Lesson video

Hello and welcome to today's video.

I'm really glad that you have decided to learn with us today.

My name is Miss Davies and I'm going to be helping you as you work your way through this lesson.

Let's get started.

The title of this lesson is Graphing Special Number Sequences Using Technology.

With that in mind, you want to have access to graphing software to get the most out of today's lesson.

And we'll talk you through how to use it, so don't panic if it's not something you've used before.

By the end of the lesson, you'll be able to plot a graph of a special number sequence.

Even better than that, we're going to explore some of the patterns that occur when we plot them on a graph.

If you're not sure what a triangular number is, we're going to use that today.

So make sure that you reread that and you are happy with those values.

So we're going to start by looking at how to plot sequences using Desmos.

This is the graphing software that I have chosen to use.

So we can plot sequences on graphs by turning the term numbers and the terms into coordinates.

Using graphing software allows us to plot more points quickly and easily.

So to start, you need to go to desmos.

com and press the graphing calculator button.

Your screen should look like this.

So we think about this arithmetic sequence to start, we've got 3, 5, 7, 9, 11.

We can do that by creating a table of values.

So our first term is 3, our second term is 5, our third term is 7, and that's how we've formed this table of values.

What we can now do is enter that table of values on Desmos.

So if you click the plus button, then you want to click on Table.

Lovely.

So now we can enter values into our table.

For now, it doesn't matter what the headings of your table are.

Mine say x1 and y1, that's absolutely fine.

So type in the term numbers into the x column and then the term values into the y column.

Give that a go and then come back and look at the next bit.

Make sure you've got that into your table before you move on.

Lovely.

So there's the first five terms, and Desmos will plot your table as coordinates.

Why do you think these coordinates lie on a straight line? Good.

We should be really comfortable with this idea now that linear sequences follow a linear relationship.

Because this was a linear sequence, those points are going to lie on a straight line.

Now this is where Desmos becomes really useful.

If we know the nth term rule for the sequence, we can generate more coordinates really quickly.

Okay, so we have a common difference of two.

So our sequence is increasing by two.

That means it's related to the two times table.

The two times table is 2, 4, 6, 8, 10.

So our sequence is one more than the two times table.

We can write that then as 2n + 1.

So let's put that into our table then.

So if you put your cursor over the headings to the table, you are able to click on them and change them.

So if you change them now, so that instead of having x, we now have n, and then put that nth term rule in the y column.

So we've got n and then we've got 2n +1.

Right, now we can generate more terms. So if you click on 5 and then press enter, what should happen is term 6 should now appear in your table.

Okay? So if you've got them going up in your order, 1, 2, 3, 4, 5, if you then select 5 and click enter, Desmos should know that you now want 6.

It's spotting that linear pattern in your n values.

If you press enter again, you should get 7, and if you keep pressing enter, you should get more terms. Try this now until you reach 20 terms. Pause the video and check you can get that far.

Lovely.

So you should find then that the 20th term is 41.

What happens then if you do this for 30 terms? Have a go and see what happens.

This might depend a little bit on your computer screen, but for mine, Desmos chooses to hide some rows so the table fits on the screen.

The 30th term is 61.

So what we can do now is we can use our graph to see if a term is in the sequence.

So is 50 in the sequence 2n + 1? So we now need to find the number 50 on our y-axis.

So find 50.

Sometimes it's easier to see if you plot the horizontal line.

So if I want the horizontal line at 50 that has equation y = 50.

If you just go into Desmos, click on the next row down, type in y = 50, you now get that horizontal line.

Now I need to look whether that line goes through any terms. We can see that it doesn't.

So 50 is not a term in our sequence.

Let's try 45 then.

So same graph, type in y = 45 to make it easier, and then this time we can see that it does go through a coordinate.

If you actually click on the point, it'll bring up the coordinate of that point so you don't have to read it off yourself.

And yet we can see that 45 is in the sequence and it's the 22nd term in the sequence.

That x coordinate represents the term number.

I would like you to check that you are happy with plotting the first 30 points of a sequence with the nth term, 3n - 4.

