Lesson video

Hi there, my name's Miss Darwish and for our maths lesson today, we're going to be exploring properties of palindromic numbers, quite exciting stuff.

But before we start our math lesson today, if I could just ask you to take yourself away from any distraction, just so you're ready for the lesson.

Okay, for this lesson, we're going to have a bit of a recap and then look at, and then we're going to be looking at some palindromic numbers and then taking a look at some conjectures of palindromic numbers.

And then there will be a quiz for you to complete.

So let's get started.

So the items you'll need for this lesson is just a pencil, a sheet of paper or a notepad, and a ruler.

If you want to grab those things, we can come back and get started.

Okay, let's start with a quick recap.

121, 2002, 3443 Oh dear, let's say these together 121, 2002, 3443.

What do we call these numbers? Palindromic numbers, why do we call them palindromic numbers? Because if we reverse the digits, it reads the same 121.

If I read it starting, so if I sort the ones in the hundreds round is still reads as 121, 2002 can still read as 2002, 3443 if I read it back to front, it would still read 3443.

Can you come up with your own palindromic number? You have 6006, we can have 77, we can have 777.

Okay, so these are of course palindromic numbers.

Now my friend says that all palindromic numbers are divisible by 11.

That means all palindromic numbers are actually multiples of 11, is my friend, right? Is this comment true or false? What do you think? What, I'd like you to do is I'd like you to write down with your, a sheet of paper on your notepad a few palindromic numbers and I want you to have a thing I'm going to do the same as well.

Do you think my friend is right or wrong and then come back and you can tell me why.

So choose three or four palindromic numbers first of all.

Can be any number of digits.

Okay, have you written three or four numbers down? Okay, read them out to me.

Okay, now I want you to divide each one by 11.

And what we want is to make sure that when we divide them by 11, we don't have any remainders no decimals.

They are all divisible by 11 'cause that's the theory that's the hypothesis we're testing.

That's the conjecture we're seeing.

So I'll give you some time to have a look.

Have you done the first one? What have you found? Hmm, let's have a look.

So here's one example maybe you have the same 121.

Is that palindromic number? Yes, it is, 'cause it also reads as 121.

It's a multiple of 11, 11 times 11 is 121.

I think my friend might be correct you know.

121 is a palindromic number and if I divide it by 11, I get 11 is divisible by 11.

121 is a multiple of 11 or we can say 11 is a factor of 121.

Let's have a look at some more examples.

So for my friend's proving to be correct let's see.

99, is 99 a palindromic number? Yes, is it a multiple of 11? Absolutely, 99 divided by 11 or 99 divided by nine is equal to 11.

99 is a multiple of 11 or 11 is a factor of 99.

So that's two examples where my friend's proving to be correct.

Should we keep exploring? What did you get? Let's keep exploring, hmm.

What about 111? That's a palindromic number, right? One, one, one, 111.

But is that divisible by 11? I know that 11 times nine is equal to 99.

But I also know that 11 times 10 is equal to 110 and 11 times 11 is equal to 121.

And 111 lies between 110 and 121.

So they are also must be between 10 and 11.

No, 111 is a palindromic number, but I don't think is divisible by 11.

Hmm, is not a multiple of 11.

So I was beginning to think, I don't know about you that my friend was actually correct.

But this example here 111 shows that actually that might not be the case.

Okay, let's have a look at some multiples of 11 first of all and see what we recognise.

Let's just go back quickly.

My friend says all palindromic numbers are divisible by 11.

Well, the examples that we've seen is that some of them are, but also some of them are not.

Okay, let's move on and see what we recognise about multiples of 11.

What can you, what do you notice? So, 11 22, 33, 44, 55.

Two of the digits are the same, right? Okay, are all multiples of 11 palindromic numbers? 11 is, 22 is, 33 is, 99 is.

Okay, so my friend is not right.

However, there are actually a lot of palindromic numbers, which are visible by 11.

So if we were saying, is this always sometimes or never true? It's sometimes true because a lot of the examples actually have shown us that it is true.

Do you remember the example the first two examples that I showed you? I don't know what your examples were maybe the examples you did showed you that they were.

99 was a palindromic number and it was a multiple of 11 and 121.

So actually a lot of them are multiples of 11, but not all of them.

We can't say that all of them are because we saw that at 111 was not.

So we can't say is always true, but we can say sometimes true.

Okay, so my friend is not right.

However, there are lots of palindromic numbers which are divisible by 11.

So you can see that, right? A lot of these palindromic numbers are actually divisible by 11.

Okay, let's have a bit of a sorting activity now, you ready? So, which palindromic numbers below are multiples of 11 or which are not multiples of 11? So we've got 343, read the next one for me.

11411, and the next number, 4224.

So first of all, let's just check 343.

Is that a palindromic number? Yes, and the next one 11411? Yes, and the next number, 4224 absolutely.

These three are palindromic numbers.

Now, are they multiples of 11 or are they not multiples of 11? Have a think.

Okay, what did you find? Were they multiples of 11? Which ones were and which ones weren't? Or were none of them? Or were all of them? What do you notice? So the only number which is palindromic and a multiple of 11 is actually 4224.

There's actually a reason for that, but I'm not going to tell you that reason just yet.

There is a way of working out if a palindromic number is going to be a multiple of 11 or not but I'm not going to tell you just yet.

I want you to have a guess and see if there's something that you notice and write it down for me.

So there is a reason that 4224 is a palindromic number and the other two are not.

What do you think this reason is? Okay, did you write it down? Okay, that's something for you to explore.

We'll come back.

I'm going to leave you with the independent task.

We'll come back and then we'll go through it together and I'll give you that reason, good luck.

Okay, how did you get on with them? Did you find the reason, let's see.

So you said all palindromic numbers with an even number of digits are divisible by 11.

Is this statement true or false and explain by showing examples.

What did you find? How many different examples did you try? Hopefully you tried at least maybe six different examples.

So look at palindromic numbers with an odd number of digits and palindromic numbers with an even number of digits and then we'll see if this is always sometimes or never true.

So they're saying that all palindromic numbers that have an even number of digits, so it could be a two digit palindromic number, a four digit palindromic number, a six digit palindromic number is divisible by 11.

It's true, and do you remember that example when I told you I wasn't going to tell you, that's why.

So let's get back and 4224, how many digits there are in that number? There are four digits in that number and is divisible by 11 'cause it has an even number of digits.

What about 343? Three digits in that number and 11411? Five digit number.

Okay, so now you know that every single palindromic number that has an even number of digits is divisible by 11.

Hopefully you have some examples to prove that theory as well, well done.

So here are some examples.

Okay, if you would like to share your work with us here at Oak National, then please ask your parent or carers to share your work for you on Twitter, tagging @OakNational and to use the #LearnwithOak.

I just want to say well done on all the fantastic brilliant learning that you have done today.

Now it's time for you to go and complete the quiz on today's session, good luck.