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Hello, and welcome to today's lesson on number pyramids.

For today's lesson, all you'll need is a pen and paper or something to write on and with.

If you could please take a moment to clear away any distractions, including turning off any notifications, that would be brilliant.

And also, if you can, please try and find a quiet space to work where you won't be disturbed, okay? When you're ready, let's begin.

Okay, so the try this task, and let me just explain it before we start.

So, what's going on? Well, you see the numbers in the middle row, we get them by multiplying the numbers in the bottom row.

So, two times four gives us eight.

So the one above it, is the product of the two below it.

20, 20 is a product of four and five, okay? So four times five is 20, so 20 goes above it.

Now, for the top row, we add these two together.

So, eight plus 20 is 28.

And that's how these number pyramids work.

So what I would like you to do, is I would like you to pause the video and see if you can complete them.

But before you do so, I just want to tell you that the last one, there may have been more than one solution, okay? I would like you to pause the video and have a go in, three, two, one.

Okay, so welcome back.

Now, here are my answers.

And I have wrote e.

g.

here for the last one, because there is definitely more than one answer.

Okay, so you may need to pause the video to mark your work.

Okay, now, just let me talk about this one, just because, here, we were multiplying them.

So when we're working backwards, we're dividing them.

So 60 divided by four gives us 15.

Binh and Yasmin try filling the bottom row of the pyramid with three, five and eight.

They come up with the following conjectures.

Now, let me just explain to you what conjectures means in case you haven't heard that word before.

Conjectures is like, it's kind of, you see something, can you notice something? And then yeah, you make a statement.

And the statement might not be true, but it's a statement of what you think is going to happen, or what you think.

Some think, a property that something has.

So for example, if you'd only ever saw birds that could fly, you might make the conjecture that all birds can fly.

That conjecture is false.

Penguins can't fly, okay? So that is a counter example.

So, if you have a conjecture, and you find a counter example, an example that shows it's not true, then that disproves your conjecture.

But sometimes conjectures are true.

So for instance, all even numbers are divisible by two.

And there's lots of conjectures that are true.

And there's lots of conjectures that can be proven false.

And it doesn't matter if your conjecture is right or wrong.

It's the working out whether it's right or wrong, that's the important part.

So Binh says, the order in the bottom row doesn't make a difference to the answer in the top brick.

So, no matter what order we put them in, we're going to get the same answer in the top.

Yasmin says, the top brick will be a multiple of the middle brick on the bottom row.

So whatever got lost in the middle there, she thinks that the top brick will be a multiple of that.

Are their conjectures right? Can you find any counter examples? Or if you can, can you find a way to explain why they are always right? And we're kind of going to look at proof today, on how we could potentially prove whether they are right or wrong.

But for now, I just want you to try some different, try this task, okay? So try Binh's statement, see if it's always true.

Try Yasmin's statement, see if it's always true.

So try different orders, decide whether the conjectures are always, sometimes, or never true.

And what other conjectures can you come up with, okay? So can you come up with any predictions, or anything you notice that you want to check whether that's always true, okay? So I'd like you to pause the video and have a go.

Pause in three, two, one.

Okay, so welcome back.

Hopefully, you've had a play around, tried lots of different numbers.

And hopefully, you've come up with some counter examples if you think they're not always true, or hopefully, you've got a good explanation as to why you think they always work.

Now, I'm going to tell you whether they are always, sometimes, or never true.

Binh's is sometimes true.

Yasmin's is always true.

Did you get that? What happened if you change your numbers? Because what I'd like you to do now, is I'd like you to try three different numbers.

And you might want to do this a couple of times.

In fact, I think it's a good idea to do it a couple of times, and check if their conjectures, are sometimes true, or always true, or never true for other sets of numbers.

So I'd like you to pause the video and do that now.

Pause in three, two, one.

Okay, welcome back.

Now, hopefully, you've tried for some different numbers.

Now, we're going to have a look at, well, for what situations, Binh's is true, okay? So why is it sometimes true? When is it true? Okay, now, I could work it out as 56 and 72, and then add them together.

