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# fractions - PowerPoint PPT Presentation

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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### Fractions

G. Donald Allen

Department of Mathematics

Texas A&M University

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

NCTM(2000)

• 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

• Conceptual understanding

• Procedural fluency

• Strategic competence

• Productive disposition

Adding it Up, - National Research Council

• 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

• 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)

• When it comes to fractions there are multiple interpretations.

• What are they?

• What do students think they are?

• 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”.

• 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.

Equivalent Fractions

Subtracting Fractions

Multiplying Fractions

Dividing Fractions

Comparing Fractions

Converting Fractions

Reducing Fractions

Relationships

Subtracting Fractions

Fractions – Basic Syllabus

• Equivalent Fractions

• Comparing - Like Denominators

• Comparing - Unlike Denominators

• Comparing – Unlike numerators and denominators

• Comparing Fractions and Decimals

• 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

• 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.

• 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.

• Why is

• It’s just arithmetic!

?

Productive disposition

• For comparing fractions

• For subtracting fractions

• For resolving proportion problems

• For scaling problems

• For calculus and beyond

• Why is

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

?

• Answer. It’s just arithmetic! We know…

• So,

Where??? College

• Definition of addition. In some sources we see…

What’s wrong with this??

• Definition of addition. In other sources we see…

lcm = 8

• 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?

• 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

• Informal surveys among teachers consistently reveal that many of their students simply give up learning fractions at the point of the introduction of addition.

• 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.

• 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.

• 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

• 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

• Division

• Division by Integers

• Multiplication

• Multiplication by Integers