# Lesson video

In progress...

Hello and welcome to another lesson.

In this lesson, we'll be looking at number grids and how we can represent different numbers on those number grids.

Remember, my name is Mr. Maseko.

And before you start this lesson, make sure you have something to write with and something to write on.

If you don't have those things, pause the video here and go get those things now.

Okay, now that you have those things, let's get on with today's lesson.

So the first step was to cross out the number one.

And then it says circle the smallest remaining number.

So that's why the number two was circled.

And then we have to cross out the larger remaining multiples of the number we just circled.

So you see, that's why all the multiples of two were crossed out.

Now, once you've done that, once you've done, just crossed out the larger remaining multiples what does step three say? It says, repeat step two.

Repeat step two.

What would you have to do? Circle the smallest remaining number.

So what's the smallest remaining number? Oh it is three.

And then what are you going to do next? Cross out all the larger multiples of three.

And then once you've crossed out the larger remaining multiples of three, you do what? You repeat step two again, and you do that until you have no numbers left to cross out or circle.

Okay? Pause the video here and give that a go.

Alright.

So what should have happened is you should have circled the numbers three, five, seven, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53 and 59.

Now those are all the numbers that should have been circled and every other number should have been shaded in.

Now if you look at this, what numbers are circled? What are all those numbers that are circled? Two, three, five, seven, 11, 13.

What are all numbers? Good.

Those are all the prime numbers.

Good.

So all these numbers that were left that were circled after you followed those instructions, those were all the prime numbers.

Now if you look at the column, columns A, B, C and D, so those columns, why are there no prime numbers in those columns? No prime numbers that are greater than three.

Well, if we look at column A, if you look at the numbers in column A, all those numbers are even.

In column A all those numbers are even.

And because they're even, they all have a factor of two.

They all have, that's why they're not prime numbers.

If you look at column B, all those numbers, those ones all have a factor of three.

Like nine, nine is three times three, 15 is three times five, 21 is three times seven, 27 is three times nine, 33 is three times 11, 39 is three times 13, 45 is three times 15, 51 is three times 17, 57 is three times 19.

And then in column C, where all those are even numbers, so they have a factor of two.

And then in column D the even numbers, they've a factor of two, and they're also, they all have a factor of three also.

'Cause all those numbers are also in three times table.

That's why there are no prime numbers greater than three in those columns.

Now we're going to look at different ways of representing numbers in this grids.

Now we have a student that says she can write multiples of six as the product of six and an integer.

So the product of six times one, six times two, six times three, six times four, six times five, six times six, six times seven and so on and so forth.

So you can represent all the multiples of six as a product of six multiplied by another number.

So that's six, 12, 18, I should say 18, 24, 30 and so on and so forth.

Now is there a way we can generalise this? We say that the six times table is six multiplied by some integer.

So to generalise this so we can, we don't be writing out the entire six times table.

So we can say that the six times table, well, that is six multiplied by an integer and will hold this integer M.

Now we could have used any letter.

Now this, this M we can say M.

Now M has to be an integer, otherwise it won't represent a number in the six times table and it can't not be an integer 'cause let's think of an example.

If M wasn't an integer what would happen? What would happen if M wasn't an integer? Well, let's see.

If we have six multiplied by 0.

5, where M is not an integer, well, this would give us three.

And you see three, this, this is not, that's not in the six times table.

So M cannot be a number that's not an integer.

So M has to be an integer.

And that way we've generalised all the numbers in the six times table.

Here is an "Independent task" for you to try.

Pause the video here and give this a go.

Okay, now that you've tried this, let's see what you've come up with.

Well, write an expression to represent all of the following multiples of the following numbers.

Well, all the multiples of four.

So that's four times one, four times two, four times three.

So four multiplied by some number, where we can say that is four multiplied by you know we can give the integer any letter.

So we can say, let's call it P.

Where P is an integer.

Remember P has to be an integer.

Well, what about five? Well five could be five times, well, let's stick with P, five times P.

'Cause we know that P is an integer.

Seven will be seven times P.

Well, instead of writing four times P, five times P, seven times P, we could write four times P as 4P, five times P as 5P, seven times P as 7P.

So all these, these represent multiples of four, five, and seven when P is an integer.

So the next question was, write an expression to represent all the common multiples of three and four.

Well, what are the common multiples of three and four? Well, the first one would be 12.

And then what we have after 12? And there's 24 and there's 36, then there's 48 and there's 60 and so on and so forth.

Now, if you look there's 12, 24, 36, 48, 60.

Those are all numbers in what timetable? Those are all numbers in the 12 times table.

So the common multiples of three and four are all numbers in the 12 times table.

So we can represent that as 12P where P is an integer.

A really well done if he got this.

Now for this "Explore" task, we're going to practise generalising a lot more.

So pause the video here if you want to get on with this without a clue.

So pause the video in three, two, one.

Okay.

Now if you want a clue, well, let's read the first thing.

It says, which of the numbers in the table can you find by substituting positive integers into the following expressions? Well, let's think so.

If we do, anything can go here.

So let's call the six times one.

Well six times one gives us six, six times two that's 12, six times three.

You see, see what's happening? So which numbers can you find if you do six multiply it by something? So that will be numbers in there.

Okay.

Keep going.

Now you pause the video and give this a go.

Three, two, one.

Alright, now that we've all tried this let's see what we've come up with.

Well, for this first expression where we multiply stakes by any integer, we generate numbers in the six times table.

So that is all of these numbers.

So that would be six multiplied by any integer, we'll call this P.

And that would be in algebraic notation, 6P.

So this there, that six multiples of P.

And then which numbers if we do six multiply by integer take away one, or let's say six multiplied by if we do six multiplied by one and then take away one will six multiplied by one that gives us six, take away one that gives us five, So we got this number.

And then six multiply, then we'll do six multiplied by two, take away one.

Well, six times two is 12, 12 takeaway one is 11.

And if we keep doing this, we'll notice that we'll be getting this integers in this column here.

So this is 6P 'cause P is any integer take away what? So this would be 6P take away one.

And now the integers for 6P add one.

Well, six times one gives you six, add one is seven.

So that'd be seven.

Six times two is 12 add one is 13.

So all of these here would be 6P add one, where P is any integer.

So all of those, those are 6P add one.

Now if P was zero, so six times zero is zero add one, that would give you one.

Okay.

So, there's an algebraic notation of how we can represent those following expressions and what numbers they represent on that grid.

Now, you'll do much more of this later on in your math education when you start looking at sequences.

But for now, thank you very much for participating in today's lesson.