Mathematics Concept & Skill Development Lecture Series:
Webvideo consolidation of site
lessons and lesson ideas in preparation. Price to be determined.
Bright Students: Top universities
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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 >> Arithmetic and Number Theory Skills >> 8 Arithmetic with Signed Numbers
Arithmetic with signed numbers includes arithmetic with integers,
rational numbers (signed fractions) and beyond them, real numbers.
Lesson 1 has just become the last lesson 11. Writing is an iterative
affair. In retrospect, being last appears to the best place for it.
Lesson 2 signed and unsigned numbers as
coordinates introduces signs as prefixes to provide coordinates along
a full number number. This use of signs + and - as prefixes to numbers in
service of providing coordinates for a full line provides an initial
context and motivation for signed numbers.
Optional Reading: Lesson 3 signed coordinates
for maps and planes and Lesson 4 signed coordinates for regions in
space describe the further use of signed numbers in pairs or triplets
to locate points.
Lesson 5 lengths and signs of numbers. Signed
numbers have a sign prefixed to a unsigned part. The latter part may be
called its magnitude, absolute value or length of the signed
Lesson 6 adding signed numbers introduces methods
for adding signed numbers - those with a common signs and those with
diferent signs. To add two or several numbers with a common sign, use the
slogan prefix the common sign to the sum of their lengths. To add
two numbers with unlike signs, prefix the sign of the longest [or
largest] to the difference, the longest length minus the shortest
length. Lesson 6 describes these methods for adding with words and
with algebra, and then gives many, many examples.
Lesson 7 negative and additive inverse for each
number IT, identifies its additive inverse - a second number which when
added to IT gives a result of zero. People who think algebraically may
think x in place of IT. Lesson 7 is preparation for lesson 9. The
accompanying slogan for computing a negative or additive is simple: keep
the length, but change the sign prefixed to it. In that change, a plus +
becomes a minus -, and a minus becomes a plus.
Lesson 8 multiplying signed numbers is based on
the slogan, multiple the signs, multiple the lengths to compute
the produce of two or more sign numbers. Twenty or so multiplication
examples employing integers, proper and improper fractions, and symbols
denoting real numbers are given.
The case of signed mixed numbers is not
covered here, but they can be rewritten as signed improper fractions.
How to multiply mixed numbers without this conversion must wait mastery
of the distributive law and perhaps associated column multiplication
methods to exploit.
The multiply the lengths, multiply the signs
slogan provides a prequel to and slogan and rule multiply the
lengths, add the angles for multiplying complex numbers.
Lesson 9 subtracting signed numbers shows how to
subtract a number by adding its negative or additive inverse. Multiple
examples are given. Those examples are followed by two interpretations of
subtraction, the more-than interpretation [i], that the length of the
difference between two numbers gives the number of units, one is more
than another; and the geometric distance interpretation [ii], that the
length of the difference of two numbers gives the distance between two.
There-in lies a prequel to the discussion of length calculation along a
coordinate line using absolute values.
Lesson 10 dividing signed numbers presents slogan
multiply the signs, divide the lengths to say how to divide signed
Lesson 11 What are real lengths and numbers
describes how numbers may describe length and position along straight
lines - number lines. It easy to understand the associated use of whole
numbers and fractions - proper and improper, but it may come as a
surprise that there are points on straight lines whose distance to the
origin is not a whole and/or fractional multiple of a unit length. Thus
more numbers to describe lengths appear. Thus extra numbers - the
irrationals - together with proper and improper fractions form the
unsigned real numbers. The latter provide coordinates along a
Remark: If a number is written without a sign, its
sign is deemed to the plus sign. With that convention, the arithmetic
operations described below can also be applied to expressions involving a
mix of signed and unsigned numbers.
Location of Signs: The use of signs + and - in
the super-prefix position, examples +5 and -3,in
the introduction of integers appeared in modern mathematics secondary
and college education 1967-75 say. But the but was not used in practice
with rational numbers given by unsigned fractions (a/b). With the
latter, signs were employed as prefixed position but not in the
superscript position. In the following lessons, signs appear in prefix
position at normal or superscript hieght, or somewhere in between.
Whether or not the symbols + and - serve as number signs or as the
number operations - here addition and subtraction, or calculating a
negative inverse - is usually well indicated by the context, with any
ambiguity being harmless. For example -5 may indicate negative 5 - the
number - or the calculation of negative inverse of 5, a calculation
that has value negative the number.
whyslopes.com >> Arithmetic and Number Theory Skills >> 8 Arithmetic with Signed Numbers
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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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