|
|
// _ _ \\
/\ /\
<| (o) (o) |>
\ | | /
|
Have your gifted students read logic
chapters 1 to 5 in online volume Three
Skills for Algebra for greater skills & confidence
in work
and study.
tell students to read notes and textbooks like a lawyer, so that no nuance, no
subtlety and no clause escapes their attention. |
-/[]\-
||
/ \_
||||||||||||||||||||||||||||
Logic
chapters 1 to 5 re- appear not in sequence, as is or longer,
in Volume 1A, Pattern Based
Reason, Bon Appetite.
Logic
Mastery
Amazing, Amusing, Amorous, Delicious, Delightful, Edifying,
Strengthening Elixir.
It eases work & learning difficulties Makes the hard easier. Opens eyes.
Leads to greater precision.
in reading and
writing
Tell students that Logic
mastery makes the hard, easier. Logic
mastery leads to better, stronger and richer comprehension. Logic
mastery improves reading and writing. Logic
mastery ease learning difficulties. Logic
mastery gives a headstart. In sum, logic
mastery will develops critical thinking, improve reading and writing,
and give a firmer base for work and studies at many levels. Good luck.
After logic,
(a) continue reading Three
Skills for Algebra, chapters 8 to 14 and do so alongside site area on solving
liinear Equations ; or (b) see this calculus
starter lesson and Volume 3, Why
Slopes & More Math, chapters 2 to 6. In Volume 2, Three
Skills for Algebra, a 4th skill for algebra appears in Chapter 14. It
provides a unifying theme for high school mathematics - equations and formulas
may be used forwards and backwards, directly and indirectly, numerically in arithmetic
solutions & with literals in algebraic solutions.
|
// _ _ \\
/\ /\
<| (o) (o) |>
| |
| |
\
/
\ = /
|
Caution: Site advice is approximately
correct, for some circumstances, not all. That leaves room for thought |
-/[]\-
||
_ / \
||||||||||||||||||||||||||||
What may be learnt and when depends on how skills
and concepts are developed. Making the hard easier and clearer will allow
earlier & richer development of skills and concepts.
Try the Twiddla
Whiteboard. In principle, it allows
to people to draw and chat together online on a copy of this webpage or a clean
sheet. The chat may be via text or audio. Visit www.twiddla.com
to set up whiteboards to work with the webpage of your choice.
For online automated help in senior high school maths & calculus,
visit quickmath.com For Automatic
Calculus and Algebra Help with derivatives, integrals, graphs, linear equations,
matrix algebra, visit calc101.com
With overlap, each site quickmath
& calc101offers a different range of
services, some free, some not, all based on webmathematica. Good luck.
| |
Chapter 7
Two Treatments of Geometry
There are two approaches to
Euclidean geometry, and the more recent one has precedence[1].
[1] This perspective is noted in the foreword by Harry
Goheen, page 1, in the reprint of the work Foundations of Geometry of
David Hilbert, Open Court Publishing Company, ISBN 0-875-164-7. This
perspective is also expressed in College Geometry by David C. Kay,
Holt, Rhinehart and Wilson, 1969 ISBN 0-03-073100-3.
1. Euclidean geometry in the plane can be presented and based on assumptions
about points and lines in the plane, and associated geometric constructions
based on circles and line segments[2]. The axioms of
Euclidean geometry dealt with points, lines and loci — the curves traced out
by points. This represents the synthetic (constructive), non-analytic
perspective begun in the work of Euclid, two thousand years before anyone
thought of axioms for real numbers or set theory. It represents the first
codification of geometry, and the first model for rule-based thought. It is
coordinate-free. Its original form was developed before coordinates were even
dreamt of.
[2] The word line itself originally referred to a taut rope
or string. Geometry (land measurement) in the first instance physically
employed taut ropes to measure and mark rectangular and circular regions on
the earth surface and in construction. On paper, the use of taut lines or
strings may be replaced by a compass and a straight edge. The question of what
points could be reached in the plane via ruler- and compass-based
construction was the initial object of Galois theory in algebra.
2. The axioms for real numbers provide an newer framework for Euclidean
geometry. The framework is called analytic geometry. It is based on coordinates.
Within this framework, the drawing of lines and circles and the location of
points correspond to the (parametric) solutions of equations and the formation
of sets of ordered pairs or triplets. The latter serve as coordinates of points
in a plane or in space. Note that solutions of equations can be obtained without
reliance on the physical senses and without ruler and compass. Paradoxes due to
imprecisely drawn diagrams and reliance on the physical senses (except for the
use of pencil and paper to record thoughts) are thus avoided in the analytic
approach or codification.
The older coordinate-free, synthetic, approach gives motivation (but no
warranty) for the definitions and calculations of the newer analytic,
coordinate-based, exposition of Euclidean geometry[3].
