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 gonna be helping you as you work your way through this lesson.

Let's get started.

Welcome to our problem-solving lesson with non-linear relationships.

You are gonna be using your knowledge of non-linear relationships to solve problems today.

If you're not confident with a geometric sequence or what a triangular number is, just pause the video and read through 'cause we're gonna use those in the lesson.

We're gonna start by looking at problem solving with geometric relationships.

So the game of chess was invented many, many years ago, and there's a story behind its invention.

So the story goes that the finished game was presented to a mean king who refused to share his vast stores of rice with the people despite there being a famine.

The king liked the game and offered a prize of whatever reward this person wished for.

What this person said was, "I would like a single grain of rice on the first square of the board." Doesn't sound too much, does it? "Two grains on the second square, four grains on the third square, and then doubling each time." Think about it.

Four grains of rice doesn't seem that much.

Even eight grains of rice is not gonna make a dinner.

The king was startled at such a small price for such a wonderful game, and he began to count out the rice.

There we go, we've got one, two, four.

How many grains will there be on the 64th square? You might wanna make a prediction, and then we're gonna work through it.

Let's look at that top row.

We've got 1, 2, 4, 8, 16, 32, 64, 128.

Right, we've got to 128, which again does not seem too many grains of rice.

However, that's gonna take a while to count out, isn't it? Let's look at the next row.

Wow, so by the end of that next row, we're up to 32,768 grains of rice.

In that third row, there's over 8 million grains by the 24th square.

Let's look at how many there'd be on the 64th square.

You're gonna need your calculators for this.

If we have a look at a rule, 1 is 2 to the power of 0.

2 is 2 to the power of 1.

3 is 2 squared.

4 is 2 cubed.

We're multiplying by 2 each time, so that exponent is gonna increase by 1 each time.

So for the 64th square, we're gonna end up with 2 to the power of 63.

You might wanna type that into your calculators.

Your calculators have probably given you an answer in something called standard form because it's so big, so that is roughly 9 times 10 to the power of 18.

So that's 9 times 10 times 10 times 10 18 times.

That is the size of the number we are looking at, 9 then with 18 zeros.

It's the number 9 quintillion.

Obviously, the king was unable to pay his debt.

This story, we don't know if it's true or not, but it's a good example of the power of geometric sequences, and it's also the limitations of using this element in modelling.

So we know that things can double, and that is a sequence we can follow for a while, but real-life examples may not follow that pattern as the numbers get too big, too fast.

Let's have a look at this example.

There are 2,000 penguins living on an island.

Every decade, their population doubles.

So there we've got 2,000, each penguin's representing 1,000.

Then after a decade, there'll be 4,000 penguins, then 8,000, 16,000, 32,000, 64,000.

I'm sure you can see a problem here.

The island will eventually run out of space.

The waters will run out of food.

When penguins have to swim further for food, they spend more time in the water with their predators, so there'll be a natural limit to the population on the island.

Let's have a look at what this looks like when we graph it.

So we're doubling each time on the y-axis, and the x-axis is showing each decade.

But what we know is somewhere there'll be a natural limit.

There'll be the amount of penguins that can live comfortably and survive on that island.

So growth cannot be endless.

The line would flatten out somewhere.

Time for you to apply this.

When the sun comes out in spring, flowers grow very quickly.

The flower's height is following a geometric pattern.

It's shown in the table to your left.

What could we assume the flower's height will be next week? And what problems do you foresee with this growth model in the future? Spend a bit of time, come back to the answers.

So if we follow this geometric pattern of multiplying by 3, the plant will be 27 centimetres tall in 4 weeks.

Hopefully you said something like growth is not usually endless.

This model will have a natural limit when the plant reaches its natural height.

If it were to continue, then in 8 weeks, the flower would be over 20 metres tall, which does not sound realistic.

Time for you to have a practise.

So something else that often forms a geometric sequence is the passing of information.

So in this scenario, the bus is gonna be late back from a school trip, and only one pupil phones their parent.

