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Protest: The site author, a McGill University,
1983 Ph. D in mathematics, failed a McGill Faculty of Education B. Ed pgm 2003-5
due to
YOU are better than YOU think. Show
yourself how:
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Read logic
chapters 1 to 5 in online volume Three
Skills for Algebra for greater skills & confidence
in work
and study
Learn to read notes and textbooks like
a lawyer, so that no nuance, no subtlety and no clause escapes your
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
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;
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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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Maps, Plans and Drawings
January 13th, 2008
1. Similarity by Observation and Design
Innate Ability: Our perception and recognition of
figures and objects in the environment is based on likeness of shape, not
size. Even before the concept of similarity arises in mathematics courses,
similarity is met and employed in daily life. We may recognize a figure or
object by its shape, independent of actual and apparent size. Apparent size
depends on distance. Reading and writing come from the innate the ability to
recognize and duplicate letters, symbols, digits and basic geometric shapes
such as squares, rectangles, circles and triangles. Primary school students
have the ability to recognize like or similar shapes even before such likeness
or similarity is put into words and even before course on geometry offer
metric definitions detailing the scaling of lengths and the preservation of
angle measures for corresponding line segments and angles. Whence
foregoing explains how primary school teachers and students can talk about and
discuss like shapes or similarity before secondary school mathematics on the
subject. There-in lies an innate ability whose first use does not need to be
logically developed in any formal way.
Students may see or be shown maps, plans and drawings, if not plans at home
and in primary and secondary school. Through the use and drawing of maps
and plans to scale, students over a few to several years may obtain an
operational or empirical command of geometry of maps, plans and drawings. Site
treatment of Euclidean Geometry - from drawing triangles to identifying and
characterizing parallelograms should be included here or followed in parallel.
- Properties of Maps, Plans and Drawings: Basic geometric shapes are
preserved - the drawing or image of a triangle, square, rectangular, circle,
regular polygon or odd-shape polygon is still, respectively, a scaled
triangle, square, rectangular, circle, regular polygon or odd-shape polygon.
- Working With Maps: Real world lengths can be measured or calculated
on a map or plan through the use of rulers and strings and then mutliplied
by a scale factor, if need-be. So making or using a diagram to scale can be
a tool for solving for missing lengths by measuring in the diagram
instead of the real world. There-in lies a base for the discussion of
similar triangles and finding missing angles and lengths there-in. In the
introduction of right-triangle trigonometry, the tabulation of trig
values may be presented as providing a tool for avoiding drawing a diagram
and measuring the missing quantities on a similar triangle.
- Scale Figures in 2D and Models in 3D. (1) The number of
square units needed to cover a real-world region is the the same as the
number of scaled squared units needed to cover the map or drawn image of the
region. The foregoing can be implied for several geometric figures, several
formulas and for regions whose areas are obtained by approximation. In the
latter case, the real-world approximation and the map approximation should
correspond. (2) Whence the ratio area of the real world region to the
area of the image equals the ratio of the area of the real world unit square
unit to the map unit square. The foregoing sets the stage for a
discussion of scale factors in 2 and 3 dimensions, and the cost of model
building, or the ratios of surface areas, volume and mass in models, make
building them worthwhile - economic for testing concepts, and for making
toys.
- Navigation: Journeys (paths, routes) can be planned or drawn on
maps as exercises in navigation. Distances between points on the path (along
the path, or as the crow flies) and between points on and off the path can
then be measured. Journeys can be planned in a zig-zag, piecewise linear way
via the head to tail addition of displacement arrows (or vectors). All that
can be introduce without the use of coordinates. Displacement arrows
drawn on a plan or map need no description. They can be seen. However, they
can also be described via length and direction. Direction may given by
compass heading once a North Direction is defined or given. Direction
may also be given and measured using polar coordinates. Lengths themselves
can be described as a multiple of a unit length. After the introduction of
the resultant of the head to tail addition of a pair or sequence of
displacement arrows (vectors), the direction of a single displacement arrow
may be described or given by a sum of horizontal and vertical component
displacement vectors. Here we may assume that the head-to-tail addition of
displacement vectors commutes, or use some Euclidean geometry to imply that.
- Adding Collinear Displacements: Of special interest is the addition
of two collinear displacements with the same or opposite directions. In the
case of the same direction, head to tail addition leads to a resultant
displacement the same direction as the addends, and length equal to the sum
of their lengths, relative to a choice of unit length. In the case of
opposite directions and same length, the resultant will be zero. One
displacement may be view as the opposite of the the other. In the case of
opposite directions and unequal lengths, one will be shorter of
length a units and the other will be longer of length b = a + c or c+a.
units. So the longer is equal to a the opposite of the shorter plus a
remainder in the direction of the longer, and equal to the remainder plus
the opposite of the shorter. Whence head to tail addition and employment of
the associative law leads to a cancellation of the shorter with its
opposite, and a resultant equal to the remainder, in the same direction of
the longer segment.
