Lesson video

Hello, Mr. Robson here.

Great choice to join me for maths again, representing sequences graphically.

This is going to be awesome.

So, our learning outcome for today is that we'll be able to represent arithmetic sequences graphically and appreciate the common structure.

Key words that we're gonna need, we're gonna be using the words arithmetic sequence and a linear sequence.

Arithmetic sequence is also known as linear sequences.

They're sequences whereby the difference between successive terms is a constant.

For example, six, 11, 16, 21 has a constant difference of positive five.

Therefore, it's arithmetic or linear, as we might also call it.

A non-example would be one, two, four, eight.

It does not have a constant difference; therefore, it's not arithmetic.

It's not linear.

The difference between arithmetic sequences, and non-arithmetic sequences will be important today.

Two cycles in today's learning.

The first will be plotting arithmetic sequences, the second will be looking at the features of them once we've represented them graphically.

Let's start by plotting them.

Multiple representations in mathematics can help us to solve problems, understand the structure of some mathematics, or expose connections to other areas.

It's an important practise in the learning of mathematics to use multiple representations, not just the numerical ones in the terms of sequences.

So, we can look at the sequence 2n, and we know it goes two, four, six, eight, 10, and we've seen tables like this used to generate the terms of a sequence before.

What we'll do today is represent that graphically.

One representation of an arithmetic sequence such as 2n could be to turn the term numbers and values into coordinates, and plot them on a graph.

We'll start with a table representing a few terms in our sequence and then we'll plot them in the form of term number, term value.

You've plotted coordinates before in the form of x-coordinate, y-coordinate.

In this case, we'll be plotting the term number versus the term value.

So, our grid will look something like this with the horizontal axis labelled as term number, and the vertical axis labelled as term value.

We'd start with the sequence 2n by plotting one, two.

That is a term number of 1 and a term value of two.

We'll then plot 2, 4, term number of 2, value of 4, and so on and so forth.

Once you've plotted those points, it looks like that.

That is the arithmetic sequence 2n represented graphically.

Notice the points align.

I've already said the word linear, and the fact that arithmetic sequences are also known as linear sequences.

Linear means that they align.

Can you see them in a straight line? Maybe tilt your head a little and you'll notice it.

Arithmetic sequences are also known as linear sequences.

They increase or decrease in a straight line.

Andeep spots this linear pattern, draws a line, and says, the 1.

5th term is 3.

The 1.

5th term is 3, Andeep.

What have you done? Oh, I see.

You've read from the term number x is 1.

5, gone up to your straight line and across to 3, the 1.

5th term number is 3.

I can see what you've read there.

But what's wrong with his statement? Pause this video, tell the person next to you what's wrong with Andeep's statement.

The 1.

5th term is 3.

Did you say, well, we wouldn't draw a line when we're plotting sequences? Why wouldn't we draw a line? Because there's nothing between 2 and 4 in our sequence.

The sequence 2n starts at 2 and the next term is 4.

3 is not a term, so we wouldn't draw a line connecting those coordinates.

Also, there's no 1.

5th term.

Term numbers have to be whole numbers.

We've got the first term, the second term, the third term.

There's no 1.

5th term number.

So, we don't plot a line, because we can't take readings like this when we plot arithmetic sequences.

Connecting what you see numerically to what you see graphically will be important today.

What do you notice about the sequence numerically and the sequence graphically? What's the connection, two, four, six, eight, 10, and what you're seeing on those points once they're plotted.

Pause this video, tell the person next to you.

Did you talk about steps that the sequence is going up in? As the term number increases by one, the term value increases by two.

You can see that in our numerical sequence from two to four to six to eight to 10.

You can see it on the graph too.

So, term number increases by one, the values increase by two.

The same thing that happens in the table is happening on our graph there.

Just to check that you've got that now, a true false.

You draw a line once you've plotted a sequence.

Is that statement true or false? I'd like you to justify your answer with one of these two sentences.

It allows us to read any point anywhere or we only plot the points when graphing a sequence? Is it true? Is it false? And select a sentence to justify your answer.

Pause this video and do that.

The answer is it's false.

We only plot the points when graphing a sequence.

So, it won't look like that with a line through it; it will just look like that when we plot sequences.

