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
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
Arithmetic with units - Skills of value in daily life and in the
further study of rates, proportionality constants and computations in
science & technology.
a Variable? - this entertaining oral & geometric view
may be before and besides more formal definitions - is the view mathematically
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.
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
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.
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 >> Volume 2 Three Skills For Algebra >> Chapter 9 Talking about Numbers or Quantities Next: [Postscript - What is a Variable.] Previous: [Chapter 8 Three Skills For Algebra.]                                         
Chapter 9. Talking about
Numbers or Quantities
Volume 2, Three Skills for Algebra
We can identify and speak about numbers and quantities without doing any
arithmetic. Words from everyday speech can be used to talk about them.
Quantities are also called amounts. Words have been missing to introduce
and describe the algebraic way of writing and reasoning.
The emphasize on words here before or beside symbols
introduces a new dimension or topic in understanding and explaining
mathematics in general, the deliberate and hopefully earlier description
of numbers and amounts apart from arithmetic and algebra.
Words have been missing to introduce and describe the algebraic way of
writing and reasoning. The following sections of this chapter offer
words to begin learning or teaching the algebraic way of writing and
reasoning. Enter one section. Then use the next, previous links in
these pages to move between them..
1. Identifying Numbers and
We first identify some numbers and quantities. After this, perhaps, we
can speak about them, or describe them, all without doing any arithmetic.
There is more to mathematics than just doing arithmetic.
Here are a few not-too-serious examples of numbers and quantities. Height
is a quantity. A building has a height. So has an elephant. The elephant
also has a weight and a width or a girth. A rectangle has a length, a
width and an area. A closed box has a width, a length, a height and a
volume. The people in a room or in a town can be counted. This gives us a
number. The difference between a number and a quantity will be explained
later. More examples of numbers and quantities follow.
- The amount of money in a bank account (measured in dollars, pounds,
- the depth of a swimming pool (measured in inches, feet, yards,
centimeters, meters, etc, whereever these units are in used).
- The height of an airplane (measured in feet or meters).
- The radius of a wheel (measured in whatever units you like).
- The number of goats in a field (a count - no units).
- The number of feet in your height.
- The number of meters in your height - not the same as the number of
- The amount of money you have (in your local currency).
- The speed of a car now (measured in miles per hour, feet per second,
meters per second, or kilometers per hour, etc.)
- The radius, area and perimeter (distance around) of a circle
(measured in feet, inches, centimeters, kilometers, etc.)
- The height, width and length and volume of a box (measured in various
- The rate of interest your savings get - compounded or simple,
measured in percent or given by a decimal number, etc.
- The number of days in this month - whatever month it is, a whole
number depending on the month and, in the case of February, depending on
the year as well.
- The distance between you and your home (measured in miles,
- The time required for a journey (measured in seconds, minutes, hours,
days, weeks, etc.)
This list could continue. We have identified several
numbers and quantities. We can talk and think about these numbers and
quantities although we have not seen and we have not measured
2. Using Everyday
Our next aim is to show how everyday words should be used
in mathematics to describe numbers and quantities - their use here is
close to their everyday meanings. For example, we can say if a number or
quantity is known or not, changing or not, constant or not, increasing,
decreasing, shrinking, growing, confidential or embarrassing, top-secret
or simply forgotten. Everyday words give the descriptive vocabulary of
mathematics. Describing and talking about quantities and numbers is a
part of mathematics after arithmetic. More examples follow.
We can speak about the height of an airplane above the
ground. We can speak about it without measuring it and without knowing it
exactly. The height will be zero when the airplane is on the ground. This
height increases as the plane takes off. The height will then remain almost
unchanged and nearly constant when the plane has reached its maximum height
or cruising altitude. Then at the end of the trip, the height of the plane
will decrease (get smaller) until the plane, we hope, gently lands.
We can also speak about the number of people in a room.
When nobody enters or leaves, this number remains constant. When somebody
enters or leaves, this number varies. This number or count is usually a
whole number or zero. When someone is just leaving and partly in and
partly out of this room, we cannot count or we have to allow
When we speak about the number of people in a room do we
mean completely in, do we include fractions, or do we just say the count
cannot be done at those moments when someone is partly in or out, moving
or not? This number or count needs to be clearly defined. Words are
needed to say precisely how it is computed, otherwise ambiguity
When a building is being constructed, its height is
increasing. The construction and the increase in height of the building
may take place over one or two years. While the building is used, say
seventy years, its height may be constant - unchanging. At the end of the
building's useful life, the building is left to fall down or it is
demolished - torn down. Here over a long or short time, the height
The height of the building varies. This height is
therefore a variable during the construction and the demolition (collapse
or falling down) of the building. The height is usually a constant,
unchanging and invariable quantity during the seventy or so years that
the building is used.
