Lesson video

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Hello, I'm Mr. Langton and today we're going to look at some algebraic proofs.

All you're going to need is something to write with and something to write on.

Try and find a quiet space where you won't be disturbed and when you're ready, we'll begin.

We'll start with a try this activity.

Fill in the blanks to each one with either odd or even.

Pause the video and have a go.

When you're ready, un-pause it.

We can go through it together.

You can start in three, two, one.

Here are the answers.

How did you get on? It's rather easy to show that adding two odd numbers together gives you an even number.

We can do it by example.

Three plus five equals eight.

There's an example of two odd numbers add together to make an even number.

Now what I'd like to do is prove that's always the case and I'd like to do that with some algebra.

Now, in a previous lesson, we looked at how we can represent odd and even numbers.

Now an example of an even number is any number that can be divided by two.

So we can represent an even number algebraically as two times by a number.

So for example, four is two times two.

10 is two times five.

50 is two times 25.

Every even number can be written like that.

Now odd numbers are always either one bigger or one smaller than an even number.

So if an even number is two lots of something, then an odd number can be written as two lots of something plus one.

For example, seven is two lots of three, add one.

Nine is two lots of four, add one.

Now, what we're going to do is use these algebraic expressions with these words here.

So if an odd number can be written as two n plus one, and we're going to add to that another odd number.

That gives us four n plus two.

Now four, we can split that up into two equal parts.

We can write four n like this.

And we can write two, two ones like that.

So that's four n plus two.

And I can split that into two equal groups with nothing left over.

Which means that this bit here, four n plus two, must be even.

So we've now got a formula.

We've proven that any time we add together two odd numbers the answer must be even.

Zacky's made a statement.

He says, "If I add together four consecutive numbers "then the answer will be even." Do you agree or do you disagree? Try it for yourself.

Pick four consecutive numbers.

For example, three, four, five and six.

Let's add those together.

Three and four is seven.

Seven and five is 12.

12 and six is 18.

That's even.

So pause the video, try it for yourself.

Pick some different numbers, see if you can come up with the same answer.

Can you always get an even number if you add four consecutive numbers? You can pause in three, two, one.

Could you find anything where it doesn't work? It seems to me like I'm going to have to agree.

Seems like it always works.

But what I'd like to do is prove it algebraically.

Now, my four numbers, each one is one bigger than the last, isn't it? So whatever I pick as my first number, I'm going to call that n.

And the number that comes after that, the next consecutive number, is going to be one bigger.

So that next number would be n plus one.

The number that comes after that will be one bigger than that, n plus one plus one.

Or n plus two.

And the fourth number that comes after that is going to be one bigger than that, n plus three.

So we've got here a formula that will list any four consecutive numbers if we want.

So, for example, if n was three, we'd add one to that to get four or three plus two is five or three plus three is six.

So what happens when we add together these consecutive numbers? We're going to do n, add n plus one, add n plus two, add n plus three.

So in this case, I've got one, two, three, four ns.

And I've got one, two, three, four, five, six.

So six.

Now I can split that into two equal groups, can't I? So we can show that this must always be even.

I'll draw this diagram just to help.

That's four n.

That's one, two, three, four, five, six.

And I've got two equal groups.

So I've shown I can pick any four numbers, any four consecutive numbers, and they will always add up to make an even number.

This has been really tricky so far, but now it's your turn to have a go.

Access the worksheet, see how many you can do and then we'll go over them together.

If you're not very confident, just leave the video going a little bit longer and I'll give you some hints on our first one.

Good luck.

I'm going to start off by giving you some hints.

First question is an even number adding an even number.

We can represent an even number, we've already said, as two n.

So can you show me what happens when we add those together? What about an even number plus an odd number? An even number can be shown like that, and an odd number can be shown like that.

Now, what about question two.

Use algebra to show what happens when you add three consecutive numbers together.

What do you notice about your answer? So here's a hint.

That's the first number, the next number is n plus one and the next number is n plus two.

I'm going to leave that there.

If you've already finished, just leave the video going.

But if this is your hint, if you were using this as a hint, pause the video now, see how many you can do.

Pause in three, two, one.

Okay, let's go over the answers.

So two n plus two n would make four n, which we can easily show splits into two equal groups.

And if it splits into two equal groups, then it must be even.

Question B, two n plus two n plus one equals four n plus one.

Now follow from what we just said, if four n is even, then four n plus one is one bigger.

