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…

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, 2001National 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 K–8: The teaching of fractionsThe 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 troubleTirosh (2000)

Polyvalence, again

When it comes to fractions there are multiple interpretations.

What are they? What do students think they are?

Multiple meanings1. Parts of a whole: when an object is equally divided

into d parts, then a/b denotes a of those b parts.

2. The size of a portion when an object of size a is divided into b equal portions.

3. The quotient of the integer a divided by b.

4. The ratio of a to b.

5. 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.

ab

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.

ab

kakb

ab

cd

cd

kakb

This principle is the single most important fact about fractions.

Equivalent fractions

Why is

It’s just arithmetic!

a cb c a

b

a cb c 1c c a

b 1 a

b a

b

?

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?

ad

bd

a bd

?

Addition - Like Denominators

Answer. It’s just arithmetic! We know…

So,

d a bd

a b

a bd

1da b a

d b

d

Common mistakes

Where??? College

ab

ac a

b cab

cd

a cb d

How to add fractions, #1

Definition of addition. In some sources we see… a

b cd

pa qbm

where m lcmb,d

and m pb qd

What’s wrong with this??

How to add fractions, #2

Definition of addition. In other sources we see… a

b cd

ab

dd

cd

bb

a db d c b

d b a d c b

b d

Example – no lcm

12

25 1

2 55 1

2 22

1 52 5 2 2

5 2 1 5 2 2

2 5 910

Example – with lcm

38

14

38

11

14

22

3 18 1 1 2

4 2 3 1 1 2

8

58

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.

Add two fractions

Add the equivalent

fractionsReduce

Find the product of the

denominators

Create equivalent fractions

Basic add fractions process

Adding fractions process, #1

Adding fractions process, #2

Add two fractions

Add the equivalent

fractionsReduce

Find the LCM of denominators

Create equivalent fractions

Advanced add fractions process

A division step here to use the lcm

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_mathematics.html#3

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