Original Site Title: Appetizers and Lessons for Mathematics and Reason, June 1995 to April 2012. New site title:
Logic and Mathematics Skill & Concept Development with How-TOs Français: 26 pages
for college students, gifted teens, home-tutoring and K1-12 schooling; and for avid readers in school and out. See site volumes.

Logic 5 Chapters Arithmetic 10 Steps Algebra 12 Starter Steps & 5 Advanced Steps
Work & Study 23 Tips Geometry 15 Steps Calculus 70 Lessons. See Site Map

Ages 15+: Why study slopes Polynomials Quadratics Why factor polynomials Logarithms Functions
What is similarity Euclidean geometry leanly Coordinates + complex no.s Vectors DC Electric Circuits
Ages 12+: Prime factorization Written work formats Decimal place value Extend arithmetic skills orally
What is a variable 5. Fraction Operations by Raising Terms Solving Linear Equations: Take I Take II

Online Volumes: 1 - Elements of Reason, 2 - 3 Skills For Algebra, 3 - Why Slopes and
More Math, 1A - Pattern Based Reason, 1B - Skill Development Principles + Troubles

Welcome: Site content may develop critical thinking, improve reading and writing, and build mathematics skills. See online chapters on on logic and pattern based reason.

Teachers: This December 2011, 5-phase framework offers a context for mathematics & logic instruction. Phases 1 to 3 focus on skills with actual or potential value for adult & daily life. College-oriented phases 5 & 4 focus on calculus & preparation for it. Phases 1 to 4 may also serve trades & professions not dependent on calculus.

Site Review: Math resources ... span ... arithmetic, logic, algebra, calculus, complex numbers, and Euclidean geometry. Lessons and how-tos .... provide a good foundation for high school and college ... mathematics. Read more.

Home < Algebra Starter Lessons << 2 What is a Variable

[1] [2] [3][4] [5] [6] [7]

What is a Variable?
İAlan Selby, August 2000.

Introduction

Words and examples to clarify what is a variable follow. We may talk about and describe numbers and quantities as being variable or constant before and then  besides the use of letters to stand in, represent or denote them and before talking about functions.

Look in a dictionary, encyclopedia and a mathematics text for a definition of what is a variable, a non-technical explanation that is understandable to you and easily repeated to others. Likely, you will not find one. The explanation below provides a remedy, one that may arrive sooner or later in dictionaries and mathematics courses.  

Variation in a Single Example

variation = amount of change

The next diagram shows the height of a bird during its journey from one tree to another.  The flight  is over the ground intervals 

[a,b], [b,c], [c,d], [d,e], [e,f]

original form by Alan Selby

Alternate form, courtesy of Sumit Paranjpe

Flight of a Bird

Letters on  horizontal axis end ground intervals where the height behavior changes. If height is measured above or below sea level, and the tops of both trees were below sea level, then increasing height would correspond to make the height relative to sea level less negative. 

Identify the intervals where the height of the bird is constant, where this height is increasing (becoming more positive or less negative) and where this height is decreasing (becoming less positive or more negative). The height may have different behaviors on different ground or time intervals. This exercise could be redone on a graph of height versus time. In this case, the ground intervals would correspond to time intervals. 

To vary means to change. Identify the ground intervals where the height of the bird is constant (not variable) and where it is variable. 

Conclusion: Whether or not a number or quantity is constant or not, variable may depend on the interval in which is observed or examined or remembered. We can talk about numbers and quantities being variable without or before the use of letters to represent them.

The following diagram shows the speed of a car along a straight road.  

Piecewise linear graph of speed versus time

Identify the time intervals where the speed of the car is constant and where it is variable. 

Challenge (a hard exercise):  From the above diagram, how would you find the distance traveled by the car in a constant-speed interval and in the variable speed intervals. Find a solution without the use of calculus. Hint: See an old high school physic text.

Variation between Examples 

In the following diagram are rectangles with different areas, heights and width. 

Rectangles B, C and D

For each rectangle, its area, its height  and  its width is constant, at least while the rectangle is not being stretched.  But each of the three quantities area, height  and width  change or vary when we shift our attention from one rectangle to another. So while our attention is fixed on one rectangle, these three quantities are constant.  Yet these three quantities change,  are variable, when we shift our attention from one rectangle to another.  These three quantities do not have the same value for each rectangle shown in the diagram. 

Conclusion: A number or quantity may have a constant or fixed value in a single situation or a single circumstance, but the number or quantity in question may vary or be variable between different circumstances. 

