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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.

30 pages en Francais || Parents - Help Your Child or Teen Learn
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 1A Pattern Based Reason >> Postscript C Consistency-as-a-Tool-for-Reason Next: [Postscript D Reflections-on-Law-of-the-Excluded-Middle.] Previous: [Postscript B More-on-Story-Telling-and-Reason.]   [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]

Appendix 3, Consistency
as a Tool for Reason

While we may not know that a theory or set of assumptions is consistent, or free of contradiction, we may use the requirement for consistency as part of the reasoning process without loss of generality or harm we hope. That is a gamble. Logician may have more say.

Example One: A detective in solving a crime may have a suspect. Then he may found the suspect has an alibi which directly or indirectly implies she did not committed the crime. So the alibi and suspicion are inconsistent - that is incompatible. The detective may drop the suspicion or challenge the alibi. Lawyers for the prosecution and defense may erect competing chains of reason, and leave it to a jury or judge to decide which one, if any, appears to be true. A conclusion may follow or not.

Example Two: In developing a story or theory, we may require its elements to be consistent. We writing or developing a story, we note that a situation A is inconsistent with what has so far been written or determined, we will not add situation A to the plot or theory. We will instead accept the situation did not occur and may employ that in the further composition of the plot, story or theory. Likewise, if that the non-occurrence of a situation A is inconsistent with what has so far been written or determined, we will be forced to add the occurrence of situation A to the plot or theory. Finally, if the occurrence of A or its non-occurrence has is independent of the plot, story or theory, we extend the story or theory to include A or not as we like, if we are writing fiction or describing what may be possible. On the other hand, if we are trying to write non-fiction then the addition of statements of patterns A to plot depends on their correspondence with reality.

In telling a story or developing a theory, we may look at the consequences of our assumptions - the situations we tend to assume as holding or being true. If a chain of reason implies that a situation C occurs and does not occur, then the story or theory is inconsistent - becomes absurd. For the sake of consistency, the story or theory needs to be revised or abandoned.

That being said in developing a theory of how matters work, we hope that the theory will be logically consistent. That we hope that there will be no contradictions as a consequence of our assumptions. The consistency of a theory is hard to prove or test, and impossible in a mathematical theory large enough to include counting with whole numbers 1, 2, 3, 4, ...

In telling a story or developing a theory, there is an inconsistency if a situation A and its negation Not A both occur. While we are developing a theory from assumptions, we cannot be certain that an inconsistency A and Not A will not occur. However, the assuming the law of excluded middle in the development of a theory or is equivalent to the statement that the situation A and Not A does not happen. It equivalent to the assumption that the theory under development is consistent. That be said, when we are developing a theory from logic and underlying assumptions, we may not be able to prove that the theory is consistent. If the theory is consistent, the law of excluded middle holds and so we may used in our logical development of theory without loss of consistency. But if the theory under development is inconsistent, assuming or using the law of excluded middle in its development may lead to the discovery of the inconsistency, sooner rather than later, if at all. Assuming the law of excluded middle, simply adds another inconsistency. .

Example Three: Assume any infinite decimal expansion locates a point or distance on a real number line. Assume further that each ratio of two whole numbers can be expressed a ratio of two whole numbers with no common divisors? The Pythagorean theorem then suggests in an isosceles right triangle, the ratio of the hypotenuse to each of the others sides, the legs, by length given by the square root of 2. Is that square root equal to a rational number? The suspicion or assumption that YES, the square root of 2 equals a rational number implies an inconsistency. Namely, that in any ratio or fraction that represents the square root of 2, the denominator and numerator will both be multiples of 2. So the square root of two cannot be rational.

The Pythagoreans in finding the inconsistency in example 2 had a problem. They assume lined segments in the plane represented numbers and they assumed all such lengths were rational multiples of each other. When these assumptions or their consequences clashed, reconciliation was not obvious. Their view of numbers collapsed and a replacement was not available. That was a serious problem for the Pythagorean school in their theory of knowledge was based on and assumed rational numbers and only rational numbers.

Today, however, we have an advantage or two. One advantage is a our assumption that infinite decimals expansions represent rational and irrational numbers. Physically, if you imagine ruler with a unit length and its division into tenths, hundredths, thousandths and so on, then you can count the maximum number of units, tenths, hundredths, thousandths and that may fit in a line segment. The result is a sequence of numbers which provide a better and better approximation to the line segment length. You can further imagine that sequence of approximations given by two, three, four, five and more decimal places in a decimal expansion locate the end of a line segment. The decimal representation of numbers does not depend on nor require the "numbers" to be rational.

Aside: The number 1.0000 represent a single unit of length. It also represents the limit of the sequence 0.9, 0.99, 0.999, 0.9999 and so on. The sequence of decimal approximations 0.99999 (9 recurring) is shorthand for a sequence of lengths L(j) =1 - 10-j < 1 where the lengths are increasing, and where the differences d(j) = 1 - L(j) = 10-j is getting smaller and smaller as j increases. So the sequence approaches 1. Because the difference tends to zero, the number 1 has several decimal expansions

  • 1, 1.0, 1.00 (0 recurring finitely many times)
  • 1.000 (0 repeating indefinitely)
  • 0.99999 (9 recurring)

where the last one represents 1 as the limit of a sequence of approximations 0.9, 0.99, 0.999, 0.9999, 0.99999, ... .


Selby A, Volume 1A, Pattern Based Reason, 1996.


www.whyslopes.com >> - Volume 1A Pattern Based Reason >> Postscript C Consistency-as-a-Tool-for-Reason Next: [Postscript D Reflections-on-Law-of-the-Excluded-Middle.] Previous: [Postscript B More-on-Story-Telling-and-Reason.]   [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]

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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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