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
|
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.
|
|
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.
|
|