# Fractions

Document Sample

```					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
fractions covered in lower grade levels
NAEP, 2001
National Assessment of Educational Progress
Mathematics Proficiency
 Conceptual understanding
 Procedural fluency
 Strategic competence
 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
 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.   Parts of a whole: when an object is equally
divided into d parts, then a/b denotes a of
those b parts.
2.   The size of a portion when an object of size a
is divided into b equal portions.
3.   The quotient of the integer a divided by b.
4.   The ratio of a to b.
5.   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            Comparing Fractions
   Equivalent Fractions       Converting Fractions
   Adding Fractions           Reducing Fractions
   Subtracting Fractions      Relationships
   Multiplying Fractions      Subtracting Fractions
   Dividing 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.
a
 This means that if b  d then d  ka
c        c
kb

 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
b
c
c
     a
b          ?

   It’s just arithmetic!
a
b
c
c
1
c     c         a
b
1    a
b
   a
b

Productive disposition
Why are equivalent fractions
important?
 For comparing fractions
 For subtracting fractions
 For resolving proportion problems
 For scaling problems
 For calculus and beyond
   Why is
a b  a  b                 ?
d  d     d

   It is by Pie charts? Fraction bars?
Spinners? Blocks/Tiles?
   Answer. It’s just arithmetic! We know…

d    a
d
b
a b

   So,
a
d
b
 1
a b
d
a
d
d
b
Common mistakes

a a  a
b  c   b c
a c  a c
b  d   b d

Where???    College
   Definition of addition. In some sources we
see… a         c     pa   qb
  m
b    d
where m lcm d
b,
and m pb qd
What’s wrong with this??
   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  bc
b d
Example – no lcm
1 2  1  5 1  2
2  5   2    5    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  2
1
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.

Find the product
of the
fractions                                     fractions
denominators

equivalent               Reduce
fractions

Add two          Find the LCM of       Create equivalent
fractions         denominators             fractions

A division step here
equivalent            Reduce
to use the lcm
fractions
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
   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

```
DOCUMENT INFO
Shared By:
Categories:
Tags:
Stats:
 views: 25 posted: 8/12/2012 language: pages: 39
How are you planning on using Docstoc?