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 Ms. Davies and I'm going to be helping you as you work your way through this lesson.

Let's get started.

Welcome to our lesson on extrapolating a sequence.

You've probably explored lots of different types of sequences already by the end of this lesson, you'll appreciate that there are other number sequences and the limitations of only seeing a few terms in a sequence, if you're not already confident with the different types of sequences, pause the video and have a read through our keywords.

We're going to start by exploring some other sequences.

So Andy is writing a sequence.

This sequence is going to follow some kind of rule.

The first three terms are two, four, and eight.

What do you think the next term is going to be? Alex says, I think it's a geometric sequence with a common multiplier of times two.

Sophia thinks it's quadratic 'cause it's added two, then four and next it'll add six.

Quadratic is a special name for sequences that have a common second difference so that the differences are increasing by the same amount each time.

It's not a word you need to know at the moment, so I wonder if any of your ideas were the same as Alex or Sophia's.

Did you come up with any other ideas? So if it was Alex's suggestion or Sophia's suggestion, the next term could be 16 if we're multiplying by two each time, but 14, if we're adding two, then four and six.

So getting the right rule is really important if we want to get the correct next term.

Andeep says, actually I'm multiplying the first two terms to get the next term.

Two times four is eight.

So four times eight is 32.

That's absolutely fine.

A sequence is any set of numbers often following a rule doesn't always need to follow a rule.

So this rule that Andeep has come up with will form a sequence.

It's just not one that Alex or Sophia recognised.

Alex wants to try one.

My sequence starts three, five, seven, nine.

What do you think the next term is going to be? I wonder if like Sophia, you said it's just a linear sequence, so the next term would be 11.

Actually, Alex has been a bit tricky.

It's actually the odd, odd numbers not containing the digit one.

So one wouldn't count, but 3, 5, 7, 9, 11 then does not follow Alex's rule.

What are the next three numbers in Alex's sequence? Pause the video and take a moment to do this one.

Okay, so the next odd number that doesn't have a digit of one is 23, then 25, then 27.

So there are many more sequences than the ones defined by key features such as arithmetic, geometric, or sequences you might have looked at like the square numbers or the triangular numbers that had a common second difference.

So they're not the only rules that sequences can have.

And sequences don't actually have to follow a mathematical rule.

They could be formed from numbers, words, pictures, and patterns.

But we are going to focus on numerical sequences in this lesson.

If we know the rule, we know we can work out the next term.

But without knowing the rule, it's impossible to know for sure what the next term will be.

Sophia's going to try one, my turn.

I'm going to start my sequence.

2, 3, 5.

What rules or patterns can you think of for Sophia's sequence and what would the next term be? See if you can come up with a couple of different ones.

Pause the video, give it a go.

I wonder if any of your suggestions were the same as these.

So it could be adding one, then adding two, then adding three.

So a common second difference of one.

If that was the case, the next terms would be eight and then 12.

I wonder if you thought you could add the previous two terms to get the next one, 'cause two, add three is five.

The next one then would be eight but then 13.

Really good if you spotted that all three of those are prime numbers.

So at the moment that could be the prime numbers in ascending order.

The next numbers then would be seven and 11.

And then I thought you could possibly double and then subtract one, 'cause double two is four, subtract one is three, double three is six, subtract one is five.

If that was the rule, then the next terms would be nine and 17.

Sophia says, good guesses.

I was going for prime numbers, but I like all of those ideas.

I wonder if you came up with even more than that.

So some rules for numerical sequences only work in a certain language.

Can you work out where this sequence of numbers comes from? 3, 3, 5, 4, 4, 3, 5, 5, 4.

So my hint is write the numbers one to nine out in English.

There are nine terms there.

Write the numbers one to nine in English.

This is a bit of a strange sequence, so if you can't work it out, don't worry.

Come back for the answer.

Did you get it? It's the number of letters in the integers greater than zero and I've put them in numerical order.

So one has three letters, two has three letters.

