Understanding Fractions

The idea of Fractions originated from the daily activities of cutting, partitioning & sharing. They emerged as parts of bigger things, or wholes. Parts of a whole are always less than the whole. The term fraction is related to the word fracture which is about “breaking”.

Humans must have started thinking of fractions, when they were sharing things with others. The earliest fractions humans started grappling with seem to be half, quarter and three quarters. Initially these must have just been daily language words without a math connection.

Why Were Fractions Needed?

We have seen that humans learnt to deal with magnitudes in two different ways - by measuring (or estimating) and counting.

A flock of sheep or the number of trees in a farm can be counted. But most magnitudes, like the distance between two points, could not be counted. For that some units were invented.

For example, the distance between two points could be estimated in terms of "number" of steps needed to cover the distance. But most distances could not be measured in exact number of steps. There was always a bit of distance at the end, which was left out.

For most practical purposes, this small bit of distance could be left out. But in cases like building a house, these small bits could not be ignored.

This possibly led to the idea of "parts". We could say that the bit left out was about "half" or a "quarter" of the step with which we measured the distance.

The idea of fractions originated from the idea of trying to quantify these bits which were less than a whole.

Analyses of large, nationally representative, longitudinal data sets from the United States and the United Kingdom reveal that elementary school students’ knowledge of fractions and of division uniquely predicts those students’ knowledge of algebra and overall mathematics achievement in high school, 5 or 6 years later. Hence they are a critical part of the math curriculum in primary school.

Part & Whole are Relative Concepts

What we take as the whole or part, purely depends on the context. What is a part in one context could be whole in another context? For example, a single student is part of a class of students (whole). However, a class of students is a part of all students of a school. All students of a school are a part of the citizens of a city and so on.

Importance of the Whole

At the same time, the idea of the “whole” is very important in understanding fractions. A fraction has no meaning, unless we also refer to the whole, of which it is a part. A toffee which a child receives from a bag of 4 toffees, is seen by the child as a “whole” toffee. Only if he sees both original bag and his toffee, he realises that it is part of a bag!

Introducing Fractions

Fractions have always been difficult areas of math. We will see this aspect in some details in the next chapter. Hence it is best to introduce them to children through concrete activities.

The activities can involve sharing items like rotis, sandwiches, sheets of paper and collections of tokens, crayons etc. Most children have a rudimentary idea of half from their home environment. They can be introduced to sharing equally between two or 4 persons. They can also be introduced to the terms whole, half, quarter and the relations between these terms.

Representation of Fractions

Using the above activities, teachers should introduce children to 3 different visual ways of representing fractions. These are Area, Line & Set representations.

The fourth and the most abstract way is representing fractions with numerals like ½ or 2/3. This should be done only after children get comfortable with the idea of fractions and representing them through activities using the first 3 representations.

Each of these representations has its advantages & disadvantages. We will see these in detail in a subsequent chapter.

Multiple Rules

Fractions & operations with them are difficult to understand. Hence teachers have a tendency to reduce the multiple operations used in dealing with fractions into a number of rules, which have to be remembered or memorized by students. Without “understanding”, these rules become too many to remember and apply correctly, leading to anxiety in quite a few students.

Hence it is better to let students work out the rules with visuals or aids and then facilitate them to try to derive the rules themselves.