Mathematics Concept & Skill Development Lecture Series:
Webvideo consolidation of site
lessons and lesson ideas in preparation. Price to be determined.
Bright Students: Top universities
want you. While many have
high fees: many will lower them, many will provide funds, many
have more scholarships than students. Postage is cheap. Apply
and ask how much help is available.
Caution: some programs are rewarding. Others lead
nowhere. After acceptance, it may be easy or not
to switch.
Are you a careful reader, writer and thinker?
Five logic chapters lead to greater precision and comprehension in reading and
writing at home, in school, at work and in mathematics. 
1 versus 2way implication rules  A different starting point  Writing or introducting
the 1way implication rule IF B THEN A as A IF B may emphasize
the difference between it or the latter, and the 2way implication A IF and ONLY IF B.

Deductive Chains of Reason  See which implications can and cannot be used together
to arrive at more implications or conclusions,

Mathematical Induction  a light romantic view that becomes serious. 
Responsibility Arguments  his, hers or no one's 
Islands and Divisions of Knowledge  a model for many arts and
disciplines including mathematics course design: Different entry
points may make learning and teaching easier. Are you ready for them?
Deciml Place Value  funny ways to read multidigit decimals forwards and
backwards in groups of 3 or 6. 
Decimals for Tutors  lean how to explain or justify operations.
Long division of polynomials is easier for student who master long
division with decimals. 
Primes Factors  Efficient fraction skills and later studies of
polynomials depend on this. 
Fractions + Ratios  See how raising terms to obtain equivalent fractions leads to methods for
addition, comparison, subtraction, multiplication and division of
fractions. 
Arithmetic with units  Skills of value in daily life and in the
further study of rates, proportionality constants and computations in
science & technology.
What is
a Variable?  this entertaining oral & geometric view
may be before and besides more formal definitions  is the view mathematically
correct? 
Formula Evaluation  Seeing and showing how to do and
record steps or intermediate results of multistep methods allows the
steps or results to be seen and checked as done or later; and will
improve both marks and skill. The format here
allows the domino effects of care and the domino effects of mistakes
to be seen. It also emphasizes a proper use of the equal sign. 
Solve
Linear Eqns with & then without fractional operations on line segments  meet an visual introduction and learn how to
present do and record steps in a way that demonstrate skill; learn
how to check answers, set the stage for solving word problems by
by learning how to solve systems of equations in essentially one
unknown, set the stage for solving triangular and general systems of
equations algebraically. 
Function notation for Computation Rules  another way of looking
at formulas. Does a computation rule, and any rule equivalent to it, define a function? 
Axioms [some] as equivalent Computation Rule view  another way for understanding
and explaining axioms. 
Using
Formulas Backwards  Most rules, formulas and relations may be used forwards and backwards.
Talking about it should lead everyone
to expect a backward use alone or plural, after mastery of forward use. Proportionality
relations may be use backward first to find a proportionality constant before being
used forwards and backwards to solve a problem.
Early High School Geometry
Maps + Plans Use  Measurement use maps, plans and diagrams drawn
to scale. 

Coordinates 
Use them not only for locating points but also for rotating and translating in the plane.

What is Similarity  another view of using maps, plans and
diagrams drawn to scale in the plane and space. Many humanmade objects
are similar by design.