The next diagram shows or indicates the number of people in a home during a day

[Diagram showing 4 people from midnight to 8 am., 2 people from 8 am to 9 am, 1 from 9 am to 4 pm, 3 from 4 pm to 7 pm and 4 again from 7 pm to midnight.]

Diagram showing 4 people from midnight to 8 am, 2 people from 8 am to 9 am, 1 person from 9 am to 4 pm, 3 from 4 pm to 7 and 4 again from 7 pm to midnight.

During each hour the number of people is constant. But the number of people is not constant for a full day because of departures and arrival at 8 am, 9 am, 4pm and 7pm. So the number of people is variable. During the small time intervals where people are leaving or entering,  you may have a person not fully in the house. During these small time intervals, how to count or define the number of  people is a matter of taste.  Food for thought: How would you count or define the number of people in the house during these small transitions, time intervals? When you have 4 people in the house, and 1 is leaving, my thought is that you should say there are 3 to 4 people in the house, but it may impolite to talk about fractions when speaking of people.  Saying you had 3.45 people to a party might lead to a criminal investigation :)

Variation of Letters

Letters have not been used in the above discussions of what numbers and quantities are variable, including when and in what sense. In the next diagram, letters and symbols appear in formulas for the calculation of areas and of perimeters for a circle and a rectangle.  


Correction: For the circle: Area A = p r2 and Perimeter  s = 2 p r 

In the  formulas, for precision (ad nauseum) we say

  1. the lowercase Greek letter   p is constant given by 3.1416 (approximately).
  2. the uppercase Roman letter A stands for the area of the circle or rectangle (depending on which one you are looking at), 
  3. the lowercase  Roman letter r stands for the radius of the circle, 
  4. the uppercase  Roman letter H stands for the height of the rectangle, '
  5. the uppercase Roman letter W stands for its width,  
  6. the lowercase Roman letter p stands for the perimeter of the rectangle, and
  7. the lowercase Roman letter s stands for the perimeter of the circle. 

The phrase "stands for" could be replaced by the phrase "is shorthand for" or "is placeholder for" or "stand-in for", or by the word "represents" or "denotes".  Some help with the English language follows.

  • denotes: to mark, signify, mean,  indicate, to be the name of.
  • placeholder: keeper of a portion of space for an number or quantity or object in general.
  • represents: stand for, symbolize, act as the embodiment of, 
  • shorthand: a method for rapid writing and abbreviation
  • stand for: act in the place of another.
  • stand-in for:  a deputy or substitute, for another actor.

You may meet other phrases that indicate the shorthand role of letters as placeholders or notation  or abbreviations for numbers and quantities in calculations. 

When does a letter denote a variable?

Letter as shorthand symbols for numbers and quantities appear in the above formulas.  
  1. When should we say that a letter or shorthand symbol is variable? 
  2. When should we call a letter or symbol a variable. 

Answers for both questions follow.

In the case of variation in a single example,  when a symbol or letter represents or stands for a number or quantity that may vary, we will say that that symbol or letter is a variable, and we will call it a variable as well.  Think here of the height h of a bird or the number n of people in the house  in the diagrams given above and reproduced below.

In the case of variation between examples, when when a symbol or letter represents or stands for a number or quantity that may vary, we will also say that that symbol or letter is a variable, and we will call it a variable as well.  Think here of the area A, height H and width L of the rectangles in the next diagram.

For each rectangle, the numbers or quantities denoted by A, L and W are constant, but between the rectangles, these three quantities vary.  So we say the symbols or placeholders A, L and W are constant or variable, according to whether or not we are thinking about their lack of variation for a single rectangle or their variation between rectangles. 

Old dictionaries and old algebra texts may be half-right when they indicate without further explanation that variable is letter used in mathematics, at least when we add the thought that a letter denotes a number or quantity that may vary.  Beyond this, the number or quantity need not have a physical meaning. Think for instance of a number that may be written by someone else and placed in an envelope for safe keeping or privacy. Denoting that number by x allows us to describe calculations with that number hidden in the envelope, with x as shorthand for it.  Calculations with a number placed in an envelope could also be described with the abbreviation x before the number is actually placed in the envelope.

Cases of Double Variation

Ten people have ten piggy banks to which they add and subtract spare coins. The value V of coins in each piggy bank depends on the person and on time. So  there here is an example of double variation: variation over time for each piggy bank, and variation between piggy banks at each moment.  


Diagram of rectangles with width constant over columns, but varying along rows.

     

   
 

 

   

Height too varies in one direction but not another. The notion of varying or not can be understood before the use of symbols.

