Lesson video

Thank you for joining me for today's lesson.

My name is Ms. Davies and I'm gonna be guiding you as you explore some of these new and exciting sequences that we are looking at today.

Make sure that you've got everything you need, before you start watching this video.

It's always a good idea to have a pen and paper so that you can jot things down and explore things in your own time.

Let's get started then.

Welcome to this very exciting lesson on Fibonacci and alternating sequences.

We're gonna get the opportunity to play around with some really interesting patterns and actually draw some of these at the end of today's lesson.

I'm gonna talk today about the absolute value to help us describe some of our alternating sequences.

So the absolute value of a number is its distance from zero.

So, five and negative five are both five away from zero.

So, have an absolute value of five.

The absolute value of negative three for example, would be three.

So, a new type of sequence, an alternating sequence is a sequence where consecutive terms change sign.

We're also gonna look at Fibonacci sequences.

We're gonna define this more in the lesson, 'cause you're gonna explore this yourself.

So, let's start by looking at alternating sequences.

There are lots of different types of sequences.

Often they're formed according to a rule and sometimes these rules are not easy to describe mathematically.

Alex and Jun are exploring a sequence which starts negative one, two, negative three, four, negative five.

Alex says the numbers are increasing by one each time.

So, this is arithmetic.

Do you agree? Pause the video, think about your answer.

Well, from negative one to two, we're adding three.

Then from two to negative three, we're subtracting five, then we're adding seven and we're subtracting nine.

So no, there is not a common difference.

This is not an arithmetic sequence.

"Oh yeah," says Alex.

"I wasn't paying attention to the signs of each number.

"They are changing between positive and negative." And that's gonna make a big difference to the rule with our sequence.

Jun says, "Multiplying by negative one changes the sign of a number.

"Maybe this is a geometric sequence "with a negative multiplier." How could we check if Jun is correct? We need to divide each term by the previous term and see if there's a common multiplier, sometimes called a common ratio.

So, to get from negative one to two, we multiply by negative two.

Then we multiply by negative 1.

5 and actually we don't need to go any further.

We can see it is not geometric.

There's not a common multiplier.

Right, can you describe this sequence? If it's continued in the same way, what would the next term be? There is definitely some kind of mathematical pattern, but how would you describe it? So, the easiest way is to use the word absolute value.

So, the absolute values are increasing by one each time.

So, if we ignore the signs one, two, three, four, five, it's increasing by one each time.

However, the signs are alternating, between positive and negative.

If we follow this rule, the next term would be six.

Now, this type of sequence is called an alternating sequence.

In an alternating sequence, each term has the opposite sign to the previous term.

We're gonna look at a few examples.

So, which of these could be alternating sequences? What do you think? Well, if you said B and D.

We're looking for sequences where each term has the opposite sign to the previous term.

You might wanna pause and just have a look at those two examples.

We've now seen three different examples.

Let's return to our previous alternating sequence.

Can you work out what the hundredth term in the sequence would be if it continued with the same pattern? What do you think? Right, hold onto your answer.

Alex says, "I don't know what the Nth term rule is "and writing out a hundred terms will take forever." We definitely don't wanna do that.

Maybe you saw a quicker way for Alex to do this.

Let's investigate.

Well, those are our first four terms and we can see that the absolute value of the term is the same as the term number.

So, we know that the hundredth term is either gonna be a hundred or negative a hundred.

Can you see how we're gonna know which one? Right, well even term numbers have positive terms. So, the hundredth term is going to be positive at a hundred, right? The first two terms in a different alternating sequence are half and negative a quarter.

What could the next term be? Give a suggestion.

Right, anything as long as it is positive.

Sequences, do not have to follow a mathematical rule.

Alex says he's going for one eighth and Jun for a sixth.

Can you describe a possible rule for each choice? So, Alex say's "You might have said it's an alternating sequence "where the denominator doubles "and the fraction alternates between positive "and negative values." You might have been able to describe that in an even better way.

For Jun, you might have said an alternating sequence where the denominator increases by two and the fraction alternates, between positive and negative values.

Let's look at Alex's in more detail.

What would the sixth term in this sequence be if it continued with this pattern, what do you think? We could keep generating the next term until we got to the sixth term or you might be able to spot a pattern which is more efficient.

We look at the denominators.

We've got two to power of one, two to power of two, two to power of three.

So, the sixth term will be two to the power of six, which is 64.

Even term numbers this time are negative, so it'd be negative one 64th.

There'd be nothing wrong in this case of generating the fourth and then the fifth and then the sixth to get your sixth term.

Well done if you also noticed that for this example, Alex's alternating sequence could be geometric with a common ratio of negative a half.

Geometric sequences with negative common ratios, will produce alternating sequences.

However, there are lots of other alternating sequences which are not geometric.