It must be odd.

We can draw it.

Like so.

So we can see we can't get two equal groups, can we? That and that are not the same.

So in this case, even plus odd is odd and we've proven it.

Now an even number multiplied by an even number.

That's going to be two n multiplied by two n, which is four n squared.

So what would that look like? I've got for lots of n squared.

So I've got n squared, n squared, n squared and another n squared.

They can be split into two equal groups.

So we've proven that when we multiply together two even numbers, our answer is even.

Finally, even times odd.

That means we need to do two n multiplied by two n plus one.

I'll put some brackets around that.

Two n times two n is four n squared.

Two n times one is two n.

Now, I'm just going to pick a different colour.

Just make some space over here where we can draw.

Four n squared plus two n.

So I've got four n squareds, one, two, three, four.

And I've got two ns.

So we'll just add those, so that's an n and that's an n.

And we can see we've got two equally sized groups.

Each of those groups contains two n squared plus n, which means if we do an even number multiplied by an odd number, the answer is always even.

And finally, question two.

What happens when we add these three consecutive numbers together? So we have n, we've added n plus one and n plus two.

So we've got three n plus three.

Now, is it odd? Is it even? Not quite sure.

I'll tell you one thing, though.

It's going to be a multiple of three.

If you add together three consecutive numbers, it will always be in the three times table.

And I can show you that like this.

N, n, n, that's my three n.

And that's my three.

And they can be split into three equal groups, can't they? So you can pick any three consecutive numbers you'd like and when they add together, they will always make a number in the three times table.

You can do a quick example and just check.

What about 10, 11 and 12.

If I add those numbers together, that makes 33, doesn't it? And 33, obviously, is in the three times table.

Finally, we've got the explore activity.

And this is really, really tricky.

You can certainly have a go at it, but if it comes to using algebra, it's going to get a little bit nasty.

Binh has got an idea.

She says, "If you pick any five numbers, "she can chose three of those five, add them together "and they will always make a multiple of three." So, Antony had a go.

He says, "What about eight, seven, four, five and three?" Have a look at those numbers and from those five that Antony's picked, Binh's picked out seven, three and five, which add together to make 15.

And 15 is in the three times table.

So why don't you investigate Binh's idea and see if she's always right.

Have a go by yourself and when you're done, un-pause it and we'll have a look at it algebraically.

See if we can figure out what's going on there.

You can pause in three, two, one.

How did you get on? Hopefully, you found that Binh was correct.

Any five numbers that you choose, three of them can be added to make a multiple of three.

And we'll start to show you how that works and I'm not going to go all the way into proving it, because it's going to get quite tricky.

But I just want to show you just an idea of how we start off, just by doing that.

I've listed the first numbers from zero to 12 and obviously, this list goes on and on forever.

All the integers keep going on and on and on and on.

I'm going to start off just by looking at the numbers in the three times table.

Three, six, nine, 12 and so on.

Each of these numbers can be made by multiplying a different number by three.

Three times something will give me any of those numbers.

And in fact, three times zero is zero, isn't it? So I've got a formula that can make an awful lot of the integers that exist.

Next up.

Let's look at four.

It's one bigger than three.

Seven is one bigger than six.

10 is one bigger than nine.

So I'll come up with a formula for those.

Those numbers are one bigger than a number in the three times table.

That's going to include one, because three times zero is zero, add one is one.

And there's only one gap between each bit now.

So let's have a look there.

Five is two bigger than three.

Eight is two bigger than six.

11 is two bigger than nine.

So I've got a new formula now.

Those numbers are all two bigger than a number in the three times table.

And actually two goes with that as well, doesn't it? Which is two bigger than zero.

Now, this one is already a multiple of three.

These two aren't, but we can do some maths to show that whichever combo, you can pick any sorts of combinations that make multiples of three.

For example, if I add them together, one and two is three.

Three n, add three n, add three n is nine n plus three.

And that is a multiple of three, isn't it? Because three goes into nine and three goes into three.

And whichever combination of five numbers you pick, you must pick three numbers that will make a multiple of three.

The are other ways that you can do it.

There's lots more ways, I'm not going to show you them all now, because it will take quite a while but that's how you'd start to prove it.

That's it for today's lesson.

I'd love to see any ideas and any work that you've done so far.

If you'd like to, please ask your parent or carer to show your work on Twitter, tagging @OakNational and #LearnwithOak.

See you later, goodbye.