The Number Line 1

Understanding the Number Line

The number line is mostly represented, to start with, as a discrete representation of whole numbers and used as a tool for counting, comparing, addition & subtraction operations with them.

The number line can lead to a deeper understanding of numbers and the number system itself.

It is essentially a model representing the "measurement metaphor". Students should be encouraged to think of it as a “continuous” line. The idea of a continuous number line should be revisited as and when new types of numbers are invented.

The number line is mostly conceptualized as a "straight" line. But it can also be thought of a a "curved" line. The important idea is the linear sequence of the numbers on the line. The space between any two consecutive numbers is also equal.

Though we start with whole numbers, even fractions, decimals, integers & irrational numbers can be represented on the number line.

The number line which extends “forever” in both directions beautifully captures the idea of “real umbers”. Ultimately the number line becomes the “real number line”. Please refer to Chapter 23.04 "Number Line 2".

It also becomes the x-axis of the two-dimensional cartesian plane.

This in turn facilitates the idea of representing imaginary & complex numbers on the "number plane"!

Three Aspects of a Number Line

Students need to understand three aspects of a number line, in increasing order of abstraction.


 * 1) As a pedagogical tool for
 * 2) Ordering and positioning of natural numbers & later fractions, decimals. Integers and eventually all real numbers
 * 3) Forward/ Backward & skip counting
 * 4) Developing numeracy skills
 * 5) Calculation strategies got addition & subtraction
 * 6) As an aid in thinking 
 * 7) While using the number line for counting, children are actually counting the spaces and not the numbers written at intervals
 * 8) Identifying wholes when they mark fractions on the number line
 * 9) Of the real number system as an integrated one.
 * 10) As a representation for the number system
 * 11) We start with marking the whole numbers from 0 onwards
 * 12) Then we add negative numbers to the line
 * 13) Then fractions
 * 14) Then irrationals
 * 15) Finally reaching real numbers

The Open Number Line

An Open Number Line is just a line without any markings. Children should be able to mark the points and use the lines for comparing, addition or subtraction.

It helps students to visualise many concepts for building number sense - breaking numbers & regrouping for addition & subtraction.

See https://www.k-5mathteachingresources.com/empty-number-line.html

Double Open Number Line

A closed number line has all of the tick marks, every whole number has a tick mark. An open number line is one where we get to choose, we put the tick marks where we want them, we sort of label the numbers that are helpful for us in solving a problem.

A double-open Number Line is a proportional reasoning model. A double open number line keeps track of corresponding quantities.

For example in a fraction- related problem, the same entities may be expressed in numbers as well as fractions. In a double-open number line, one line can represent the numbers & the other can represent the fractions. The length of both the lines would be same and points where both the number and the fraction represent the same entity would be coinciding.

An example of such a problem would be - "John read 30 pages of a book on Monday. On Tuesday he read 1/8th of the book. On Wednesday he completed the remaining 1/4th of the book. How many pages did the book have?"

It can also be used to represent two entities which are proportional. For example, the first line shows the distance & the second line shows the time. Both remain in proportion,"