The height of the building may or may not be known to us
during the lifetime of the building. Yet we can still refer to the height
of the building, and to its other dimensions, even if we have not
measured these quantities and even if they are unknown to some or all of
Here are some more questions, just for fun. What do we
mean by the height of the building? Before the building is built, can we
talk about its height? Can the height be taken to be zero? When the
building is being built, is the height of the building equal to the
height of its walls as they are being put up? If the building has a
basement or a foundation, do we say the height of the building is
negative or is it undefined while the basement is being dug, or the
foundations being built? When the building is being demolished, does it
have a height? What is it?
What do we mean by height? Better yet, we can speak of
the height of a building whenever we can say what it represents (means)
and/or how we might measure it. This permits us to speak of the current
height, the planned or intended height, the past height, the future
height. Is the height of a demolished building zero, or undefined? Is the
planned height of a building equal to its actual height before
construction, during construction, during its use or during demolition? A
definition or identification of the height we want to speak about, is
Usage of Words
The above examples show how everyday words are used to
describe numbers and quantities. Our next task is to say further or more
precisely how the words variable, constant, known and unknown are used to
both describe and refer to numbers and quantities.
See the postscript: What is a
A. Variables Versus Constants
To say that a number or quantity is variable means that
the number or quantity may vary or change. To say that a number or
quantity is constant means that its value remains unchanged. For
The Greek letter π= 3.14159... (approximately) stands for or denotes a constant - a value
or number which will never change.
The time of day is always changing. So time is varying.
It is an example of a number or quantity which is always increasing and
therefore variable. When you ask what time it is, you will get an
To complicate matters further, numbers and quantities may
change in one period and not in another. The height of house increases
slowly as it built, remains constant while it used, and decreases rapidly
if it torn down. So this height may be variable in some situations and
constant in others. In everyday life and in mathematics, when a number or
quantity is called a constant, we expect its value not to change in the
situation at hand. Similarly, when a number or quantity is called a
variable, we should expect or suspect that its value may change.
More examples: Your height is a variable or it was
a variable while you were growing. The speed of a car or a bicycle is an
example of a variable (a variable number or quantity that is). The speed
of a car can be almost constant. The zero speed of a stationary car or a
parked car is constant - in one reference system at least. Note that a
number or quantity can be variable in one situation, and constant in
another. We can further talk about a previously constant or a previously
variable number or quantity.
In summary, the terms constant and variable can be used
to talk about and describe numbers and quantities. A constant is a number
or quantity whose value is expected not to change - whose value should
not change. A variable is a number or quantity whose value does or might
change. The use of these terms is flexible and context dependent. What is
constant in one situation may be varying or changing in another.5
5In some algebra texts and in some
dictionaries, the term variable means or refers to the letters that
appear in formulas. That use of the term variable departs from the use
and meaning given above. In my view, the mathematical usage of everyday
words should be in the first instance linked and extracted from their
ordinary usage. Where the mathematical usage has departed from the
everyday usage, we need to ask if that departure is necessary, and
whether or not the departure should be corrected. Documenting
reasons or possibly causes for such departures could be material for a
thesis in linguistics.
B. Known Versus
Numbers and quantities can be known or unknown. You may
know your own height, age and weight, but I don't know your personal
measurements. To you these quantities are known. To me they are unknown.
Whether they are known or not depends on the company you keep - that is
to whom you speak. When you see the instruction find the unknown,
you should ask the question: unknown to whom? Note further in solving an
equation, the solution of the equation goes from being unknown to being
This is a note mainly for people who know how to solve equations.
See the following chapter or
chapters to learn how.
- There is only one number x solving the equation
2x = 10. Before you solve this equation, its solution,
the number x is unknown to you. The solution is x
= 5. When or as you solve the equation (or see the solution),
the number x becomes known.
- When you are only speaking about the solution x of
the equation 2x = 10, the solution is given by a
constant. The letter x stands for the constant,
non-changing number 5.