So, if the denominators in this alternating sequence continued to increase by two each time, this was Jun's pattern, what would the 10th term be? Give it a go.

So, you could keep generating the next term, until you've got to term 10.

Or you might notice the denominators are double the term number.

Even term numbers have negative terms. So, this is gonna be negative one 20th, well done if you've got that.

Time to practise them.

The following sequences are all alternating sequences.

I'd like you to describe a possible rule for each and find the next three terms according to your rule.

Whatever rule you come up with is absolutely fine as long as it produces an alternating sequence.

Give those a go.

So, the following sequences are all alternating sequences as well.

I'd like you to describe a possible rule and find the 10th term according to your rule.

Give this one a go.

And finally, a sequence starts with the terms a half, a negative a half.

Alex says, "I can make this into an alternating arithmetic sequence "with common difference, negative one." What is the problem with Alex's statement? Then Jun says, "I can make this into an alternating geometric sequence "with common ratio negative half." What is the problem with Jun's statement? When you're happy with your answers, we'll look at those together.

My other rules could be possible.

I've come up with a suggestion for each.

If you went with an alternating sequence where absolute values increase by two each time, you'd get negative 10, 12 and negative 14.

For B, if you went with an alternating sequence where absolute values double each time, you'd get 48, negative 96, positive 192.

For C if you also went with alternating sequence where absolute values are multiples of five, then you'd get 25, negative 30, 35.

For D if you had an alternating sequence where absolute values are increasing by three, you'd get 16, negative 19 and 22.

And this last one was easier.

If you change one into three over three, then you can see we've got negative two thirds, three thirds, negative four thirds, five thirds.

So, you might have also gone with an alternating sequence where absolute values are increasing by a third.

With that in mind, you get negative six thirds or negative two, seven over three and negative eight over three.

You could have of course written them as mixed numbers if you preferred.

And again, for question two, it'll depend on the rule you went for.

If you went for an alternating sequence where absolute values are multiples of three, the 10th term would be positive 30.

If you went with an alternating sequence where values alternate between one and negative one, that actually is a geometric sequence with a common ratio of negative one.

The 10th term would be negative one.

For C if you went with an alternating sequence where absolute values double each time, the 10th term will be negative 1024.

Using your powers of two might help you here, particularly if you're allowed a calculator.

The first term is choose the power of one.

The second term is negative two to the power of two.

So, the 10th term will be negative two to the power of 10 and your calculator can help you get there.

For D, I went with an alternating sequence where absolute values are the square numbers.

So, your 10th term will be positive a hundred.

And for E, I went with an alternating sequence where absolute values are increasing by three, so the 10th term will be 32.

Quickest way to do that is to the signs and notice that five, eight, 11, 14 has an N term rule of 3N plus two.

The 10th term would be 32 or negative 32.

And then we can see from the pattern that even term numbers are positive.

So, it'd be positive 32.

If you came up with different rules, then you'd have different 10th terms. And finally, the problem with Alex's statement is that if the sequence arithmetic, the next term would be negative three over two and then the sequence would not alternate.

In fact, it's impossible for arithmetic sequences to be alternating.

The problem with Jun's statement is that he's got the wrong common ratio.

You can have an alternating geometric sequence, but in this case, the common ratio would be negative one, 'cause a half times negative one is negative a half.

And now we're gonna look at Fibonacci sequences is another type of sequence and this is a fantastic sequence to know and use.

I know you're gonna enjoy some of the things we explore in this part of the lesson.

So, although there's evidence that this sequence originated in other cultures much earlier than Western Europe, it is named after an Italian mathematician whose name was actually Leonardo of Pisa, but he was known by the name of Fibonacci.

It was a bit of a nickname.

Now, Fibonacci conducted a thought experiment, based on the ideal growth of a rabbit population, starting with a male and female pair.

The parameters of his experiment led him to the sequence of Fibonacci numbers, which we're gonna look at now.

Now, we are gonna visualise the exact same thought experiment, but with the branches of a tree, it's easier to see.

So, each row of this diagram starting at the bottom, is representing the next phase of growth of a tree.

The tree starts with one branch, you might call it the trunk.

After one phase of growth, a new branch splits off from the original branch.

So, now I've got two.

The new branch will grow for another phase, without splitting again.

The original branch will split again at the next phase of growth.

So, now we've got three.

Now this pattern will continue if a branch is new, it takes another phase before it splits again.

But if it wasn't new in the last phase, it will split again in the next phase.

I'll draw it for you.

There's the next phase.

There's the next phase, and so on.

Now here are the first seven phases.

Pause the video and have a look at that pattern in more detail.

What I'd like you to do is count the number of branches including the trunk in each row.

Right, got tricky at the top.

So, you should have one and two and three and five and eight and 13, then 21.

Now, these form a sequence of Fibonacci numbers.

