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Mathematics and Logic - Skill and Concept Development

with lessons and lesson ideas at many levels. If one site element is not to your liking, try another. Each one is different.

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Online Volumes: 1 Elements of Reason || 2 Three Skills For Algebra || 3 Why Slopes Light Calculus Preview or Intro plus Hard Calculus Proofs, decimal-based.
More Lessons &Lesson Ideas: Arithmetic & No. Theory || Time & Date Matters || Algebra Starter Lessons || Geometry - maps, plans, diagrams, complex numbers, trig., & vectors || More Algebra || More Calculus || DC Electric Circuits || 1995-2011 Site Title: Appetizers and Lessons for Mathematics and Reason

Mathematics Concept & Skill Development Lecture Series: Webvideo consolidation of site lessons and lesson ideas in preparation. Price to be determined.

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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 2-way implication rules - A different starting point - Writing or introducting the 1-way implication rule IF B THEN A as A IF B may emphasize the difference between it or the latter, and the 2-way 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?

Early High School Arithmetic

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.

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?
- 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 human-made 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 >> 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 Reason


This 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:

  1. how to directly use a single implication rule to get conclusions, and
  2. how to link, chain or thread implication rules together to obtain or derive more rules and more conclusions.

The examples are not important (and are perhaps ridiculous) but they illustrate some rule-based methods in reason. Examples which involved real-life situations might distract from mastering these methods. That is, in real-life 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
Direct and Indirect Usage

Pretend the following implication rule is never disobeyed.

Each time Suzy the cat is on the ground and Suzy sees a dog, Suzy climbs a tree and stays in it for at least five minutes.

Direct Usage

What can we say for sure when Suzy the cat sees a dog? One possible answer is that Suzy the cat stays in a tree for at least five minutes. Another possible answer is that Suzy the cat climbs a tree. A more complete answer is that Suzy the cat climbs a tree and stays there for at least five minutes. Each of these answers or conclusions is correct. The last conclusion or result is fuller and more complete than the others. It gives more information. Which answer or conclusion is wanted here depends on who is interested in what. When many conclusions are possible, we state only those conclusions of interest to us. We do not have to state the most complete conclusion. The choice is ours.

Indirect Usage

What can you say for sure if Suzy the cat has not climbed nor stayed in any tree for at least five minutes? To check your answer, you might have to remember or revisit the questions in the chapter Implication Rules. But you should do this after you have read the following words.

Linking and Chaining Two Rules Together

The examples below and in the next page show how to chain, link or connect implication rules to get information and conclusions. The examples in themselves are not important. The information in them is silly. But these examples just show how to put implication rules together. So read on, with patience.

  • Every time Suzy the cat climbs a tree, it gets stuck in the tree
  • Every time Fred the dog visits the park in which Suzy the cat lives, Suzy climbs a tree.

By linking or chaining these implication rules, we can make three conclusions:

  1. Whenever Fred the dog visits the park where Suzy the cat lives, Suzy climbs a tree.
  2. Whenever Fred the dog visits the park where Suzy the cat lives, Suzy gets stuck in a tree
  3. Whenever Fred the dog visits the park where Suzy the cat lives, Suzy climbs a tree and gets stuck.

Each of these conclusions is correct. Each conclusion gives a new implication rule which we could use in our reasoning process. The third implication rule is the most informative. It contains the most information. When we view each correct conclusion as a possible destination for our reasoning process, we may sometimes select our destination.

Putting Several Rules Togethe

We can chain or link not only two but also several implication rules together. This sometimes yields useful, new information. As an exercise, we ask the question: What happens whenever Fred the dog visits the one-tree park? Several answers are possible. Some have more details than others. All are correct. To answer the question, assume or pretend the next five implication rules are never disobeyed. Further, assume that Suzy the cat lives in the one-tree park.

  1. When Suzy the cat climbs the tree in the one-tree park, Suzy gets stuck in the tree.
  2. Each time Fred the dog visits the one-tree park, Suzy the cat climbs the tree.
  3. Every time Charles the human visits the park, Charles sits on a bench for one hour.
  4. Whenever a cat climbs the tree in the one-tree park, the five birds living in the tree fly around in the park.
  5. Each time birds fly around in the park, sensible worms go underground.

All the information has been stated. We start our reasoning process. That is, we will answer the question: What happens whenever Fred the dog visits the one-tree park?

To answer the question, suppose or assume Fred the dog visits the park. Then from the implication rule (2), we see that Suzy the cat climbs a tree. Next, from the implication rule (1) we see that Suzy the cat gets stuck and from the implication (4) we see that birds fly around the park. Finally from the implication (5), we note sensible worms go underground.

We could list all that occurs when Fred the dog visits the park. Or, we could state only those results of Fred's visit to the park which are of most interest to us. The choice is ours. For instance, one of our possible conclusions follows:

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 rule-based 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 one-tree 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) never-disobeyed 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, experience-found, 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. Self-deception needs to be avoided here.

Inductive reason inside mathematics refers to another process, namely, the extraction or drawing of conclusions from ladder-like 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 experience-based origins of mathematical reason.

Criteria for the recognition of reliable, non-accidental 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]

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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 head-to-tail 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 pattern-based 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 story-telling 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; pattern-based 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 cross-products, 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 how-tos 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.
- 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 trig-formulas for dot- and cross-products.
Lines-Slopes [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.

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