Lesson video

Well done for making decision to learn using this video today.

My name is Ms. Davis and I'm gonna be helping you as you work your way through this lesson on proof.

We've got lots of exciting things coming up, so make sure you've got everything you need and let's get started.

Welcome to today's lesson where we're gonna be making conjectures about patterns and relationships.

So today is a fantastic lesson where we're absolutely fine to be incorrect.

We're gonna have a look at making some guesses about things and then seeing if we are right or not.

And it's just as interesting when we are not correct first time as when we are.

So we're gonna look at making and testing conjectures about the generalisations that underpin certain patterns and relationships.

So our knowledge of mathematical patterns today is gonna be really put to the test.

So there's a few key words that we're gonna use in this lesson.

We're gonna talk a bit about prime numbers.

So we need to be happy that a prime number is an integer greater than one with exactly two factors.

We might also come across the triangular numbers today.

A triangular number is a number that could be represented by a pattern of dots arranged into an equilateral triangle.

The term number is the number of dots in a side of the triangle.

The first five are written for you.

And our main focus today is conjecture.

A conjecture is a mathematical statement that is thought to be true but has not yet been proven.

So I'm gonna start by testing some conjectures.

There are lots of interesting patterns and relationships in mathematics.

So some pupils are investigating prime numbers.

Laura says seven and three are both prime.

When I multiply them I get an odd number.

Seven times three is 21.

Andeep tries them too.

11 times five is 55, 13 times 17 is 221.

We have tried a few and I think that when we multiply two prime numbers, the result is always odd.

So this is a conjecture, it's a prediction or a conclusion, but it's not been verified or proven to always be true.

So it's only a conjecture at the moment.

In mathematics there's lots of things that are still conjectures 'cause mathematicians can't prove that they're always true.

So Andeep said, I conjecture that the product of two prime numbers is always odd.

What we can do then is we can test the conjecture to see if it halts.

So there's four more examples.

So Laura says we have tried seven examples now, how many is enough to know for sure if Andeep is correct? Well, we cannot know for sure if this conjecture is true by testing examples.

There's some conjectures you can prove true by testing examples, but most of them we won't be able to be sure if they're always true.

Because what if two prime numbers existed, which have an even product and we haven't thought of those ones yet.

We haven't tried those ones yet.

Laura says, I've just remembered two is a prime number and five times two is 10.

This does not give an odd product.

So it's possible to disprove this conjecture by testing and any conjecture it is possible to disprove if it's not true by testing.

We just need to find the one example where it does not work.

In this case, Andy's conjecture did not hold.

Quick check.

It is possible to disprove a conjecture by testing it.

What do you think? Can you justify your answer? Yeah, true.

All we need to do is find one example that does not work.

We don't have to test every single example to disprove a conjecture.

We only need to find one that doesn't work.

Sam says, I've been exploring square numbers.

Nine and four are consecutive square numbers and nine subtract four is five.

I conjecture that the difference between consecutive square numbers is always odd.

I'd like you to pause the video and test another three examples to see if Sam's conjecture holds for a few more examples.

Right, you could have tried all sorts.

These were the three that I tried and so far Sam's conjecture seems to hold.

I wonder if you found the same for your examples.

But, we can't claim that Sam's conjecture always holds just by testing the examples 'cause we'd have to test every pair of consecutive square numbers and it's not possible to do that because they are infinite.

Andeep says one and nine are square numbers and nine subtract one is eight and that's not odd.

Why does this not disprove Sam's conjecture? Well, I dunno if your brain was working for this one.

Sam's conjecture is about consecutive square numbers and one and nine are not consecutive.

So you need to be really clear what it is that you are trying to prove or disprove.

They are our consecutive square numbers or the first six anyway and actually Sam's conjecture will hold for all cases.

It is a mathematical fact that the difference between consecutive square numbers is hot.

However, we would need a more formal method to prove this and if you carry on with your proof skills, that may well be something that you can prove.

