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Mathematics and Logic - Skill and Concept Development

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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.
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- 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 1 Introduction to Chapters 2 to 6 Next: [Chapter 2 Implication Rules - Forwards and Backwards.] Previous: [Foreword.]   [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]


Chapter 1

Elements of Reason

Previous: Foreword

To reason with someone often means to persuade them of the need for an idea or action. Persuasion and reason can take many forms. Methods for arriving at conclusions and judgments in all skills disciplines are, or should be where possible, based on the use and recognition of reliable rules and patterns, and also data to use with them.

Each of us needs to understand fully or as much as is possible, whatever we might be doing or learning. In reasoning, some rules and patterns are reliable. Others are guidelines. Each of us needs to know which is which.

Mathematics, in and besides arithmetic, depends on rules and patterns, which are used one at time or one after another, to obtain conclusions. The aim of the next chapters is to introduce and provide an image of rule and pattern thought in mathematics and other disciplines.

When ideas in mathematics or another discipline are described instead of being drawn from implication rules, the role of rule-based reason or logic may be forgotten. But in every discipline including mathematics, signs of rule- and pattern-based reason are given by the word and phrases such as from this, then, if, therefore, thus, because, since, as, gives, yields etc.

About the Next Chapters

The next chapters describes some basic elements of rule- and pattern-based thought. In particular, four chapters, Implication Rules, Chains of Reason, Longer Chains of Reason and Islands and Divisions of Knowledge describe basic ideas, keys for reason and logic.

  1. The chapter Implication Rules presents two logic puzzles. Each consisting of a rule and five questions. Answers are also provided. The puzzles show the difference between one- and two-way implication rules.
  2. The chapter Chains of Reason describes how to directly use rules one at-a-time or chained together, one-after another, for arriving at conclusions and judgments.
  3. The chapter Longer Chains of Reason starts to indicate the special role of reason in mathematics. It describes, in a very non-mathematical fashion, the concept of induction, a method used in mathematics to arrive at conclusions. This concept of induction is an example of a method of reason employed mainly or only in mathematical subjects.
  4. The chapter Islands and Divisions of Knowledge describes how rule and pattern-based bodies of thought may be organized. Here different starting points, first principles or assumptions, may lead to the same body of rule-based knowledge.
    In philosophy, the discipline that is literally the love of knowledge, perhaps an infatuation, Euclid's logical or rule based arrangement of geometry provided a model for reason. This chapter with words and images apart from geometry describes the model and the variations possibly within it.
These chapters develop thinking and reasoning skills needed in daily life. They provide a standard or model for arriving at conclusions and making decisions: how to argue politely if you must. They also strengthen basic skills needed in mathematics, science, technology, writing, persuasion and communication. Reason and persuasion touch all skills and all disciplines.

The chapter A Change of Language introduces the conventional if-then and iff forms for writing one- and two-way implication rules. The one- and two-way implication rules in this work have been identified with condition and bi-conditional statements. But the terminology one and two-way employed here draws on the present-day common experience of one and two-way roads.


Next: Chapter 2, Implication Rules

To Learn More About Logic, see chapters 26 to 31 in this Volume

Chapters 19 to 24 in Volume 1A,. Pattern Based Reason . Volume 1A describes the benefits, origins of rule and pattern based thought, deeds and hopes in greater detail, and still leaves room for thought. Online postscripts in the Volume 1A site area discuss further the methods and context for indirect reason in and outside of mathematics. Finally, Appendices Story Telling includes a theory of knowledge based on our ability to collect, invent and tell stories with words and symbols, written, spoken or drawn.

Multi-Level Logic Skill Mastery - What it Means

Postscript: August 28th, 2011.

Before the study of implication rules of the form A IF B and the form A IF and ONLY IF B, older children and young adolescents may be shown how written work can be done and recorded in steps that can be seen as done or later by the doer, a peer or teacher. Steps observed as done or as recorded allows work to checked and in that, the domino effect of mistakes to be seen. In advanced mathematics, the concept of checking goes further: In derivations and proofes, steps are not only done and recorded for later inspection, reasons for them, where not obvious, are recorded as well, again for later checking. However learning to do and record steps, minus reasons for them, provides the first form of proof - empirical and preductive - in which steps are checked. Empirical reason in science and technology goes further by calling for results to obtained by procedures that are recorded for the sake of confirming that the results are repeatable and reproducible. So logic occurs in many different ways.

To repeat, two kinds of logic children and adolsecent may accept or understand consists of the following:

  • Learning to do work that be done and recorded in steps or pieces that can be seen and thus checked as done or later.
  • Learning to follow methods or processes where the steps are not necessarily recorded, but the results are repeatable and reproducible, and hence verifiable.
Both kinds of logic are empirical. The former is a special case of the latter. In it, advanced mathematics and further disciplines which employ deductive reason in theirs proofs, records both steps and reasons for them for later checking - that adds some rigour of the deductive kind.

Beyond and besides the foregoing mechanics of empirical and deductive methods for arriving at conclusions, the underlying rules and patterns may form smaller and large islands and bodies of rule and pattern based practices and theory. Different entry points into the bodies and islands will make entry and exploration easier or difficult. Some trial and error, or experience, may be needed to find the easiest routes to follow. For example, site logic and mathematics material includes some easier entry point or easiet routes to follow, but simpler or clearer routes may be yet be found. The latter is not surprising. Only time and comparision will tell. The composition of site material was an iterative affair. Another iteration will likely improve it. Bon Appetit.


www.whyslopes.com >> Volume 2 Three Skills For Algebra >> Chapter 1 Introduction to Chapters 2 to 6 Next: [Chapter 2 Implication Rules - Forwards and Backwards.] Previous: [Foreword.]   [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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