www.whyslopes.com
Appetizers and Lessons for Mathematics and Reason 
a calculus, preparation for calculus and math ed reform website

Online Volumes (Book Orders)
1,  Elements of Reason.
1A. Pattern Based Reason 
1B. Math Curriculum Notes
2. Three Skills for Algebra
3. Why Slopes & More Math

 Mathematics Course Designers: LAMP offers food for thought.
More Site Areas 
1. Help Your Child or Teen Learn 
2. Solving Linear Equations
3. Fractions Ratios Rates Proportions & Units
4. Euclidean Geometry
5. Analytic Geometry/Functions 
6. Number Theory
7. More Calculus
More Site Areas 
8. Complex Numbers 
9. Qc Maths  Education  
10. Secondary IV(?) maths
11. Real  Analysis 
12. LaTeX2HotEqn:
13. Electric Circuits Etc  
14.  Français
15. Algebra, Odds & Ends, Etc
More Site Areas 
16. Math Education Essays
17. Telling & Working with Time
18. Maps, Plans & Drawings
19. Quantitative Skills for  home, shopping and work 
20. Statistics Useful, or Not.

||Définition d'une variable || Algèbre || Arithmetique || Logique ||La raison basée sur les règles et modelés||
Back ] Up ] Next ]


YOU are better than YOU think. Show yourself  how:  

      |      
//  _   _ \\
/\             /\
  <|  (o)   (o)   |> 
 \     | |      / 

Learn to read notes and textbooks like a lawyer, so that no nuance, no subtlety and no clause escapes your attention.

Read  logic chapters 1 to 5  in online volume Three Skills for Algebra  for greater skills & confidence in  work 
and study

 -/[]\- 
||
   / \_ 
 ||||||||||||||||||||||||||||

 Logic chapters 1 to 5  re- appear not in sequence, as is or longer,  in  Volume 1A,  Pattern Based Reason, Bon Appetite.

Logic Mastery
 Amazing, Amusing, Amorous,  Delicious, Delightful, Edifying, Strengthening Elixir. 
It eases work & learning difficulties Makes the hard easier. Opens eyes. Leads to greater precision.
in reading and
writing

Logic mastery makes the hard, easier. Logic mastery  leads to better, stronger and richer comprehension.  Logic mastery  improves reading and writing.  Logic mastery ease learning difficulties.  Logic mastery gives a headstart.  In sum, logic mastery  will develops critical thinking, improve reading and writing, and give a firmer base for work and studies at many levels. Good luck.


After logic  (a) continue reading Three Skills for Algebra, chapters 8 to 14  and do so alongside site area on solving liinear Equations ; or (b) see this calculus starter lesson and Volume 3, Why Slopes  & More Math, chapters 2 to 6;

      |      
//  _   _ \\
/\             /\
<|   (o)   (o)  |> 
     | |     |
   \             /   
\    =   /

Caution: Site advice is approximately correct, for some circumstances, not all. That leaves room for thought

 -/[]\- 
||
  _ / \     
 ||||||||||||||||||||||||||||

What may be learnt and when depends on how skills and concepts are developed. Making the hard easier and clearer will allow earlier & richer development of skills and concepts.


Try the Twiddla Whiteboard. In principle, it  allows to people to draw and chat together online on a copy of this webpage or a clean sheet. The chat may be via text or audio.  Visit www.twiddla.com to set up whiteboards to work with the webpage of your choice.

For online automated help in senior high school maths & calculus, visit  quickmath.com  For Automatic Calculus and Algebra Help with derivatives, integrals, graphs, linear equations, matrix algebra, visit calc101.com  With  overlap, each site quickmath & calc101offers a different range of services, some free, some not, all based on webmathematica. Good luck.

 What is a Variable?
©Alan Selby, August 2000.

www.whyslopes.com/freeAccess/sample_problem_set.html

This webpage presents the text of sample chapter 
for a problem book that was solicited, but not written. 
problems remain to be added.

Goal:   Master the mathematical use of the word variable

Introduction

Look in a dictionary, encyclopedia and a mathematics text for a definition of what is a variable, an introduction that is understandable to you and easily explained to others. If you find such a definition or introduction clear enough to help in mathematics after arithmetic, the rest of this essay need not be read.

Alice in Wonderland if she could speak today, would observe that  the  view of a variable as a function begs the question of how to explain the notion of a function without using the concept of a variable. The essay or chapter before put the concepts of what is a variable first and before the use of symbols and notation in mathematics for numbers, amounts, quantities and functions.

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]

    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

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.  

Postscript for essay What is a Variable.

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

 


To learn more, see

Three Notions of a Variable

Constants,Parameters,Variables

Talking about numbers
Dependent or Independent
Variable, a Matter of Choice

www.whyslopes.com
Algebra, Odds & Ends,


1. Hints for Exams
2A. Exact Arithmetic
2B. Fractions Briefly
3. What is a Variable?
4.. Square Roots
5. Straight Lines
6. Problem Solving Methods
7. Trig and Complex No.
8. Complex Applet
9. History of No.s
10. ln(x) and exp(x)
13. Rename the > Sign
14. Problems: Quadratics
15. Problems: Algebra Test
16. Problems: Linear Eqns I
17. Problems: Linear Eqns II
18. Problem Solving Hints
19. Functions & Sets
20. Independent Variables
21. Why Logic
22. Why Math
23. The 15 Times Table
24.  The  20 Times Table
25. Algebra Formulas
26. On Learning Maths
27. Maths in Biology
28. Navigation +Time
29 Quibble-What is Algebra
30. Logic in Maths


Odd and Ends, Essays

Constant Retirement Rate
Road Safety
3 Strikes Law in California.
Math HOW-TOs
9 Steps in Maths

Twiddle this page! Study With Others: twiddla.com has developed a collaborative whiteboard with audio & text chat that allows students, tutors & teachers to explore & scribble on blank pages and copies of  webpages together,  If scribbling is awkward with one browser, try another.


In Volume 2, Three Skills for Algebra, Chapters 8  to 14 and postscript What is a Variable point to a greater & clear use of words in algebra. Chapter 14 introduces a 4th skill for algebra, an  elaboration of  the third: - The direct and indirect use of formulas, numerically and algebraically, is unifying theme that should be mentioned aloud, with words, in each and every use of formula. 

 


www.whyslopes.com
[Top of this Page] [Site Exit] Back ] Up ] Next ]
[Comments, Reactions, Feedback][ Road Safety Message ]
: Favourite SitesBBC News  and mathematics portion of  English National Curriculum  

All trademarks and copyrights on this page are owned by their respective owners.
Copyright to comments & contributions are owned by the Poster. 
The Rest © 1995 onward by site author,   Alan Selby,
a 1983 McGill. Ph. D. in mathematics
All Rights Reserved.