Multiple Geometry Islands
Geometry Island: Euclidean Geometry
Site logic chapters
set the stage for Euclidean Geometry. The site development of
Euclidean Geometry is based on the direct use of implication rules. and
give another logical structure in mathematics for students to see
parallel to those seen in algebra. The site development provide a
stand-alone-island of implication rules and patterns that in turn prepare
the way for the logical development of of geometry, trigonometry and
complex numbers.
The
site development of Euclidean geometry stands on the earlier exposure
and practices we hope, with maps, plans and diagrams drawn to scale. The
latter makes the assumptions and practices of Euclidean geometry easier
to motivate and accept. Following that, more Euclidean geometry may be
obtained deductively and directly without the use of indirect reason. The
avoidance here of indirect reason makes the account more acessible. This
logical account of Elucidation provides an island and body of skills and
concept that has intellectual value in demonstrating how chains of reason
may occur in mathematics, largely apart from the algebraic reasoning
process. So it can standalone besides and studied apart from algebra.
The site account of Euclidean Geometry is self-contained and logical
according to my standards, but in stands on my limited memory and
exposure. Readers with a better command of mathematics may refine and
adjust the site development to provide an approach more consistent with
past developments of the subject, subject to the constraint of covering
enough Euclidean Geometry to support or act as background theory for the
further development of analytic geometry, complex numbers and
trigonometry here or elesewhere.
Geometry Island: Drawing to Scale, Similarity silently mastered.
Earlier instruction may show how drawing maps, maps and diagrams to scale
may help solve geometric problems and in that help find missing angles
directly and missing lengths or areas via proportionality relations. In
that we may emphasize the number of diagram unit lengths a drawn length
and number of units in the "original" length are equal. Thus the ratio of
the foregoing unit lengths equals the diagram scale factor. Likewise, we
may emphasize the number of diagram unit squares in covering a region of
a map and number of square units in the real region are equal. Thus the
ratio of the unit lengths, squared, equals the area scale factor.
Related scale or proportionality factor for length and area in plane,
and more generally for length, area and volume in space, appear when or
where different objects are but instances at different scales of of a
2D or 3D drawing. That may lead to a discussion of similarity by design
for man-made objects. Note the site development of similarity via
the concept of two objects in the plane being similar if there be
coordinate systems and unit length attached to each, so that the
corresponding set of coordinates are equals provides a unifying view of
the otherwise disconnected views of similarity that arise in the
discussion of solid and hollow circles, triangles, rectangles and
further polygons or shapes in the plane. Course designs may decide
when or if to introduce that view. The view is implicit in computer
representations of maps and plans, and in interactive video game
scenes.
Geometry Island: Trigonometry with Right Triangles
Earlier practices show how maps-plans-diagrams drawn to scale may solve
geometric problems before the explicit mention of similarity and before
any mention of trigonometry. At this level, trigonometry for right
triangles may be introduced as an analtyic, that is numerically, means to
solve geometric problems via a sketch and numerical computation instead
of drawing all to scale carefully. The two solution methods should be
interchangeable. The study of unit circle trigonometry may be left to the
next phase. Course designer may place this before or after the the
development of complex numbers.
The key element element is a two-part geometric proof that
multiplication distributes over addition. With that the polar
coordinate computation rule for multiplication products is equivalent
to a Cartesian, that is, real and imaginary part, computation
rule.
The development envisioned here, of complex numbers may begin as soon as
students have mastered Cartesian and polar coordinates. The development
may clarify and strengthen comprehension of real numbers and the absence
of real square roots of negative numbers. Proofs of the Distributive law,
if not assumed, are available to provide a full and quick treatment,
easily understood and repeated in class.
The square roots of negative numbers and the arithmetic methods with
integers and rational numbers - signed prefixed fractions, may be met in
the first phase or earlier of mathematics education. With the aid of
Cartesian and polar coordinates, we may show how to add and multiply
"points" in a the plane. With that the square roots of negative numbers
can be visualized geometrically, and the complex numbers geometrically.
Operations on complex numbers demonstrate and provide a context for the
introduction of translations, rotations and reflections in the plane. All
the latter may be done in the first phase, this phase or the next. The
polar coordinate definition of multiplication is consistent with the law
of signs for real numbers, and may be used to re-inforce the latter. See
site coverage.
After a simple development of Euclidean and analytic geometry, see site
steps, a simple proof of the distributive law which does not depend on
mastery of unit circle trigonometry may be given. While multiplication
above is introduced with the help of polar coordinates, the distributive
law proof combined with properties of "real numbers" implies product
formula for complex numbers in terms of Cartesian coordinate, or "real"
and "imaginery" parts. That may come with this introduction of complex
number or in later lessons at the calculus precalculus level.
Remark: The early and full geometry development of Complex Numbers
before the study of unit-circle trigonometry provides a simpler starting
for the latter. With that trigonometric identities met in the next phase
follow easily and more accessible from complex number mastery due to the
equivalence or equality of two ways to multiply complex numbers. Further
more, formulas for dot- and cross-products of vectors in the next phase
are easily implied, again due the equivalence or equality of two ways to
multiply complex numbers. Thus an operational command of complex numbers
and trigonmetry is made easier and stronger. Students for whom the
geometric reasoning implying the distributive law is too early or too
overwhelming may assume the distributive law instead. Here again an
operational command of mathematics does not require a minimal set of
assumptions, only a consistent set. On the other hand, presenting simple
proofs of the distributive law adds to the logical structure of
mathematics - a structure with intellectual value to some if not all.
