Fractions

- G. Donald Allen
- Department of Mathematics
- Texas AM 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

- 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

- Conceptual understanding
- Procedural fluency
- Strategic competence
- Adaptive reasoning
- Productive disposition

Adding it Up, - National Research Council

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

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

- When it comes to fractions there are multiple

interpretations. - What are they?
- What do students think they are?

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

- 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
- Relationships
- Subtracting Fractions

Comparing Fractions

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

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

- Prime and Composite Numbers
- Factors
- Greatest Common Factor
- Least Common Denominator
- Least Common Multiple
- Simplifying

Relationships

- Relating Fractions To Decimals
- Relating Decimals to Fractions
- Relating mixed fractions to improper fractions
- Relating improper fractions to mixed 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 fractions

- Why is
- Its just arithmetic!

?

Productive disposition

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
- Addition - Like Denominators
- Addition - Unlike Denominators
- Addition Mixed Numbers

Addition - Like Denominators

- Why is
- It is by Pie charts? Fraction bars? Spinners?

Blocks/Tiles?

?

Addition - Like Denominators

- Answer. Its just arithmetic! We know
- So,

Common mistakes

Where??? College

How to add fractions, 1

- Definition of addition. In some sources we see

Whats wrong with this??

How to add fractions, 2

- Definition of addition. In other sources we see

Example no lcm

Example with lcm

lcm 8

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.

Adding fractions process, 1

Adding fractions process, 2

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?

- 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

- 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 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 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

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

- Division
- Division by Integers

Multiplying Fractions

- Multiplication
- Multiplication by Integers

Division of fractions

Mixed fractions

Multiplication of fractions