The spelling of three though has five letters.

So that's why we've got 3, 3, 5.

There are possibly other things which give the same values, but that is the sequence that I was going for in this case.

Geometric sequences have a common ratio between successive terms, arithmetic sequences have a common difference between successive terms. So Alex says, I wonder what will happen if we combine these ideas.

If we start with a first term of one and then we multiply, but we also add.

So we're going to double and add one.

The next term would be three.

What are the next three terms? So we'll have seven, 15, and 31, let take a bit of time now, see if you don't spot any patterns in this sequence, you might want to write the numbers out and have a little bit of a play around.

Can you see anything that's happening that might help us work out how this sequence grows? Okay, so obviously now we know that you can double the previous term and add one to get the next term.

But I wonder if you also explored the differences between the terms. So we're adding two, then four, then eight, then 16.

That's interesting then because the differences follow a geometric pattern pattern, I want you to see if you can use that then to calculate the next term.

Can you find two different ways to work out what the next term would be? Off you go.

So you can use our original rule, so double the previous term and add one.

That would give us a next term of 63.

We could also use what we've spotted here.

So the differences are following a geometric pattern.

So the next difference would be 32.

That also gives us 63.

So it looks like we've spotted a couple of patterns that are forming with this sequence.

Pause the video and spend a bit of time seeing if you can generate another sequence where the differences form a geometric sequence.

Well done for spending some time exploring that idea.

Actually geometric sequences have this feature.

So if you wrote out a geometric sequence, you'll see that the differences between terms follow a geometric pattern.

Here's an example, we're multiplying by three each time we look at the differences, we increase by 10 and by 30, then by 90.

So our differences also form a geometric pattern.

You might have come up with some other sequences that has a turn to term rule with a multiplicative or an additive step, like our previous example.

So 2, 4, 10, 28 we're multiplying by three and subtracting two.

And then you'll see that our differences are 2, 6 18.

They form a geometric pattern with a common multiplier of times three.

I've given you some numerical sequences down the left hand side, you've got five of those and I'd like you to match it with a rule on the right hand side.

So which of those rules could work for each sequence? I suggest you read all the rules first before you start matching them up.

Off you go, well done.

Don't worry if that took you a little bit of time.

There was a lot going on in that activity.

So the first one, you could treble the previous term and subtract one.

The second one you could add the two previous terms to get the next term.

C, this was a strange one.

It was the square numbers in ascend order spelt in English and it was the number of letters.

So you've got, 1, 4, 9, 16, 25 and so on spelt in English.

D, this was a really strange one.

We're adding one, then multiplying by one, adding two, then multiplying by two, adding three, then multiplying by three.

And the last one we were multiplying by one, then by two, then by three, then by four.

So we had a linear pattern in the multipliers time to have a go yourself.

So I've given you starting two terms and I'd like you to find three different ways to continue that sequence.

The important thing is that you explain the rule you've used for each way.

Give it a go, come back for the next bit.

Well done hopefully now you've thought of some of those crazy rules, you'll be able to apply some of those to this question here.

So I have started some sequences and I would like you to suggest what the next two terms might be.

At this stage, there's no right or wrong answer as long as you have a reason.

Lovely, there were some sequences that was really hard to work out any kind of pattern.

So well done if you came up with some of the next terms. Oh, I've continued the sequences like this, so they're the same at three terms, but I've added a fourth term now and I've told you what the rules are for each sequence.

All you now have to do is match them up.

I wonder if you'll have a moment when you go, oh, that's what that sequence was about.

Now that you've seen some of the ideas, give those a go and come back when you're ready for the answers.

Well done.

There are loads of different answers for these questions.

I've put some of my suggestions up.

If you had difficulty coming up with ideas, then you might want to just pause the video and have a look through my suggestions.

And the same for C and D.

Interesting one with D, I went for prime numbers greater than 10 as an option 'cause they're both prime.

So you could have prime numbers greater than 10 or the first two triangular numbers are one and three.