So Laura says, I conjecture that every multiple of five has an even number of factors.

I'd like you to pick three examples and give this a go.

But you could have gone with all sorts of multiples of five.

Those are the examples for five, 10 and 20.

If you haven't already, can you find an example which disproves her conjecture? Give this one a go.

The smallest one is 25 'cause the factors of 25 are one, five and 25.

The piece of information that's helpful here is that square numbers have an odd number of factors.

So you need a square number that's a multiple of five and the smallest one is 25.

Time to give this a go.

So Lucas is exploring prime numbers.

He wants a way of generating numbers which are definitely prime.

He does two squared subtract one, two to power five subtract one, two to power seven subtract one.

And his conjecture is that numbers of the form two to the N minus one where N is an integer greater than one are always prime.

Why can Lucas not make this statement at the moment? I'd like you to rewrite his statement as a conjecture.

I'd like you to test another two examples, see if it holds, give that a go.

So for question two, Laura has written the first six square numbers.

She says, I conjecture that all square numbers have a ones digit of zero, one, four, six or nine.

I'd like you to test another five square numbers and then write a conclusion to explain what you have found.

Give that one a go.

So Andeep is looking at square and triangular numbers.

He's written the first four square numbers and the first four triangular numbers.

Andeep says I conjecture that the only number which is square and triangular is one.

How would you go about testing his conjecture? Give it a go and write a conclusion.

Come back when you're ready for the next bit.

Question four, we are gonna study the collapse conjecture.

This may have been something you've seen before, it might not.

So this is a mathematical conjecture and it says that any sequence of positive integers which follow the rule below will eventually reach the number one.

So what that means is if you pick any positive integer and follow the rule that I'm about to explain, you'll eventually reach one, that's the conjecture.

So if the number is even you half it to get the next number, but if that's odd, you multiply it by three and add one to get the next number.

And you follow that rule and see if you can reach the number one.

I'll give you an example.

If you start on 10, that's even so half to get five, that's odd.

So we need to time it by three and add one that's 16.

That's even so we'll half to get eight that's even so half to get four, half to get two, half to get one.

And for that example we have reached the number one.

So our conjecture holds for the number 10.

What I'd like you to do is test the first nine positive integers, 'cause I've done 10 for you.

And then test an integer of your choice greater than 10.

Anyone you like.

Part C, can we say for sure if this conjecture holds, can you give your answer as a full sentence? When you finish playing around with that conjecture, come back and we'll look at the answers.

So for Lucas, he's only tested a few examples.

Even if he tested loads of examples, that's not to prove it always works to be better off writing it as a conjecture.

So I conjecture the numbers of the form two to the power of N minus one where N is an integer greater than one are always prime.

Now this holds for the three values that he chose.

Now again, it's gonna depend on which example you have chosen.

So two cubed minus one is seven, which is prime, but two to the four minus one is 15, which is not prime.

And two to the six minus one is 63, which is not prime.

So this conjecture does not hold.

If you only found values that did hold, then it's okay to write the statement this conjecture holds for the tested values, but we've managed to find an example that doesn't work.

So I can say that does not hold.

It's important to note that mathematicians have not yet discovered a systematic way to generate all the prime numbers.

So if even if you think you found a rule that works for some prime numbers, may not work for all prime numbers unless you found something that mathematicians have not yet discovered.

So for Laura it'll depend what square numbers you tested.

I carried on and did seven squared, eight squared, nine squared, 10 squared and 11 squared.

My conclusion was that this conjecture holds for the examples chosen.

So all square numbers up to 11 squared ends in a zero, one, four, five, six or nine.

Interesting to note here, that in this case we have tried squaring numbers ending in all 10 digits, zero to nine and we know that when we multiply digits together, it's only the ones digit that determines what the ones digit and the answer's gonna be.

So actually we've tested all relevant cases so we can confidently say that this will always hold this time because we've tried every type of case that we could possibly come across.

It doesn't happen very often, but as I say, there are conjectures that you can prove true just by testing as long as you cover all the relevant cases Right, Andeep it is square and triangular numbers.

