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Fractions

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Title: Fractions


1
Fractions
  • G. Donald Allen
  • Department of Mathematics
  • Texas AM University

2
From the NCTM
  • Middle school should acquire a deep understanding
    of fractions and be able to use them competently
    in problem solving.
  • NCTM(2000)

3
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
4
Mathematics Proficiency
  • Conceptual understanding
  • Procedural fluency
  • Strategic competence
  • Adaptive reasoning
  • Productive disposition

Adding it Up, - National Research Council
5
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

6
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)
7
Polyvalence, again
  • When it comes to fractions there are multiple
    interpretations.
  • What are they?
  • What do students think they are?

8
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.

9
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.
10
Fractions Basic Syllabus
  • Basic Fractions
  • Equivalent Fractions
  • Adding Fractions
  • Subtracting Fractions
  • Multiplying Fractions
  • Dividing Fractions
  • Comparing Fractions
  • Converting Fractions
  • Reducing Fractions
  • Relationships
  • Subtracting Fractions

11
Comparing Fractions
  • Equivalent Fractions
  • Comparing - Like Denominators
  • Comparing - Unlike Denominators
  • Comparing Unlike numerators and denominators
  • Comparing Fractions and Decimals

12
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

13
Reducing Fractions
  • Prime and Composite Numbers
  • Factors
  • Greatest Common Factor
  • Least Common Denominator
  • Least Common Multiple
  • Simplifying

14
Relationships
  • Relating Fractions To Decimals
  • Relating Decimals to Fractions
  • Relating mixed fractions to improper fractions
  • Relating improper fractions to mixed fractions.

15
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.
16
Equivalent fractions
  • Why is
  • Its just arithmetic!

?
Productive disposition
17
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

18
Addition
  • Addition
  • Addition - Like Denominators
  • Addition - Unlike Denominators
  • Addition Mixed Numbers

19
Addition - Like Denominators
  • Why is
  • It is by Pie charts? Fraction bars? Spinners?
    Blocks/Tiles?

?
20
Addition - Like Denominators
  • Answer. Its just arithmetic! We know
  • So,

21
Common mistakes
Where??? College
22
How to add fractions, 1
  • Definition of addition. In some sources we see

Whats wrong with this??
23
How to add fractions, 2
  • Definition of addition. In other sources we see

24
Example no lcm
25
Example with lcm
lcm 8

26
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.

27
Adding fractions process, 1
28
Adding fractions process, 2
29
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
30
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.

31
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.

32
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.

33
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
34
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
35
Dividing Fractions
  • Division
  • Division by Integers

36
Multiplying Fractions
  • Multiplication
  • Multiplication by Integers

37
Division of fractions
38
Mixed fractions
39
Multiplication of fractions
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