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.
Return to Page Top
www.whyslopes.com >> Arithmetic and Number Theory Skills >> 5 Integers
These lessons and three appendices include exercises to consolidate
and extend understanding of integers.
The lessons provide a hands-on, thought based development. Lessons
assign three geometric roles to integers to develop and explain rules for
- Role I: Integer are first introduced as coordinates for points on a
line, where adjacent points are a unit distance apart.
- Role II. Integers then serve as multipliers in the definition of
integer multiples of a unit movement, integer multiples that can be added
and multiplied by whole numbers and then integers.
- Role III. Integers themselves may describe movements, how many steps
to the left or right, along a straight line, and so can be identified
with movement, integer multiples of a unit movement, now called a step.
That third role or identification leads allows integers to be added and
Exercises are included in most lessons.
Three appendices cover an associative law, division quotient and
remainder options for integers, and how remainder arithmetic explains the
alternating sum of digits test for divisibility of decimals by 11.
Extension: This three role, geometric
development and explanation of integers starting with unsigned whole
numbers provides an example to follow for the development and
explanation of (i) rational numbers starting with unsigned fractions;
and (ii) real numbers starting from unsigned (positive) real numbers or
their decimal representation. The site exposition of complex numbers
continues the geometric development of number theory.
The Main Lessons
Saying how to do an operation defines it.
Lesson 1: 1 introduces the first role
of integers a signed whole or natural numbers serving as coordinates
for points a unit apart along an infinite straight line.
Lesson 2 introduces the second role of
integers as multipliers in providing integer multiples of a movement,
a unit movement along a line.
Lesson 3 shows how to add integer
multiples of a unit movement when those multiplies have the same
Lesson 4 shows how to add opposite and
same direction integer multiples of a unit movement.
Lesson 5 introduces the idea of
additive inverses for integer multiples of a unit movement. That
answer the question what movement added to another will result in a
zero net movement.
Lesson 6 says how to multiple integer
multiples of a unit movement by a natural number, zero or a whole
Lesson 7 shows how to apply the
positive and negative signs + and - to a vector. The application of
the plus sign + to a movement is the identify operation while the
application of the negative sign - to a movement yields its additive
inverse, that is, a movement with the same direction and opposite
Lesson 8 shows how to multiply an
integer multiple of a unit movement by an integer. Since each nonzero
integer may be written as a plus or minus sign prefixed to a whole
number, we take multiplication by a nonzero integer to be the
operation of multiplying by its whole number part (also known as a
length or magnitude) followed by an application of its sign to the
latter product. There is a further refinement or extension of the
second role of integers as multipliers of movements. The first role
is to provide signed coordinates for points a unit distance apart
along a line.
Lesson 9 identifies integers with
movements - so many steps in one direction or another. There-in lies
a third role for integers - a movement role. That leads to an
extension of lesson 8 in which multiplying integer multiples of a
unit movement turns into multiplying integers by integers.
Lesson 10 continues lesson 8 and 9 to
obtain rules for products of signs and products of integers that be
easily applied and learnt by rote, or seen as a consequence of
lessons 1 to 9.
Lesson 11 applies the lessons 3, 4 and
5 on the addition of integer multiples of a unit movement to the
third movement role of integers. That leads to methods for addition
Lesson 12 echoes lesson 11 in
providing rules for adding integers, rules that are easily understood
and repeated, with or without explanation of why they work.
Appendix A (optional reading)
continues lesson 11in an algebraic manner (a departure from the
presentation in the other lesson) states an associative law for
multiplication of integer multiples of unit movements by integers.
That may - proof to be written - imply an associative law for
Appendix B (more optional reading)
looks at the long division method for pairs of whole numbers for
pairs of nonzero integers - long division of their whole number parts
(lengths, magnitudes) implies several dividend = quotient times
divisor plus remainder relations, where the sign of the remainder may
be like or unlike the sign of the dividend and/or divisor.
Appendix C links to the alternating
sum of digit test for divisibility by 11 to remainder arithmetic,
modulo 11, with the aid of integers. The link here is given without
proof. Indeed the whole treatment of remainder calculations provides
rules to apply and follow to repeatable and reproducible results,
without full (algebraic) explanation of why these practices work. The
explanation why is or or is to be included in the site coverage of
www.whyslopes.com >> Arithmetic and Number Theory Skills >> 5 Integers
Return to Page Top
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.
Return to Page Top