Then I want you to check if 65 is in the sequence and if 60 is in the sequence, if they are, which term in the sequence is it? And then tell me what type of sequence it is and how you know.

Off you go and then I'll talk you through how to do it.

Okay, so what you needed to do is get a table.

You need to name your table n and 3n - 4.

Then if you enter 1, 2, 3 in the n column, it'll start generating the y coordinates.

Then you can start just pressing enter and get up to the 30th term.

The 30th term is 89.

I want to know if 65 is in the sequence.

So I've plotted y = 65 and I can see that yes, there is a coordinate where y is 65 and it's the 22nd term.

Let's look at 60.

60 is not in our sequence.

What you might have spotted is that it's between the 20th and the 21st term, but is not actually on a term in our sequence.

So 60 not a term in the sequence.

Lots of ways for you to see that this is a linear or an arithmetic sequence.

You could have just said the nth term is a linear expression.

You could have said that successive terms have a common difference.

You might have started writing out terms and gone, oh, they're increasing by three each time.

And the points form a straight line, which is also telling us that it's a linear sequence.

Okay, we're going to do the same with some geometric sequences.

I'd like you to start by plotting the first five terms on Desmos.

So use that table function to plot 1, 3, 9, 27, 81.

Do that before coming back.

Okay, so it should look like mine, but you might want to stretch the x-axis scale to get a better idea of the shape.

If you press shift and then use your cursor, hover over the x-axis and then you can click and drag it right, and it'll stretch that scale.

You can do the same to the y-axis if you want at any point as well.

So now we can see a bit clearer what it looks like.

Jacob says, "I think the nth term could be 2n - 1." Why is Jacob definitely not correct in this case? 2n - 1 is a linear expression, so it'll be a linear sequence, which means our point should lie on a straight line.

And you can see very clearly that they don't.

I'm going to plot the line y = 2x - 1 to show that's definitely not the right equation.

And you'll see that our line does not go through our points.

It starts going off through the first two but then doesn't follow the rest of our coordinates.

Jun says, "Let's try the nth term rule of n squared." Right, give it a go.

Type in y = x squared and then see if that rule follows our points.

Let's have a look then.

No.

Again, it goes through two of our points.

It goes through the first term and the third term, but doesn't go through the rest.

So it's also not the correct nth term rule.

We have got a curve this time.

Actually the nth term rule is 3 to the power of n - 1.

You don't need to be able to work that out at the moment, but you do need to be able to type that into Desmos.

What you need to make sure you type in is y = 3 to the power of x - 1.

You might need to put brackets around x - 1 'cause that whole expression is the exponent.

What I've done here is I've put them in a table so that I can use that enter key to get a few points.

Then I've plotted the equation as well and checked that it goes through the points.

Pause the video and get your screen looking like mine.

Okay then.

So drawing the graph of the nth term rule allows us to check if larger term values are in the sequence.

So let's see if the number 2000 is in this geometric sequence.

So type the equation y = 2000 so we can have the horizontal line in the right position and then click on the point where this line intersects the curve.

Take a second to do that.

Okay, so I've got the coordinate.

7.

919 is possibly a rounded value because Desmos does round off the values.

So it's 7.

919, 2000.

This is not in our sequence 'cause the x value is not an integer.

For this sequence, remember the n values are 1, 2, 3, 4, 5 and so on.

We can't have a term number of 7.

919 for this sequence.

What that number does tell us is that 2000 is between the seventh and the eighth terms. Be aware though that when plotting a sequence on a graph, we do not draw a line or a curve through the points.

The term numbers in our sequences are the positive integers greater than our equal to 1.

So 1, 2, 3 and so on are our term numbers.

A line or a curve shows all the values that fit the rule including non integer x values, which doesn't make sense for a sequence.

However, as we've just seen using graphing software, it could be quite useful to draw the line or the curve to help us see patterns.

That's okay as long as you're aware that the line or the curve does not show all the values in the sequence, but it shows all the values following that rule.

Let's see then.

So we can limit our graph if we want, so it only displays the part where the term number is 1 or greater.

So what you can do is where you have an equation, if you click on it and then we can restrict what x values we have by using curly brackets.

