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
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?
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
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.
Early High School Geometry
Maps + Plans Use - Measurement use maps, plans and diagrams drawn
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.
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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whyslopes.com >> Secondary Mathematics - A Practical Approach >> Chapter 3 - Algebra Starter Lessons Next: [Chapter 4 - Logic for Reading Writing and Geometry etc.] Previous: [Chapter 2 - Why Sets.]         
Chapter 3. Algebra Starter Lessons, Effective
The algebraic way of writing and reasoning with letters and further
symbols is a great mystery and great source of discomfort for many
students and teachers. The mystery invites rote learning and
computational approach to secondary mathematics. Many students do not end
their studies of mathematics follows from immersion. The textbooks of my
youth employed the algebra was of writing and reasoning but did not
introduce it. Mastery was left to chance. As a student, I was left to
provide my own rationalization for the shorthand role of letters and
symbols. With it I was able follow the logic in mathematics and science
texts, the mastery made the latter possible, but I lacked the verbal
ability to discuss and share my rationalization with others.
Tackling the algebra problem in secondary mathematics and calculus has
been a long-term aim. Success can be reported in the form of methods,
online here this, for introducing the algebra shorthand way of writing
and reasoning using words and gemometry. To learn how, explore the
Skip those on 1
Working With Sets. The presentation as is is best left to the
last years of secondary school or the first year of college.
Correct, but needs to be made more accessible.
2 Formula Forward Use - Evaluation is also valuable for
the evaluation of arithmetic expressions. The section gives
instruction and multiple examples of how evaluate formulas and
arithmetic expressions, so that each step of the calculation can be
seen and checked as done or later. In evaluation and in the written
work needed for solving equations, using them forward and backward,
the aim is to show students how present their work for the efficient
confirmation or correction of skill mastery. The latter has to be
seen to believed.
3 Solving Linear Equations begins with a three column
stick diagram method for solving linear equations ax+b = cx+d, a
method intended to reinforce fraction skills while leading students
from an easy manipulative approach with sticks, that is line
segments, to the standard solutions of the latter equations, one that
involves no sticks. After that this section introduces systems of
equations in essentially one-unknown and trianglar system of
equations to introduce students to the solution of systems. Most word
problems in earlier secondary school are best formulated as systems
in essentially one unknown, systems that are easily solved by
transformation into one equation in one unknown. In the early
secondary school treatment of word problems, students have to pass
through mental gymastics to arrive one equation in one unknown.
Formulating the word problem as system in essentially one unknown by
passes the mental gymnastics - provides a mechnanical alternative,
one in which algebraic skills are directly developed. Including
systems easy to solve early secondary school aims to provide skill
and confidence. In all, examples here show how to write and check
solutions so that skill can be seen or corrected as needed. The
example are accompanied by message: when a check fails, the mistake
or mistakes will be found between the start of the solution and the
end of a check. Not all checks are fault-free. Chapter 15 on Solving
Linear Equation in site volume contains more examples. The latter
include a neat example of the algebraic or literal solution of
equations ax + b = c, all to illustrate and introducte the algebraic
way of writing and reasoning.
4 Computation Rules and Function Notation:
covers the following.
The subsequence dicussion of equilavent computation rules sets the
stage for the explanation of axioms or rules for algebra as
statements that two computation rules with different forms may give
the same result. The latter in turn sets the stage for applying
axioms or rules of algebra to changing the form of arithmetic and
algebraic expressions in ways that leaves their values unchanged.
Jump to lesson 7 below and return to the next two lessons as needed.
The subsection 5.
Real Numbers lesson span the properties of real numbers and
their employment as coordinates along a line and in the plane. The
lesson on multiplication of real numbers echoes earlier lessons on
sign numbers. In practice, computations with real numbers are
approximately. The use of calculators allows computations to be done
to 3+ decimals, the more the better, for the sake of accuracy.
Calculators should be used to show students the decimal approximation
of the PI - the ratio of circle perimeter to its diameter, to 5+
decimal places. To many students in primary school do not learn that
PI is 3.14 approximately, to 2 decimal places - they and some of
their teacher may think the approximation is exact. The use of
calculators to display approximations to PI to 4 and more decimal
places may correct that mistaken belief. The value of PI can be
physically approximated by measuring the perimeter of a circle with a
given diameter, and then calculating the ratio. At this level, the
latter can be taken as its definition - an ideal for empirical but
not mathematical verification - oops! The mathematical fact that PI
has an infinite, non-repeating decimal expansion has to be given. At
this level, the proof of the latter is not for all. The proof is
reserved for undergraduate or graduate students in advanced
The axioms for Real Numbers etc lesson describes the arithmetic
properties of addition and multiplication only. Subtraction and
Division are not mentioned. However, subtraction of number can be
recast as adding the negative or additive inverse of the number,
while division by a number can recast as multiplying by it
multiplicative inverse or reciprocal. Once the recasting is done in
an arithmetic expression or computation rule, here an algebraic
expression, the algebraically described computation properties in the
axioms and earlier oral rules for adding and multiplying multiple
addends and terms can be applied to the recast form of the expression
or computation rule.