Best way is to keep following the sequences until you find a number in both.

If you did that you would've seen that 36 is both a square and a triangular number.

And this Collatz conjecture then.

So those are the first nine pause video if you need to check your sequences.

Now if you wanted an efficient answer for B, choose a number that's already come up in one of the previous sequences.

So 16, 20, 40, 13, all of those we've seen in other sequences.

So if you pick one of those you'll quickly get down to the number one.

Now we cannot say for sure that this conjecture holds, just testing examples is not enough.

We haven't covered all possible cases and this is an interesting one 'cause in fact mathematicians are still trying to find a way to formally prove or disprove this conjecture.

Computers have tried this for many, many, many numbers, but that's still not enough to say it will always work.

We need a formal proof and mathematicians cannot work out how to do that for this pattern at the moment.

So let's have a go at making our own conjectures.

We can form our own conjectures based on any branch of mathematics we like.

Lucas is playing around with prime numbers again.

He notices the numbers either side of six are both prime.

Five and seven are both prime.

Then he tries 12, 11 and 13 are both prime.

Then he tries 30.

29 and 31 are both prime.

What conjecture could he make from this information? Notice that he says numbers either side of nine and 15 are not prime.

So he they cannot form part of his conjecture.

What do you think? Can you work out what you think he's discovered? Right, you might have said the numbers either side of a multiple of six are prime, six, 12, and 30 are all multiples of six, nine and 15 aren't.

We're gonna have a go at writing this conjecture using algebra.

So any multiple of six can be written as 6n where N is a positive integer.

So we could write this conjecture as any number of the form 6n plus one or 6n minus one where N is a positive integer is prime.

And writing using algebra can sometimes make our conjectures a bit shorter or a little bit easier to prove or disprove later on.

So we could use our calculator checklist for larger values.

Let's try six times 32 minus one.

That would be one less than a multiple of six.

Well that's 191.

And your calculator may have a button which does the prime factorization of different numbers for you.

If you do have that function on your calculator, it is an easy way to see if this is prime.

So on mine you press format, change to prime factor and it'll write it as the product of prime factors.

Now this time it's just written 191.

So that must mean that there aren't any prime numbers that multiply to 191, therefore 191 is prime.

So do you agree with Lucas's conjecture? Take some time.

Do you think this is true? Can you find an example that doesn't work? There are lots of examples that don't work.

Well done if you found one.

36 is a multiple of six but 35 is not prime and that's one less than a multiple of six.

So this conjecture does not hold.

You might have tried a larger number to practise using your calculator.

So you could have done six times 20 minus one and that's 119 and we can use the prime factorization function on our calculators to show that that's seven times 17.

So not a prime number.

So Sam is exploring the powers of two.

Sam says, I have noticed something about the ones digits of all of these numbers.

What conjecture do you think Sam could make? Right, you might have said that all powers of two end in either a two, four, six or eight.

We could write this using algebra.

So we can say numbers of the form two to the power of N where N is a positive integer, have a ones digit of two, four, six or eight.

And writing these numbers as two to the power of N where N is a positive integer will help us make sure we're exploring the right cases.

Alright, I'd like you to pause the video and test this conjecture.

What conclusion could you come to? So I can tell you that this conjecture holds for all positive integer powers of two.

However, it's important to remember that just testing for a couple of values of N, not enough to prove that this is always true.

We'd have to use a more mathematical method to show that this is the case.

For now though, you've tested some examples and you can say that it holds true for the examples you tested.

Right, Sam says that numbers of the form two to the power of 4n where N is a positive in integer have a ones digit of six.

What do you think they mean by this? Right, if you have the powers of two handy, it'll be easier to see what they mean.

If the exponent of two is a multiple of four, so four, eight, 12, 16, then the value will end in the number six.

I would like you to test this conjecture.

What conclusion could you come to? Right, this conclusion should hold four tested values.

Laura's trying one.

