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
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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 >> 1 Decimal Place Value
Food for thought. The SI lesson below would be
sufficient for upper secondary schools in many countries , and
independent of the use of local naming systems for big and small
numbers and measures.
The exercise of reading decimal aloud with several places before and
after the decimal point is not a practical skill. It is a skill which can
provide comic relief in a mathematics class while developing and
reinforcing place value comprehension. A possible modification of the
following lessons to favour UK-terms meaning of billions and trillions is
Place Value in Three Digit Whole Numbers describes the ones,
tens and hundreds place value of the digits and, more importantly,
says how 3 digit number aloud, digit by digit.
Groups of Three Place Value for Multidigit Decimals gives a
single example to introduce or review the ones, thousand,
millions, billions, trillions and quadrillions place value of
groups of three in multidigit decimals. Here place value is
determined by reading a number backwards (from right to left) while
the normal way of reading identifies the groups of three and their
place value from left to right.
More on Groups of 3 Multi-Digit Place Value gives more
examples to example to introduce or review the ones, thousand,
millions, billions, trillions, quadrillions and quintillions
place value of groups of three in multidigit decimals. Here again
place value is determined by reading a number backwards (from right
to left) while the normal way of reading identifies the groups of
three and their place value from left to right. Recognizing place in
groups of three backwards has to be done before the number is read
Groups of 3 Place Value in Decimal Fractions gives a single
example to introduce or review the thousandths, millionths,
billionths, trillionths and quadrillionthss place value of groups
of three in multidigit decimal fractions. Here place value is
determined by reading a number forwards(from left to right and can be
done while the decimal fraction is read aloud.
More on Groups of 3 Place Value in Decimal Fractions gives a
more examples to introduce or review the thousandths, millionths,
billionths, trillionths, quadrillionths and quintrilionth place
value of groups of three in multidigit decimal fractions. As just
said, place value is determined by reading a number forwards(from
left to right and can be done while the decimal fraction is read
Groups of 3 Place Value in Mixed Decimal Fractions gives a
single example in which place value of groups of three in a mixed
decimal number or fraction is based on place value of groups of three
determined backwards, that is, contrary to the normal direction of
reading in English from a decimal point while place value of groups
of three after the decimal point is determined in accordance with the
normal reading direction.
More on Groups of 3 Place Value in Mixed Decimal Fractions
gives more examples of the appearance of place value from
quintillions to quintillionths in a multidigit decimals that included
a decimal point.
Review Lesson 1 2 4 and 6 - All in One present four examples,
one from each lesson.
Place Value Review - Decimal form of Avogrados number
included. Students in science courses may appreciate the order of
magnitude of Avogrado's number after this stand-alone extension of
the previous lessons introduces them to 602 septrillions.
Names for Big Numbers and Powers of 1000 Expansion goes from
the power of ten expansion of multidigt decimals to powers of 1000 =
103 groups of three expansion of multidigit decimals with
digits before and after a decimal point. A reference is given.
Place Value SI Standard International way. The SI (System
International) systems of units and measures provides "standard
names" for big and small numbers from yota to yocto for powers of
103 and it reciprocal 10-3 that can occur in
discussing group of three place value in multidigit decimals. The
alternative names are given along side dare-we-say common US-style
names for numbers, big and small. A reference is given. for
Modification for UK big and small number naming.
In the UK, a billion is not a thousand million, is it is a million
million. Further, a trillion is not a thousand million, it is a million
billions. That suggests a variation of the foregoing in which multidigit
decimals may be read aloud in full or partial groups of six, where each
group of six consists of two groups of three. Thus
56 789 345 230 456 678 . 560 678 999 12
might be read aloud as
56 thousand and 789 billions
345 thousand and 230 millions
456 thousand and 678 ones or units,
560 thousand and 678 millionths, and
999 thousand and 120 billionths
That being said, some variation of this ideas may work better. Again, the
skill of reading multidigit decimals aloud in groups of three or six is
not a practical. It is a skill aimed at developing and re-enforcing place
Is this better? For reading in full or partial groups of three
digits, UK instructors might emphasize the use of SI terms yota to yocto
for powers of 103 and it reciprocal 10-3 because
groups of three are easier to digest than groups of six, and groups of
three may lead to better comprehension of the SI naming system for large
and small numbers and measure.
www.whyslopes.com >> Arithmetic and Number Theory Skills >> 1 Decimal Place Value
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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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