[3] Technical Note: In the set theoretic framework
for arithmetic-based mathematics, the axioms for real numbers provide a
foundation for analytic geometry including the theory of surfaces. Some
examples of non-Euclidean geometry are given by study of curved surfaces.
Particular examples are given by the surfaces of a sphere, donuts or torii,
ellipsoids (or footballs), and Mobius strips. On such curved surfaces, paths
followed by taut strings yield the smallest distances between points. These
taut strings define line segments which are curved — not necessarily
straight. Moreover, for triangles drawn on these surfaces using three taut
strings, the sum of interior angles need not be 180 degrees. The sum in fact
depends on the curvature, the departure from flatness, of the surface area
enclosed by the taut strings. The mathematical adept can modify this statement
to include surfaces formed by the bending without stretching of flat surfaces
— the ruled or developable case of zero Gaussian curvature.
The physical or geo-measurement assumption that ordered pairs and triplets of
real numbers correspond to points in the plane or space gives the correspondence
between the newer analytic, coordinate-based approach and the older synthetic
approaches to Euclidean geometry. The applications of analytic or coordinate
geometry in physics, engineering, technology, etc., all depend on this
assumption. In particular, this physical assumption is required for the drawing
of diagrams and graphs.
Mixing the analytic and the older synthetic approaches to and representations
of Euclidean geometry or not distinguishing between the two, is convenient in
the relaxed or elementary development of mathematics and its applications in
computational disciplines. The mix may also be present in the initial exposition
of trigonometry and calculus [4].
[4] The high school exposition of sines and cosines relies
on the physical identification of angles of triangles of angles spanned by
sectors of circles. One proof of the angle sum formula for cosines relies on
the rotation of an isosceles triangle and the physical or geometry assumption
of rigidity (preservation of lengths and angles) under rotation. The geometric
constructions here departs from the purely analytic treatment or codification
of mathematics. In college calculus, geometric arguments leading to formulas
for the slope or derivative of the cosine function may fall into the same
category. (For a purely analytic treatment without diagrams, a very succinct
one understandable to a college student specializing in mathematics, see the
text Principles of Mathematical Analysis by W. Rudin, McGraw Hill,
second edition 1964. It gives analytic definitions and treatments of the
exponential function, the natural logarithms, sines and cosines.)
Such mixing departs from the modern mathematics ideal of having a smallest
possible set of axioms for geometry, trigonometry, calculus and other parts of
arithmetic-based mathematics, at least when how to supplant or replace the
coordinate-free approach to Euclidean geometry is not indicated.
Next: Chapter 8, Modern
Mathematics
| |
www.whyslopes.com
Volume 1B, Mathematics Curriculum Notes,
Foreword + Chapters 1 to 10 + 12
Book Entrance
Inductive Principles
1 Introduction
2 For & Against Math
3 Algebraic Thought
4 Why Slopes & SQRT of -1
5 Books & Articles to Read
6 Unruly Origins of Algebra
6. Axiomatic Civilization
7 Geometry, 2 Ways
8 Modern Instruction
9 The Two Ends
10 The Transition
10 Explaining Logic
10 Explaining Algebra
10 Why Sets in Math.
12 Four Phases
Links
Chapter 11: Primary School Mathematics
11 Primary Math
11 Cue Cards
11 Counting
11 Decimals - Addition
11 Decimals -Times
11 Decimals & Subtraction
11 Fractions and Division
11 Notational Conflict
11 Reciprocals Etc
11 Decimals - Ratios
11 Size Comparison
11 Numbers, +ve or -ve
11 Rename < Sign
11 Complex Numbers
Will provide an alternative to Chapter 11 later, most likely in the Parent's
Area: Help Your Child or Teen
Learn
Most students in high school are not heading for calculus,
but most topics in high school mathematics are present due to calculus.
Preparation for calculus demands their coverage at full strength.
See too, this site 55+, Math
Education Essays. Site areas and pages provide pieces of the a Mathematics
Education, Jigsaw Puzzle, in formation.
-Inductive
principles for course design & delivery require a clear
description of where and how skills and concepts may rest on earlier ones, so
that difficulties may be explained and remedied by looking for what was
missed or forgotten in earlier studies.
Mathematics is a demanding subject. All errors in notation
and comprehension need to be identified and corrected. In
reading, spelling and writing, students have to learn all the letters in the
alphabet, not just some. and memorize spelling. Anything less implies
difficulty.
Likewise in mathematics, students have to master key skills
and concepts, one at a time and one after another. Anything less implies
difficulty.
Modern mathematics curricula introduced an inconsistency
into course design and delivery. They did not sanction the use of decimals nor
the use of diagrams in skill and concept development but decimal arithmetic
and diagrams are needed for student comprehension and for an operational
mastery of quantitative skills. That implies the need for an mixed-math
curricula based on a systematic development of operational skills, sufficient
for applications and sufficient to provide a base & context
for the very optional study of pure mathematis.
|