  • Width is a constant for each column, a constant that differs or varies between columns. That may give a variable constant.
  • Height is variable for each column, but this variable is constant along rows. That may give a constant variable :)

If you change the width of this page (resize your browser window), the width may also vary over time.

Conclusion or recapitulation

Numbers and quantities may vary

  • in one or more spatial directions
  • over time
  • between examples

all at once or separately.

Numbers and quantities may vary in different directions (spatial or temporal) and between discrete instances

 

Teachers & Tutors: Site pages offer better or best practices for providing skills - simpler than expected & comprehensive but for exercises. For your charges, your duty is to study them alone or in groups and develop skill building exercises & activities to share. Start now. The effort here is the best I can do. Others are welcome to refine or exceed it. Please do.

Secondary Mathematics for Ages 11+, A Practical Approach for home-tutoring or -schooling, or for schools & colleges with local curriculum control. Study how to include site content - its skill development how-TOs and innovations into present or future lesson plans - some reading required.

Road Safety Messages and Questions: When and why should you face traffic when walking along a road or cycle path? Is it a good idea to hang limbs outside of cars etc? What gives more protection in a crash: a car, motorbike or bicycle? See too, the BBC-Belgium story Texting and Driving - texting & the impossible test - the article links to a gruesome utube video on the subject

The Logic of Injustice: How Texas sent an innocent man to his death - The wrong Carlos. Some judgments are irreversible. Procescution: Where and when prosectors play to win rather than for justice, guilt beyond a reasonable doubt goes unrespected due to prosecutors who putting winning first, those innocence before the law may be convicted. Some procescutors offices in continuing to accuse after a pardon due to reasonable doubt or innocent being shown, may sucessfully oppose compensaton for false convictions by asserting a pardon individual is still under suspicion. Then the pardoned individual or the latter's estate is not compensation for years or decade of improper or false imprisonment, or for execution. Site chapters on Logic
and some in Pattern Based Reason may slowly lead to greater precision in reading, applying and writing laws.

May 2012, Composition Starting: Pre-School and Primary Mathematics - Quantitative Skills, An Intellectual View, Feedback Welcome:

The 8 Most Popular Site Inlinks

20 Times Table - the most popular site page - popular pages - unexpected.
Fractions & Ratios - with lesson on raising terms to introduce & justify times, division & comparison as well addition & subtraction
Parent Center - See below
Volume 1, Elements of Reason - Intro to all site books.
What is a Variable - best for ages 13+
Written work formats for Arithmetic and Algebra - a skill method and standard!
Complex Numbers Visually - best for ages 13+
Natural Logs, Exponentials, Powers, Roots

Division of Labour: This site offers advice and directions with pointers to resources elsewhere, if known, when they help or lessen the need to write more.

Parent Center: Help your child or teen learn:

Parent-friendly Work Booklets for ages 3+ to 13 Use these or others to check or build skills. Other booklets are available but these booklets allow parents unsure of themselves in mathematics to help their children. The selection acquired in Canada is published in the USA. So it has a US orientation. In retrospect, the selection shows parents what to check with the booklets or by other ways, the choice is theirs. But in retrospect, the selection does not cover integral and fractions liquid weights and measures - ask the publishers to correct that! For ages 9 to 12 say, parents may compensate by showing boys and girls how to use weights or mass, and further measures in food preparation. Beyond that children may be shown how to measure and calculate angles, lengths and areas [proportional amounts too] directly or by using maps and plans drawns to scale. Learning how to gather and measure all the ingredients, pots and pans for a dish or a meal, along with cleaning up sets the stage for like activities or experiments in science courses, and in developing organizational skills, gives boys and girls a head start. Good luck. At the other extreme, more comprehensive than light, if your motto is McCainian: drill, drill, drill then Toronto mathematician and actor John Mighton's jump math organization has jump math workbooks for at least grades 3 to 8 for at-home and in-school use - training sessions for teachers available. Jump math has been expanding to cover older students. Jump Math Samples: plus Fractions for Grades 3-4 & Grades 5-6 [Read] Free Resources grades 1 to 8 [unread - likely to be good]. and

Mathematics Skills For Ages 3 to 14 - technical!

Skills with take home value - A few ideas

Basic skills include time-date-calendar Matters; money matters; map, plan and scale diagram matters;counting, measuring and figuring; decision making with logic and likelyhood; being careful and being aware of the domino effect of mistakes; reading and writing with precision.

Is your child able to add, subtract and multiply amounts of money, work with fractions, work with clocks and calendars, work with maps and plans, and measure length, weight-mass and volume? Schools may promote your son or daughter without providing basic skills in reading, writing and arithmetic.