She says, I think the numbers of the form two to the power of 3n will have a ones digit of eight.

I'd like you to test Laura's conjecture this time.

Right.

I've gone two to the power of three times two, two to the power of six, which is 64.

That doesn't have a ones digit of eight.

So this conjecture does not hold.

I'm sure you found a value as well.

So let's look at those first eight powers of two again.

There's definitely a repeating pattern, but it is not the pattern that Laura said.

Can you work out what the correct conjecture would be? This takes a bit of thinking, so pause the video, have a look through the powers of two, which ones ended in eight, how could we generalise that? Right.

The pattern repeats every four numbers, two, four, eight, six, two, four, eight, six and so on.

So it's numbers of the form two to the power of 4n minus one.

It's one less than a multiple of four that have a ones digit of eight.

Equally, numbers of the form two to the power of 4n minus two have a ones digit of four.

So we need to use that multiple of four because the repeating pattern is every four.

You are gonna explore this a little bit in the tasks, so make sure you're happy with that before you move on.

So Andeep is dividing square numbers by five.

I've tried five examples and square number divided by five always has a remainder of zero, one or four.

Write a sentence to explain why Andeep could not yet make this claim.

Well, if you said he is not shown that it's true for all possible cases.

He could make a conjecture and say it holds true for the examples he has tried or he could be more mathematical and see if he can cover all cases or prove this algebraically, which is a skill you might develop if you go on to do it more proof.

But for now he can't say that it's always true, only that it holds true for the examples he tried.

Right, Andeep has rewritten his example as a conjecture.

What information does he need to add to make sure he's only talking about square numbers divided by five? Read his conjecture, what does he need to add? He needs to define his variable.

We need to know what N is.

N has to be an integer for N squared to be a square number.

He doesn't need to say positive integer.

Integer is fine as any value squared is positive, even negative integer squared will be positive.

The only thing he might need to consider is zero and whether he's including zero as a square number.

If he's happy with zero as a square number, then N as an integer is absolutely fine.

If he's not happy with zero being included as a square number, he might need to say where N is not equal to zero, or he might then want to say positive integers.

It's just worth thinking about what values are included once you've defined your variable.

Right, your turn.

The first three powers of three are shown, this is similar to the one we just did with Sam and Laura with the powers of two.

I'd like you to write out some more powers of three and make a conjecture about numbers that form three to the power of N.

I'd like you to test your conjecture and write a conclusion.

Then I'd like you to do the same for numbers of the form four to the power of N and when you are happy exploring those, come back and we'll look at a different number pattern.

Right, Andeep is investigating something called factorials.

He writes the first four factorials.

So I'll show you some examples.

So one factorial is just one, two factorial means two times one, two times the integer below it.

Three factorial is three times two times one.

So three multiplied by the integer one than it multiply by the in integer one less of it until you get to one.

Four factorial is four times three times two times one.

And so what? Five factorial will be five times four times three times two times one.

Right now we're happy with factorials.

Let's look at his conjecture.

I've noticed something about the number of factors of each factorial.

So one has one factor, two has two factors, six has four factors, 24 has eight factors.

What do you think Andy has noticed? Can you write a conjecture.

When you've done that, test your conjecture and write a conclusion.

All right, Lucas is investigating other number patterns.

If I square an odd number greater than one, subtract one, then divide by eight, I've noticed something about my results.

I'd like you to try out some examples.

What do you notice? Write a conjecture.

How could you express the numbers Lucas is investigating using algebra? See if you can include that in your conjecture.

Give this a go.

And finally you've got some freedom.

So for this question, A is even, B is odd and C is prime.

Can you test the conjecture that C squared is always odd? And then I want you to make your own conjectures.

So use the basis that I've already given you and add in values into those gaps to make your own conjectures.

Make sure you test your conjectures and write a conclusion.

Doesn't matter if they're incorrect or correct, come up with something, test it out.

See what you find.

Once you finish playing where I'm with that, we'll come back and look at the answers Well done.

