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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….

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G. Donald Allen

Department of Mathematics

Texas A&M University

from the nctm
From the NCTM…
  • Middle school should acquire a deep understanding of fractions and be able to use them competently in problem solving.


from the naep
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
Mathematics Proficiency
  • Conceptual understanding
  • Procedural fluency
  • Strategic competence
  • Adaptive reasoning
  • Productive disposition

Adding it Up, - National Research Council

bottlenecks in k 8
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
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)

polyvalence again
Polyvalence, again
  • When it comes to fractions there are multiple interpretations.
  • What are they?
  • What do students think they are?
multiple meanings
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”.
definition of a fraction
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.

fractions basic syllabus
Basic Fractions

Equivalent Fractions

Adding Fractions

Subtracting Fractions

Multiplying Fractions

Dividing Fractions

Comparing Fractions

Converting Fractions

Reducing Fractions


Subtracting Fractions

Fractions – Basic Syllabus
comparing fractions
Comparing Fractions
  • Equivalent Fractions
  • Comparing - Like Denominators
  • Comparing - Unlike Denominators
  • Comparing – Unlike numerators and denominators
  • Comparing Fractions and Decimals
converting fractions
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
Reducing Fractions
  • Prime and Composite Numbers
  • Factors
  • Greatest Common Factor
  • Least Common Denominator
  • Least Common Multiple
  • Simplifying
  • Relating Fractions To Decimals
  • Relating Decimals to Fractions
  • Relating mixed fractions to improper fractions
  • Relating improper fractions to mixed fractions.
equivalent fractions
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.

equivalent fractions16
Equivalent fractions
  • Why is
  • It’s just arithmetic!


Productive disposition

why are equivalent fractions important
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 - Like Denominators
  • Addition - Unlike Denominators
  • Addition Mixed Numbers
addition like denominators
Addition - Like Denominators
  • Why is
  • It is by Pie charts? Fraction bars? Spinners? Blocks/Tiles?


addition like denominators20
Addition - Like Denominators
  • Answer. It’s just arithmetic! We know…
  • So,
common mistakes
Common mistakes

Where??? College

how to add fractions 1
How to add fractions, #1
  • Definition of addition. In some sources we see…

What’s wrong with this??

how to add fractions 2
How to add fractions, #2
  • Definition of addition. In other sources we see…
go with the flow
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.
is this too difficult
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
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
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 fractions32
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 fractions33
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

engaging students
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
Dividing Fractions
  • Division
  • Division by Integers
multiplying fractions
Multiplying Fractions
  • Multiplication
  • Multiplication by Integers