Arithmetic and Number Theory Skills

Algebra Starter Lessons

1 Working With Sets
2 Formula Forward Use - Evaluation
3 Solving Linear Equations - Skip first step with students able to solve 1 eqn in 1 unknown.
4 Computation Rules and Function Notation
5 Real Numbers
6 More Less Greater Than Inequalities and Comparison
7 Axioms Logic and Equivalent Equations
8 Unifying Theme For Algebra
9 Proportionality Backwards and Forwards
10 Examples of Algebraic Reasoning
A Origins of Counting and Figuring Methods
B Real Numbers Extrinsic Development


Site coverage of formuala evaluation format, of computation rules and axioms, and of the forward and backward use of formulas and proportionality relations lessens the amount of natural talent needed to understand and explain algebra.

Geometry - maps plans trigonometry vectors

1 Maps Plans Measurement
2 Euclidean Geometry - Constructions + extras
3 Cartesian and Polar Coordinates
4 Lines and Slopes Take 1
5 What is Similarity
6 Trigonometry first steps
7 Complex Numbers
8 Unit-Circle Trigonometry
9 Lines and Slopes Take 2 with tangent function
10 Intersecting Straight Lines and Transversals
11 Parallel Straight Lines and Transversals
12 Function Translating and Rescaling
13 Vectors
14 Degrees to Radians and Radians to Degrees
15 Arc or Inverse Trigonometric Function

Pre-Teen and young teen mastery of skills and practices which should be common with map-plans-diagrams drawn to scale, contour interpretation included, has actual or potential take-home value for daily- and adult-life in solving routine problems. Elevating some practices to principles, axioms or postualates, provides a base for analytic and Euclidean geometry, an analytic view of similarity, and an efficient mastery of trigonometry and complex numbers. Right triangle trigonometry provide an analytic alternative to solving geometric problems by drawing diagrams to scale.

More Algebra

Natural-Logarithms Exponentials Powers Roots
Five Polynomial Operations
Quadratics Geometrically
Functions
5 Factored Polynomial Sign Analysis Examples
Rewriting algebraic substitution as function substitutions

The first topic leads to a full high school level theory for the forward and backward mastery of growth and decay models and for definition, range and domains of radicals, roots and powers. The next two topics make quadratics and polynomials easier to learn and teach. Site coverage of functions turns vertical and horizontal line rules into computation methods for evaluating functions.

70 Calculus Starter Lessons

Calculus Lessons Elsewhere:

  1. How to Ace Calculus: Street Wise Guide - Mostly Text.

  2. Flash Video for Calculus Phobics

They cover basic topics in ways likely to complement your notes, your textbooks and site material. When Goldilocks trespassed in the house of the three bears, she found three bowls of porridge, two not to her liking, and one just right. Different bears have different tastes. As invited guest here and elsewhere, if one or more explanations is not to liking, try another. It may be better or just right.

Unsolicited Advice

Learning to do and high marks if it comes to easy is often deceptive - light rather than deep. For that reason, students with learning difficulties determined not to let it get in their way may go deeper and farther than those with none. High marks, if the come easy, may be deceptive - provide a too light and not a deep mastery. That could have been your problem in secondary school, one that leads to comprehension shock or difficulties in calculus and more generally in the first year of college. Bon Appetite.

Return to Page Top

Home < Algebra Starter Lessons << 2 What is a Variable

[1] [2] [3][4] [5] [6] [7]

Logic-Reason for all
Careful Thinking
Chains of Reason
Mathematical Induction
Responsibility
Bodies-of-Knowledge

Arithmetic - Ages 10+
1. Deciml Place Value - fun
2. Decimals for Tutors
3. Prime Factors - quickly
4. Fractions + Ratios
5. Arith with units - science

Geometry
1 Maps + Plans Use
2 Euclidean Geometry
3 Rct +Polr Coordinates
4 Lines-Slopes [I]
5. What is Similarity 		Algebra Starters - the base
1. Better Work Format
2. Solve Linear Eqns
3. Computation Rules
4. Axioms, Item 3 Viewpnt
5. Formulas Backwards 		More Algebra
Logarithms-ax & m/nth roots
Five Polynomial Operations
Quadratics Geometrically
Functions || Vectors too
Arith. Skill Check+Answers 		Calculus Prep/Preview
What is a Variable
Why study slopes
Why factor polynomials
Complex Numbers
Limits + Continuity

All trademarks and copyrights in this are owned by their respective owners.
Copyright to comments & contributions are owned by the Poster.
The Rest © 1995-2011, by site author, Alan Selby, Ph. D., Montreal,
All Rights Reserved --- Skype or Email to contact.