So I thought they could be the triangular numbers add 10.

Okay, so this one, there are loads of answers for each question.

I've given you an example for each.

If you'd like to read through them, pause the video and then we'll have a look at the ones on the next slide which had the examples of some of the rules to match it up with.

So pause the video if you want to read any of my ideas.

Okay, so A was actually the numbers containing the letter T when spelt in English.

So 2, 3, 8, 10.

So the next one would be 12, B was actually the number of days in each month in a leap year.

So January has 31, February has 29 in a leap year.

March has 31, April has 30, so May has 31.

C, multiply the previous term by 10 and then subtract 10.

So that would give you 18,890.

D.

It was square the digits of the number and then add them together.

So two squared is four, four squared is 16 one squared add six squared is one, add 36, which gives you 37.

The next one would be three squared, add seven squared or nine, add 49, which gives you 58.

And finally it's the digits in alphabetical order according to their English spelling.

So if you wrote the digits 1, 2, 3, up till nine in alphabetical order you'd have eight and five and four, then nine, then one, then seven, and then six.

Both starting with S, there were some crazy sequences in there.

I'm hoping you had a little bit of fun thinking of some of these interesting rules.

We're now going to have a look at extrapolating a sequence and some of the difficulties that can occur.

You've already seen that a little bit when trying to guess the rule for those previous sequences.

So it's possible for a rule to look like it works for a certain number of terms but then fail for the following terms. Sofia asks a good question, how many terms in a sequence do we need to be able to work the rule out for sure? What do you think? All right, hold onto your answers and we're going to explore this a little bit.

Andeep has found a sequence for us to try.

It starts with a number six.

We can see clearly that one term is not enough to determine a rule.

We could pick any term to term rule and apply it to the number six.

He's given us some more information.

Now the second term is seven.

Can we now determine a rule? What do you think? I mean we can find rules that work for six and seven, but there's going to be more than one option.

There's going to be loads of different rules that work for six and seven.

And remember that sequences don't necessarily have to follow a mathematical rule.

He gives us a third number.

So the third number is nine.

Our sequence now is 6, 7, 9.

Sofia says this cannot be an arithmetic or geometric sequence.

Alex says it's a quadratic sequence.

It adds one then two.

So next it'll add three.

Do you agree with either of those statements? Perfect.

Sofia is correct.

There is no common difference or common ratio.

So these cannot be the first three terms in a linear or a geometric sequence.

Alex is correct that it could be quadratic, it could have a common second difference add one, then two, it could then add three, then four and so on.

But I'm sure you're getting this idea now.

We do not know that it definitely is quadratic anything could happen for the next term.

So the next term is 12.

So Andeep's sequence so far goes 6, 7, 9, 12.

This makes Alex feel good 'cause Alex says it is quadratic then.

Add nine plus three is 12, we've now added one, then two, then three.

So far it looks like it's worked for all terms. Is that enough to say for sure what the rule is? We've got four terms now.

Can we confidently say what the rule is? What do you think? No, just like we've already said, there's still no guarantee that Alex's rule will continue working.

In fact, without knowing a rule we cannot assume we know how a sequence will continue even if we're given lots of terms and we can see that that so far there's a pattern, we do not know that that's necessarily going to continue Andeep's told us now where he is got these numbers from.

These are actually the average daytime temperatures in my hometown for each month of the year and he's done them in degrees Celsius.

Let's plot them on a graph and see if we can spot a trend.

There we go.

We've got the months of the year starting on January.

So January we had six degrees, February seven degrees, then nine, then 12.

What do you now think the fifth term will be? See if you can explain your answer.

So we can see that there is a general trend in the values, so far they are increasing and they're also increasing at a faster rate each time.

So you can see that we end up with a curve as the difference between terms is increasing.

So the next month is May.

So we'd imagine the temperature would increase again.

I wonder if you thought that it would carry on following Alex's pattern.

In fact, the average temperature was 16 degrees.

Alex's rule still works.

