Equivalent Fractions

There is one property which is unique to fractions; that of Equivalent Fractions.

The concept of Equivalent Fractions is a very important one. It impacts both understanding of fractions as well as operating with fractions.

Two related perspectives about fractions

1. Fractions as “multiplicative relations” between numbers.

   1. This is the idea through which the idea of a fraction is usually introduced in schools.

   2. It makes it easier to think of an entity which requires 2 separate numbers to describe it.

      1. 3/4 as a relation between numbers 3 & 4. A relation which shows the relative sizes of 3 & 4 in relation to each other. In this sense the relation between 3 & 4 can also be seen as being same as the relation between 6 & 8.

      2. It is similar to the idea of “brother” in social relations. Two boys may not be related to each other at all. But both could be a ‘brother’ of two different unrelated individuals.

2. Fractions are numbers

   1. This says that 3/4 is also a number, on par with whole numbers.

   2. This is a mathematically sophisticated idea and takes many years to form and be accepted.

   3. This idea becomes easier to accept when represented linearly. We can plot fractions on the number line.

The first idea "fractions as relation" is easier to understand. The second idea "fractions are numbers" takes many years to understand.

Equivalent Fractions

Equivalent fractions are connected to the idea of a “relation”.

They are also used to emphasize the idea that all equivalent fractions represent the “same number or value”.

The value of all equivalent fractions is same. They are all equal in value. But since they are represented with different numbers, they are called “equivalent” numbers.

The term “equivalent” conveys the idea that they are equal in value but different in appearance.

The arithmetic of equivalent fractions is very easy, but the concept is very difficult.

If teachers quickly jump to numerical manipulations (multiply numerator & denominator by the same number), without clarifying the concept, students are likely to hot a roadblock later.

Teachers need to spend a lot of time using area, line & discrete sets representations for children to get used to the idea of equivalent fractions.

Here are some ways in which the idea of “a third” is demonstrated by dividing a rectangular cake.

Diagram here

Standard Form of an Equivalent Fraction

Equivalent fractions are infinite ways of expressing the same fraction using different digits. “A Third” can be represented as 6/18 or 4/12 or 18/54 or 1/3.

The common convention is to use the numeral representation using the smallest numbers. Here it is 1/3. So “a third” is usually represented as 1/3. It is also called the “standard form” of “a third”.

Teachers wrongly call this process as "simplifying". It is rewriting a fraction with the simplest possible numbers.

It is representing a fraction with a numerator & denominator which are co-primes. This means that the Numerator & Denominator do not have a common factor and hence cannot be represented with smaller numbers.

Once this concept is understood, the pattern uniting the various numerical representations is easy to see. That each equivalent fraction can be derived by any other equivalent fraction by dividing or multiplying the numerator & the denominator by the same number.

That number can be any type of number; whole number or fraction or any real number.

6/18 becomes 1/3 if both the numerator & the denominator is divided by 6.

4/12 becomes 18/54 when both the numerator or denominator is multiplied by 9/2.

A thorough understanding of equivalent fractions and how they are formed, is important for further understanding of operations with fractions.