Let's look at those numbers again.

Can you spot any patterns? Can you describe how this sequence is growing? What do you notice? Hopefully you spotted that one add two is three, then two add three is five, then three add five was eight, then five add eight was 13 and so on.

Each term is produced by adding the two previous terms. Pause the video.

What would the eighth term be following this rule? So, if we do 13 plus 21, we get 34.

Now this sequence grows really quickly.

If we were continuing to draw our tree, the next row would have 34 and then 55.

So, you can see why it stops at the seventh phase.

This is quite an interesting sequence, 'cause if you keep generating terms, it ends up looking quite similar to a geometric sequence.

If you were to plot it on a graph, you'll see it gets very, very steep, very, very quickly, like geometric patterns.

Although Fibonacci started with the number one then too.

Some mathematicians start the Fibonacci sequence on an earlier term.

If the second term was one, what would the first term be? So, we need something that when you add one, you get two.

So, the first term would be one.

So, sometimes you'll see the sequence written as one, one, two, three, five, eight.

Equally, you can sometimes see it starting on zero.

So zero, one, one, two, three, five, eight.

Regardless of the start number, we call this sequence of numbers the Fibonacci sequence.

Now, Fibonacci numbers are seen a lot in nature and this is where this is really interesting.

Many species of flower will grow to have a number of petals, which is a Fibonacci number.

So, if you went out to explore this, you'd see a lot of flowers have five petals or eight or 13 or so on.

Not always, but this is something you do see as a pattern in nature.

For some species of plant, the optimal growing patterns of seeds will be in spirals, of which there'll be a Fibonacci number.

It's to do with how well they fit together.

So, here's a pine cone.

You'll see the scales of the pine cone grow in spirals.

If you had your own pine cone, you'd be able to see this a lot better by turning the pine cone round.

If you count the clockwise, an anti-clockwise spirals, you should see some Fibonacci numbers.

I'm gonna do that for you now.

So, those are the anti-clockwise spirals.

There's eight of them and those are the clockwise spirals.

And there's 13 and those are two of our Fibonacci numbers.

Let's look at this flower.

There are clockwise spirals.

They're quite hard to see this.

It's a lot easier in real life and are anti-clockwise spirals.

And this time if you counted those, you see there's 13 clockwise and 21 anti-clockwise.

Now these numbers don't just appear in nature.

You also see them in graphic design, computing and other disciplines.

Have a think about this next time you're doing any drawing of things found in nature, right? What are the missing terms in this Fibonacci sequence? Can you get all three? You should have two, five and 89.

Sequences that follow the same rule, so each term is the sum of the two previous terms can be called Fibonacci sequences, even if they don't start on one-on-one or one and two.

So, we can start with any two numbers.

So I've got five and seven.

What would the next three terms be? 12, 19 and 31.

Jacob says, "Would it be possible to create a Fibonacci sequence "that decreases?" So, giving you three ideas here, can you continue these three sequences? What do you notice? Those are your three sequences.

And for the first one, you could see this one started increasing, after the second term.

So, the second term could be less than the first term.

They're both positive.

They will then continue to increase.

The second one was interesting.

It decreased for a while, 'cause the second term was negative.

The third term was less than the first term and then it got smaller again, 'cause four add negative one was three.

Then because the third and the fourth terms were positive, that will then continue to increase in the future.

And the last one is possibly the most interesting.

It alternated for a while, but because you can see that we've got a fourth and fifth term, which are both negative, this is now going to continue decreasing into the future.

With enough information we can fill in missing terms in Fibonacci sequences.

What could we do then to work out the missing term in this sequence if it was a Fibonacci sequence? Or we need four add something to be 15.

So, if you do 15, subtract four, that tells you we'll need 11 and we can just double check four add 11 is 15 and then we need to do 11 add 15 to get the next term.

Right, what do you think we can do this time? Again, this is not too tricky.

We've got two consecutive terms. We just need something add 10 to be 13.

So, that would need to be three.

And then we need something add three to be 10 and that would be seven.

And we've seen these sequences before where the second term is less than the first term.

That's absolutely fine.

Pause the video.

What are the missing terms in each of these Fibonacci sequences.

If you've got seven and 16, 10 and five, negative four and four, that one was a little bit trickier, the four was not too tricky to get, then we need something, add eight is four.

So, what number when you add eight did you get four? It's negative four.

It's always a good idea to run through and check your sequence is a Fibonacci sequence.

Once you've got your missing terms. Right, this is the challenge.

If we know two terms, we can find the missing terms in between.

But if there's two gaps, it's a little bit trickier.

Jacob says we can use trial and improvement, until we find the correct second and third terms. This is a good method.

So, he's gone with three.

So four ad three is seven, three ad seven is 10, not 20.

If we need a bigger number, he's gone with four and five.

