now with a number line the students will hopefully
be familiar with a number line using other
numbers, using whole numbers, they're used
to counting along the line and you know, they
know that's how they do it. We usually have
a 0, and we'll have a 0 here, and we're moving
in this direction, and we're going to add
one every time, only this time we're not going
to add 1 whole, but we're going to add a fraction.
So let's do this in eighths, and if we say,
"Here's a fraction, it's more than nothing,
what number are we up to?" Well we're up to
"1/8" so let's make this point on the number
line equal to "1/8". Let's put another one
there, "What will we call that?" of course
that will be "2/8", and we continue on. When
we do this the student should notice that
there's something not changing here, it's
the denominator of course, the 8 on the bottom
does not change, "Why not?" Because that only
tells us how big the fraction is, it tells
us, what sort of fraction it is, it gives
us the name of the fraction that of course
is where the word denominator comes from,
it comes from a word meaning, "The name for
something". And we just keep on doing this,
and then we keep adding pieces and we'll do
5/8, 6/8, 7/8, "What will happen when we get
all eight of them?" This of course is the
interesting bit; it's also the tricky point
in the process for the students. When we get
all eight of them we could give that another
name or we could write it down in another
way, and of course that is to say that it's
the same as a whole, and "What do we write
down for 1 whole?" if we have one object we
write down one. So what we could write here
is just plain "1". Now there's an alternative,
we could say, "8/8" and that is that is still
true this are both the same, they're equal
in value and so we'll teach that at some point,
we're going to teach the difference between
improper fractions and mixed numbers. Let's
say we're not doing improper fractions at
this point, let's just say we're now up to
a whole, so this is where we're up to, we've
counted up in eights, we now up to a whole,
"How much will I have if I add another eighth?"
so "What's the next number on the number line?"
of course it's going to be "1 1/8" and the
next one will be, "1 2/8". Clearly this is
more complicated when you get past the one;
we've been fine up until now, I've chose eighths
deliberately, because it takes quite a while
to get up to the difficult part in the sequence
and we can count up to 8 quite easily. When
we get past this, it's a sort of transition,
it's like regrouping when we're counting in
ones in base ten, when you get to the 10 you
have to change the name of then 10, you've
got 0 on the end and then you into a new decade.
Here we're moving into a new whole, it's a
very similar idea mathematically, it's basically
the same, just using different size pieces
and so we progressively add more and more
eighths, and then we can say, "What happens
when we get to 1 8/8?" of course that will
be 2 and then we'll move on. So this will
give us plenty of scope for developing students
understanding, but the key part is this part
here, where we reinforce this idea that these
pieces are parts of a whole and if you collect
enough of them together they equal a new whole
and then you can continue from there onwards.
