G'day, I'm Dr Peter Price of Classroom Professor.
Welcome to this week's video in the Free Math
Worksheet Series. This week we'll go on with
another fractions topic, and the topic is
"Counting Fractions along a Number Line".
So this is an introductory topic when students
are just starting to learn about fractions
and they understand a bit about how to write
them down and so on and we're going to help
them to develop an understanding of the sequence
of fractions and what happens when you get
up to the next whole and that sort of thing.
So here we go, so we've got a resource here,
which is quite helpful for showing fractions
as a model, as a physical quantity. I'm going
to start with quarters, let me take one of
those away, it's important that students realise
that a fraction is a part of a whole, so we
have to emphasis the fact that this is just
one, this is one object so if we had counters
or blocks, or some other sort of object, this
is just one object. And so the important transition
the students have to make in their thinking
when they start doing fraction work is, we're
now talking about smaller bits than 1, so
the one is the, you know the whole countable
object, but the fractions are bits of that
object. And of course the parts come in different
sizes and the whole topic of fractions is
very complicated, so we want to take our time
with this, I do like these resources, they're
bright and colourful, they stick to the board
easily, and they will really help the students
to get an idea of, for one thing the different
sizes of fractions. So you can show them,
here's a quarter, a twelfth and an eighth,
"What do you notice?" Well they're all different
sizes, "What do you notice if I put them in
order like this?" So this is one part out
of 4, this one part of 8, this is one part
of 12, they're getting smaller as they go
on, and as the denominator gets bigger and
so on. We're going to be able to talk about
a whole range of topics about fractions, but
I want to focus on the ways that we record
fractions, and they are 3 principle ways of
doing that, one is with a picture, or some
sort of resource or something like this. Now
I've deliberately drawn this curve line here,
so we can see the rest of it, because if we
just have those 3 pieces it could be that
shape, rather than parts of something else,
alright. So we got those parts as a pictorial
or physical model, then we have a symbol,
where we write two numbers with a line in
the middle, and of course we're going to have
to explain all that to students and what it
all means. Technically we call this the Numerator,
the Denominator and the Vinculum, but you
almost certainly won't choose those words
with young students because they, they're
very big words they're not used in other context
and so, I feel students aren't ready for that
until they're quite a bit older. But we will
explain what the 3 means and what the 4 means
and so on, so the students can understand
the rules, for writing a fraction. Let me
move on to the third one and that is, sorry,
I'm going to use a better pen, three quarters
or I should say "Fourths", ok. So we have
this two different words, it depends largely
on where you are from, you know, and the word
that you used when you're in school and so
on. In the United States you very sensible
call them "Fourths", I think that's a great
word, in a lot of the rest of the world we
call them "Quarters", either way we have a
name for the fraction. So with these 3 representations
we can then ask a range of questions for students
to say, "If I give you one of these, can you
give me the other two?" So you can say, "Can
you write this fraction down as a symbol,
as a fraction", "Can you tell me what this
is called in words, what fraction is this?"
or we could provide a student with this representation
and say, "Can you show me that with materials?"
or "Can you colour in that much of a circle?",
"Can you tell me what that's called in words?"and
so on. Ok, so that whole idea of moving around
this triangle of a visual representation,
the symbolic notation and the verbal words
are to name the number. We use the same thing
exactly for whole numbers and for decimal
fractions and so on, when we are teaching
those topics. Let's move on to the number
line, so I'll just tidy up a little bit here
and remove this, 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.
So there's a lot of work that can be done
with this, and the worksheets really just
get things started I guess, on this topic
and I hope that you find them useful. That's
it for this video, I look forward to seeing
you next time, and that's it for me.
