G'day, I'm Dr Peter Price of Classroom Professor.
Welcome to this Video.
The topic this week is "Counting Fractions
along a Number Line" and the worksheets that
go with the video come from this book, which
is our Classroom Professor Gadgets Worksheets
Book for the gadgets called, "Fraction Counter".
So the gadgets are a set of software tools
that we have on our website that are interactive
online software tools.
So the idea behind them is to help students
to develop their thinking, and to develop
conceptual understanding for the various topics,
I found most of them are about fractions,
because we know how difficult those are.
And this particular piece of software has
a screen that looks; there we go, a bit like
that, so it has a number of whole shapes and
fractional shapes and a number line underneath,
and the students can count along the number
line.
So the worksheets themselves this week are
a sort of follow up exercise, they look like
this, so there are number lines with symbols
and the students have to fill-in the blanks,
there are no pictures, no regional models
or anything like that, that are represented
here, but of course you could supplement what
the students do with these particular worksheets
with pictures and diagrams, and I'm going
to talk about that on the screen.
So what we're trying to do here is to familiarise
students with common fractions and specifically
with sequences and how the numbers change
as we go on adding fractional parts, so we'll
have done number lines with whole numbers
and numbers with tens, and hundreds and probably
decimal fractions at some point as well, but
when we're looking at common fractions of
course the pieces are different depending
on the size of the fraction.
So the example I've chosen for this, this
time on the screen is "Sixths", so here we
have some numbers along a number line, one
of the activities for students is to work
out the size of the pieces, and then the name
for the pieces and then the symbols for the
pieces, so if you look at this here from 0
to 1, we've got a number of lines here, between
them and the same up to 2 and to 3 and then
there's two extras we move on.
Notice of course, and this is obvious to an
adult, if you simply count the lines you'll
get the wrong answer if you say, that's the
number of pieces, so there are five vertical
strokes between the 0 and the 1, but there
are six even spaces.
So there's no way around that, it's as what
I like to call the "Fence post question".
If you have a problem solving question, when
you say, there's a fence, a certain, that
needs to go along a boundary and it's a certain
length and the posts are this far apart, "How
many post will there be?" if you simply divide
the two numbers you get one less then you
should, because there are posts at both ends.
So we have in some mathematical context, we
need to count the spaces and not just the
dividing line between them.
I found even adults find it difficult sometimes
when doing graphing and things like that.
So that's a particular point that you'll need
to emphasize with students I suspect, because
often times students get that part wrong.
So we need to say, "How many spaces are there,
what are we going to call them?"
there are six spaces, so we could even write
it like this, "1 whole = six spaces" and we're
going to call them "Sixths".
"Why do we call them Sixths?" because there
are six of them, "How do we write them?"
well they look like that, they got a 6 on
the bottom.
Alright, so we could, I not going to because
they're too small but I could write the symbols
that are in there, in the exercises you'll
find that there are some symbols that are
here and some that are missing, actually I
will write some, so let's say, "1/6, 3/6,
4/6" ok, and that's not very neat, I apologies,
and up here we might have "1 and 2/6, that's
a bit neater, 1 and 4/6, 2 and 1/6".
As I've said, there's not a lot of space here
to fill them all in.
So then the exercise in the worksheets is
to have the students fill-in the blanks, we
should point out immediately that you could
do this without understanding fractions, because
this is 1/6 and this is 3, this is 4, then
there's a gap, if we just fill-in the blanks,
we just see there's sixths on the bottom of
each one, and if all we did was just to put
sixths there, and sixths there and there's
going to be something with a sixths there
and something with a sixths there, and then
fill-in the gaps, yeah this is 1 something,
3, 4 something, it must be 2 and 5, we don't
want the students to simply fill-in the blanks
by looking for the pattern in the symbols,
we want them to think about the size of the
numbers.
So what I would much rather do is say, "There's
1/6, what's it going to look like when we
have another one, there's 2/6, let's write
the symbol for that one, what's this one going
to be?"
"Oh look, what other fraction is that equal
to?"
"That's equal to 1/2" and you can see that's
half way between the 0 and the 1.
So we'll talk about the fractions as we go,
"There are 4/6, what's the next one going
to be?"
"5/6" "Why don't we have one here that says
6/6?"
"Isn't it what that is?"
"Yes it is, but have a look, it's the same
as the whole, that why we have the whole and
it doesn't actually say 6/6", but we could,
so here we could write "6/6" this will be
5/6 and 2/6.
Alright so all along the way we'll be talking
to our students about what it means and especially
talking about what happens when we get to
the transition from one number of wholes to
the next one.
So for example, "1 and 4/6, 1 and 5/6, what's
the next one?"
"2, then we have 2 and 1/6, 2 and 2/6, so
on, so on, 2 and 5/6, 3."
So that, in whole numbers we call that regrouping,
so you get you know, "57, 58, 59, what's the
next one?"
"60, why 60?" because we've got a group of
10, we make it into a group, we've got 10
ones, we make it into a group of 10, call
it "6 tens / 60" and we count on, and then
have another transition at 69 to 70, it's
exactly the same thing here, mathematically
the same process, but the number of pieces
is different, and the tricky bit I think about
common fractions is not that we've got to
do that, but that it changes every time, so
that when we're doing fourths, we get up to
3/4 and the next one is a whole number, if
we're doing twelfths, we get up to 11/12 and
then there's a whole number, and so on, and
it varies obviously for each denominator.
Alright, so the exercises, I just want to
reiterate what I said before, they can look
a little bit mechanical, they can, the students
can almost guess what goes in the middle before
you even teach them about it, but we want
to avoid that sort of guessing, we want them
to think about it and we're going to need
to ask questions, you know, "How do you that's
that one?"
"What happens when you do this?"
"Where's the transition?" you know, questions
of that style.
Alright, so we finished this video, I hope
you've enjoyed it and I'll talk to you next
time.