But I'm going to actually write it like this, okay? Add them two together.

Times them two together, I'm sorry.

Times them two together.

Now, I'm going to add them.

Okay, now, you may be thinking, why have I done it like this? Why haven't I worked them out? And the reason why, is because leaving it like this, it kind of reveals the structure of what's going on.

It shows it better, I think, than working out.

And let's do this, okay? Let's do this the other way.

So I'm actually going to write, I'm going to write nine times eight, rather than, I'm going to write eight times nine, rather than nine times eight.

Because multiplication is commutative, the order doesn't matter.

And I'm going to write this as seven times eight, rather than eight times seven.

And that's fine, because in multiplication, it's commutative.

And the order doesn't matter.

So, here, I've got, eight times nine, plus seven times eight.

Now, addition is also commutative.

Two plus five is the same as five plus two.

Seven plus three is the same as three plus seven.

So, seven, eight times nine, plus seven times eight is the same as seven times eight, plus eight times nine.

And look, they are the same there.

So because addition is commutative, and because multiplication are commutative, we get the same answer when we have, when we swap around the two corner bricks.

But if we were to change, for instance, if we were to change the eight and the nine around, so we had seven, eight, nine here, and here, we had eight, nine, seven, they would be different, okay? So the middle brick has to be the same, and the outer bricks have to just be swapped around.

So that's why Binh's statement is sometimes true.

It's not always true, because I can swap that nine and eight around and get a different answer.

So that would be called a counter example.

Okay, and we're going to look at Yasmin's statement in a minute.

But before we do, we're just going to get a sense of the distributive property.

So this is something we've looked at in previous lessons.

But if you need a reminder, okay, we've got seven times eight, plus eight times nine.

Well, there's a times eight in common, okay? There's a factor of eight here and a factor of eight there.

So we can take out the factor of eight, eight times seven plus nine.

Okay, so I'd like you to pause the video, and try and fill in the blanks.

So pause in three, two, one.

Okay, welcome back.

Now, here's what I've got.

Well, there's a 13 in common, so I can take out a factor of 13, and I can write that like that.

Now here, what we're going to do is, we're now going to distribute that four.

So we got a four times three.

Hmm.

What's in common between these two questions? What's the same? What's different? Well, they all just swapped, of these two.

Does that change our answer? No, it does not.

So, the one, this one, and the one above, they are equal to each other.

These are equal, and these are equal.

Why? Because addition is commutative.

Okay, what's the factoring common here? The factoring common is 12.

So I can take out 12.

What's the factoring common here? 11.

Now, ooh, look, a triangle.

Did you get this? Oops, sorry.

That's the times.

Did you get that? It doesn't matter if it's a number or no.

If it's a shape, if it's a letter, the same work with the same rules are true.

What about this last one? Again, it doesn't matter if it's a letter or a symbol.

The same rules are still true.

The distributive property still holds.

Okay, and why have I done this? Well, because we're going to try and prove Yasmin's statement here.

Okay.

So I'd just like you to just take a second to read this.

So, what's going on? Well, we're going to try and fill this with symbols.

Which of these three would go here? Square times triangle.

Which of the three would go there? Square times star.

Standing on the top, it's that plus, that, which we can write like this.

So we've done so much now, you've probably forgot what Yasmin's statement was.

But Yasmin's statement was that the top is always a multiple of the middle brick.

And can you see why that is true now? We've added them together, we've used the distributive property, and we've got square times something.

So of course, that is a multiple of square.

So what we've actually done now, is we have proved Yasmin's statement is always true, which I think is pretty powerful.

And it's something that I really, really like about maths.

And something I love about maths, to be honest, that we can prove something is true, always.

Okay, and you've probably noticed that I've got this other diagram.

And we can, I just wanted to show you this, that we can use symbols with the array model as well.

So we've got here, we'd have, which brick would it be? It would be square times triangle.

And here, we'd have square times star.