- Now in two different problems in which you solve for
x, their solutions x are often given by different
numbers (constants). Thus the value of the solution x
may change as you go from one problem to another. From this
perspective, the solution x can be also called a
For the sake of variety in our speech, numbers and
quantities are also called parameters. A parameter is another name for a
number or a quantity. When we say a number or quantity is a parameter, we
have no immediate expectation that the number or quantity in question
will be constant nor that it will be variable. The term parameter gives a
vague expectation somewhere between constant and variable. We can talk
about numbers and quantities in precise and imprecise ways.
4 Approximate Knowledge
Numbers and quantities are known, given,
measured or estimated with varying precision. For instance, the cost of a
hot dog could be 2.25 dollars. This cost is given exactly. In contrast,
the height of a man might be between 5[1/2] and 6 feet and the weight of
a truck could be between one and ten tons. In these two cases, the
quantities in question are sandwiched or bracketed between two extreme
values: the least and greatest possible. (The term sandwiched is
preferred. It is more graphic.) The distance between the bracketing
values measures the uncertainty in our knowledge.
7NOTE FOR ADVANCED
STUDENTS: More precisely, if x is a number whose value is known
to be between two positive number a and b with a
£ b, then the mean value c =
[(a+b)/2] gives an approximation to x. The
absolute error in this approximation is £
[1/2]|b-a|. The percentage
error in this approximation is £
The relative error in this approximation is £ [1/2][(|b-a|)/(a)].
To say that the percentage error is at most 1% indicates a better
approximation than a percentage error of at most 5% or even 100%. In
the above examples, note for instance the following: The height of the
man is known within 100[(0.25ft)/(5.5ft)] = 4.55%
£ 5%, a small (?) uncertainty. The weight of
the truck is known within 100[(4.5tons)/(1ton)] = 450%.
The uncertainty in the latter is large.The symbol £ is shorthand for the expression less than or equal
Knowledge of numbers and quantities may
be exact or approximate. But we can still speak about them. We can also
use approximate values in calculations and then hope the resulting error
is not too large. Estimating errors in calculations is a useful topic
which cannot be fully explored here. Error estimation is limited by the
observation that perfect knowledge of the error in a computation would
provide a means for removing the error. So error estimates must remain
8Significant Digits etc: When you
say that the height of a building is 10.47 meters (approximately)
without giving any further information, the uncertainty in the last
digit 7 should be £ [1/2]. When a single
decimal is used to approximate a number or quantity, the digits in it
are said to be significant when and only when the uncertainty in the
last digit written is £ [1/2] of a unit.
Digits which are uncertain by more than [1/2] should not be written
when we report the result of a measurement or calculation.
Exception: When a single quantity x is
bracketed between two others, say a and b, their mean
value c = [(a+b)/2] provides an approximation to
x with an error of at most d = [(|b-a|)/2]. In this case
we may write x = c±d
and keep some digits in the decimal expansion of c with an
uncertainty in them of more than one half unit. Writing x =
(10.472±0.003) meters for example provides
more information about x than the single estimate x =
In some situations, the location of the last
digit with an uncertainty of less than [1/2] of a unit may be unknown and
this convention may be difficult to follow. Errors in long calculations
may be minimized if rounding-off is postponed as long as possible, for
instance done at the end of all calculations and not for intermediate
Another Example: In crossing a
toll bridge with one rate for trucks weighing under 10 tons and with a
higher rate for trucks over 10 tons, the knowledge that the truck is
between one and ten tons means that the lower rate is used. But in
crossing a bridge with a higher toll rate for trucks over five tons, the
knowledge that the truck is between one and ten tons is not accurate
enough. The truck has to be reweighed.
5. Numbers Versus Quantities
When you ask how tall I am, you may get
the answer: 5 feet and 10 inches or 1.75 meters. The answer in either of
its forms involves both numbers and units. A number times a unit of
measurement gives you a quantity. Quantities can be added together: 5
feet plus 10 inches is 5 and 5 sixths feet.
To further understand the difference
between numbers and quantities, you may ask how many pennies (or cents) I
have in my pocket. The answer could be the number 10. For the same
pocket, if you asked how much money I had in it, the answer would be the
quantity 10 cents or even 0.10 dollars, a tenth of a
Numbers are given by counts - whole
numbers, proper and improper fractions, decimal numbers. Quantities are
given by a count (a whole number or fraction) times a unit of
measurement. Any object that can be counted can serve as a unit of
measurement. Examples of units of measurement are: meter, foot, $ or
dollar, square foot, square meter, second, hour, meters per second,
kilometers per hour, dollars per hour, miles per hour and so
Numbers include no units. You get a
number when you ask how many units there are, and you have specified the
unit. You get a quantity when you ask how much there is. Saying a length
is given by the number 5 is meaningless, if no units of measurement are
given. Saying a length is 5 raises the question 5 what?
