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Mathematics and Logic - Skill and Concept Development

with lessons and lesson ideas at many levels. If one site element is not to your liking, try another. Each one is different.

30 pages en Francais || Parents - Help Your Child or Teen Learn
Online Volumes: 1 Elements of Reason || 2 Three Skills For Algebra || 3 Why Slopes Light Calculus Preview or Intro plus Hard Calculus Proofs, decimal-based.
More Lessons &Lesson Ideas: Arithmetic & No. Theory || Time & Date Matters || Algebra Starter Lessons || Geometry - maps, plans, diagrams, complex numbers, trig., & vectors || More Algebra || More Calculus || DC Electric Circuits || 1995-2011 Site Title: Appetizers and Lessons for Mathematics and Reason

Mathematics Concept & Skill Development Lecture Series: Webvideo consolidation of site lessons and lesson ideas in preparation. Price to be determined.

Bright Students: Top universities want you. While many have high fees: many will lower them, many will provide funds, many have more scholarships than students. Postage is cheap. Apply and ask how much help is available. Caution: some programs are rewarding. Others lead nowhere. After acceptance, it may be easy or not to switch.

Are you a careful reader, writer and thinker? Five logic chapters lead to greater precision and comprehension in reading and writing at home, in school, at work and in mathematics.
- 1 versus 2-way implication rules - A different starting point - Writing or introducting the 1-way implication rule IF B THEN A as A IF B may emphasize the difference between it or the latter, and the 2-way implication A IF and ONLY IF B.
- Deductive Chains of Reason - See which implications can and cannot be used together to arrive at more implications or conclusions,
- Mathematical Induction - a light romantic view that becomes serious.
- Responsibility Arguments - his, hers or no one's
- Islands and Divisions of Knowledge - a model for many arts and disciplines including mathematics course design: Different entry points may make learning and teaching easier. Are you ready for them?

Early High School Arithmetic

Deciml Place Value - funny ways to read multidigit decimals forwards and backwards in groups of 3 or 6.
- Decimals for Tutors - lean how to explain or justify operations. Long division of polynomials is easier for student who master long division with decimals.
- Primes Factors - Efficient fraction skills and later studies of polynomials depend on this.
- Fractions + Ratios - See how raising terms to obtain equivalent fractions leads to methods for addition, comparison, subtraction, multiplication and division of fractions.
- Arithmetic with units - Skills of value in daily life and in the further study of rates, proportionality constants and computations in science & technology.

Early High School Algebra

What is a Variable? - this entertaining oral & geometric view may be before and besides more formal definitions - is the view mathematically correct?
- Formula Evaluation - Seeing and showing how to do and record steps or intermediate results of multistep methods allows the steps or results to be seen and checked as done or later; and will improve both marks and skill. The format here allows the domino effects of care and the domino effects of mistakes to be seen. It also emphasizes a proper use of the equal sign.
- Solve Linear Eqns with & then without fractional operations on line segments - meet an visual introduction and learn how to present do and record steps in a way that demonstrate skill; learn how to check answers, set the stage for solving word problems by by learning how to solve systems of equations in essentially one unknown, set the stage for solving triangular and general systems of equations algebraically.
- Function notation for Computation Rules - another way of looking at formulas. Does a computation rule, and any rule equivalent to it, define a function?
- Axioms [some] as equivalent Computation Rule view - another way for understanding and explaining axioms.
- Using Formulas Backwards - Most rules, formulas and relations may be used forwards and backwards. Talking about it should lead everyone to expect a backward use alone or plural, after mastery of forward use. Proportionality relations may be use backward first to find a proportionality constant before being used forwards and backwards to solve a problem.

Early High School Geometry

Maps + Plans Use - Measurement use maps, plans and diagrams drawn to scale.
- Coordinates - Use them not only for locating points but also for rotating and translating in the plane.
- What is Similarity - another view of using maps, plans and diagrams drawn to scale in the plane and space. Many human-made objects are similar by design.
- 7 Complex Numbers Appetizer. What is or where is the square root of -1. With rectangular and polar coordinates, see how to add, multiply and reflect points or arrows in the plane. The visual or geometric approach here known in various forms since the 1840s, demystifies the square root of -1 and the associated concept of "imaginary" numbers. Here complex number multiplication illustrates rotation and dilation operations in the plane.
- Geometric Notions with Ruler & Compass Constructions :
1 Initial Concepts & Terms
2 Angle, Vertex & Side Correspondence in Triangles
3 Triangle Isometry/Congruence
4 Side Side Side Method
5 Side Angle Side Method
6 Angle Bisection
7 Angle Side Angle Method
8 Isoceles Triangles
9 Line Segment Bisection
10 From point to line, Drop Perpendicular
11 How Side Side Side Fails
12 How Side Angle Side Fails
13 How Angle Side Angle Fails

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www.whyslopes.com >> Algebra Starter Lessons >> 2 What is a Variable Next: [3 Adding Words To Arithmetic.] Previous: [1 Three Skills For Algebra.]   [1] [2] [3][4] [5] [6] [7]

What is a Variable?
İAlan Selby, August 2000.