And that length there is, so we've got, I'm sorry, that length there is triangle, plus star.

So the total area is either square times triangle plus star, like that written there.

Or, like this.

Oops, sorry.

And we know that those two are equal.

We know they are equal because of the distributive law.

Okay.

So let's try and generalise the distributive law.

Let's bring in some math.

Let's bring in some letters.

Let's bring in some algebra.

This we've got here, we've got 11, and we've got A.

The area of this, we can say is 11 times A, okay? That's exactly what we do if it was a number, we times them together.

Here, we've got 11 times seven.

So we can say that the area is 11 times A, or we can say, it's that length, the full length, times that length.

And those two statements, they are equal, they are the same, they are equivalent to each other.

So now, I'd like you to pause the video and see if you can do the same for the awesome one, okay? So pause the video and have a go.

Okay, welcome back.

Hopefully, you've had a go.

Now, next would be, A times two.

And it doesn't matter if you wrote two times A.

They are both the same.

They're equivalent because of the commutative law.

So we've got A times two, plus A times 1.

5.

Or, that is the same as A, times by two, plus 1.

5.

And it's actually not just equal to, it's identical equal to.

These two are always true.

And sometimes, we actually use for something that is always true, we call an identity.

And we can actually use like, it's like a triple equal sign, or an equal sign with an extra line under it, okay? And don't worry too much about that now, but I just wanted to introduce you to that notation.

Okay.

So now, it is time for the independent task.

So, I would like you to pause the video to complete your task.

Resume once you've finished.

Okay, and here are my answers.

Hopefully, you got the same.

Hopefully, you managed to not get fooled or tricked by any of the algebra.

I'm really well done.

In fact, I'm massive well done.

If you actually wrote this symbol there, the identity symbol, that would be, I'd be very impressed if you're doing that.

Okay.

So now, it is time for the explore task.

We can see that Yasmin's conjecture will always be true, okay? So this is what we did before.

What other conjectures and generalisations can you come up with? Okay, so here we can see some statements.

And I want you to explore these statements, to start with numbers, okay? And see what you think.

Try a few different times with different numbers.

And then, try and come up with a logical argument.

And if you can do that, then maybe try and even go and try and do it with symbols, okay? So, I'd like you to pause the video, and resume once you've finished.

Okay, so here are some possible answers.

Now, the top brick is even if the square is even.

Why is that true? Well, because if that's even, that's got a factor of two, so the whole thing will have a factor of two.

Or, it's even if those two add together to get an even number.

Because if they add together to get an even number, it will have a factor of two.

Or, if all three are odd.

So, if these two are odd, two odd numbers, so seven plus nine, 16, five plus 11, 16, 21 plus three, 24.

Two odd numbers add to get an even number.

So that's one of the ways why the top brick won't be even.

The top brick will be the greatest when the largest number is placed there.

Why? Why is that? Well, let's just imagine here, and let's make it ridiculous.

Let me make this one, two.

And let's make this one, three.

And let's make that one, 100.

So here, we'd have 200.

And here, we'd have 300.

But if it was the different way around, if it was, if we had 100 here, added a two there, here, we'd only have six, and there we'd have 200.

So you can see because the square is in both products, we want our biggest number in the middle.

The top brick will be odd if the square is odd, and one of this is odd, only one though, because an odd and an even add together to get an odd number.

So we need this to be odd, and this to be odd.

This sum here.

The top brick will be a square number if those two add to get the square, because then it'd be square times square, which is a square number, lots of squares in that sentence.

Swapping the triangle and star has no impact on the value of the top brick.

And that is true.

We saw that before.

That was to do with Binh's statement.

Swapping those two, it doesn't matter.

So hopefully, you had the, an opportunity there to think and cope with these conjectures and test them out, and test if they were true.

And that is, for me, is one of my favourite things about maths, when I get the opportunity to make conjectures and test if they're always true.

So, I hope you enjoyed today's lesson.

So that is all.

Thank you very much for all your hard work, and I look forward to seeing you in the next lesson.

Thank you.