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.
For students of reason in society, science and technology:
Pattern Based Reason describes
origins, benefits and limits of rule- and pattern-based thought and
actions. Not all is certain. We may strive for objectivity, but not
reach it. Postscripts offer
a story-telling view of learning: [
A ] [
B ] [
C ] [
D ] to suggest how we share theories and practices.
These online chapters may amuse while leading 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. Site Theme: Different entry
points may be easier or harder for knowledge mastery.
Deciml Place Value - funny ways to read multidigit decimals forwards and
backwards in groups of 3 or 6, US-CDN, UK-German and Metric SI style.
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.
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.
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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.
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]
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.]](images/sample4.gif)
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
- the lowercase Greek letter p is constant
given by 3.1416 (approximately).
- the uppercase Roman letter A stands for the area of the circle or
rectangle (depending on which one you are looking at),
- the lowercase Roman letter r stands for the radius of the circle,
- the uppercase Roman letter H stands for the height of the rectangle,
'
- the uppercase Roman letter W stands for its width,
- the lowercase Roman letter p stands for the perimeter of the
rectangle, and
- 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.
- When should we say that a letter or shorthand symbol is variable?
- 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.
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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
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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]
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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?
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. ...
Maps + Plans Use - Measurement use maps, plans and diagrams drawn
to scale.
Euclidean Geometry - See how chains of reason appears in and
besides geometric constructions.
Coordinates - Use them not only for locating points in the plane
or space.
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 assumes no
knowledge and some knowledge of the tangent function in
trigonometry.
What is Similarity - another view of using maps, plans and
diagrams drawn to scale in the plane and space. May buildings in
space are similar by design.
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 - this 1995-96 lesson introduces calculus
skills and concepts. It may also may be given to introduce further function maxima
and minima both inside and at the ends of closed intervals.
Check Arith. Skills - too many calculus and precalculus
students do not have strong arithmetic and computation skills. The
exercises here check them while numerically hinting at
equivalent computation rules.
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