs.

The first one was 3n + 10 because it's an increasing sequence with all positive terms. The second graph was 10 - 3n.

It's a decreasing sequence which starts with positive terms. And the third graph, 3n - 10 as an increasing sequence, which started with negative terms. Moving on now from looking at just arithmetic sequences, we'll start in the second half of the lesson to plot coordinates that follow a rule.

The arithmetic sequence 2n + 1.

We generate the term numbers by using the rule, multiply n by 2 and add one.

We get those term numbers.

We can also plot Cartesian coordinates related by such rules.

For example, if I said plot Y = 2x + 1, you'd need a table of values.

And you need to generate your coordinates with the rule, multiply x by 2 and add 1, and that would give you your Y values.

You would complete the table like so.

So we've got the arithmetic sequence, 2n + 1.

We've got the rule, y = 2x + 1.

Do you notice any similarities? Pause this video.

Tell the person next to you.

They're the exact same numbers.

For the arithmetic sequence 2n + 1, we've got term values of 3, 5, 7, 9, 11.

For the rule, y = 2x + 1, we've got y values of 3, 5, 7, 9, 11.

They both generated an arithmetic or linear sequence, and they've both got a common difference of +2.

We can represent both of these things graphically.

That's how we represent the arithmetic sequence 2n + 1.

So is that how we represent the rule y = 2x + 1? And are they the exact same thing? What do you think? What's your mathematical intuition tell you? Pause this video, have a conversation with the person next to you.

Are they the exact same thing? They're not.

Because n, the term number, takes positive integer values when plotting the sequence, whereas x can be any value.

What do I mean by that? Well, if I take those first five values that we calculated for the rule y = 2x + 1, and I put them in this table.

and give myself lots of space to add some extra x values, x can take on any value.

So I could say, for example, x is 1.

5.

2 lots of 1.

5 + 1 would give a y output of 4, and I could plot that coordinate.

What other x values could I have? I could have 1/2, x might be a fraction.

2 lots of 1/2 + 1, that'll give us a y output of 2, so I plot that coordinate, I could do x = 2.

1, which will give me a y value of 5.

2, which would plot there.

x = 2.

2, a y value of 5.

4, which would plot there.

I could go into a second decimal place, x = 3.

61.

Give a y output of 8.

22, which I could plot there.

I could even say x = 0.

Well, that's nice and easy to do.

2 lots of 0 + 1 is 1, and I plot (0,1).

I could keep going with this infinitely.

Because of that, when we're graphing the rule linking x and y on a Cartesian coordinate grid, we do draw a line.

And that line represents the infinite number of x inputs and their respective y outputs.

That's the crucial difference between when we're plotting a rule relating x and y and when we're plotting an arithmetic sequence.

In this case, we do draw a line.

True or false, when plotting the rule y = 3x + 2 we only plot positive integers.

Is that true, or is it false? And can you justify your answer with either of these statements? It's the same as plotting an arithmetic sequence, and has to be positive integer, so x has to be 2.

Or, when plotting rules that link x and y coordinates, there's infinite inputs and outputs, not just positive integers.

Which is it? True or false? And which of those statements justifies your answer? Pause this video and have a think about that.

This statement's false.

We don't just plot positive integers.

When we're plotting rules that link x and y coordinates, there's infinite inputs and outputs, not just positive integer values.

What happens when we fill in this table of values and graph the rule 3x -2? The difference with this table of values is we've got negative x inputs because the X coordinate can be negative.

We can have negative X values input into our rule and we can plot those.

My recommendation though, is that you'll find it easier to calculate the 0 and positive values first.

If we substitute x = 0 into there, 3 lots of 0 - 2, well, that's -2.

And then I'll try the next easiest value, which, in my opinion, is 1.

If x = 1, 3 lots of 1 -2 is 1.

I'm going to substitute in x = 0, x = 1, x = 2, x = 3, and I'll get those values out.

Do you notice anything? Have you spotted a pattern? I hope so.

Like an arithmetic or linear sequence, we've got a common difference between -2 and 1.

A common difference of +3 between 1 and 4.

Common difference of +3 between 4 and 7.

A common difference of +3.

If there's a common difference of +3, then I don't even need to substitute in -1, -2, -3 into my rule.