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.

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.  

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

www.whyslopes.com >> Algebra Starter Lessons >> 2 What is a Variable Next: [3 Adding Words To Arithmetic.] Previous: [1 Three Skills For Algebra.]   [1] [2] [3][4] [5] [6] [7]

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Road Safety Messages for All: When walking on a road, when is it safer to be on the side allowing one to see oncoming traffic?

Play with this [unsigned] Complex Number Java Applet to visually do complex number arithmetic with polar and Cartesian coordinates and with the head-to-tail addition of arrows in the plane. Click and drag complex numbers A and B to change their locations.

Pattern Based Reason

Online Volume 1A, Pattern Based Reason, describes origins, benefits and limits of rule- and pattern-based reason and decisions in society, science, technology, engineering and mathematics. Not all is certain. We may strive for objectivity, but not reach it. Online postscripts offer a story-telling view of learning: [ A ] [ B ] [ C ] [ D ] to suggest how we share theory and practice in many fields of knowledge.

Site Reviews

1996 - Magellan, the McKinley Internet Directory:

Mathphobics, this site may ease your fears of the subject, perhaps even help you enjoy it. The tone of the little lessons and "appetizers" on math and logic is unintimidating, sometimes funny and very clear. There are a number of different angles offered, and you do not need to follow any linear lesson plan. Just pick and peck. The site also offers some reflections on teaching, so that teachers can not only use the site as part of their lesson, but also learn from it.

2000 - Waterboro Public Library, home schooling section:

CRITICAL THINKING AND LOGIC ... Articles and sections on topics such as how (and why) to learn mathematics in school; pattern-based reason; finding a number; solving linear equations; painless theorem proving; algebra and beyond; and complex numbers, trigonometry, and vectors. Also section on helping your child learn ... . Lots more!

2001 - Math Forum News Letter 14,

... new sections on Complex Numbers and the Distributive Law for Complex Numbers offer a short way to reach and explain: trigonometry, the Pythagorean theorem,trig formulas for dot- and cross-products, the cosine law,a converse to the Pythagorean Theorem

2002 - NSDL Scout Report for Mathematics, Engineering, Technology -- Volume 1, Number 8

Math resources for both students and teachers are given on this site, spanning the general topics of arithmetic, logic, algebra, calculus, complex numbers, and Euclidean geometry. Lessons and how-tos with clear descriptions of many important concepts provide a good foundation for high school and college level mathematics. There are sample problems that can help students prepare for exams, or teachers can make their own assignments based on the problems. Everything presented on the site is not only educational, but interesting as well. There is certainly plenty of material; however, it is somewhat poorly organized. This does not take away from the quality of the information, though.

2005 - The NSDL Scout Report for Mathematics Engineering and Technology -- Volume 4, Number 4

... section Solving Linear Equations ... offers lesson ideas for teaching linear equations in high school or college. The approach uses stick diagrams to solve linear equations because they "provide a concrete or visual context for many of the rules or patterns for solving equations, a context that may develop equation solving skills and confidence." The idea is to build up student confidence in problem solving before presenting any formal algebraic statement of the rule and patterns for solving equations. ...

Senior High School Geometry

- Euclidean Geometry - See how chains of reason appears in and besides geometric constructions.
- Complex Numbers - Learn how rectangular and polar coordinates may be used for adding, multiplying and reflecting points in the plane, in a manner known since the 1840s for representing and demystifying "imaginary" numbers, and in a manner that provides a quicker, mathematically correct, path for defining "circular" trigonometric functions for all angles, not just acute ones, and easily obtaining their properties. Students of vectors in the plane may appreciate the complex number development of trig-formulas for dot- and cross-products.
Lines-Slopes [I] - Take I & take II respectively assume no knowledge and some knowledge of the tangent function in trigonometry.

Calculus Starter Lessons

Why study slopes - this fall 1983 calculus appetizer shone in many classes at the start of calculus. It could also be given after the intro of slopes to introduce function maxima and minima at the ends of closed intervals.
- Why Factor Polynomials - Online Chapter 2 to 7 offer a light introduction function maxima and minima while indicating why we calculate derivatives or slopes to linear and nonlinear curves y =f(x)
- Arithmetic Exercises with hints of algebra. - Answers are given. If there are many differences between your answers and those online, hire a tutor, one has done very well in a full year of calculus to correct your work. You may be worse than you think.

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