We've added one, then two, then three, then four.

Spotting a pattern early did allow us to predict what was going to happen next.

Sequences can be really useful for predicting future values.

What do you think's going to happen in June? So if Alex's role was to continue the June, average temperatures would be 21, we'd then add five.

I wonder if you went with 21 or if you made a different prediction.

In fact, it was 19 degrees C.

So you can see that it is not increased by quite as much.

It has increased but not by the same amount.

So our sequence is now 6, 7, 9, 12, 16, 19.

Last chance have a prediction for July and August and then we'll see the rest of his graph.

I wonder if you then went back to the terms and had a look for a pattern.

So we've added one, then two, then three, then four, then it's added three.

So I wonder if the difference is going to decrease.

If you were following that rule, you might have said that next time it's going to increase by two degrees, it would be 21, but there's no way to know for sure that that's how the sequence grows.

In fact, it was actually 20 and then 19.

Were they what you expected? Here are all the numbers in Andeep's sequence.

So those were all the average temperatures in in his hometown for the months from January through to December, you'll see that we've ended up with a a curve which was increasing and then it was de decreasing.

It's not perfectly symmetrical though.

Hopefully what you've seen from this is there's no number of terms which are definitely enough to determine a rule for a sequence.

After a few terms, you might be able to predict patterns that might continue and that is called extrapolating.

So when you use what is already happened to make a prediction for the future, we say that's extrapolating.

This is not always guaranteed to work.

So for example, if you thought that that was a quadratic sequence and then you added six and then seven and then eight, that's not going to work, 'cause of course the temperatures are going to start dropping at a certain point.

Temperatures as well are not uniform.

So anything could happen.

You might have a year whether summer was particularly warm or particularly cool, but this can be the case with mathematical sequences too, not just real life examples.

Let's have a look at the prime numbers, ignoring two, we'll have a look at the odd prime numbers.

So we've got 3, 5, 7 at the moment, that could be an arithmetic sequence, but the next five terms are 11, 13, 17, 19, and 23.

If we ignore that three for the moment, we've now got this pattern.

We're adding two, then four, then two, then four and two and four.

So there might be a new pattern emerging.

Let's see if this continues.

The next number, if we followed this sequence would be 25, but 25 is not prime.

It's divisible by five.

So there was a pattern that seemed to form for a little while but then eventually it failed.

This is something that people who work with prime numbers do use to explore different structures.

There are other rules that happen for prime numbers further on in the sequence of prime numbers, which seem to hold for a really long time, but again, they eventually fail.

They can be really useful tools therefore to work with we need to be aware that when patterns form we can't say for certain that's always going to be the case.

Have a look at this example.

This table shows the amount of money in a savings account at the end of the month, your August, September, October, November, that's the amount of money in pounds.

What do you think is happening every month? See if you can put it into words and what do you predict the value in the account to be at the end of December? Give this a go.

So you might have noticed that 190 pounds is being saved.

Often people have a set amount of money that they will save each month.

They might have it on a standing order.

Some people would like to save a certain percentage of their earnings.

I wonder what you predicted would be in the account at the end of December.

Obviously whatever value you went for could be correct.

If we assumed this trend continues, it would be 2,013 pounds.

Actually the value is 1,900 pounds.

Do you think you can think of a reason why that might be? Spend a bit of time and then we'll see if we've come up with any of the same ideas.

So I thought that in December people might save less money because it's around the holidays and they might be spending money on other things.

It might be that that person has changed jobs and therefore can't save the same amount of money.

We cannot assume that the trend will continue, for your task, we are going to investigate patterns found when points marked on a circle are joined with straight lines.

I'm going to explain how this is going to work.

Then you are going to have a go at exploring this idea yourself.

So we've got a circle and every point on the circle is going to be joined with a straight line.

For example, there are four points there on the circle and we need to join each point to every other point with a straight line.

We call those bits on the circle, the points And then the lines connecting them we're going to call lines.

We've put them all in now.

So here are all the possible lines.