So if you type curly brackets and then x is greater than or equal to 1, close curly bracket and that needs to be on the same line as that equation.

If you're not sure how to get that greater than or equals symbol, if you put in the greater than symbol and then press the = button, it'll turn it into a greater than or equal to.

Try that and then see if your screen looks like mine.

Your graph now should not have any values less than 1, 1.

Okay, quick check then.

So which of the following is the correct graph of the sequence 7 - 2n? And explain your answer.

So you should have picked B.

When plotting a sequence, no line should be drawn.

The first term of the sequence is when n = 1.

So A can't be correct 'cause A has coordinates where n is less than 1.

So it's got to be B.

We've got all the values on integer x coordinates where x is 1 or greater.

I'm saying n and x interchangeably.

'cause in this case we're plotting n on the x-axis, so they're the same thing.

Okay, a bit of a chance to use Desmos then.

I'd like you to use Desmos to determine which of these is the correct nth term rule for the sequence 25, 125, 625, and we've got a geometric sequence where we're multiplying by five.

You'll need to stretch the scale on your x-axis and zoom out.

Have a play around, come back when you think you've got the right one.

Lovely then.

Have a look at my graph.

If you didn't get this far, you might just want to pause your video and see if you can work out the correct nth term from my diagram.

Lovely.

So you can see from my diagram, y = five to the power of x + 1 gave us the correct curve.

Therefore the correct nth term is 5 to the power of n + 1.

Perfect chance for a practise.

Lucas wants to know if negative 10 is in this sequence.

Have a look at his diagram.

He has plotted the lines y = 100 - 3n, and the line y = negative 10.

He says they intersect.

So negative 10 is a term in the sequence.

I'd like you to explain to me what Lucas has done correctly and then explain to Lucas what mistake he has made.

It's a really good skill in mathematics to be able to explain things to other people.

So take your time to make your answer coherent.

And for this second part, you're going to need to use Desmos.

I'd like you to use Desmos to match these sequences to their nth term rules.

You might want to colour them so that you can see the separate sequences clearly, or you might want to do one at a time so you don't get yourself in a muddle.

Give those a go and then there's one final question to look at together.

Final question then.

I'd like you to use graphing software to help work out if these numbers are in the sequences.

You'll see that I have given you the nth term rules for all these sequences so you don't have to work them out yourself.

And then you need to tell me if those numbers are in the sequence.

If it's in the sequence, see if you can tell me what term number it is.

Off you go.

Well done.

I hope you're really starting to see the benefits of using technology to help us with our graphing skills.

So you might have said something like Lucas has plotted the correct lines and he knows he's looking for an intersection point.

He's actually got a really good start on this question.

The mistake however, is that the x coordinate needs to be an integer for it to be in our sequence.

The coordinate shows that negative 10 is actually between the 36th and the 37th term.

So not in our sequence.

Okay, so our matchup A is 2 to the power of n, B is 2 to the power of n + 2, C is 3 X 2 to the power of n, D is 2 X 3 to the power of n, and E is 4 X 3 to the power of n.

You absolutely do not need to know how to get those nth terms yet, but it's good to be able to use that graphing software to explore these features.

And finally, yes, 30 is the 16th term in that sequence.

No, 2044 is between the sixth and the seventh.

So not a term in the sequence.

Yes, 1,280 is the eighth term, and finally, no, 4,004 is between the fifth and the sixth term.

Right, what we're going to do now is use all those things we've learned about technology to explore some special number sequences.

So we're going to have a look at what the square numbers look like when plotted.

So we can build a sequence out of square numbers.

We're going to start with the first five square numbers.

We're going to use our table function to plot these.

If you need some time, pause the video and get that plotted.

And again, we're going to stretch the x-axis to get a better idea of the shape of the graph.

Make sure your graph looks like mine.

This is only the beginning of the sequence of all square numbers.

We've only got the first five terms so far.

So in order to generate more terms, we need to know the nth term.

Well this sequence is generated by squaring the term number.

That's how we get the square numbers.

So Jun is going to use the nth term to draw this graph.

You might want to do the same on yours.

