Mathematics Concept & Skill Development Lecture Series:
Webvideo consolidation of site
lessons and lesson ideas in preparation. Price to be determined.
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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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www.whyslopes.com >> Arithmetic and Number Theory Skills >> 10 LCM GCD and Euclid GCD Algorithm
Least Common Multiples [LCM] Introduction. This video lists
the first 14 multiples of 6, and the first 6 multiples of 14 to see
if there is a smaller common multiple that 6 × 14 = 14 × 6. The video
provides a hint of the role of primes in find the LCM of the two
numbers. ??? KILL
Least Common Multiple LCM intro via list method. This video
answers the question what is a LCM, explains the motivation for LCM
calculation, and introduces the list method for finding the LCM of a
pair of small whole numbers, here 6 and 8. For these two numbers, the
list method begins by writing or listing the first 6 multiples of 8
and the first 8 multiples of 6 to be list
LCM 60 45 Avoid List Method Use Primes. This video explains
why the use of prime factorization requires less work than the list
method to find the LCM for two numbers, namely 60 and 45. Includes a
clear introduction of the prime factorization based method for
LCM of 8 and 10 via Primes. This video shows how to find the
least common multiple of 8 and 10 using their prime factorizations. The video
explains the method. The video includes the list method as well for
Common Divisors 60 45 via Primes. This video employs the prime
factorizations of 60 and 45 - obtained in the previous lesson - may
be used to generate common divisor and to identify the greatest
Optional Question: How many common divisors are their. Master
section on Combinatorics to answer.
GCDs from Primes. This video shows how prime factorization of
whole numbers may be used to find the greatest common divisors of the
GCD and LCM from prime factorization. This video gives
examples of how to compute Greatest Common Divisor and Least Common
Multiples of a pair of numbers, each equal to product of primes -
their prime factorizations.
GCD from Euclid's Algorithm. This video gives a
first example of Euclid Algorithm for find the greatest
common divisor of two numbers, here 875 and 300. It then
simplifies the fraction 875 over 300. Finally, it shows how
to construct a small - in fact the least - common multiple of them
for use in addition of two fractions with denominators 875 nad 300.
GCD of 360 110 via Primes and Euclidean Algorithm. This video calculates
the GCD of 360 and 110 with Euclid Algorithm and then verifies
the same result can be obtained from prime factorization. Euclid Algorithm
may be quickest - proof of that or discovery of that is left to further
studies in mathematics.
Euclid Algorithm for 129 125 and for 45 14. This video provides two
more examples of greatest common divisor calculation with Euclid's algorithm.
The GCD in both examples is 1. Thus implies that in each pair of numbers,
the pairs are relatively prime - their prime factorization share no common
GCD 2700 288 via Euclid's Algorithm. This video calculates the greatest
common divisor of 2700 and 288 via Euclidean Algorithm. Then it employs
the GCD to simplify a fraction where one is the numerator and the other
is denominator. Lastly, it employs number obtained from the algorithm
to obtain a Least Common Multiple - LCM
GCD 2700 288 via Primes.This video calculates the greatest
common divisor of 2700 and 288 using their prime factorizations
GCD from given Prime Factorizations. This video shows how to calculate
GCD for numbers given as products of primes. Three products are given. The products
are consider in pairs. Question: What the GCD of all three numbers?
GCD of 650 110 via Primes. Then LCM via Product Rule. The product of two numbers
equals the product of their GCD and LCM. We call that relation, a product rule. If the
product GCD × LCM is known along with one of the factors, then the other
factor can be calculated. That represents a backward use of this product rule.
GCD of 650 225 via Euclid Alg. Then LCM via Product Rule. This video
calculates the GCD of the two numbers, and then uses the product rule introduced
in the previous lesson to obtain the LCM. The next video confirms the GCD and LCM
computed here by deriving them from prime factorizations.
GCD and LCM of 650 225 via Primes. This video confirms the GCD and LCM
computedin the previous video using prime factorizations.
GCD LCM of 85 and 60 via Primes. This video calculates the GCD and LCM
of the two numbers 85 and 60 with the aid of their prime factorizations.
www.whyslopes.com >> Arithmetic and Number Theory Skills >> 10 LCM GCD and Euclid GCD Algorithm
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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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