Intersections are where the two lines cross, so we're not counting where the lines meet on the points, we're just counting where they cross inside the circle.

The regions are the areas that are created by the lines and the circle.

So you've got a region there.

And then I'm going to show you the other regions.

So in this case we have one, two, three, four, five, six, seven, eight regions.

In this example, there are four points, six lines, one intersection, and eight regions.

In the examples you are going to try, the points are going to be equidistance round the outside of the circle.

They are drawn for you so you don't need to do that part.

So start with a circle with one point marked.

As there is only one point, we cannot join the points with a line.

So there are zero lines, there are no lines, so there's no lines to cross each other.

So there are no intersections, but there is one region.

Let's have a look at two points.

We've got one line, no intersections and two regions.

I've given you a table to record what you find.

I would like you to fill in the table for the two points and then try it with three points and four points.

Stop at four points and describe any patterns you notice.

Pause the video and do that bit now.

Okay, so now you've done the first four.

I'd like you to make a prediction for the fifth column.

Make sure you make that prediction before you work it out.

Then fill in the fifth column using the circle and I want you to write a sentence about whether you are correct and whether any of your patterns have changed.

Okay, now I'd like you to have a prediction for the sixth column before you try it.

And again, write a sentence explaining what you have found.

And lastly, I'd like you to think about whether any of the patterns you've found will continue, write a sentence explaining why you think they will or won't or whether you've got enough information.

Reflect on what you've discovered so far and then we'll look at what we've done together.

Well done.

I'm going to show you the full table on the next slide.

So for two you should have 1, 0, 2 for three, 3, 0, 4 and for four, 6, 1 8.

I wonder what predictions you made at this point.

So you might have discovered that the number of lines increased by one, then by two, then by three.

So you might have then said that they're going to increase by four, then by five, then by six.

You might have even said that they look to be the triangular numbers.

If we ignore 0, 1, 3, 6 are triangular numbers.

You might have seen that the number of regions we've got 1, 2, 4, 8.

That seems to be doubling each time.

At the moment it's hard to say anything about intersections.

We haven't really got enough information.

Your fifth column there should be 10, 5, 16.

I wonder if that followed any of your predictions.

And again that the number of lines and regions have continued with the same pattern we identified and still probably not enough information for the intersections.

You might make a prediction, you might say, well it might increase by four again and be nine next time.

Or you might think something else is going to happen.

Let's have a look.

Okay, so for six we've got 15, 13, 30.

This is where things got interesting.

So your answer might have included this idea, the number of regions looked geometric, but 16 multiplied by two would be 32.

The way we've drawn this shape, we actually end up with 30 regions.

If your hexagon doesn't end up being symmetrical, it is actually possible to get 31 regions.

That's still not 32, which will be following the geometric sequence.

The number of lines for pattern two onwards did seem to be the triangle numbers and that so far has continued.

The number of in sections is not an arithmetic sequence, it's not adding four each time.

We've still probably not got enough terms to make any prediction.

I wonder whether you think then that any of these patterns will continue.

The number of lines will continue to be the triangular numbers and this is because the triangular numbers is the same as the number of connections between two and three or four or five objects.

There's lots of different activities and lots of different patterns that form the triangular numbers and that will continue in this case.

It is unclear how the other two patterns will continue.

Well done today, I hope you had a little bit of fun exploring those patterns.

We've seen that not all numerical sequences follow a mathematical rule.

They're just a set of numbers written in an order.

However, if we know a few terms in a sequence, we can predict how the sequence might continue, but it's possible that it could continue in multiple ways.

Even if a pattern applies to many terms, there's no guarantee that will continue.

Think about that regions one that we just looked at.

It was a mathematical rule.

It looked like it was going to be doubling each time, but that at some point it didn't.

Finding the patterns is still really useful 'cause it can help us make predictions.

We just got to be aware that we can't assume they're going to be correct.

Really enjoyed having to play around with that today and I hope you join us for more lessons in the future.