- Multiplication by Unsigned and Signed Whole Numbers and Fractions
(mixed numbers too). Note too whole number and fractional
multiples of a unit vector and other vectors may be defined or developed due
to the possibility of adding collinear vectors or fractions thereof, with
the same direction. There-in lies an issue of repeated addition. Negative
multiples of a vector may be introduced as unsigned multiples of the
opposite of the vector. For positive multiples, drop the sign to get an
unsigned multiple.
- Introduction of Coordinates and Signed Numbers/Signed
Coordinates/Signed Coefficients of unit vectors. For rectangular
maps with origin at one corner (lowest, leftmost), unsigned coordinates
suffice to locate points but not to indicate the direction of horizontal and
vertical components of displacement vectors. For rectangular maps with the
origin located in the interior, signed coordinates may be introduced for
location of points. A displacement vector is declared to be drawn in
standard position if and only if it tail is situated at the origin. Each
point in a rectangular map may be identified with a vector in standard
position. Points with one coordinate zero, can be identified
coordinate can be identified with vectors in standard position collinear
with a coordinate axes. In particular, points on the horizontal axes can be
identified with horizontal vectors in standard position and with signed
coordinate multiple of a unit length. The head to tail addition of
horizontal (collinear) vectors implies how to describe such additions with
coordinates alone - drop the unit length. Since addition of
displacement vectors is commutative and associative, the addition of
coefficients (signed coordinates) is also commutative and associative.
Optionally: Multiplication of these horizontal vectors by Unsigned and
Signed Whole Numbers and Fractions (mixed numbers too) implies rules for
multiplication of signed numbers/coordinates which are serving as
coefficient of a unit vector along the horizontal axes. Note later on,
consistent with the option, we will define multiplication of points in
the plane with the aid of polar coordinates and the rule, add angles,
multiple (unsigned) lengths. Digression: The choice of unit length
and direction of the axes (unit vectors) determines the coordinate
system and hence the geometric manifestation of this multiplication. In
essence, we define the operation geometrically, and use that to define a
multiplication of coefficients - the coordinates.
- Again, real world lengths can be measured or calculated on a map or plan
through the use of rulers and strings and/or coordinates, and then
multiplied by a scale factor, if need-be. The use of real-world
coordinates may obviate need for the latter. So making or using a diagram to
scale can be a tool for solving for missing lengths by measuring in
the diagram instead of the real world. There-in lies a base for the
discussion of similar triangles and finding missing angles and lengths
there-in. In the introduction of right-triangle trigonometry, the
tabulation of trig values may be presented as providing a tool for avoiding
drawing a diagram and measuring the missing quantities on a similar
triangle.
- Extension/Continuation/Digression: (1) Projective and Perspective
Drawing for art and for technical drawings. Projection = another way of
forming a map. To be more precise, projection of plane region onto a
parallel surface gives a map with some (angle and scale) distortion if not
parallel. Projection of objects at different distance leads to
perspective drawings in which size of the object depends on distance. (2) Geometric
optics with parallel rays and provides examples of dilations with
positive and negative scale factors (reversal of orientation).
- Introducing Complex Numbers: The starter
lesson for complex numbers shows how rectangular coordinates can be
employed to define addition of points in the plane and how polar coordinates
can be employed to define multiplication of points in the plane (and real
numbers too). The starter lesson shows how hat multiplication
distributes over addition for complex and real numbers. The demonstration of
the special case of real numbers might be given first - it is simpler and
may motivate the second case. See the site area on number theory. This
item steals the thunder or gives the technical element of the
following. Further Reading: See the easy consequence of the starter
lesson in the area of trig and vector analysis (dot & cross-products).
Extrinsic Operational Viewpoint: Every map has a top side
(written on) and a bottom side- hence orientation is defined.
2. A Hand-waving, Accessible, Geometric- Decimal Development of Real and
Complex Numbers
Reference: The two site areas on Number
Theory. and on Complex
Numbers, the starter
lesson for complex numbers (outside the latter site area), and the
forthcoming site section on Maps, Plans and Drawings - Similarity by
Observation and Design
From Foreword
of Volume 3.
The physicist Richard Feynman (1918-1988) gave three
public lectures at McGill University in 1976. His work on physics has
been followed by many scientists and students.
In the lectures, partly tongue-in-cheek, he suggested that physics was
based on two easily described operations, namely the addition and
multiplication of arrows in the plane. His description of arrow
addition and multiplication for a general, non-mathematical audience
was a model for the informal, very visual, most adequate, presentation
of mathematical ideas. But he gave it under the guise of describing
physics. And he avoided panic among the mathematically shy by not
saying that the arrows, with their addition and multiplication,
represent what pure and applied mathematicians (since Gauss) regard as
the complex numbers.
No mastery of the algebraic way of writing and thinking was required
to understand his live description of addition and multiplication.
When I attended Feynman‘s lectures, I thought his description of
arrows in the plane could be an excellent way to introduce complex
numbers. The chapters on complex numbers elaborate on Feynman’s live
presentation, although their on-paper presentation employs the
algebraic way of writing and reasoning
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There are at least two general routes to developing the theory of numbers. One
is to start with counting or Peano's axioms as is or in set theory form, and
from their define and construct rational and irrational numbers in terms of
ordered pairs, sets (Dekedind cuts), and sequences The latter provides the
intrinsic route of pure mathematics. In contrast, Richard Feynmann
in a 1976 guest lecture at McGill described physics as the addition and
multiplication of arrows in the plane.
