Rectangular Coordinates for lines and planes
Summary: This lesson and the next
offers motivation for the introduction of signs.
In elementary school, people learn about whole numbers n
and fractions p/q before the use of signs. Ordered pairs of
unsigned numbers may be introduced as coordinates in the first
quadrant. Introducing signs + and - gives ordered pairs of
numbers with signs as prefixes to provide coordinates for four
quadrants.
For a Line Segment or Line
Unsigned and Signed Coordinates For Line Segments
finite or infinite
For a line segment, we can measure distance from one end using unsigned
numbers.
Figure 1: Line segment with origin at left end.
The choice of which end to start the numbering or measuring gives an
orientation.
For a line segment, we can also measure distance and direction from a
point in the middle using signed coordinates.
Figure 2: Line segment with origin in middle.
Here the positive sign (+) indicates distance and direction to the right
while the negative sign (-) indicates distance and direction to the
left.
One Dimensional Assumption
Geometric Assumption: An infinite line can be
described by signed coordinates with an origin (the zero point)
anywhere on the line, and the and +1 point anywhere on the line except
the at the origin. Note for later: the directed line
segment from 0 to +1 defines the line orientation and the unit length
for the coordinate system.)
Thus every coordinate (real number) determines a point on the line and
vice-versa. For sake of argument, we assume infinite (straight) lines
exist.
For Plane Geometry
Unsigned Coordinates for Rectangular Maps
origin at corner, Unsigned Coordinates
Figure 3: Rectangular Coordinates for Maps.
Ordered pairs of numbers without signs such [1,4] or [3,2] may be
used to locate points on a map when the origin or reference point
is at the bottom left corner.
On such maps there is no need for signs. More generally, you
use coordinates such as [1.5, 3.27] or [a, b] to locate points on
the map -- provide their rectangular coordinates. Here a and b stand for
any pair of unsigned numbers including zero that may be used as
coordinates.
In the above map, the left edge of the map region give the vertical
coordinate axis while the bottom edge gives a horizontal vertical axis
for coordinate use.
The word rectangular is used above as "polar coordinates" will be
introduced later. Rectangular coordinates are also called Cartesian
Coordinates.
Signed Coordinates in the Plane
This lesson and the previous one offers motivation for the
introduction of signs. In elementary
school, people learn about whole numbers n and fractions p/q before the
use of signs. Ordered pairs of unsigned numbers may be
introduced as coordinates in the first quadrant. Introducing
signs + and - gives ordered pairs of numbers with signs as prefixes to
provide coordinates for four quadrants.
If our first map extends to the left and/or below the origin,
the horizontal and vertical coordinate axis's may be
extended. These extensions divide the map into four regions call
quadrants. To get coordinates for all four regions or quadrants we
may place signs in front of numbers. See the diagram below.
Figure 4. Plane with Signed Coordinate System
Here axes are perpendicular (orthogonal, at right angles) to each
other..
In the above map, identify the points with coordinates [+2,+1], with
coordinates [+2,-4], with coordinates [-2.5, -3] and lastly with
coordinates [-4, +3]. By convention, + signs in front of
numbers are optional. So +2 = 2 and +1 = 1.
Remark: Descartes employed pairs of unsigned numbers to locate
points in rectangular region with the origin of the coordinate system
at a corner of the rectangular region (possibly a map or a plan). The
use of signed coordinates came later. Non-negative coordinates
are sufficient for finite regions. The current mathematical habit
is to write the pair of coordinates as an ordered pair (x,y) or (a,b)
where the variables a, b, x and y will be given by real numbers.
This applet
illustrates the use of rectangular coordinates. Play with it. Move the
points A and B on it and see how their coordinates change. The
coordinates are displayed in the top left corner.
Two Dimensional Assumption
Geometric Assumption: An extended or infinite flat plane can be
covered by a rectangle coordinates using infinite lines parallel to a
horizontal and vertical axis and real numbers as coordinates to locate
points in the plane, regardless of the choice of unit length and
orientation & placement of the horizontal and vertical
axes.
The foregoing assumption extrapolates our comprehension or familiarity
with finite rectangular regions. It gives a powerful numerical or
algebraic model of \work with points, lines, circles and further
geometric objects in the plane.
By describing geometry with numbers, alone, in ordered
pairs or in ordered triplets, the arithmetic properties of numbers can
be used to arrive at conclusions about geometry in one, two and three
dimension via chains of reasons as coordinates provide a
precision missing in useful but suggestive and approximately drawn
diagrams on a small or large scale.
Geometric Model *(Assumption): An extended or infinite flat
plane can covered by a rectangle coordinates using infinite lines
parallel to a horizontal and vertical axis and real numbers as
coordinates to locate points in the plane.
The foregoing assumption extrapolates our comprehension or familiarity
with finite regions regions. It gives a powerful numerical or
algebraic model of \work with points, lines, circles and further
geometric objects in the plane.
By describing geometry with numbers, alone, in ordered pairs or in
ordered triplets, the arithmetic properties of numbers can be used to
arrive at conclusions about geometry in one, two and three dimension
via chains of reasons as coordinates provide a precision missing
in useful but suggestive and approximately drawn diagrams on a small or
large scale.
|
|
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.
|
|