Fractions

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Fractions

• G. Donald Allen
• Department of Mathematics
• Texas AM University

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From the NCTM

• Middle school should acquire a deep understanding
  of fractions and be able to use them competently
  in problem solving.
• NCTM(2000)

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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
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Mathematics Proficiency

• Conceptual understanding
• Procedural fluency
• Strategic competence
• Adaptive reasoning
• Productive disposition

Adding it Up, - National Research Council
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Bottlenecks in K-8

• It is widely recognized that there are at least
  two major bottlenecks in the mathematics
  education of grades K8
• The teaching of fractions
• The introduction of algebra

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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 numbers and run
  into trouble

Tirosh (2000)
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Polyvalence, again

• When it comes to fractions there are multiple
  interpretations.
• What are they?
• What do students think they are?

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Multiple meanings

• 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.

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Definition of a fraction

• A rational number expressed in the form
• a/b --- in-line notation, or
• --- traditional "display" notation
• where a and b are integers.

This is simply the division of integers by
integers.
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Fractions Basic Syllabus

• Basic Fractions
• Equivalent Fractions
• Adding Fractions
• Subtracting Fractions
• Multiplying Fractions
• Dividing Fractions

• Comparing Fractions
• Converting Fractions
• Reducing Fractions
• Relationships
• Subtracting Fractions

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Comparing Fractions

• Equivalent Fractions
• Comparing - Like Denominators
• Comparing - Unlike Denominators
• Comparing Unlike numerators and denominators
• Comparing Fractions and Decimals

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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

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Reducing Fractions

• Prime and Composite Numbers
• Factors
• Greatest Common Factor
• Least Common Denominator
• Least Common Multiple
• Simplifying

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Relationships

• Relating Fractions To Decimals
• Relating Decimals to Fractions
• Relating mixed fractions to improper fractions
• Relating improper fractions to mixed fractions.

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Equivalent fractions

• Two fractions are equivalent if they represent
  the same number.
• This means that if then
• The common factor k has many names.

This principle is the single most important fact
about fractions.
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Equivalent fractions

• Why is
• Its just arithmetic!

?
Productive disposition
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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

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Addition

• Addition
• Addition - Like Denominators
• Addition - Unlike Denominators
• Addition Mixed Numbers

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Addition - Like Denominators

• Why is
• It is by Pie charts? Fraction bars? Spinners?
  Blocks/Tiles?

?
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Addition - Like Denominators

• Answer. Its just arithmetic! We know
• So,

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Common mistakes
Where??? College
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How to add fractions, 1

• Definition of addition. In some sources we see

Whats wrong with this??
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How to add fractions, 2

• Definition of addition. In other sources we see

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Example no lcm
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Example with lcm
lcm 8

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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.

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Adding fractions process, 1
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Adding fractions process, 2
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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
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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.

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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.

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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.

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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_ma
thematics.html3
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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
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Dividing Fractions

• Division
• Division by Integers

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Multiplying Fractions

• Multiplication
• Multiplication by Integers

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Division of fractions
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Mixed fractions
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Multiplication of fractions