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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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Fractions l.jpg

Fractions

G. Donald Allen

Department of Mathematics

Texas A&M University


From the nctm l.jpg
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 l.jpg
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 K–8:

    • 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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Basic Fractions

Equivalent Fractions

Adding Fractions

Subtracting Fractions

Multiplying Fractions

Dividing Fractions

Comparing Fractions

Converting Fractions

Reducing Fractions

Relationships

Subtracting Fractions

Fractions – Basic Syllabus


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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

  • It’s 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. It’s 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…

What’s wrong with this??


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How to add fractions, #2

  • Definition of addition. In other sources we see…




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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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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_mathematics.html#3


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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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