This graph represents which sequence? Okay, I see term numbers, term values, I see plotted points.

Do those points represent the sequence n, 3n, or 6n? Pause this video, tell the person next to you.

They represent the points of the sequence 3n.

If you think about the term numbers and term values, the first term is 3, the second term is 6, the third term is 9, those are our coordinates one, three, two, six, three, nine.

They're the sequence 3n.

The visual representation of 3n demonstrates that.

Does it demonstrate the fact that the sequence is linear, the sequence increases by 3 each time, or that the sequence starts at the origin coordinate 0,0? Pause this video.

Which of those is true? We should have said the top one.

It shows that the sequence is linear.

I can see those points aligning again a little tilt of my head, I see that line.

The coordinates align is a nice way to phrase that.

The second one is true.

The sequence increases by three each time.

You can see that in the steps between the points as a term number goes up by one.

Our term value goes up by three.

C was not true.

There is no zero-term number.

Zero is not a term value in this sequence.

Next, this is the sequence 5 plus 2n, represented graphically.

The sequence 5 plus 2n, we could generate the first five terms of that sequence in a table.

The sequence goes seven, nine, 11, 13, 15.

And when we plot them, the sequence looks like that.

What will be different if we plot the sequence 5 minus 2n? Give me some suggestions.

Pause this video and shout some things back at me.

If you said something along the lines of, that's a decreasing sequence, so we'll see the points going in a different direction, super! 5 minus 2n, the sequence decreases.

And not only are we going to see those points decreasing as the term number increases, we're going to see them get into negative values.

What was that going to look like when we plot it? Start with a table of values.

I'm going to substitute in n equals 1, 5 minus 2 lots of 1, that's 3.

I'm going to substitute in n equals 2, 5 minus 2 lots of 2, that's 1.

And I get those term values.

The coefficient of n is negative 2, hence I see that sequence changing by negative two each time.

What's that going to look like once we've plotted it? It'll look like that.

Those negative terms are now in the fourth quadrant.

True or false? When we plot sequences, we'll only need the first quadrant.

Is that true? Is it false? I'd like you to justify your answer with one of these statements.

Term numbers are always positive.

Or we need the fourth quadrant too because term values can be negative.

Pause this video, answer true or false and justify that answer.

We should have said false.

As an example, we've just seen with five minus two n, we needed that fourth quadrant too because we have negative terms in our sequence.

So, we can't graph in just the positive quadrant.

We need our graphs to look like this if we're gonna have any negative terms. Practise time now.

Question 1.

I'd like you to find and plot the first 5 terms of these arithmetic sequences.

You're gonna need to populate that table of term values for 3n- 9 and plot on an axis like that.

And then I'd like you to do the same for the sequence 3 minus n.

Pause this video and give that a go.

Question two is a matching exercise.

Match these sequences to their graph.

I've omitted a lot of things from those graphs.

You haven't got scales, labels on the axes, but does that mean we can't recognise which of those sequences matches 4n plus 10? We can.

Have a think about that.

Match those three pairs up.

Feedback time.

Your table of values for the sequence 3n-9 should have looked like this.

We should have started with the first term of negative 6 and then increasing by 3 each time to negative three, zero, three, six.

How does that look graphically? Like that, starting at one, negative six, two, negative three, three, zero, et cetera, et cetera.

For the second one, three minus n, three take away one, three take away two, three take away three, you should have generated these term values, two, one, zero, negative one, negative two.

How would that look when we plot it graphically? Like that.

Matching sequences to their graph, that one is 4n plus 10.

How do we know? Well, it's an increasing sequence with all positive term values.

No matter what n value you plug into there.

You're not gonna get a negative term out.

The second one is 4n minus 10.

How did we know? Because the coefficient of n is positive 4.

It's an increasing sequence, whereas that third graph shows a decreasing sequence.

So, the middle one must have been 4n minus 10.

Finally, 10 minus 4n, because that was the only decreasing sequence.

What's next? Features of sequences represented graphically.

This is where we're gonna take a really deep look at the structure of an arithmetic sequence.

Let's start with this.

What is the same, what is different about these two sequences? There's the first five terms of the sequence 2n, there's the first five terms of the sequence 2n plus 3.

