Fractions
G. Donald Allen
Department of Mathematics
Texas A&M University
From the NCTM…
Middle school should acquire a deep
understanding of fractions and be able to
use them competently in problem solving.
NCTM(2000)

From the NAEP…
Reports show that fractions are
"exceedingly difficult for children to
master. "
 Students are frequently unable to
remember prior experiences about
fractions covered in lower grade levels
NAEP, 2001

National Assessment of Educational Progress
Mathematics Proficiency
Conceptual understanding
 Procedural fluency
 Strategic competence
 Adaptive reasoning
 Productive disposition

Adding it Up, - National Research Council
Bottlenecks in K-8

It is widely recognized that there are at
least two major bottlenecks in the
mathematics education of grades K–8:
 The
teaching of fractions
 The introduction of algebra
Student mistakes with fractions
Algorithmically based mistakes
 Intuitively based mistakes
 Mistakes based on formal knowledge.


e.g. Children may try to apply ideas they
have about whole numbers to rational
Tirosh (2000)
numbers and run into trouble
Polyvalence, again
When it comes to fractions there are
multiple interpretations.
 What are they?
 What do students think they are?

Multiple meanings
1.
2.
3.
4.
5.
Parts of a whole: when an object is equally
divided into d parts, then a/b denotes a of
those b parts.
The size of a portion when an object of size a
is divided into b equal portions.
The quotient of the integer a divided by b.
The ratio of a to b.
An operator: an instruction that carries out a
process, such as “4/5 of”.
Definition of a fraction

A rational number expressed in the form
 a/b

--- in-line notation, or
a
b --- traditional "display" notation
where a and b are integers.
This is simply the division of integers by integers.
Fractions – Basic Syllabus






Basic Fractions
Equivalent Fractions
Adding Fractions
Subtracting Fractions
Multiplying Fractions
Dividing Fractions





Comparing Fractions
Converting Fractions
Reducing Fractions
Relationships
Subtracting Fractions
Comparing Fractions
Equivalent Fractions
 Comparing - Like Denominators
 Comparing - Unlike Denominators
 Comparing – Unlike numerators and
denominators
 Comparing Fractions and Decimals

Converting Fractions
Converting to Mixed Numbers
 Converting from Mixed Numbers
 Converting to Percents
 Converting from Percents
 Converting to Decimals
 Converting to Scientific Notation
 Converting from Scientific Notation

Reducing Fractions
Prime and Composite Numbers
 Factors
 Greatest Common Factor
 Least Common Denominator
 Least Common Multiple
 Simplifying

Relationships
Relating Fractions To Decimals
 Relating Decimals to Fractions
 Relating mixed fractions to improper
fractions
 Relating improper fractions to mixed
fractions.

Equivalent fractions
Two fractions are equivalent if they
represent the same number.
c
a
c
ka


 This means that if b
kb
d then d
 The common factor k has many names.

a
b

ka
kb
This principle is the single most important fact about fractions.
Equivalent fractions


Why is
ac
bc

a
b
?
It’s just arithmetic!
ac
bc
1
c
c
a
b
1 
Productive disposition
a
b

a
b
Why are equivalent fractions
important?
For comparing fractions
 For adding fractions
 For subtracting fractions
 For resolving proportion problems
 For scaling problems
 For calculus and beyond

Addition
Addition
 Addition - Like Denominators
 Addition - Unlike Denominators
 Addition Mixed Numbers

Addition - Like Denominators


Why is
a b  a b
d
d
d
It is by Pie charts? Fraction bars?
Spinners? Blocks/Tiles?
?
Addition - Like Denominators

Answer. It’s just arithmetic! We know…
d

So,
a
b
d
a
b
d
a b
 
a b
1
d
a
d
b
d
Common mistakes
a a  a
c
b
b c
a c  a c
b
d
b d
Where???
College
How to add fractions, #1

Definition of addition. In some sources we
see… a
pa qb
c
  m
b
d
where m lcm
b, d
and m pb qd
What’s wrong with this??
How to add fractions, #2

Definition of addition. In other sources we
see…
a c  a  d c  b
b
d
b
d
d
b
 a d c b
b d
d b
 a d c b
b d
Example – no lcm
1 2  1  5 1  2
5
5
2
2
2
2
 1 5 2 2
2 5
5 2
 1 5 2 2
2 5
 9
10
Example – with lcm
lcm = 8
3 1  3  1 1  2
8
4
8
1
4
2
 3 1 1 2
8 1
4 2
 3 1 1 2
8
5
8
Go with the flow
Flow charting a process can reveal
unnoticed complexities.
 The difference between using the lcm and
simple denominator multiplication is not
insignificant.

Adding fractions process, #1
Basic add fractions process
Add two
fractions
Find the product
of the
denominators
Add the
equivalent
fractions
Create equivalent
fractions
Reduce
Adding fractions process, #2
Advanced add fractions process
Add two
fractions
Add the
equivalent
fractions
Find the LCM of
denominators
Create equivalent
fractions
Reduce
A division step here
to use the lcm
Is this too difficult?
Remember this can be regarded as strictly
a skill.
 It will always be used as a skill – when it is
used.
 At what point – we may ask – is
fundamental understanding suppose to
kick in?

Consider calculus – the accepted wisdom
Is this true?

Informal surveys among teachers
consistently reveal that many of their
students simply give up learning fractions
at the point of the introduction of addition.
Tips for teaching fractions
Engage your students’ interest in fractions.
 Stress the importance of fractions in the
world around them and in successful
careers.
 Emphasize that fractions are used in a
variety of ways.

Tips for teaching fractions

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Practice understanding of fractions by using
math manipulatives.
Practice basic words or phrases by giving
students a problem and a list of relevant terms,
e.g., "numerator," "denominator,“
Practice fractions by having students observe
their surroundings, e.g., what fraction of
classmates have black hair, have brown eyes.
Tips for teaching fractions
Practice fraction problems by having
students write their own fractions based on
their own experiences.
 Practice fraction problems by having
students work in small groups to create
their own surveys around fractions based
on classmates' preferences

http://www.meritsoftware.com/teaching_tips/tips_mathematics.html#3
Engaging students…
Pallotta, J. (1999). The hershey's milk
chocolate bar fractions. Cartwheel Books.
 Adler, D. A., & Tobin, N. Fraction fun.
 Ginsburg, M. Gator Pie.
 Leedy, L. Fraction Action.
 Mathews, L. Gator Pie.

Mostly elementary
Dividing Fractions
Division
 Division by Integers

Multiplying Fractions
Multiplication
 Multiplication by Integers

Division of fractions
Mixed fractions
Multiplication of fractions