Chapter 18 in the site Volume 2, Three Skills for Algebra, tries
to describes and illustrate the underlying concepts. The chapter was
written in 1995-6 when I was struggling with how to make algebra
skills and concepts clearer through the use of words and numerical
Chapter 18 in the site Volume 2, Three Skills for Algebra, also
describes and illustrate the properties of real numbers given here in
a step by step, complementary manner. Two views are better than one,
6. More, Less and Greater Than Inequalities - Comparison
includes several lessons on comparing real numbers not by their
lengths or magnitudes, but essentially by their position when used as
1D coordinates along a "real number" line. Here I try to bend
previous treatments by introducing the notion of 10 be more than -5
by 15, or -1 be less than 6 by 7. I would keep the name of the less
than sign as is, but in the context of Real Numbers [as coordinates
along a line] rename the greater than sign and call it the more than
sign in order to step away from the comparison of size, magnitude or
length implicit in the comparison of unsigned numbers. This section
may be covered ligthly to introduce the less than and more than signs
in the context of signed numbers, that is Real Numbers.
7. Axioms, Logic and Equivalent Equations includes five
1 Equivalent Computation Rules: This first amd key lesson
views the distributive law a(b+c) = ab+ac as the equivalence of
two different computation rules, one provided by h(a,b,c) =
a(b+c) and the other by h(a,b,c) = ab+ac. In the context of area
computations, the equivalence of computation rules h and g is
implied by the presume equality of two different ways of
computing the area of a rectangle with sides of length a and b+c.
The notion that h and g denote two different forms of the same
function can be discussed later. The geometric context allows the
letters a, b and c to denote the length of line segments. The
equivalence can also be verified by hand-computation, by
calculator-computation and by writing short computer programs to
evaluate each side. A counting context is also offered. Examples
of other computation rules are included in the lesson.
2 Addition and Multiplication Axioms: The second lesson casts
Addition and Multiplication Axioms for Real Numbers etc in terms
of equivalent computation rules. The distributive law again
3 Product Axioms - Two Forms: The third lesson present two
forms of the product rule. First, we observe from the
consideration of decimals that the product of two non-zero
numbers or lengths is nonzero. The word lengths here is used to
indicate a possible extension or variant of this consideration:
The area of a rectangle with two nonzero sides, the lengths, is
"clearly" non-zero. The contrapositive form of this product rule,
it can be written as an implication rule IF A THEN B, is the
statement or axiom that if a product of two real numbers is zero
than at least one of the factors must be zero.
4 Subtraction and Division Axioms: The fourth lesson - more a
lesson idea than a lesson, adds subtraction and division axioms.
The aim here is to imply how the addition and multiplication
axioms may be applied by converting subtraction and division
operations into addition and multiplication operations through
the use of additive inverses - the negative of a number, and
through the multiplicative inverses - the reciprocal of number,
5 Equality in Algebra: The fifth lessons discusses equality
axioms or practices. The same number may have several different
representations. Substitution Practice: All can be used
interchangeable in computations. Indeed, one representation may be
replaced by a more convenient one for the sake of aiding a
computation, while results remain repeatable and reproducible.
Equality axioms and practices show how one equation may imply
another, and vice-versa. That leads to the concept of equivalent
equations or equivalent systems of equations.
6 Equations and Systems - Equivalent or Implied: The sixth
lessons slowly introduces concept of equivalent equations or
equivalent systems of equations just mentioned.
8. Unifying Theme For Algebra: In primary and secondary
level mathematics, and calculus too, many tables, rules, patterns and
formulas may be employed not only directly - in the forward manner,
but also indirectly in a backward manner. Addition and times tables
may be used backwards to answer subtraction and division questions,
some - not all. In logic, the contrapositive of an implication rule
represents an indirect or backward use or formulation. In calculus,
differentiation rules will be reversed to provide
anti-differentiation or integration methods. In secondary algebra,
the study of formulas entails not only their forward or direct use,
but also their indirect or backward use. In the UK, a quantity given
by a formula is termed the subject of the formula. The formula itself
may involve one or several other quantities or variables. Using the
formula for the original quantity to find an algebraic expression for
one of the other quantities is termed changing the subject. It
entails mastery of the algebraic or literal [with letters] way of
writing and reasoning. The aim of section lessons is gradually and
systematically develope this algebraic way of writing and reasoning.
The lessons here are unique. They endeavour to fill a gap I have seen
in the development of algebra since my own school days in the 1960s
and 1970s where I saw or sensed the algebraic ways of writing and
reasoning was required but not systematically introduced. Here is a
remedy. It may be easier and simpler than you expect. Good luck.