7
Complex Numbers Appetizer. What is or where is
the square root of 1. With rectangular and polar coordinates, see how to
add, multiply and reflect points or arrows in the plane. The visual or geometric approach here
known in various forms since the 1840s, demystifies the square root of 1 and the associated concept of
"imaginary" numbers. Here complex number multiplication illustrates rotation and dilation operations in the plane.
 Geometric Notions with Ruler & Compass Constructions :
1 Initial Concepts & Terms
2 Angle, Vertex & Side Correspondence in Triangles
3 Triangle Isometry/Congruence
4 Side Side Side Method
5 Side Angle Side Method
6 Angle Bisection
7 Angle Side Angle Method
8 Isoceles Triangles
9 Line Segment Bisection
10 From point to line, Drop Perpendicular
11 How Side Side Side Fails
12 How Side Angle Side Fails
13 How Angle Side Angle Fails
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www.whyslopes.com >> Geometry  maps plans trigonometry vectors >> 3 Cartesian and Polar Coordinates >> 3 Rectangular Coordinates  Review Next: [4 Polar Coordinates  to and from.] Previous: [2 Cartesian Coordinates with signs.] [1] [2] [3] [4][5] [6] [7] [8] [9] [10] [11] [12] [13] [14]
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
viceversa. 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. Nonnegative 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.
www.whyslopes.com >> Geometry  maps plans trigonometry vectors >> 3 Cartesian and Polar Coordinates >> 3 Rectangular Coordinates  Review Next: [4 Polar Coordinates  to and from.] Previous: [2 Cartesian Coordinates with signs.] [1] [2] [3] [4][5] [6] [7] [8] [9] [10] [11] [12] [13] [14]
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Road Safety Messages
for All: When walking on a road, when is it safer to be on
the side allowing one to see oncoming traffic?
Play with this [unsigned]
Complex Number Java Applet
to visually do complex number arithmetic with polar and Cartesian coordinates and with the headtotail
addition of arrows in the plane. Click and drag complex numbers A and B to change their locations.
Pattern Based Reason
Online Volume 1A,
Pattern Based Reason, describes
origins, benefits and limits of rule and patternbased reason and decisions
in society, science, technology, engineering and mathematics. Not all is certain. We may strive for objectivity, but not
reach it. Online postscripts offer
a storytelling view of learning: [
A ] [
B ] [
C ] [
D ] to suggest how we share theory and practice in many fields of knowledge.
Site Reviews
1996  Magellan, the McKinley
Internet Directory:
Mathphobics, this site may ease your fears of the subject, perhaps even
help you enjoy it. The tone of the little lessons and "appetizers" on
math and logic is unintimidating, sometimes funny and very clear. There
are a number of different angles offered, and you do not need to follow
any linear lesson plan. Just pick and peck. The site also offers some
reflections on teaching, so that teachers can not only use the site as
part of their lesson, but also learn from it.
2000  Waterboro Public Library, home schooling section:
CRITICAL THINKING AND LOGIC ... Articles and sections on topics such as
how (and why) to learn mathematics in school; patternbased reason;
finding a number; solving linear equations; painless theorem proving;
algebra and beyond; and complex numbers, trigonometry, and vectors. Also
section on helping your child learn ... . Lots more!
2001  Math Forum News Letter 14,
... new sections on Complex Numbers and the Distributive Law
for Complex Numbers offer a short way to reach and explain:
trigonometry, the Pythagorean theorem,trig formulas for dot and
crossproducts, the cosine law,a converse to the Pythagorean Theorem
2002  NSDL Scout Report for Mathematics, Engineering, Technology
 Volume 1, Number 8
Math resources for both students and teachers are given on this site,
spanning the general topics of arithmetic, logic, algebra, calculus,
complex numbers, and Euclidean geometry. Lessons and howtos with clear
descriptions of many important concepts provide a good foundation for
high school and college level mathematics. There are sample problems that
can help students prepare for exams, or teachers can make their own
assignments based on the problems. Everything presented on the site is
not only educational, but interesting as well. There is certainly plenty
of material; however, it is somewhat poorly organized. This does not take
away from the quality of the information, though.
2005  The
NSDL Scout Report for Mathematics Engineering and Technology  Volume 4,
Number 4
... section Solving Linear Equations ... offers lesson ideas for
teaching linear equations in high school or college. The approach uses
stick diagrams to solve linear equations because they "provide a concrete
or visual context for many of the rules or patterns for solving
equations, a context that may develop equation solving skills and
confidence." The idea is to build up student confidence in problem
solving before presenting any formal algebraic statement of the rule and
patterns for solving equations. ...

Euclidean Geometry  See how chains of reason appears in and
besides geometric constructions. 
Complex Numbers  Learn how rectangular and polar coordinates may
be used for adding, multiplying and reflecting points in the plane,
in a manner known since the 1840s for representing and demystifying
"imaginary" numbers, and in a manner that provides a quicker,
mathematically correct, path for defining "circular" trigonometric
functions for all angles, not just acute ones, and easily obtaining
their properties. Students of vectors in the plane may appreciate the
complex number development of trigformulas for dot and
crossproducts.
LinesSlopes [I]  Take I & take II respectively assume no
knowledge and some knowledge of the tangent function in
trigonometry.
Why study slopes  this fall 1983 calculus appetizer shone in many
classes at the start of calculus. It could also be given after the intro of slopes
to introduce function maxima and minima at the ends of closed intervals. 
Why Factor Polynomials  Online Chapter 2 to 7 offer a light introduction function maxima
and minima while indicating why we calculate derivatives or slopes to linear and nonlinear
curves y =f(x) 
Arithmetic Exercises with hints of algebra.  Answers are given. If there are many
differences between your answers and those online, hire a tutor, one
has done very well in a full year of calculus to correct your work. You may be worse than you think.
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