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 2 Three Skills For Algebra >> Chapter 3 Chains of Reason Next: [Chapter 4 Longer Chains of Reason.] Previous: [Chapter 2 Implication Rules  Forwards and Backwards.] [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] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] Chapter3, Chains of ReasonIntroductionThis chapter shows how reliable rules and patterns can be directly employed one at a time, or one after another, to get conclusions or further reliable rules and patterns. The question of what rules are reliable is considered in the following chapters. Rules used to get or suggest conclusions are called implications. Just as there are methods for adding and multiplying numbers carefully, there are also methods for using implication rules by themselves to get conclusions. There are also methods for linking, threading and chaining implication rules together to get more implication rules. This chapter uses examples to explain two basic ideas:
The examples are not important (and are perhaps ridiculous) but they illustrate some rulebased methods in reason. Examples which involved reallife situations might distract from mastering these methods. That is, in reallife situations, each of us may have opinions or prejudices about what should occur. That could spoil an explanation of the use and linkage of implication rules. There is a need for neutral examples to illustrate the use of implication rules one at a time or one after another. Arithmetic, algebra and geometry give many neutral examples for this. The examples below involve no mathematics. Bon Appetite. Conclusions From a Single Rule

If Fred the dog visits the park then sensible worms go underground. 
This conclusion is not of interest unless you are a fisherman (or woman) looking for worms, sensible or not, for use as bait. The conclusion selected and stated here hides the reasoning process. That is, it hides the chain of implications leading to it. Our last conclusion does not mention the intermediate events where a cat climbs a tree and birds fly around the park.
The long path by which we get conclusions shows that implication or rulebased thinking can lead to surprising results. These surprising results are true if the initial implications are also true.
In the long path by which we got the conclusions, the information in the third implication (3) about Charles the human is not used. The conclusion we reached is independent of implication (3). In fact, without further information, I see no way of linking the rule about Charles with the other rules. The third rule is extra information. It can be ignored.
In answering questions, we often have extra information. Indeed, you can imagine the five rules given above are stated in random positions among a list of twenty, or hundred and twenty rules. An answer to the question
What happens when Fred the dog visits the onetree park?
now depends on finding the rules in the list which can be used. This is a game of hide and seek. So we have to be selective, observant or fussy in deciding or seeing what information leads to our conclusions.
The scenery or route by which a conclusion is reached may contain as much useful information as the conclusion itself. A conclusion may contain a fraction of the information we could have stated or written. Being aware of the route or proof by which a conclusion is attained will sometimes suggest how more conclusions can be reached. This awareness is often more important that any conclusion we state because it allows us to state more conclusions, as needed.
Mathematics students take note. Remembering the route taken in solving a problem is worth more to the development of skills than remembering the solution.
Deductive, Inductive or Empirical Reason
Deductive reason uses or chains together supposedly (or preferably) neverdisobeyed implication rules to suggest, to make or to reach conclusions. See the examples above. The implication rules in question may come from assumptions. The assumptions may be tentative.
The phrase inductive reason has one role in mathematics and another outside of mathematics. To induce (or induct) literally means to draw or extract. When you see a rule or pattern that no one has suggested, you are extracting or drawing that pattern from your observations. This process of recognizing rules and patterns that may hold, accidentally or not, is called inductive reasoning. Inductive reason outside of mathematics refers to the identification and recognition of rules and patterns from data and observations. Here rules and patterns may hold accidentally.
Reason which relies on a single or several, experiencefound, rules and patterns to arrive at conclusions is called empirical. The underlying problem of inductive, empirical reason is to extract (infer, draw, induct or identify) from experience, in particular, data and observations, rules and patterns not satisfied merely by accident and which appear to be reliable. Selfdeception needs to be avoided here.
Inductive reason inside mathematics refers to another process, namely, the extraction or drawing of conclusions from ladderlike chains of reason. See the next chapter for a more precise image or explanation. The rules or assumptions here are usually so certain, that we deliberately ignore the experiencebased origins of mathematical reason.
Criteria for the recognition of reliable, nonaccidental rules and patterns are described later in chapter 16, Origin of Rules and Patterns .
Selby A, Volume 1A, Pattern Based Reason, 1996.
www.whyslopes.com >> Volume 2 Three Skills For Algebra >> Chapter 3 Chains of Reason Next: [Chapter 4 Longer Chains of Reason.] Previous: [Chapter 2 Implication Rules  Forwards and Backwards.] [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] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42]
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?
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:
2001  Math Forum News Letter 14,
2002  NSDL Scout Report for Mathematics, Engineering, Technology  Volume 1, Number 8
2005  The NSDL Scout Report for Mathematics Engineering and Technology  Volume 4, Number 4
Senior High School Geometry

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.
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.

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.