There's all sorts of conjectures you could have had.

You could have said numbers of the form three to the N where N is a positive integer.

Have a ones digit of one, three, seven or nine.

You might have been more specific and said numbers of the form three to the power of 4n have a ones digit of one or numbers of the form three to power of 4n minus one or numbers of the form three to power 4n minus two and so on.

You could have explored any of those ideas.

Depending on what conjecture you chose will dictate what your conclusion's going to be.

There is a pattern in the powers of three.

They do have a ones digit of three, then nine and seven, then one, then three, then nine and seven, then one.

So make sure that you've said whether your conjecture held or did not hold for those tested values.

The four to power of N again, it's gonna depend which conjecture you chose.

You could have said every power of four alternates between ending in a four and a six.

So numbers of the form four to the 2n have a ones digit of six, four to the 2n minus one have a ones digit four.

Make sure with your conjectures that you are defining your variables.

So where N is a positive integer, we are not looking at four to the power of negative one or four power of negative two.

It needs to be a positive integer.

Again, depending on what you chose would dictate your conclusion.

This is a mathematical truth.

They will alternate between four and six and it is something that you can show thinking about multiplying one's digits together and what that does to your answer.

So these factorials then, so Andeep has probably noticed that the number of factors doubles each time.

We've got one, then two, then four, then eight.

We've got a geometric sequence.

Your conjecture could have been that five factorial will have 16 factors and six factorial will have 32 factors.

You may have gone with other conjectures as well.

That's absolutely fine.

Now if we test the conjectures, I came up with five factorial is 120 and that does have 16 factors.

Quite a lot, it's true.

So the conjecture holds for five factorial.

Now six factorial is 720.

Now if you're trying to find all the factors of 720, prime factorization does help here, takes a while to write out your prime factorization and think about all the possible combinations.

You'll find that it only has 30 factors.

So that conjecture does not hold for six factorial.

Well done for persevering with that one, particularly finding all those factors.

So with Lucas, hopefully you tried out a few values and you might have noticed that they're all triangular numbers, right? In this shoes in algebra, the numbers could be written as N squared minus one over eight.

So our conjecture could be numbers of the form N squared minus one over eight.

Where N is an odd integer greater than one are always triangular numbers.

Now you could go one step further and this would be the way to prove this.

Because odd integers can be written as 2n minus one or 2n plus one.

You could have written numbers of the form 2n minus one or squared minus one over eight.

Where N is an intergrade than one are always triangular numbers.

The key bit here is we need to capture odd numbers, not including one, so 3, 5, 7, and nine.

So you could dictate that N has to be odd and greater than one.

Or a better way to write it would be to write it as two M minus one where N is greater than one.

So if N was two, for example, two times two is four minus one is three, and that's the first number we want to start on.

Just for your interest, if you're going on to prove this, you need it in that form because you need to be able to manipulate that and show why it works for odd numbers.

'Cause it doesn't work for even numbers.

Right.

So we're gonna test that C squared is always odd.

Well done if you remembered that prime numbers are all odd except for two.

So it will work for any of your odd numbers, 'cause an odd number squared is odd.

But if you remember that two is prime and two squared is four, then the conjecture does not hold.

If you didn't find that example, you might have written that your conjecture does hold.

Well done if you remembered that example.

For B, you could have come up with all sorts of ideas.

You might want to work with a partner to check through what you have come up with, hope you discover some interesting number facts or manage to show why some things are not the case.

Thank you for joining us today.

You have worked incredibly hard.

I hope you've had some fun looking at some of these patterns and relationships with number.

Sometimes when you're doing lots of maths you forget to stop and just look at some of these interesting results.

So I hope some of the things such as the powers of two, three, and four and some of those patterns, some of those things like what happens when you multiply odd numbers together and even numbers together are things that maybe you haven't thought about for a while.

Right, if you want to read through what we've looked at today, pause the video and do so now.

Otherwise, thank you for joining us and I really look forward to seeing you again.