Hello and welcome to Maths with Jay..
In this video we're going to look at the pattern formed by colouring in
multiples of 3 in Pascal's Triangle.
First, let's remind ourselves
of how to recognize a multiple of 3. Well,
a number is a multiple of 3 if 3
divides into it exactly. So
all the numbers in the 3 times table are multiples of 3,
that means 3, 6, 9, 12,
15 and so on. Another example that's a multiple
of 3 is 462; when we divide that by 3
we get 154 exactly.
So no remainder. Now, if you don't have a calculator to hand,
you can still work out whether or not
a number is a multiple of 3: instead of dividing though,
it's easier to do a bit of addition: all you need to do
is work out the "digit sum"
of the number. And if that works out to be 3,
6 or 9, then the number IS a multiple of 3.
So let's look again at 462
and show that that IS a multiple of 3, without
working out the actual division. So the digit
sum of 462 is simply what it says:
we add together all the digits - we find the sum of the
digits in 462 so we add 4
and 6 and 2, which gives us 12
and then either we can say, well,
12 is a multiple of 3, so we know that 462 is a multiple of 3,
or we can apply the digit sum rule again
to 12, so we add together the digits in 12, the 1 and the 2,
& we get 3. So we know that
462 IS a multiple of 3.
You may like to try this method out on the
bigger numbers in Pascal's Triangle. Whichever method you use,
THIS is the pattern you should get in Pascal's Triangle,
when you shade in the multiples of 3. You may like to compare this with
the pattern you got
when shading in the even numbers. Other things to look at...
are... what would happen in the next row
of this triangle here? - Where where would you be shading in?
One way to think about this, would be to think about if you're adding together
two numbers which are multiples of 3,
would the answer also be a multiple of 3?
And if you're adding together one number which IS a multiple
of 3 and one which ISN'T, would the answer be a multiple of 3?
- That will help you predict what's going to happen next.
Something else to think about,
is to look at which rows have no shading in at all:
So look at the triangle we've got here,
think about which rows don't have any shading in, which rows have
NO multiples of 3 in them, and can you work out which row
will be the next row that has no multiples of 3?
Once you have finished looking
multiples of 3 in Pascal's Triangle, and you've looked at
multiples of 2 in Pascal's Triangle, then you might like to look at Pascal's
Triangle again,
but this time shade in multiples of 6.