Slopes and a Calculus Preview
The study of straight
lines and their equations in the plane should not be a great challenge. Site steps on solving linear equation in one, two or more unknowns, and
site steps on the forward and backward use of formula, in arithmetic and
algebraic solutions, that is, numerically and literally, make the
algebraic study of equations for straight lines, and the algebraic
question of when such lines intersect simpler to learn and teach. The product of slopes of two
lines, each perpendicular to the other, is -1. That can be shown later
directly, or as part of trigonometry, unit-circle, style.Aside: In the
case where proportional denominate numbers with unlike units are graphed
against each other, straight lines also appear.
The study of straight
lines and their equations in the plane
allows a light
why slopes calculus preview to be given. That would the stage in this phase or the next
for location of intervals where functions are increasing, intervals where they are decreasing, with easy
implications for the location of maxima and minima. See the preview. Calculus is the subject of
slope related computations forwards and backwards for linear and nonlinear functions.
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Teachers & Tutors: Site pages offer better or best practices for providing skills -
simpler than expected & comprehensive but for exercises. For your charges, your duty is to study them alone or in
groups and develop skill building exercises & activities to share. Start now. The effort here is the best I can do.
Others are welcome to refine or exceed it. Please do.
Secondary
Mathematics for Ages 11+, A Practical Approach for home-tutoring or -schooling, or for schools & colleges
with local curriculum control. Study how to include site content - its skill development how-TOs and innovations
into present or future lesson plans - some reading required.
Road
Safety Messages and Questions: When and why should you face
traffic when walking along a road or cycle path? Is it a good
idea to hang limbs outside of cars etc? What gives more
protection in a crash: a car, motorbike or bicycle?
See too, the BBC-Belgium story Texting and
Driving - texting & the impossible test - the article links to a gruesome utube video on the subject
The Logic of Injustice:
How Texas sent
an innocent man to his death - The wrong Carlos. Some judgments are irreversible. Procescution: Where and when prosectors play to win rather than for
justice, guilt beyond a reasonable doubt goes unrespected due to prosecutors who putting winning
first, those innocence before the law may be convicted. Some procescutors offices in continuing to accuse after a pardon
due to reasonable doubt or innocent being shown, may sucessfully oppose compensaton for false convictions
by asserting a pardon individual is still under suspicion. Then the pardoned individual or the latter's estate
is not compensation for years or decade
of improper or false imprisonment, or for execution. Site chapters on Logic
and some in Pattern
Based Reason may slowly lead to greater precision in reading, applying and
writing laws.
May 2012, Composition Starting:
Pre-School and Primary Mathematics - Quantitative Skills, An
Intellectual View, Feedback Welcome:
The 8 Most Popular Site Inlinks
Parent Center: Help your child or teen
learn:
Parent-friendly
Work Booklets for ages 3+ to 13 Use these or others to check
or build skills. Other booklets are available but these booklets
allow parents unsure of themselves in mathematics to help their
children. The selection acquired in Canada is published in the
USA. So it has a US orientation. In retrospect, the selection
shows parents what to check with the booklets or by other ways,
the choice is theirs. But in retrospect, the selection does not
cover integral and fractions liquid weights and measures - ask
the publishers to correct that! For ages 9 to 12 say, parents may
compensate by showing boys and girls how to use weights or mass,
and further measures in food preparation. Beyond that children
may be shown how to measure and calculate angles, lengths and
areas [proportional amounts too] directly or by using maps and
plans drawns to scale. Learning how to gather and measure all the
ingredients, pots and pans for a dish or a meal, along with
cleaning up sets the stage for like activities or experiments in
science courses, and in developing organizational skills,
gives boys and girls a head start. Good luck. At the other
extreme, more comprehensive than light, if your motto is
McCainian: drill, drill, drill then Toronto
mathematician and actor John Mighton's jump math organization has jump math
workbooks for at least grades 3 to 8 for at-home and in-school
use - training sessions for teachers available. Jump math has
been expanding to cover older students. Jump Math Samples: plus
Fractions for
Grades 3-4 & Grades 5-6 [Read] Free Resources grades 1 to 8
[unread - likely to be good]. and
Mathematics
Skills For Ages 3 to 14 - technical!
Skills with take
home value - A few ideas
Basic skills include
time-date-calendar Matters; money matters; map, plan and
scale diagram matters;counting, measuring and figuring;
decision making with logic and likelyhood; being careful and
being aware of the domino effect of mistakes; reading and
writing with precision.
Is your child able to add, subtract and multiply amounts
of money, work with fractions, work with clocks and calendars,
work with maps and plans, and measure length, weight-mass and
volume? Schools may promote your son or daughter without
providing basic skills in reading, writing and
arithmetic.
Arithmetic
and Number Theory Skills
Algebra
Starter Lessons
Geometry
- maps plans trigonometry vectors
More
Algebra
70
Calculus Starter Lessons
Calculus Lessons Elsewhere:
-
How to Ace Calculus: Street Wise Guide - Mostly
Text.
-
Flash
Video for Calculus Phobics
They cover basic topics in ways likely to complement your
notes, your textbooks and site material. When Goldilocks
trespassed in the house of the three bears, she found three bowls
of porridge, two not to her liking, and one just right. Different
bears have different tastes. As invited guest here and elsewhere,
if one or more explanations is not to liking, try another. It may
be better or just right.
Unsolicited Advice
Learning to do and high marks if it comes to easy is often
deceptive - light rather than deep. For that reason, students
with learning difficulties determined not to let it get in their
way may go deeper and farther than those with none. High marks,
if the come easy, may be deceptive - provide a too light and not
a deep mastery. That could have been your problem in secondary
school, one that leads to comprehension shock or difficulties in
calculus and more generally in the first year of college. Bon
Appetite.
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