I can just consider that difference of 3.

That must be -5, that must be -8, that must be -11.

But rather than leave this to chance, I'm just gonna check that's right.

If I did substitute -3 into 3x - 2, that'd be 3 lots of -3 -2 give me -11, it is right.

But my top tip, work with the easiest values first when you are plotting these rules.

So when we plot the line 3x -2, we'll have those coordinates like so.

And you'll notice that unlike in our arithmetic sequence, we're now plotting in more than two quadrants.

And x has infinite inputs.

So we draw a line through all the points, and crucially, beyond the last points, to reflect that this line goes on infinitely in both directions.

Quick check now.

I'd like to complete the table of values to plot the rule y = 5 - x.

Some of the values are already done for you.

Pause this video.

Give this a go.

We should have got those values.

If you start with the easiest x values to substitute in, x = 0, 5 -0 is 5.

5 -1 is 4, 5 -2 is 3.

So you get those y values, 5, 4, 3, 2, 1, 0, -1.

And going the other direction, you'd find 6, 7, and 8 in your table.

When we plot those, the coordinates will look like so.

And we join through with a straight line, which we draw to the edges of the grid.

Did our table of values have to be in consecutive integer order for us to be able to plot the rule y = 5 - x? No.

You can have any inputs you like.

I've selected these four at random.

x = 0.

5, 5 - 0.

5, that's 4.

5.

That coordinate, I'll plot there.

What if x is -0.

5? 5 - -0.

5, 5.

5.

I can plot that coordinate there.

What if x is 3 1/2? A mixed number? Well, the y output would be 1 1/2, and I could plot that coordinate there.

x = 1.

75.

5 - 1.

75 is 3.

25, and I can plot that coordinate there.

And you'll see this same linear pattern emerge.

Any x value input into the rule will generate the same linear pattern.

Practise time now.

Question one, six rules there.

I'd like you to match the verbal rule to the simplified algebraic rule.

Pause this video and match those up.

Question two, I'd like to complete the table of values and plot the rules y = 2x - 5 and y = 1 - x.

Few values are done for you to help you fill in those tables.

Once you've filled the tables, plot those points on the grid.

And you know what to do once you've plotted the coordinates.

Pause this video, give that a go.

For question three, Jacob plots the rule y = 5 - 3x.

He has that table of values.

He draws that line.

I'd like you to mark his work and give him some feedback.

Pause this video and do that now.

Feedback for question one.

Matching these algebraic rules to the verbal rules y = x + 3.

That means add 3 to the x coordinate.

y = 3x + 5.

That means the x coordinate is being multiplied by 3, and then you are adding 5.

y = 5x - 3.

That means the x coordinate is multiplied by 5, and then you subtract 3.

y = 5x + 3 That's the x coordinate multiplied by 5.

Add 3.

y = 3 - 5x.

The x coordinate is multiplied by 5 and subtracted from 3, leaving us with y = 3 - x to be the x coordinate subtracted from 3.

Question two, table of values for 2x -5 should have looked like so.

Your table should read -9, -7, -5, -3, -1, 1, 3.

Your table of values for y = 1 - x should read 3, 2, 1, 0, -1, -2, -3.

You'd then get those two lines plotted as coordinates.

And you'll draw straight through those coordinates.

Question three, marking Jacob's work and giving him some feedback.

I hope you notice that one of the points was out of line.

It should have made a straight line and it didn't.

So, y = 5 - 3x is a linear rule.

This should make a straight line.

That coordinate, (2,1), doesn't look like it fits.

So instead, we substitute in the x value of 2 again, 5 -3 lots of 2.

That's -1, not +1.

So we should have plot the coordinate to -1 instead.

Also, I hope you noticed, Jacob didn't continue his lines to the very edge of the grid.

The rule y = 5 - 3x will go on infinitely in both directions, so he should have continued his lines to the edge of the grid.

If he'd done it correctly, it would have looked like that.

That's the end of today's lesson.

In summary, we can represent mathematical rules graphically and algebraically.

We do this graphically by plotting coordinates that fit our rule.

For example, this graph shows the algebraic relationship y = 3x - 2 I hope you've enjoyed this lesson.

I hope to see you again soon for more mathematics.