Extrinsic Routes: If we assume we can draw and locate points and
displacements on maps and plans, without and then with coordinates, we can
describe real and complex numbers geometrically, and also describe them with
signed decimal coordinates.. With the aid of some pre-coordinate Euclidean
geometry, the foregoing leads to an extrinsic view of real and/or
complex numbers in which the operational assumption that addition and
multiplication operations are essentially independent of the choice of
coordinate axes and unit length implies the field properties of real and complex
numbers. The use of unit lengths and decimals to measure lengths, and the
unfolding of the need for finite, repeating and then non-repeating
decimal expansions then provides a simple operationally, geometric &
decimal, manipulative hand waving viewpoint of real and complex numbers.
Whence the development and properties of real and complex numbers follows the
assumptions needed to use coordinates with maps and plans to model or describe
locations, figures and displacements (a.k.a vectors or arrows)
In this viewpoint, the development and properties of real and complex numbers
stems from the easily understood and presented geometrical use of maps, plans
and drawing to describe points, figures and displacements (vectors, arrows)
without and then with mention of coordinates. After a selection of the
coordinates axes (two perpendicular directed lines) and a selection of a unit
length, ordered pairs of lengths (numbers) can be used to define rectangular and
polar coordinates. Whence we can describe operations on points and displacements
first in a coordinate free manner, and then with coordinates. The Parallelogram
law (see this site's minimal treatment of Euclidean Geometry) then implies the
addition of displacements and hence coordinates is commutative. The
associatively of the head-to-tail addition of displacements that addition of
coordinates is also associative. Counting principles in the limit imply or
suggest multiplication of coordinates is commutative. The distributive law
follows as the addition of displacements in the plane is independent of the
choice of unit length or vector.
Remark: Chapter 7 in Volume 1B, Math
Curriculum Notes, raises a concern about assuming the connection of ordered
pairs of real numbers with the plane. The axioms of pure mathematics,
deliberately context-free for the sake of rigour, by themselves are not
enough to connect its constructs to the real world. Those axioms provide an
intrinsic view of coordinates, real numbers and complex numbers. Yet the
introduction of trigonometry, geometry and calculus requires an extrinsic view,
some hand waving to illustrate and explain mathematical concepts in context. The
above material shows how to develop mathematics in an operational manner and
pushes aside the concern in Chapter 7 by consistency taking an operational,
extrinsic view of the subject to develop skills and comprehensions, and to avoid
nuances which should be left to after mastery of the algebraic and deductive way
of reason. The aim is to provide an operational command of mathematics skills
and concepts in ways that directly support quantitative disciplines.
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www.whyslopes.com
Mathematics Education Essays,
57 or so
Area Entrance & Hub
Ideas for Better Instruction
4 Ways to Improve Reform
Theory of Knowledge
Peer Review
The Trouble With Algebra
Course Design and Delivery
How Letters Appear
Sit Down & Study
Modern Education
Key Notes and Themes
Site Lesson Plans
How This Site Differs
Site Origins
Math & Logic Puzzles
Comments on site content.
Words For Instructors
Inductive Principles
Fairness Principles
Apprentices & Masters
Three Remarks
For a Leaner Curriculum
Mixed Maths Curricula
Cultivating Intelligence
Reason - 3 kinds in maths
Logic in Mathematics
Science Education
Maths Instruction in General
Operational View & Values
Standards
Ends and Values
Goals & Unifying Themes
Algebra Lesson Plans
Algebra, Geometrically
Mathematics Curriculum Shifts
Teaching Tips - Fractions to Calculus
Math Ed Perils
Talk the algebra talk
Sec I - Fraction Focus
Sec II - algebra focus
Sec III - Focus on Slopes
Maps-Plans-Drawings
Math Wall Posters
Education, Empirical Art
Damage Reversal
North American Math Curriculum
Managing Reform
Essay January 2007
Educational Follies
Contructivism Incomplete
Missing the Point I
Mathematics in Context
What and When, A Challenge
Grouping Students
Teacher Certification
Education of Math Ed. Professors
Site Eurekas
Links
Help Me Learn/Teach;
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words before symbols
- direct &
indirect use of formula, numerical versus algebraic solutions - what
is a variable (more words)
- Arithmetic
- exercises
- with fractions
-
videos on primes, lcm, gcm,lcd, square roots etc
- Calculus - geometric
preview, algebraic
preview,
3 study guides,
much more
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-starter lesson with java applet - easy
consequences for trig & vectors in the plane
- Education
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Design & Delivery
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- alone
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algebra
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hindsight
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substitution -
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construction, duplication & Isometry - Failure
of ASA & the // line postulate - angle
sum in triangles -//
grams - Triangle
Similarity
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trigonometry
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and of proofs
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& (?) derivatives
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& proportions - slopes & rates included
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dot & cross - cosine
law
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