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. Early High School Arithmetic
Deciml Place Value  funny ways to read multidigit decimals forwards and
backwards in groups of 3 or 6. Early High School Algebra
What is
a Variable?  this entertaining oral & geometric view
may be before and besides more formal definitions  is the view mathematically
correct? Early High School GeometryMaps + 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 
www.whyslopes.com >>  Volume 1A Pattern Based Reason >> Chapter 15 ObjectiveProcesses Next: [Chapter 16 OriginsandLimitationsofRulesandPatterns.] Previous: [Chapter 14 DeductiveandEmpiricalViewsofMathematics.] [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17][18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] Chapter 15, Objective ProcessesReproducible ResultsArithmetic shows the main idea of objectivity. Namely, a result does not depend on who or what performs a calculation, but only on the rules for addition, multiplication, subtraction and division. Except perhaps for roundoff error, arithmetic results are repeatable and reproducible. Recipes and rulebased processes, when carefully done, give results independent of who obtains them. In this situation, the results cease to be subjective — that is dependent on the person getting them – and they depend only on the context. In this situation, the results are said to be objective. The main advantage of objective (rulebased) reason and processes is as follows. Once we have agreed upon the rules and recipes and on the evidence or ingredients to use, the results obtained are independent of who or what obtains them. The result could be a number if we are doing arithmetic. It could be a judgment or a conclusion if we are dealing with people. It may be an action or product when operating a device or machine. Rulebased reason is ideal when or if you agree on the rules and information employed. Disagreement on this, and the ensuing absence of rules or information needed by them, represents a limit of rulebased reason. Disagreement over what rules, if any, to apply makes conclusions subjective – that is, dependent on who obtains them. Objective reason and empirical processes both rely on the idea of following previously stated recipes and guidelines, preferably ones that have given good results in the past. Unfortunately, people singly or in organizations are capable of repeating and reproducing bad or inferior results as well. Still, for many problems, rules or recipes for their solution may be known. The recipes provide solutions for problems that other people have met and solved. These recipes and guidelines represent the experience and the opinion of others, those who have investigated or explored the problems before. In arithmetic, science and technology, this knowledge (recipes, tricks, procedures) is represented by written or verbal statements of rules, patterns and recipes which may work. Search For Repeatable and Reproducible MethodsDepartures From ObjectivityThe ideal or goal of objectivity is represented in the legal system by the idea of impartiality. Lawyers, juries and judges interpret evidence and laws. One aim is to obtain impartial, objective verdicts of guilt or innocence, and assignments of blame, damages and punishments. Rules and laws are subject to geographic chances. In different countries, we have different legal systems. Some are impartial. In these there is an attempt to apply previously established rules and regulations objectively. In other systems, the justice may be corrupted by bribes, prejudice, etc. Even in the moreorless impartial ones, laws and regulations differ. So what is against a law or a regulation in one location may be legal in another. Laws, including commercial ones, often have a moral or religious basis. Moral and religious ideas often define and differentiate groups. What is considered polite, or inoffensive in one group, will be impolite or offensive in others. Laws and regulations in legal systems reflect these differences. Laws and regulations are often, if not always, subject to human interpretation. Commercial laws are intended to control or regulate business. Laws may also define or remove previous obligations or liabilities. The economic needs, selfinterest and desires of people, affect laws. Lawmakers are further requested by interest groups to write laws in one way or another. Each group readily accepts laws to control and restrict the behavior of any other group but itself. Laws as they are being formulated may be changed minutely to the benefit of one group or another. All of us have different ideas of what is fair. Our laws themselves are compromises between the views, principles and interests of several groups, often satisfactory to none. So we cannot say in advance that a set of laws will be complete and not contradictory. Circumstances may occur to which the laws apply, but for which the rules are not intended. Or unforeseen circumstances will occur to which the laws do not directly apply. This points to the need for a new law or new judgments about the application of existing laws. Human laws are human creations. And humans individually or collectively may err. The formulation of laws and rules and principles by people introduces the possibility of error.
Approximate ObjectivityLaws and regulations, however obtained, may be applied in an objective manner. Objectivity may be subject to mitigating circumstances, political interference, the ability of lawyers, the opinions or morals of judges, etc. Results may vary or differ due to different laws and mores in different locations — including your hometown; or due to ad hoc departures from objective applications and interpretations of existing laws and rules, however carefully written or not. But the ideal of objectivity with humanmade and humanapplied regulation remains. Selby A, Volume 1A, Pattern Based Reason, 1996. www.whyslopes.com >>  Volume 1A Pattern Based Reason >> Chapter 15 ObjectiveProcesses Next: [Chapter 16 OriginsandLimitationsofRulesandPatterns.] Previous: [Chapter 14 DeductiveandEmpiricalViewsofMathematics.] [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17][18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] 
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 ReasonOnline 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 Reviews1996  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. ...
Senior High School Geometry

Euclidean Geometry  See how chains of reason appears in and
besides geometric constructions. Calculus Starter Lessons
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. 