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 >> Arithmetic and Number Theory Skills >> 3 Prime Factorization Skills
Quick Prime Identification and Composite Number Factorization Method
If a whole number N less than 169 = 132 is not divisible by
2, 3, 5, 7 nor 11 [the primes smaller than 13] then the number is
prime. Otherwise, it - the number N - is a product of 2, 3, 5, 7 or 11
with another factor N' less than 169.
Knowledge of times tables, divisibility rules or a calculator - one
displaying results to two plus decimals, may be used to recognize
multiples of 2, 3, 5, 7 and 11. Two decimals are sufficient because the
fractions one half to one eleventh are all more than 0.01 = one
hundredth. See Efficient Square Rule Use to quickly learn more.
In learning and applying algebra exactly, one usually needs to compute
with fractions or ratios of whole numbers less than 169 or so. Learning how to efficiently use the above method for recognizing
primes and obtaining prime factorization of of whole numbers less than
169 is sufficient for most ends and purposes purposes in high school and
college mathematics and science courses. The above method is based on the
square method below.
If a whole number N less than the square of a given prime is not divisible by
all the the primes smaller than given one then the number is
prime. Otherwise, it - the number N - is a product of one smaller prime
and another factor N' less than the square of the given prime.
Recognition of prime factors and the prime decomposition
of whole numbers speeds calculations of LCM, GCD and LCD in exact
arithmetic with whole numbers and fractions. They also lead to cosmetic
simplification of square roots. The recognition of common factors in
numerators and denominators of fractions alone or in products helps with
the reduction of fractions and via cancellation leads to efficient
methods for multiplying fractions.
Description of Folder Lessons
[video] how Products are bigger than factors. This observation
simplifies the identification of primes and composite numbers.
Prime or Composite less than 16.. Instead of saying a whole
number is prime when and only when it is not a product of whole
numbers larger than one, we say a whole number is prime when and only
when it is not the product of two or more smaller whole numbers
larger than one. The addition of the word smaller, redundant and
technically not required due to lesson 1, none the less makes prime
number identification easier to learn and teach. This webpage employs
the definition of primes and the 12 times table to recognize that the
whole numbers 2, 3, 5, 7, 11 and 13 as primes.
[video] Primes and Composites from 9 times table. Saying a
whole number is prime when and only when it is not the product of two
or more smaller whole numbers larger than one allow small primes in
the 9 times table to be quickly identified. Here is a short form
video version or variant of the previous lesson.
[video] Prime Factorization Introduction. This video lesson
introduces and illustrates prime factorization for a few whole
numbers. In the last example, the equivalence between tree notation
and the use of equal signs in the development of prime factorization
Prime Factorization and a Square Rule. This lesson provides a
more detailed discussion of prime factorization and introduces a
well-known, but I suspect hitherto nameless rules, for identifying
primes and obtaining prime factorization. The name square rule is
coined. This rule or method provides students with a quick manual
method for obtaining the prime factorization of whole numbers. This
quick method can be employed with the aid of the divisibility rules
for the small primes 2, 3, 5 and 11, when or if mastered, or with the
aid of calculators that display sufficiently many digits after the
decimal point. See the calculator usage notes and cautions below.
Sieve-of-Eratosthenes upto 100. Discussion of this Sieve (or
filter) shows how striking out of multiples of 2, 3, 5 and 7 is
sufficient by the square rule to identify all primes in a list or
table of whole numbers 1 to 100.
Calculator Usage Notes and Cautions. The square rule for
identifying primes and obtaining prime factorizations, two side of
the same coin, can be employed with the aid of calculators that
display sufficiently many digits after the decimal point. Here are
notes on how with some cautions.
[video] Prime Factorization upto 19. Given the identification
of all primes less than 100, this video is redundant. However, it is
retained to illustrate the use of the square rule. An alternative
approach would be to use the 18 times table alone or a subtable of a
larger times table.
[video] Prime Factorization upto 19 squared. The square of 19
is 361. provides examples of prime factorizaton with the square rule
for a few numbers less 361.
[video] Prime Factorization upto 23 squared. The square of the
prime 23 is 5629. provides examples of prime factorizaton with the
square rule for a few numbers less 529.
Efficient Square Rule Use. The last part of the lesson
Prime Factorization and a Square Rule provides a few hints or
directions for the efficient use of the square rule. This lesson on
efficient use provides examples.
LCD GCD and LCM calculations using Primes. Finding
least ommon ddominators, least ommon
multiples and greatest common divisors
are of sets of whole numbers, two or more, represent number theory
practices of service in simplifying, adding, comparing, subtracting,
multiplying and dividing fractions, and of service in cosmetic
conventions for the exact representation of square, cube and higher
roots of whole numbers. This webpage gives some examples of LCD GCD
and LCM calculations using Primes, or more precisely prime
Lessons 13 to 18 present and illustrate methods to count and find all
whole number factors of a given whole number using tables, products and
trees. In the study of quadratics, factoring by inspection requires the
identification of all integer factors of a given integer. The methods
here or variant of them turn that identification part of this factoring
a quadratic by inspection into a systematic art.
[video] Factors of 24 using primes
[video] Factors of 24 Take II
[video] Factors of 20 using Prime Factorization
[video] Factors of 980 using primes
Identify and Count Factors using Primes
[video] Count Factors given Prime Factorization
[video] Prime Factorization Unique. This video lesson
introduces the notion that the prime factorization of a whole number
will be independent of the route that gives the factorization.
Knowing about the uniqueness and how it implies the latter is nuance
that may included for completeness, or not.
Uniqueness of Prime Factorization. This lesson describes the
uniqueness in general - offers reason for it.
whyslopes.com >> Arithmetic and Number Theory Skills >> 3 Prime Factorization Skills
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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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