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

2-split

3-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 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.

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?

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?

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?

An important idea in this problem is that the donuts, the ones in this case, are all the same size. If they weren’t then you would have a different problem.

You could start by giving the children as many whole donuts as you can.

Next you could chop the remaining donuts in half and give each child one half.

Then you could cut the remaining half into three pieces and give each child one of those pieces.

How much donut has each child received?

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, that’s eight thirds altogether.

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 1⅔

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 1⅔

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?

4/6

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.

0 1⅔

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.

Two-thirds is eight-twelfths and five sixths is ten-twelfths.

Half way between eight-twelfths and ten-twelfths is nine-twelfths.

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)

0 1

5/6

⅔

10/12

8/12

9/12

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.

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The shares can be found using a dealing process, one at a time or 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”.

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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:

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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:

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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.

From this we can see the connection between fractions as numbers and fractions as operators.

Fractions as numbers

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.

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.

This could be written as the ratio 2:3.The ratio could be replicated to form 4:6 and 6:9.

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.

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?

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 the punch is raspberry so…Five-ninths of the punch is blueberry.

There is four-fifths as much raspberry as blueberry so…There is five-fourths as much blueberry as raspberry.

We will now expand the connections between fractions as numbers, operators and relationships.

Fractions as numbers

Fractions as operators

Whe

n a

fract

ion

oper

ates

on

one

(a u

nit)

then

the

resu

lt is

that

frac

tion

as a

num

ber,

e.g.

2 / 3 o

f 1 =

2 / 3.

If fin

ding

a fr

actio

n of

a q

uant

ity c

anno

t be

done

equa

lly b

y di

strib

utin

g on

es th

en o

nes

mus

t be

split

into

frac

tiona

l par

ts, e

.g.

2 / 3 o

f 8 =

16 / 3 =

51 / 3 o

nes.

Fractions as relationships

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.

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.