The number 5 may give the number of units
of length in a distance. Writing this number by itself does not say what
the unit of length might be. Some information, the unit, is missing. So I
repeat, in answering questions demanding how much, we need to give a unit
of measurement as well as a number. People should not have to guess your
unit of measurement when you speak. A length may be given by 5 miles (or
8 kilometers). Of course, if we are asked how many miles (or kilometers)
there are in the length concerned, the number 5 (or 8) is expected
because the unit was specified. When you are asked how many people there
are in a room, you may respond with a pure number like 7 or 10. The unit
of measurement can be worded or written as person or
In measurement and counting, a single
unit of measurement, a fraction of one or several units, may appear. For
instance, a length of time may involve 1 hour or 12.5 hours. Notice the
addition of the letter s to the unit hour here when fractions or more
than one unit appears. In mathematics, we choose to ignore the difference
in spelling between the singular and the plural. If we insisted on using
the singular form, we would have to write 12.5 hours = 12.5 ×1 hour. The
latter gives the exact meaning of 12.5 hours. In writing units in
calculations, we may and will change their spelling (or abbreviations)
according to the rules of grammar. The plural and singular forms of each
unit are declared to be equal or interchangeable. Each is allowed to
replace the other. Which one sounds the most appropriate will be written
in our formulas and calculations.
5.1 Changing Units
In responding to questions how much,
we have a choice in the units of measurement used. Quantities come with
units. Numbers come without. A quantity says how much. A number counts or
says how many. You may measure weight or mass with kilograms or with
pounds. You may measure length in centimeters (cm), meters (m), kilometers
(km) inches, feet or miles. You may measure your savings or investments in
dollars and cents, or in your favorite currency. You may also count items
(objects, things, people). This gives or yields a number. For instance, the
number of meters in a kilometer is 1000. The number of people in this room
could be 5.
In reporting quantities or measurements,
the choice of units affects the number of units. Changing the unit of
measurement will change the number. For example, the amount of time 2
hours is the same as the amount of time 120 minutes, but the
numbers 2 and 120 are different. We can say we have two units of time
when the time is measured in hours. We can say we have 120 units of time
when time is measured in minutes. But we cannot say the amount of time is
2 or 120 without saying or somewhere saying what unit of measurement is
used. The following example shows how to change the unit of
To express the quantity 10.5 hours in
terms of minutes, we replace 1 hour by
60 minutes = 60 × 1
6 Review and References
We have seen the mathematical use of
several words including: variable, constant, changing, unchanging,
non-varying, unknown, known, etc. Speaking about numbers and quantities
without doing any arithmetic is part of mathematics. It is also part of
any subject involving calculations.
The following books may help you review
or practice your mathematical skills. ( A
visit to a local book store showed me there are many, more recent books.
Still more appear everyday.)
Arithmetic Made Simple by A. P.
Sperling & S. D. Levinson, 1988, Doubleday 1960 & 1988, 666
Fifth Avenue, New York, New York 10103. ISBN
Arithmetic Refresher for Practical
People by A. A. Klaf, 1964, Dover, New York, ISBN
Mathematics for Practical Use by
Kaj L. Nielsen, Barnes & Noble 1962.
Short-Cut Math by Gerard W.
Kelly, 1984, Sterling Publishing Co, ISBN 0-486-24611-6.
Helping your child with Maths by
R. S. Harrison, 1982, Harrap Limited, ISBN 0 245-53802-X
www.whyslopes.com >> Volume 2 Three Skills For Algebra >> Chapter 9 Talking about Numbers or Quantities Next: [Postscript - What is a Variable.] Previous: [Chapter 8 Three Skills For Algebra.]                                         
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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?
1996 - Magellan, the McKinley
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,
... 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
Euclidean Geometry - See how chains of reason appears in and
besides geometric constructions.
Coordinates - Use them not only for locating points in the plane
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
Lines-Slopes [I] - Take I & take II respectively assumes no
knowledge and some knowledge of the tangent function in
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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