Transcript Fraction Personalities

Fraction Personalities
A psychological examination of
the schizophrenia of fractions.
The learning intentions of this workshop are
that you gain the following
psychoanalytic skills:
1. Recognise when a fraction is acting as a
number (quantity).
2. Recognise when a fraction is acting as
an operator.
3. Recognise when a fraction is acting as a
relationship between quantities.
4. Know the connections between all three
fraction personalities.
Imagine that
you had to
measure the
height of
Heidi using
dark green
cuisenaire
rods.
How tall is
Heidi?
In many situations
an answer of
“Nearly four rods
tall,” or “A bit more
than three rods
tall” might be okay.
If you wanted to
be more precise
you might divide
up the dark green
rod into smaller
units of equal size.
What will you call
these part-units?
You have a choice
here.
Each new unit could
be given a name of
its own but
remember that you
have to measure
Heidi using dark
green rods as your
unit.
So you could
describe the partunits as “two-split”,
“three-split” and “sixsplit” to show how
many of them fit into
a dark green rod.
3-split
2-split
6-split
Of course we call
these part units
halves, thirds and
sixths. That is our
convention. But the
words mask the
nature of the splits.
“Twoths, threeths,
and sixths” would be
better.
Write the symbols for
one half, one third
and one sixths.
How do the symbols
reflect the splits?
Now, how tall is
Heidi? (precisely)
Your answer could
have been 3 whole
rods and 2 three-splits
of a rod, or 3 whole
rods and 4 six-splits of
a rod.
Of course we would
say “Three and twothirds” or “Three and
four-sixths” (of a dark
green rod).
The symbols 3 2/3 and
3 4/6 reflect the
meaning of the
numerator (top
number) as a count
and the denominator
as the size of units
that are counted.
It is important to know why two-thirds and four-sixths are
the same amount.
Two
Split
Call
Splitthese
thirds
one
these
unit
is
into
units
the
units
same
three
thirds.
sixths
in half so
equal
because
amount
there parts.
are
as
six
twice
of
them
four-sixths.
as many.
make
one.
Why?
Nine-twelfths
Call
What
Follow
these
are
these
these
parts
is
splits:
new
the
fourths
parts
same
as
orthreequarters.
Dividecalled?
one
unit into
quarters.
Split
They
four equal
each
are called
part
parts.
into
three parts
twelfths
What
other
because
so
fractions
there
twelve
are
three
the
of
same
them
times
asfit
asin
many.
one.
three-quarters?
Measuring with increased precision is one situation in
which fractions have purpose, as numbers that are parts of
one (unit).
Sharing is another situation.
Supposing that we had to share eight donuts fairly among three
children.
How much donut do we give each child?
Then
you could
could chop
cut the
remaining
half
into three
pieces
and
Next
you
the
remaining
donuts
in
half
and
give
An important idea in this problem is that the donuts, the
give
child
one
of all
those
pieces.size. If they weren’t then
oneseach
in this
case,
are
the same
each
child
one
half.
you would have a different problem.
You could start by giving the children as many
wholemuch
donutsdonut
as you can.
How
has each child received?
In ancient Egypt they would have given the shares as:
1 + 1 + ½ + 1/6 (sum of unit fractions)
Each child will get one third from each donut, that’s eight
The modern
convention is to express the shares in the
thirds
altogether.
simplest fraction form possible.
The easiest way to think about this sharing is to divide each
donut into three equal parts as there are three children!
Eight-thirds can be transformed into two whole donuts and
two-thirds of a donut.
Note that 1 + 1 + ½ + 1/6 can be combined to make 2 2/3
since ½ + 1/6 = 2/3.
The situations that prompt the need for fractions,
measuring and sharing, require fractions to be regarded as
quantities. This means that fractions have an ordinal
relationship with other numbers on the number line and a
size relationship with one.
When the place of two-thirds is found on the number line it
is considered as two-thirds on one.
0
⅔ 1
Where would the fraction 9/3 be located?
You should have worked out that 9/3 is another name for 3
ones (three).
Recall previously that we found that 2/3 and 4/6 were the
same quantity. Consider what that means for placing
numbers on the number line.
0
4⅔
/6
1
So 2/3 and 4/6 are names for the same number and have
the same position on the number line.
How many other fractions occupy the same position?
