# Fractions - PowerPoint PPT Presentation

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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
• NAEP, 2001

National Assessment of Educational Progress
4
Mathematics Proficiency
• Conceptual understanding
• Procedural fluency
• Strategic competence
• 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
• 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
• where a and b are integers.

This is simply the division of integers by
integers.
10
Fractions Basic Syllabus
• Basic Fractions
• Equivalent 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
16
Equivalent fractions
• Why is
• Its just arithmetic!

?
Productive disposition
17
Why are equivalent fractions important?
• For comparing fractions
• For subtracting fractions
• For resolving proportion problems
• For scaling problems
• For calculus and beyond

18

19
• Why is
• It is by Pie charts? Fraction bars? Spinners?
Blocks/Tiles?

?
20
• Answer. Its just arithmetic! We know
• So,

21
Common mistakes
Where??? College
22
• Definition of addition. In some sources we see

Whats wrong with this??
23
• 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
28
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

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