Give me one thing that is the same in both those sequences and one thing that's different about them.

Pause this video, make some suggestions to the person next to you or back to say them back to me at the screen.

Right, same, both are arithmetic sequences with a common difference of positive 2.

2, 4, 6, 8, 10, common difference of positive 2.

5, 7, 9, 11, 13, common difference of positive 2.

Any 2n sequence is going to have a common difference of positive 2.

2n plus 100, 2n minus 100, they'll have a common difference of positive 2.

What was different between the sequence 2n and 2n plus 3? We could have said the terms of 2n plus 3 are a translation of positive 3 from the sequence 2n.

What do we mean by that? Well, if we compare the first terms, 2 and 5, compare the second terms, 4 and 7, compare the third terms, six and nine.

In the sequence 2n plus three, we're always three above the sequence 2n.

So, we call this sequence 2n plus three a translation positive three from the sequence 2n.

What does that look like graphically? Those similarities and differences, what do we see once we've plotted those? We're just gonna need three term numbers to show this.

There's the first three points of the sequence 2n.

Let's do the same for 2n plus three.

I'll change the colour up so we can see the difference between those two.

There we go.

2n versus 2n plus 3.

Can you see the constant difference of positive 2 in both sequences? Can you see the translation of positive 3 between the two sequences? We can.

The same, both are arithmetic sequences with a common difference of positive 2.

How does that reflect in this graph? I see that.

As for that difference, the terms of 2n plus 3 are a translation plus 3 from the sequence 2n.

That translation we see in each of these points on the graph.

True or false to check that you've got that? Could these two sequences be 5n plus three, and 5n plus eight? For that graph is that true or is that false? I'd like to justify your answer.

You can justify it with one of these two sentences.

One is above the other and eight is above three.

Or the sequences do not have the same steps of increase, so they can't both be 5n sequences.

Declare that statement true or false and choose one of those two justifications for your answer.

We should have said false, followed by the statement, the sequences do not have the same steps of increase, so they can't both be 5n sequences.

I can see the top sequence increasing far quicker than that bottom sequence.

And 5n plus anything is going to have a common difference of five.

Next, if you know the jade sequence is 5n and the translation is plus four, there's that translation.

What is the black sequence? Is it 4n? Is it 5n plus four? Or is it not possible to know? The sequence is 5n plus 4.

The terms of 5n plus 4 are a translation of plus 4 from 5n.

This is reflected in the graphical representation.

Notice when I put those arrows on you could see that every term was positive four up from the j'd terms. So, we've taken the sequence 5n and translated it by positive four, giving us the sequence 5n plus four.

Right, next our Oak students are discussing these two arithmetic sequences, the sequence 4n with those term values and the sequence 4n plus 1 with those term values.

I've got Lucas here saying the fifth term of 4n is 20, so the 10th term must be double that.

The 10th term must be 40.

Okay, logical.

Double 5 is 10, double 20 is 40.

It's logical.

Sophia joins and talks about the sequence 4n plus 1.

The fifth term of 4n plus 1 is 21, so the tenth term must be double that.

The tenth term must be 42.

Again, logical.

Double 5 to get to 10, double 21 to get to 42.

That is logical.

Do you agree with them? You might want to have a little play with those sequences and see, is Lucas, right? Is Sophia, right? Are we right to do this? Pause this video, have a little play with this mathematics and see what you think.

Right, in Lucas's case, 4, 8, 12, 16, 20, 24, 28, 32, 36, 40.

Did you just continue that sequence on in the constant steps of 4 and get to 40 and say, yeah Lucas, you're right, that is the 10th term, the 10th term is 40, it works Lucas.

But in the case of Sophia, did you notice that sequence? It didn't.

If we continued that sequence to the 10th term, we get to 41, whereas Sophia has claimed, I can just double the 5th term to get the tenth term.

Double 21 is 42.

So, let's go back a step.

It works for Lucas's.

Double the fifth term to get the tenth term.

That works for the sequence 4n, but it doesn't work for the sequence 4n plus 1.

I wonder why it doesn't work for the sequence 4n plus 1, but it did work for the sequence 4n.

If it's not double the fifth term in this case, we want to explore why.

We'll explore why by understanding the structure of what's going on in those sequences.

Graphical representations will help us to see this structure.

