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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. |
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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.
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Caution: Site advice is approximately
correct, for some circumstances, not all. That leaves room for thought |
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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.
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Chapter 11: Complex Numbers
By means of measurement coordinates locate points on line and in the plane
with both rectangular and polar coordinates. Navigational examples can be
employed to introduce, illustrate and motivate vector and displacement addition.
The earlier described operation on vectors of multiplying lengths and adding
angles can be used to define multiplication in the plane and extend the
concept of multiplication on real line from pairs of positive numbers, to any
pair of real numbers – points on a horizontal axis. The foregoing defines the
complex numbers (minus a representation of products in terms of real and
imaginary parts). The polar coordinate definition of multiplication multiply
lengths and add angles applied to real numbers provides or agrees with the
law of multiplication and law of signs.
This development is pre-algebraic, but rule-based. It is in accordance with
the expositional principle of putting the material easiest to understand first.
Before this development only a familiarity with addition, subtraction and
multiplication of positive numbers (fractions or decimal representation) is
assumed. Negative numbers need only be employed as coordinates. Operations with
them need not be defined before the exposition of complex numbers. An
elaboration of these ideas follow. This development represents an innovation for
elementary mathematics instruction, and it has some consequence for intermediate
mathematics instruction.
This new perspective introduces operations on real numbers (signed decimals,
signed fractions or points on a horizontal axis of the complex plane) without
relying on algebra or algebraically described properties of real numbers. It is
computational and pre-algebraic.
Coordinates On the Line
Before the introduction of negative numbers, the notion of positive numbers
is not emphasized. Students may have a knowledge of unsigned (positive) numbers.
These numbers can be employed as coordinates on an infinite half-line to locate
points. After this, signed numbers, positive and negative can be employed to
locate points, that is to serve as coordinates on the bi-infinite real line.
This allows students to graphically comprehend the role of positive and negative
numbers, and zero too, as coordinates or marks on a coordinate line. No
arithmetic is required. Examples of coordinate lines are provided perhaps by
temperature scales, by water levels (the signed height of tides, reservoirs or
river waters above or below a zero mark) and by bank account balances.
Accountants today employ parentheses to avoid writing negative signs. Prior to
the 15th century, negative numbers were thought to be imaginary – figments of
the imagination.
Coordinates in the Plane
Ordered pairs of positive and then arbitrary real numbers can be introduced
as rectangular coordinate for the plane after the selection of an orthogonal
pair of axes. Following Descarte, ordered pairs of positive or unsigned number
locate points in the first quadrant. Following Newton (or others before Newton),
signed coordinates can be employed to locate points in all four quadrants. This
role of signed coordinates offers another motivation for having and employing
positive and negative numbers.
Displacement and Vector Addition
Points in the plane can be identified with vectors (issuing from the origin).
The transport of these vectors and the head-to-tail addition of vectors can be
described graphically, and then in terms of rectangular coordinates. The rules
for this can be drawn from examples in an inductive fashion. Motivation can be
provided by the problem of planar navigation, and moving from point to point on
a map. Here the addition of vectors representing displacement on a map can be
introduced. The resultant of two successive displacement can be declared to be
the linear displacement between the initial point of the first displacement and
the terminal point, following the second displacement. Students will find from
examples and exercises that the addition methods appear to be repeatable and
reproducible, and thus verifiable in a pre-algebraic and pre-deductive fashion.
Restriction to An Axis
Addition of vectors or points on a coordinate axis or coordinate line can
then be viewed as a special or restricted case of the more easily visualized
situation in the plane, an application of the head to tail vector addition
method to pairs of points, alias vectors, on the coordinate line. This will lead
via examples to easily visualized rules for addition of positive and negative
numbers, that is points on the horizontal axis with positive and negative
coordinates. Rules for the addition and subtractions of numbers, vectors or
displacements in the horizontal coordinate line, can now be extracted from the
planar case: regarded as the special or limiting case of motion restricted to a
single line in the plane. This provides another means to visualize mathematics.
Multiplication of Planar Points
Both polar and rectangular coordinates with respect to a pair of axes can be
determined (measured) for points. Given or measured values of polar or
rectangular coordinates can also be used to locate points. The foregoing
geometrically suggests that polar and rectangular coordinates are
interchangeable. It provides a method, geometric measurement, for obtaining
polar coordinates from rectangular, and vice-versa. This approach is hands-on,
physically dependent and while not deductive, it is repeatable, reproducible,
and thus secure. Angles in polar coordinates can be computed, modulo 360
degrees.