Right, so he says, "That's interesting, it makes a sort of U shape." Jun's comment applies to the graph of y = x squared.

What does he need to remember about the sequence n squared? Think about what we just talked about in our previous part of our lesson.

So it's really good to be aware that y = x squared, makes a curve shape, a kind of U shape.

But remember the sequence itself only starts when n = 1.

So all that part of the graph to the left in the x = 1 is not actually part of the sequence.

Try using curly brackets so the graph no longer shows the value when x is less than 1.

Pause if you need a second to do it and then come and check mine.

Okay, so I'm restricting the x values and then my graph now looks like that.

So I've now got all the x values, where x is greater than or equal to 1.

We're going to do the same with cube numbers.

I'd like you to make a prediction.

So pause the video.

What would the cube numbers look like when plotted? Okay, so to get the cube numbers, we need to cube the term number, therefore the nth term rule is n cubed.

I'd like you now to use this nth term rule to generate the first 10 terms of the sequence n cubed on a graph and then we'll explore the features together.

I'll give you a moment to do that.

Okay, I've left my square numbers on there for comparison.

So we've got our square numbers and then our cubed numbers.

This sequence also produces a curve, but you'll see it increases at a much faster rate.

That makes sense 'cause cubing a number is going to make it larger than squaring a number, and that's if we're talking about our positive integers, which we are at the moment.

Jacob says, "I think the number 1 is in both sequences." How could he check that that's true? Is he correct? You might need to zoom in a little bit here, but we can see that both sequences have the coordinate (1, 1).

So not only is 1 in both sequences, 1 is the first term of both sequences.

If you zoom in, you can see that 64 is the fourth term in the sequence n cubed but the eighth term in the sequence n squared.

So 1 was the first term in both sequences.

So that's why the coordinates are directly on top of each other.

But you'll see that 64 is also in both sequences.

They're on a horizontal line with each other.

The fourth term in n cubed and the eighth term in n squared.

So what do you think the sequence of all triangular numbers will look like when plotted? We're going to look at some pupils' ideas to help you form your own.

Jacob says, "I think they'll make a curve again." Lucas, "I think they'll start in the same place as the square and cube numbers." and Jun says, "I think they'll be between the square and the cube numbers." Use those ideas to help you form your own idea what the angular numbers will look like when plotted.

Let's have a look then.

The nth term for the triangular numbers can be written as n (n + 1) all over 2.

You don't need to remember that.

Use this to generate the first 10 terms in the sequence.

Does it look as you predicted? Pause the video and give yourself time to do that.

Okay, so we've got another curve.

They do start at (1, 1).

This curve seems to increase at a slower rate than the square numbers and the cube numbers.

I'd like you to see if you can find a number, which is in the sequence of square numbers and the sequence of triangular numbers.

So obviously 1 is in both sequences, but you might have also found 36.

36 is the sixth square number and the eighth triangular number, whereas 1 is the first term in both sequences.

If you've not already done so, I'd like you to input the equations for the square, cube and triangular numbers into Desmos.

See if you can limit them.

So we've got it when x is greater than or equal to 1.

I've written them on the screen to help you.

So your graph should now look like this.

And that's just going to help us see some of our patterns.

So we're going to explore another type of sequence.

This sequence has the first two terms, 1, 1,.

and the next term is generated by adding the previous two together.

So we've got 1, 1, 2.

The next three terms are going to be 3, 5, and 8.

So we're adding the two previous terms to get the next term.

I'd like you to use the table function on Desmos to plot these six terms. There's no easy nth term rule for this sequence, so you'll have to just plot them in a table.

Pause the video and make sure you've got those in.

Okay.

Lucas says, "This sequence seems to be rising at the slowest rate." You can see this underneath our triangular numbers.

I'd like you to plot the next three terms to see what happens.

So you're going to have to add the two previous terms to get the next term.

Off you go.

There you go.

So now Lucas is saying, "Oh, I think it might overtake the triangular numbers" Plot the 10th term.

What do you notice? The 10th term in our sequence is 55.

That's the same as the 10th triangular number.