If we graph the sequence 4n, we can plot the points, it looks like that.

The structure of this, almost the building blocks of this maths I could represent with these blocks.

There's a block of 4 next to the first term in the sequence 4n.

Two blocks of 4 representing the second number in the sequence 4n.

There's two fours, and three fours, and four fours, and five fours.

That's the structure of the sequence 4n.

The fifth term of 4n was 5 lots of 4.

So, what is the tenth term? It will be 10 lots of 4.

And you see in those two towers there, it's exactly double the height.

So, in the case of the sequence 4n, double the 5th term made the 10th term because double 5 lots of 4 is 10 lots of 4.

So, what's different when we try to represent 4n plus 1? It looks so similar, but there's a subtle difference.

The 1.

That's the first term.

I've got 1 lot of 4 plus a 1.

Then I've got 2 lots of 4 plus a 1.

3 lots of 4 plus a 1.

4 lots of 4 plus a 1.

5 lots of 4 plus a 1.

So, there we have the structure of the sequence 4n plus 1.

As we move across the term numbers, what is changing and what never changes? Pause this video, throw some suggestions at me.

Visually, it's obvious.

What's changing? The 4s.

What's not changing? That 1.

That brick at the bottom.

The 1.

Why? Because it's an nth term expression.

4n plus 1.

4n is a variable term and 1's a constant.

1 is a constant.

I can see that in that visual representation.

The 1 remains constant, whereas for n, the variable terms, that's why the number of 4s is changing.

But the 1 does not.

So when we build the fifth term of 4n plus 1, that's our 21.

So, when we build a tenth term, double 21 to make 42, it's not going to work like that because it was only the 4s that doubled.

The 1 did not double.

Can you see why it works for 4n, but it didn't work for 4n plus 1? So, check you've got that now.

Which arithmetic sequence is represented by this structure? What structure? This structure.

First term, second term, third term, fourth term.

Very pretty, but which arithmetic sequence is it? Is it the sequence 2n? Is it 2n plus three? Is it 3n plus two? Pause this video, make a suggestion to the person next to you.

It's 2n plus 3.

Why? Because there's a constant of 3, what's varying the number of 2s.

2n is the variable term, we're going to have a varying number of 2s in the sequence 2n plus 3.

Next, which arithmetic sequence is represented by this structure? Oh, it looks so similar! But there's a crucial difference.

Have you spotted it? Is this going to be the arithmetic sequence 2n, 2n plus 3, or 3n plus 2? Pause this video, make a suggestion.

We should have said 3n plus 2.

Why? Because the 2 is constant.

What's varying the 3s? 3n, the variable term, is going to vary the number of 3s we have.

That's why that structure represents the sequence 3n plus 2.

Right, next, what's different about the sequence n squared and the sequence 2n? Shouldn't they be the same thing? There's an n, there's a 2, they're not.

The sequence 2n, 2, 4, 6, 8, 10 is very different to the sequence n to the power of 2 or n squared, which goes 1, 4, 9, 16, 25.

What is different? Or what is the difference between those two sequences? Pause this video, make some suggestions.

So, we should have spotted 2n, a constant difference of positive 2 each time.

Whereas n squared behaves very differently.

We've got a difference of 3, then 5, then 7, then 9.

It's not a constant difference.

This is why 2n we call an arithmetic sequence or linear, it's going up in a straight line, whereas the sequence n squared is not.

It's going up by x to three, then an extra five, then an extra seven, then an extra nine, we call that non-linear, it's not increasing in a straight line.

Does that mean we can't plot it? No, of course we can plot it.

We're just gonna take the first term of one and plot one, one, and then plot two, four, and then plot three, nine.

What's it going to look like when we plot it? When we plot 2n, we see that representation.

When we plot n squared, it's going to look like this.

I've kept the scale the same on both of those axes, just so you can see the difference between 2n, and n squared.

How do the two graphical representations look different to one another? Give me some suggestions.

I hope you said something along the lines of, the sequence 2n is arithmetic, it's a linear sequence, it's got a constant common difference of positive two, whereas n squared is clearly non-linear.

Those points are not aligned, they're not in a straight line.

There's an ever-increasing difference between the terms, it's as if they're steepening.

