Fraction Personalities

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Fraction Personalities

• A psychological examination of the schizophrenia
  of fractions.

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• The learning intentions of this workshop are that
  you gain the following psychoanalytic skills
• Recognise when a fraction is acting as a number
  (quantity).
• Recognise when a fraction is acting as an
  operator.
• Recognise when a fraction is acting as a
  relationship between quantities.
• Know the connections between all three fraction
  personalities.

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• Imagine that you had to measure the height of
  Heidi using dark green cuisenaire rods.
• How tall is Heidi?

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• 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?

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• 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 part-units as
  two-split, three-split and six- split to
  show how many of them fit into a dark green rod.

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• 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)

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• 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 two-thirds 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.

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• It is important to know why two-thirds and
  four-sixths are the same amount.

Split one unit into three equal parts.
Call these units thirds.
Split these units in half so there are twice as
many.
Call these units sixths because six of them make
one.
Two thirds is the same amount as four-sixths. Why?
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• Follow these splits
• Divide one unit into four equal parts.

Call these parts fourths or quarters. Split each
part into three parts so there are three times as
many.
What are these new parts called? They are called
twelfths because twelve of them fit in one.
Nine-twelfths is the same as three-quarters. What
other fractions are the same as three-quarters?
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• 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?
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Then you could cut the remaining half into three
pieces and give each child one of those pieces.
Next you could chop the remaining donuts in half
and give each child one half.

• An important idea in this problem is that the
  donuts, the ones in this case, are all the same
  size. If they werent then you would have a
  different problem.

You could start by giving the children as many
whole donuts as you can.
How much donut has each child received?
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• In ancient Egypt they would have given the shares
  as
• 1 1 ½ 1/6 (sum of unit fractions)
• The modern convention is to express the shares in
  the 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!

Each child will get one third from each donut,
thats eight thirds altogether.
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• 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.
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• 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.

Where would the fraction 9/3 be located?
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• 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.

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?
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• You should have recognised that there are an
  infinite number of fractions that are the same
  size as 2/3, 6/9, 8/12, 66/99, to name a few.
• 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.

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• 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.
• Two-thirds is eight-twelfths and five sixths is
  ten-twelfths.
• Half way between eight-twelfths and ten-twelfths
  is nine-twelfths.

9/12
5/6
10/12
?
8/12
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)
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• 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 share for one person will be two-thirds, for
the other it will be one-third.
The shares can be found using a dealing process,
one at a time or in multiples.
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• 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.

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• The calculation of 2/3 x 24 can also be carried
  out in two ways
• 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.
• Multiply 24 by two then divide by three.
• Lets see what that looks like

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• The calculation of 2/3 x 24 can also be carried
  out in two ways
• 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.
• Multiply 24 by two then divide by three.
• Lets see what that looks like

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• 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.

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• 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.
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• 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.

This could be written as the ratio 23.
The ratio could be replicated to form 46 and 69.
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• The relationship between apple and orange can be
  expressed in several ways.

Two-fifths of the punch is apple and three-fifths
is orange.
Four-tenths of the punch is apple and six-tenths
is orange.
Six-fifteenths of the punch is apple and
nine-fifteenths is orange
These fractions describe the part-whole
relationships.
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• Fractions can also be used to describe the part
  to part relationships.

There is two-thirds as much apple as orange.
There is one and a half times as much orange as
apple.
There is four-sixths as much apple as orange.
There is six-fourths as much orange as apple.
What is the relationship between 2/3 and 1 ½ and
between 4/6 and 6/4?
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• 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?

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• You might have noticed

Four-ninths of the punch is raspberry so
Five-ninths of the punch is blueberry.
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• There is four-fifths as much raspberry as
  blueberry so

There is five-fourths as much blueberry as
raspberry.
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• We will now expand the connections between
  fractions as numbers, operators and relationships.

Fractions as numbers
When a set or object is regarded as the whole
(one) then the part-whole relationship is
expressed as a fraction, e.g. 2 red and 3 blue
then 2/5 of set red.
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.
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.
The part to part relationships within a set are
described as the operator that maps one part onto
the other, e.g. There are two-thirds as many
apples as oranges.
Fractions as relationships
Fractions as operators
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