I'd like you to plot some more values and see when our sequence overtakes the square numbers.

Off you go.

You should have found that the 12th term in our sequence is 144, which is the same as the 12th square number.

After that it's going to continue to rise.

"Surely then, this new sequence will never overtake the cube numbers 'cause they increase at such a fast rate." What do you think? And then we're going to experiment.

Let's fill in the table up to term 20.

What you might need to do for this one is keep adjusting your scale to see the new points.

You are welcome to use a calculator to help you with the largest sums where necessary 'cause some of these numbers are going to get quite big.

Try this out, come back and look and see if we have the same thing.

Here we go.

So I've zoomed out quite far.

You can see that I've got the value 10,000 on my y-axis.

So you can see that although it starts off at quite a shallow curve, it gets steep really quickly.

It overtakes the cube numbers between terms 20 and 21.

Perfect.

We're going to have a go at exploring some of these sequences yourself now.

So this graph shows the sequence of square numbers, which is in green, and the sequence 5n - 1.

That's the linear sequence in black.

Which of these statements are true? You can use my diagram or you can draw your own if you want to be able to zoom in.

Off you go.

So both sequences do contain the number 4.

It's the first term in 5n - 1, and it's the second square number.

Both sequences do contain the number 9, but both sequences do not contain the number 14.

It's only in 5n - 1.

And both sequences do not have the same fifth term.

If you find the fifth term, you can see they are not the same value.

Time to put that all into practise.

This task is really exploring your use of technology.

So read the questions carefully, it'll tell you what to do, and see if you can write your answers in full sentences.

Off you go with the first one.

Well done.

So some different sequences this time for you to have a go at plotting.

And again, make sure you're writing a full sentence for E.

Off you go.

For three, you need to plot the triangular numbers.

I've given you the nth term rule so you can plot lots of terms for that.

Have a go at those questions and then we'll have a look at question four.

And finally we're looking at another one of these sequences where you add the two previous terms to get the next term.

I'd like you to plot the first 10 terms and then have a go at answering B and C.

Off you go.

Fantastic.

Lots to explore to make sure you take the time to compare your graphs to mine.

What I've done is I've added the curves and the lines in for these sequences.

Even though the the sequences should not have curves or lines, I've added them in just to help us see the patterns.

So 2n and 2 to the power of n have the same first term.

All three sequences have a second term of 4.

That's why they all intersect at 2, 4.

2 to the power of n is increasing at the fastest rate.

That's a geometric sequence.

And there's lots of terms which are in the sequence 2n and 2 to the power of n.

In fact, all numbers in 2 to the power of n are also in 2n.

So you could have 2, 4, 8, 16, 32.

That's a geometric sequence multiplying by two.

And the 15th term 32,768.

Those are our next sequences.

Again, I've drawn the curves to help us see what's happening.

N cubed - 5 has the smallest first term.

5 - n squared has the smallest second term.

And n cubed- 5 then has the largest third term.

It's rising quite fast, that one.

And 5 + n squared and 5 - n squared, you might have said something like they are reflections of each other.

Or you might have said the terms in 5 + n squared are always larger than the terms in 5 - n squared.

5 + n squared seems to curve upwards.

5 - n squared seems to curve downwards.

And for three, the 19th triangle number is 190.

Term five is where this sequence overtakes the triangular numbers.

And the 16th triangle number is larger because it's 136 and that's larger than 128.

And finally, comparing these two sequences, the second sequence after the third term is decreasing and it has this kind of curve shape.

There's a little bit of a wobble before it does that, but then it seems to end up with this nice curve shape.

The previous one that you plotted for A also has the same curve shape after the first few terms. If you zoom in, it doesn't have that curve shape to start with, but then it seems to end up with a curve.

And that's quite an interesting thing about these type of sequences, that eventually they end up looking quite similar to geometric sequences, even though they don't start off like that.

I hope you've enjoyed exploring those sequences today.

We've looked at how we can plot sequences on Desmos using the table function.

We've looked at how the nth term rule makes generating the points really easy, and we looked at using the graphing software to see if a term is in a sequence.

Thank you for joining me and I hope to see you again.