This section consists of the following lessons:
Some lessons duplicate material from site Volume 2, Three Skills for
Algebra. Upper level secondary mathematics and science is based on
the backward use of formulas. Talking about the latter recognizes and
vocalizes a commonality and so gives a unifying theme for the
learning and teaching of algebra.
9. Proportionality Backwards and Forwards This section
includes the following webpages. Some represent lessons. Other
represent lesson ideas that tutors or instructors will have to expand
and clarify for learners. The coverage is rather rich. It may be
spread over mid- and upper secondary school level instruction in
mathematics and science.
Concepts and Practices- Three plus Kinds of Proportionality
Relations, Forwards and Backwards: The lesson says what is
(defines) Direct, Joint, Inverse Proportionality and describes
how to shift or generate proportionality relations from each
others. In a proportionality relation (or equations),
algebraically interchanging the dependent quantity with an
independent one via a backward use of the relation leads to
further proportionality relations of the same or different type.
The use of proportionality relations begins with the backward use
problem of finding the value of a proportionality constant. Once
its value is known, the proportionality relation can use in the
forward direction to find values of the dependent variable, or in
the backward direction to find values of a so called independent
. This lesson overlaps the others.
Twenty or so
Examples of Proportionality and Multiple Ratios or
Proportions: Many examples of proportionality relations
appear in high school mathematics and physics. Here is a list of
some (most if not all) that may be met. Remember each
proportionality relation will be used forward and backwards in
Multiple-Term Ratios, a proportionality constant viewpoint.
Fraction and ratios are overlapping concept and have overlapping
roles in arithmetic, but they are not identical even though
fractions a/b where a and b are whole numbers may be called
ratios. In mathematics ordered pairs of whole numbers a and b may
appear in coordinate form (a,b) or [a,b]; in ratio form a:b and
in fraction form a/b.
Constants for Equivalent Fractions: The numerator is
proportional to denominators in any fractions equivalent to a
given one - a simple matter.
10. Five Examples of Algebraic Reasoning. The material
here is optional. That being said, keen or gifted students may test
their algebraic reasoning skills by reading here the fraction lesson:
2 Fraction Operations Physical Development. It provides an
algebraic perspective of how raising terms can lead and justify
methods for addition, subtraction, comparison, multiplication and
division of fractions.
Column Multiplication Methods for Arithmetic and Algebra
Duplicate Material, and Deliberately So
Column multiplication methods appear in primary school with decimals and
take advantage of place value. The appear in middle to senior secondary
school in the multiplication of polynomials. Such methods can be
introduced geometrically. See
Column Multiplication Methods in General. The latter ideas using
letters to denote lengths, subsegments, along the side of a rectangle.
The letters limited to two per side could also represents the integral
and fractional part of a mixed number. The result is a column
multiplication methods for the product of sums, one justified geometrical
in the case of positive or unsigned numbers. But operationally the method
works in general - we assume that in place of proving for the sake of
accessiblity. The multiplication method provides a simple, mechanical
replacement for the FOIL method taught in algebra. For polynomials, see
Column Multiplication Method.
Remark:Mastery and sanction of column multiplication methods
extends the distributive law given in algebra. The recommendation here
for the sake of an operational command of mathematics is to give the
methods and the law, and allow studies in pure mathematics if taken, to
explain the redundancy, or how the law implies the methods. The practical
aim of secondary mathematics here is not to give a lean axiomatic base
for mathematics, but an operational command based on consistent rules and
practices, axioms included. Lean may follow later in specialized courses
taken by students mastering the more theorectical aspects of university
mathematics, engineering or science. Lean too early is a burden.
The algebra starter lesson end with enriched material - optional
Learn More - Readings for Now or Later
Three Skills For Algebra. See Chapters 8 to 17 and optional 18.
Tutors, teachers and learners should read if enjoyable.
3 Why Slopes and More Mathematics. The algebra shorthand way of
writing and reasoning is employed at full-strength in calculus.
Chapters 2 to 7 of this work provide an algebraic light calculus
preview, one that stems from the observation that the middle part of a
calculus course is algebraically less challenging that the leading
parts. This preview may be employed at start of a first course in
calculus. It may also be used in secondary mathematics before calculus
to a context for the study of slopes and polynomial factorization, all
in a way that should advances algebra skills.
whyslopes.com >> Secondary Mathematics - A Practical Approach >> Chapter 3 - Algebra Starter Lessons Next: [Chapter 4 - Logic for Reading Writing and Geometry etc.] Previous: [Chapter 2 - Why Sets.]         
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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.
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. ...
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
Lines-Slopes [I] - Take I & take II respectively assume no
knowledge and some knowledge of the tangent function in
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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