The Number Line 1

Introducing the Number Line

The Number Line is an abstract representation of numbers. It is a versatile representation which can convey several properties & relations connected with numbers.

To start with, the number line is mostly represented, 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, from natural numbers to whole numbers, fractions, decimals, integers, irrationals and real numbers. It can be introduced from preschool itself.

Children also come across this idea in their daily experiences with rulers, tape measures, bathroom scales & thermometers.

Measurement Metaphor

We start with the idea of a number line as a sequence of discrete numbers, as a "counting metaphor".

It is mostly conceptualized as a "straight" line. But it can also be thought of as 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.

It is gradually developed into a "continuous" line on which fractions, decimals, integers & even irrational numbers can be represented.

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.

It is ultimately developed as a model representing the "measurement metaphor".

Real Number Line & Complex Plane

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 30.04 "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

   1. Ordering and positioning of natural numbers & later fractions, decimals. Integers and eventually all real numbers

   2. Forward/ Backward & skip counting

   3. Developing numeracy & number sense skills

   4. Calculation strategies got addition & subtraction

2. As an aid in thinking

   1. While using the number line for counting, children are actually counting the spaces and not the numbers written at intervals

   2. Identifying wholes when they mark fractions on the number line

   3. Of the real number system as an integrated one, which contains within itself the entire number system.

3. As a representation for the number system

   1. We start with marking the natural numbers

   2. We then add 0 and make it represent whole numbers

   3. Then we add negative numbers to the line

   4. Then fractions

   5. Then irrationals

   6. Finally reaching real numbers

The Open/ Blank Number Line

It is a further abstraction of the number line.

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

An Open Number Line is just a number 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 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."