Four a five is nine.

Nine add five is 14.

Not quite big enough.

If we carried on this way, we could find the correct terms. I'm gonna show you now a slightly more efficient way.

If we use a letter to represent the first missing term, we can then form an equation.

So, let's put A as our second term.

So, we've got four then A.

What could we call the next term? Right it would be useful to us, it needs to use a again.

So, you can write the next term as A plus four, 'cause we get the next term by doing four plus A.

Right, the following term then 20 must be made by doing A, plus A plus four.

So, let's write 20 equals A plus A plus four.

But this is all right now, 'cause we've got an equation we can solve.

2a plus four is 20.

2a is 16, so a must be eight.

And we absolutely want to check this.

So, we put eight in there.

We get eight at four is 12.

12 add eight is 20.

So, that works.

So, that algebra might just help you speed things up, 'cause trial and improvement can sometimes take quite a long time.

Time for a practise.

This first activity, I'd like you to fill in the missing terms in these Fibonacci grids.

The way these work is, each row should form a Fibonacci sequence from left to right and each column should form a Fibonacci sequence from top to bottom.

That means you should be able to check your numbers are right 'cause it should work horizontally and vertically.

Give these a go.

You've got three more to do.

Off you go, right? I'd like you to have a go at filling in the missing terms in these Fibonacci sequences.

Pay attention to sequence C.

The reason you've got so many terms to fill in is something interesting happens for a while with that sequence.

Be particularly careful with E onwards, 'cause that's when you've got more than one gap, between terms. So, you could use trial and improvement or you could use that method of using a letter for your second term and forming an equation.

Give those a go.

And now we have by far the most interesting task.

So, I'd like you to look at this pattern, which is being built out of squares.

So, you've got the first five patterns on square paper.

I'd like you to draw pattern six on the grid provided.

Be careful where you start your pattern in order to get it to fit on with the squares that you've got.

You might wanna sketch this out first.

Then square A has side length one and square B has side length one.

What are the side length of the other squares? If you do that in order, what do you notice, right? I'd like you to take everything you've noticed and now have a go at drawing the seventh pattern in the sequence.

And then using a pair of compasses.

I'd like you to draw an arc in each square.

So, you'll see I've done the first three for you.

So, the radius of each arc, should be the same length as the side of each square.

And what you should end up with is a spiral pattern.

Give this a go.

See if you can make this pattern.

Right, you need to pause the video and check your numbers for this one, off you go.

Again, pause the video and check that you've got the right values for D, E and F.

Hopefully you were really confident with those, 'cause you could check your numbers were right by making sure the sequences were Fibonacci, going horizontally and vertically, right? There's some tough answers to this one.

So for A, you should have nine, 10, 19, 29.

For B, you get five, two, seven, nine, and then that's gonna continue to increase.

C was quite interesting because you get three then negative two, then one, then negative one, then zero, then negative one, then negative one, then negative two.

So, those values fluctuate around zero for quite a while.

The next term they'll be negative three, then negative five and then it'll continue to decrease quite sharply.

For D, you get one, negative six, negative five, negative 11, negative 16.

Well then if you've got the missing numbers in E, you should have two, three, then five, then eight.

For F, you should have 15, then 22 and G seven, then 10.

Well, then if you thought about how you're gonna do H and I with three gaps.

So, if you use algebra and you have your second term is A, the third term is A plus eight and the fourth term is A plus A plus eight.

So, your fifth term would be A plus eight, plus A plus A plus eight, or a better way to write that, 3a plus 16.

So, 3a plus 16 is 25.

Then you can solve it to get A as three.

Obviously you can then check that works.

If you do something similar for I, you get the next term as negative six, then that would be two, then negative four.

And then that does give us negative two as the fifth term.

Well done for solving those last two problems if you got that far.

And finally, pattern six should look like this.

And then our side length, you'll notice will one, then one, then two, then three, then five, then eight.

And the side length of the squares form the Fibonacci sequence, which is why this gets called the Fibonacci spiral.

And there's your completed spiral.

So, for the seventh pattern, the biggest square has side length, 13 and 13.

You can obviously draw more patterns with larger pieces of paper.

However, you start needing to have lots of squares on your paper in order to fit in too much more than the seventh pattern.

Hopefully you got a nice neat arc using your compass, so you have this lovely spiral pattern as well.

I hope you enjoyed playing around with those Fibonacci sequences and learning this new type of sequence, called an alternating sequence.

If you'd like to review what we've looked at today, pause the video and read through that now.

Hopefully you can give yourself some time to go out and explore where these Fibonacci numbers appear in nature.

You might surprise yourself.

You might also want to investigate where else these numbers are used as lots of different disciplines, seem to follow this Fibonacci pattern.

Thank you for joining me today and I look forward to seeing you for another lesson soon.