As we move up the sequence, that difference gets greater and greater, and the graph looks steeper and steeper.

Interesting.

So, check you've got that.

True or false? The points of every sequence we plot will be in a straight line.

True or false? Can you justify your answer with either the sentence I plotted 3n-9, 3-n, 5-2n, 5-2n, and they were all in a straight line, so I think all sequences do that.

Or, the points of non-linear sequences such as n2 and n3 will not generate a straight line.

True or false? And one of those sentences to justify, please.

We should have gone with false, and we should have said the points of non-linear sequences such as n squared as we just saw, and n cubed if we were to plot that will not generate a straight line.

Practise time now, your final practise of the day.

Question one, match these arithmetic sequences to the correct plotting.

I've got the sequence 3n, 3 over 2n, 3 over 2n plus 11 over 2, and 3n minus 10.

And the right-hand side, I've got two graphs, each with two plots on them.

Which of those nth terms matches to which of those sequences on the graphs? Once you've matched them up, I'd like a few sentences explaining how you identified which sequence was which.

Pause this video and give it a go.

Next, Izzy's plotted the sequence five minus five sevenths of n.

Interesting sequence.

Which term number is wrong? Once you've selected which term number is wrong, I'd like you to write a sentence to justify your answer.

Pause this video and give this one a go.

Question 3 now.

Sophia has started to draw the structure of the arithmetic sequence 2n plus 5.

I'd like you to finish that diagram for her.

For part B, I'd like you to write a sentence explaining why this structure represents the sequence 2n plus five.

Pause this video and give this a go.

Question four, the 10th term of the arithmetic sequence, 3.

5n is 35 and the 20th term is 70.

That is true.

So, Laura takes that truth and then declares, I know the 10th term of the arithmetic sequence, 3.

5n minus 21 is 14, so the 20th term must be double that, it must be 28.

Explain to Laura why this does work for the arithmetic sequence 3.

5n, but it does not work for the sequence minus 21.

Pause this video, write some sentences to explain that to Laura.

Okay, feedback now.

Matching the arithmetic sequences to the correct plotting.

We should have noticed that those are the sequences 3 over 2n and 3 over 2n plus 11 over 2, which would make those ones 3n and 3n minus 10.

Why? You might have written something along the lines of the sequences 3n and 3n minus 10 have the same increases term to term, with 3n minus 10 being a translation of 3n by 10 in the negative vertical direction.

He would have noticed that 3n minus 10 was the only sequence to generate any negative numbers.

All the others remained positive.

So, that 3n minus 10 should have stood out because it was the only one with negative terms. Once he identified that that was 3n minus 10 and 3n, the other two, 3n over two plus 11 over two has the same term to term increase as 3 over 2n, but it's a translation by 11 over two in the positive vertical direction.

And that's how we differentiated the sequence 3 over 2n from 3 over 2n plus 11 over two.

For part two, how are we supposed to know which term number is wrong when we've got no numbers? We should have spotted that it's the 6th term that's wrong, and we know it's wrong because it's an arithmetic sequence, 5 minus 5, 7 to n.

So, the points should align.

An arithmetic sequence, a linear sequence, the points should align and that sixth plot there is not in line.

Question three.

I asked you to finish Sophia's diagram.

We should have drawn that for the second term, for the third term, for the fourth term, for the fifth term.

The 5 remains constant, the 2s vary.

In 2n plus 5, 5 is a constant, 2n is a variable term, that's why the 2s vary.

Write a sentence explaining why the structure represents the sequence 2n plus 5.

You might have written something along the lines of the 5 remains constant whereas 2n is a variable term which varies the number of 2s.

For the fourth question, explaining to Laura why it works for the sequence 3.

5n but not for 3.

5n-21, you might have said, a sequence 3.

5n has no constant.

So, it is just twice as many variables, whereas 3.

5n-21 contains a constant, and that constant would not double when you move from the fifth term to the 10th term.

That is the end of our lesson for today, sadly.

To summarise what we've been through, we can plot sequences graphically by plotting them as coordinates, their term number and their term value.

Multiple representations can allow us to see patterns and structure in these sequences.

Well, I've thoroughly enjoyed today's lesson.

I hope you have too.

I hope to see you again soon for more mathematics.