You should have recognised that there are an infinite
number of fractions that are the same size as 2/3, 6/9, 8/12,
66/ , to name a few.
99
The number line gets even more interesting when we think
about what numbers lie between other numbers. Think
about what number might be half way between 2/3 and 5/6.
0
1
⅔
5/
6
In the same way that thirds can be split into sixths, sixths
can be split into twelfths. Because twelfths are half the size
of sixths, twice as many of them fit in the same space.
9/ five sixths is ten-twelfths.
Two-thirds is eight-twelfths and
12
Half way between eight-twelfths and ten-twelfths is ninetwelfths.
5
10
/
0
8⅔
/12
6
12
1
What fraction lies half-way between eight-twelfths and nine-twelfths?
Are there any two fractions for which you cannot find a fraction half-way
between? (Try 98/100 and 99/100)
There are many situations in which we want fractions to
behave as operators. For example, suppose we want $24
to be shared between two people so one person gets twice
$
$
$
$
as much as the other.
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
The
Theshares
share for
canone
be person
found
using
will bea two-thirds,
dealing process,
for the
one
other
at ita will
timebeorone-third.
in
multiples.
In the money sharing activity two-thirds operated on 24. We
found two-thirds of 24 dollars (2/3 x 24= 16).
An interesting thing is that this could be worked out in two
ways. On the previous page we established one-thirds by
equal sharing. However we could have seen two-thirds as
“two for every three”.
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
The calculation of 2/3 x 24 can also be carried out in two
ways:
1. Divide 24 by three then multiply by two, just like we did
when we worked it out by equal sharing or “two for
every three.”
2. Multiply 24 by two then divide by three.
$
$
$
Let’s
see
what
that looks$ like:
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
The calculation of 2/3 x 24 can also be carried out in two
ways:
1. Divide 24 by three then multiply by two, just like we did
when we worked it out by equal sharing or “two for
every three.”
2. Multiply 24 by two then divide by three.
Let’s see what that looks like:
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
Not all fractions as operator problems are able to be
accomplished by equal sharing of ones.
Finding two-thirds of eight is not so tidy.
Previously we found one-third of eight
by solving 8 ÷ 3 = 8/3 = 2 2/3.
So two-thirds of eight must be twice as much,
that is 2 x 8/3 = 16/3 = 51/3.
From this we can see the connection between fractions as
numbers and fractions as operators.
When a fraction operates on one (a unit) then the
result is that fraction as a number, e.g. 2/3 of 1 = 2/3.
Fractions as
numbers
Fractions as
operators
If finding a fraction of a quantity cannot be done
equally by distributing ones then ones must be split
into fractional parts, e.g. 2/3 of 8 = 16/3 = 51/3 ones.
Some situations involve fractions as relationships between
quantities. To spot these relationships students need to
understand the fractions involved as both numbers and
operators.
Suppose I have a recipe for making fruit punch that has two
parts apple to three parts orange.
The ratio
This
couldcould
be written
be as
the ratio 2:3.
replicated
to form 4:6
and 6:9.
The relationship between apple and orange can be
expressed in several ways.
Six-fifteenths
Two-fifths
Four-tenths
ofof
the
ofthe
the
punch
is appleisand
punch
apple
three-fifths
and sixis orange.
tenths
nine-fifteenths
is orange.
is
orange…
These fractions describe the part-whole relationships.
Fractions can also be used to describe the part to part
relationships.
There is six-fourths
two-thirds
one
four-sixths
and a as
half
as
times as
much
apple
orange
much
asasorange
orange.
apple.
as apple.
What is the relationship between 2/3 and 1 ½ and between 4/6 and 6/4?
1½ is another name for 3/2 (three halves).
We say that 2/3 and 3/2 are reciprocals.
Operating with reciprocals results in one fraction undoing
the other, e.g. 2/3 of 15 is 10, 3/2 of 10 is 15.
What fractions as relationships can you find in this punch
mixture of raspberry and blueberry?
You might have noticed:
Four-ninths of
Five-ninths
of the
the punch
punch is
is blueberry.
raspberry so…
There is five-fourths
four-fifths asas
much
much
raspberry
blueberryasasblueberry
raspberry.
so…
We will now expand the connections between fractions as
numbers, operators and relationships.
Fractions as
numbers
Fractions as
operators
Fractions as
relationships