Given a pair of nonzero vectors issuing from the origin, that is two points
in the plane, their angles can be measured, and their lengths measured and
represented by an unsigned decimal number – a unit-free length. Adding the
angles together, modulo 360 degrees, and multiplying the unit-free lengths
together yield the angle and unit-free length of third vector, their product.
This defines via polar coordinates, the multiplication of points or vectors in
the plane.
Multiplication of Real Numbers
Following the identification of the horizontal axis with the real number
line, a polar-coordinate representation or visualization of the product of real
numbers follows. This may define for students such products. This definition
implies the law of signs for the product of real numbers. Moreover, the
identification also provides a context and location for the definition of square
roots of negative numbers. These square roots can be found on the vertical,
alias imaginary, axis.
Consequences for Intermediate Instruction
The foregoing offers in elementary instruction a computational and visual
comprehension of arithmetic with real and complex numbers. In intermediate level
instruction, there is choice of how expressions for the real and imaginary parts
of the product of two complex numbers are to be obtained. Assumption of the
distributive law of multiplication over addition in the complex plane
immediately implies expressions for the real and imaginary parts of the product
in terms of those of the factors. Ease of exposition may justify the assumption:
Intermediate instruction need only offer strands of reasoning. Threading them
together in a purely deductive fashion may be left to advanced courses. But
the distributive law can be seen or justified via geometric arguments:
The distributive law itself can be geometrically implied or suggested by
viewing multiplication by a nonzero complex number as the consequence of
multiplying by a positive length and following up by a rotation. The
operations commute. Both are distributive over addition. Reasons for the
latter follow.
For just a rotation, one can physically show the distribution law by
considering the rotation of a parallelogram.
For just a positive stretch factor, one can show this in the special case
of small whole numbers, and then proceed inductively to the case of rational
numbers. The case of irrationals now follows by an assumption of and intuitive
appeal to continuity.
The geometric argument has the appeal that it applies to real multiplication
as well.
As a third alternative, the distributive law can be assumed for real numbers
only and then later on in a trigonometry course, the distributive law for
complex numbers can be obtained from the angle sum formulas. The
rotate-a-triangle proof of these formulas may be less of a surprise and more
accessible to students who have seen the add the angles, multiple the lengths
polar coordinate method for complex multiplication. Against this third approach,
I suspect that many students on learning the distributive law for real numbers
will apply it to complex numbers without a second thought, and with little
patience for the notion that it should be derived. They may be correct.
For ease of exposition, and to provide a greater command of mathematics, the
distributive law for complex numbers can be assumed, and from it the
distributive law for real numbers obtained as a special. In the derivation of
mathematics from set theoretic foundations for arithmetic (axiomatic set
theory), both distributive laws, the one for reals and the one for complex
numbers, are almost equidistant from the axioms in terms of the work required
for their respective derivation. The first exposition of complex numbers like
that of trigonometry and calculus may mixed algebraic and geometric arguments
which illustrate the deductive aspect of mathematics.
The geometric argument, the first alternative outline above, avoids the
semantic problem of which distributive law to assume first. A second chapter on
complex numbers in the companion book Why Slopes and More Math explores
these possibilities in more detail. A fourth alternative is to present the
distribution law as a theorem and leave its proof as an intellectual IOU.
Footnotes:
- This educational writer has no first hand experience of
the elementary school classroom as teacher. The image here is based on
home-based observations of the children of others, how they learn, and this
author’s memory as a student in the classroom. This author as a child was
an adult in waiting – mentally alert and observant, if not informed,
attending the future and a reason for being.
- Numbers may be just adjectives that become objects when
discussed separately from the description of other objects.
- Another context for subtraction is provided by the
coordinate line. Addition of a number n corresponds to n steps or a movement
in a forward or positive direction. Subtraction corresponds to step in
another direction.
- flimsy evidence perhaps
- Technical detail: if a measurement is known or seen to
lie between a lower limit L and an upper limit U, the measurement can be
recorded as equal
- Technical Note: Arithmetic operations with the decimal
expansions of whole numbers is a modification of the polynomial
multiplication process which takes into account the carrying and borrowings
which avoids coefficients with values >9 the base -1.
- A Caution: Isolation in its use should be avoided.
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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.
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