That parent then phones 5 other parents, and each of those parents phone 5 other parents.

For this example, we're gonna assume it takes 10 minutes to make those 5 phone calls.

How many people would know after 40 minutes? Why might this not be realistic? If it carried on, how many people would know after 2 hours? And why might that not be realistic? Try this out, come back when you're ready for the answers.

Let's have a look then.

So if we multiplied by 5 every 10 minutes, then after 40 minutes, there'd be 625 plus the original parent who found out first.

So 626 people would know if the growth continued at the above rate.

Why might this not be realistic? Well, some parents might ring the same people.

Some parents might not ring anyone.

You might have said it might take longer than 10 minutes to ring 5 parents.

Depending on the size of the school trip, you might be thinking that 625 sounds like too many parents to be rung in the first place.

Let's have a look after 2 hours if this was to continue.

So down the left-hand side, you can see it's multiplying by 5.

Then I started to get into rather big numbers, so I thought I might think about my exponents and use my calculator.

So after 10 minutes, it's 5 people.

Then we're multiplying that by 5, so that's 5 squared.

Then we're multiplying that by 5, which is 5 cubed, 5 to the power 4, and so on.

For 2 hours then, it'd be 5 to the power of 12.

That's the amount of 10 minutes in 2 hours.

5 to the power of 12, and then I've added 1 for that original parent, gives me that value on my calculator.

That is approximately 200 million people.

Well, we know for a start that no school is that big, so there will not be that many parents to tell.

There is gonna be a natural limit to that information passing.

Now we're gonna have a look at adding sequences.

We're gonna investigate the idea of what happens when we add the values in two different sequences together.

Sofia says, "I think adding two arithmetic sequences will give a different arithmetic sequence." I'm gonna show you an example.

So if we add 3, 5, 7, 9, 11 to the sequence 4, 7, 10, 13, 16, so if we add the first terms together, the second terms together and so on, we get the sequence 7, 12, 17, 22, 27.

That's an arithmetic sequence with a common difference +7.

I'd like you to try your own example.

What do you think to Sofia's statement? So Sofia is correct for my example.

I wonder if this will always work.

We're gonna try with an increasing and a decreasing sequence.

So 5, 8, 11, 14, and 10, 8, 6, 4.

We're gonna add those together, and they're both gonna continue with a linear pattern.

When we add them, we have 15, 16, 17, 18.

So we have an arithmetic sequence with a common difference of 1.

I wonder if you started noticing any links between the common differences.

We've only tried a couple of examples.

We can't say it always works.

Even if we tried 100 examples, it would not be proof that this always works.

What we can do is we can use algebra to explore the structure and make a proof that it will definitely always work.

So let's look at the nth term rules.

The nth term rule of the first sequence is 2n + 1, and the second one is 3n + 1.

If we wanted a number in the sequences added together, you could pick a number for n, you could apply the rule 2n + 1, you can apply the rule 3n + 1, and then you can add them together.

Adding those together, we get 5n + 2.

So the rule for our new sequence should be 5n + 2.

And if we have a look at it, that seems to be true.

We're increasing by five each time, and seven is two more than the first number in the five times table.

That is the correct nth term rule.

So what we can see is adding two sequences means we can add their nth term rules.

If we have two linear expressions, adding them together will always give another linear expression.

You can prove that using algebra.

Sofia says, "What about if we add two quadratic sequences together?" A quadratic sequence has a linear relationship in the differences.

We can say they have a common second difference.

So for example, a sequence might add 2, then 4, then 6, then 8, so the linear pattern in those differences.

Try an example, see what you think is happening.

Here's my example.

So I've gone 2, 6, 12, 20, 30.

It has a common second difference of +2.

And then 5, 6, 8, 11, 15 with a common second difference of +1.

When I add those together, I get 7, 12, 20, 31, 45.

So that's increasing by 5, then 8, then 11, then 14.

So that seems to be a quadratic sequence, this time with a common second difference of 3.

We can explore this algebraically but not without knowing the nth term rules.

For now, we're just gonna try some more examples.

