Ho Math Chess Research and Articles > Why fraction is so difficult?


5 Oct 2009

 

Why fraction is so difficult?
Because it is too powerful.
 
Frank Ho, Amanda Ho
 
www.mathandchess.com
 
September, 2009
 
Many have thought that fractions are difficult to learn for students because the concepts of fractions were not explained to students. Many research papers are written on how to “teach” fractions to students. Fraction is difficult for students because it “inherently” has some powerful properties, which no other numbers can even come close when comparing to fraction. Fraction has many names to confuse students: proper, improper, mixed, equivalent, simplifying, unit, relative prime etc. The operations of multiplications require fractions to be changed to improper. Before the multiplying operations, fractions need to be “cleaned up” (simplifying) etc. The negative fractions operations require a different set of rules for doing addition or subtractions. The meaning of fraction also is very confused for students.
 
  1. A pizza is cut into 3 equal pieces. Each piece is represented by __________.
  2. 10 girls are represented in a group of 20 kids by ____________.
  3. 7 white and 3 black balls are in a basket, each white ball is represented by _______.
Each piece in the above example may or may not be “equal” size.
 
A fraction can have the following properties:
 
  1. Expanding or contracting (reducing, simplifying).
 
If these unique properties of fraction were pointed out students by comparison then students would understand better than not doing any comparison at all. The following list some of the unique features of fractions when comparing to other types of numbers like integers or decimals.
 
 
A fraction can expand or contract but not integers or decimals. The name for this property is called equivalent fractions. Often a fraction is splitter (factorization) to conduct reducing and this is normally needed in other number’s operation.
 
  1. Operating like a queen’s move
 
In chess, fraction is like a queen since it can operate in rook’s or bishop’s directions where as a decimal or integers normally operates in bi-direction.
 
3.     Turn upside down (reciprocal)
When working on division, the second fraction is turned upside down to do multiplication. The division does not involve the reverse of times table like whole number times operation, instead the concept of reciprocal is involved.
 
4.     The reciprocal concept is related to exponential number
For example, 2/3 to negative power of 1 is reciprocal but it also is an exponential number.
5.     The rule of converting to the same LCD does not apply to multiplication or division and even worse, the same rule does not work well when doing negative fractions operations. For example, 1 ½ - 2 1/3 is better done by converting to improper fractions. For this matter, all fractions can be converted to improper fractions but then why we are teaching so many different rules to students like 2 1/3 – 1 /12 to just convert the fraction part to the same LCD? Well, one rule sometimes does not cover all conditions. How about 265 ¾ + 167 2/3? It would be much “easier” to just convert the fraction parts to the same LCD and then add together.
6.     When conducting multiplications, we are asking students to convert them to improper, why? It does not have to but then students are required to do 4 multiplications and the addition, it involves distributive law. So shall we teach students distributive law? If not then shall we explain to them why we are converting fractions to improper fractions when doing fractions multiplications? 
 
We would like to thank Mr. Henry Leung’s comments on this article.

Amanda Ho

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