Okay, you're gonna try this one then.

"The square numbers form a quadratic sequence when written in numerical order." "The triangular numbers do too!" I'd like you to write the first five terms of the sequence of square numbers in ascending order.

Then do the same for the triangular numbers.

And then I'd like you to add your sequences together using the same rules we did on the previous examples.

What do you notice about your new sequence? Give it a go.

So they're your square numbers and your triangular numbers.

Adding together, we should get 2, 7, 15, 26, 40.

We have got a quadratic sequence.

It increases by 5, 8, 11, 14 with a common second difference of +3.

If you think about it, your square numbers have a common second difference of 2, and your triangular numbers have a common second difference of 1.

When we added them, we got a common second difference of 3.

Time to put that all together then.

So in each question, the sequences were created by adding the terms of two of the sequences below.

So I've given you sequences A to H, and they were added to make the sequences in questions A to I.

I'd like you to work out which two sequences were added in each question.

You might wanna spend some time looking at your sequences.

See if you can spot any linear sequences, any geometric sequences, any that have a common second difference, and then that might help you with A to I.

Well done.

So you can use the examples that you looked at previously and any of your own examples.

I'd like you to investigate the following questions.

Write a sentence about any patterns you have noticed.

Okay, so for A, you had C + D.

B was A + C.

C was E + F.

D was D + F.

E was A + D.

And F was D + H.

G was A + B.

H was C + G.

And I was E + G.

For question 2, I wanted the nth term of sequence A.

Sequence A had an nth term of 7n - 2.

Then I wanted you to look at the nth term rules for the sequences that were added to get A, which was C and D.

The nth term rule for C is 5n - 1, and for D is 2n - 1.

When you add 5n - 1 and 2n - 1, you get 7n - 2, which was our rule for A.

Essentially what we've looked at is that adding the two nth term rules will give you the nth term rule for the sequence when you've added the values together.

Okay, lots of things you could have discovered when you were exploring these.

So when you add two geometric sequences together, interestingly, you do not get another geometric sequence.

In fact, it's hard to see any arithmetic or geometric pattern in the differences or ratios between terms, which was maybe quite a surprising result.

Two quadratic sequences, often you get another quadratic sequence where the common second difference is equal to the sum of the second differences in the sequences that you have added.

So we saw that before with the square and the triangular numbers.

They had a common second difference of 2 and 1.

The new sequence had a common second difference of 3.

However, and I wonder if you spotted this, it was in question I of question 1, if the second differences are additive inverses of each other, you can add two quadratic sequences and get a linear sequence.

So if the differences are increasing by two in one sequence but decreasing by two in another sequence, when you add them together, you can end up with a linear sequence.

Most of the time, though, two quadratic sequences add to another quadratic sequence.

For C, a linear sequence to a geometric sequence.

So the resulting sequences were not arithmetic or geometric, but you might have seen a pattern.

There seems to be a geometric relationship in the second differences between terms. And finally, a linear to a quadratic sequence, you will get a quadratic sequence.

If you knew the nth term rules, you could add the nth term rules to get the new nth term.

And if you add a quadratic expression to a linear expression, you just get another quadratic expression.

None of that is stuff that you need to memorise, but it's really good that you've spent some time looking at those sequences, spotting patterns in the way things develop.

That will help your mathematical reasoning skills as you improve with your mathematics skills.

Well done today.

We have looked at how spotting a sequence makes it possible to predict behaviour.

We've looked at applying mathematical modelling to real-life situations, such as the penguins on the island.

But we've seen how that has its limitations.

Then we had fun playing around building sequences by adding them together.

We looked at how the nth term rules could be added together to get the rule for the new sequence.

If you go on to do further sequences in the future, that concept of adding the nth term rules together gives us a really nice way of finding nth terms of trickier sequences.

We have seen how there's problems in mathematics that are ongoing, that mathematicians are working to figure out.

I'm hoping you're thinking that'll be a really interesting thing to spend some time doing in the future.

That is what mathematics is all about.