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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

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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 >> Chapter 5 Deception Next: [Chapter 6 Chains of Reason.] Previous: [Chapter 4 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]

Chapter 5, Deception (and Hype)

Suggestive or Misleading Questions

Recall that one question for the one-way rule

When Aunt Jane visits her nephew Tom's home, Tom goes out to play.

asked what could be said for certain about Aunt Jane when Tom goes out to play? The answer is nothing. But the wording in this question hinted or suggested that a little bit more could be said for certain about Aunt Jane. The question was slightly misleading. A less misleading question would be what, if anything, can be said for certain. You have to be aware of misleading questions. The topic of suggestive and misleading questions is discussed next.&

Are you trusting? Are you willing to politely accept everything I or someone else says or suggests without question? The phrase what can you say for sure in the above question makes you expect something could be said for sure, not nothing. You have to watch for misleading and suggestive questions in and outside of this book.

When someone tries to convince you with a suggestive chain of reasoning, you need to recognize the weak and strong links in that chain. Then you can decide for yourself whether or not to accept the suggestions or conclusions obtained. Faulty logic may hide some deliberate deception or some reparable chains of reason. In particular, you may see where the chain fails and is broken, or where the chain can be strengthened or repaired. In our thoughts, we need to identify or keep track of what is certain, what is almost sure, what is guessed, what is probable, and what is only suggested.

The next example is far-fetched in most worldly locations, but it illustrates a situation that you need to recognize. Suppose I asked how long have you been beating your elephant? This question suggests you own a mistreated elephant. A gullible, too trusting, person overhearing this question could believe (assume) you own an elephant. A gullible person overhearing the question could believe this unless you say the question is absurd because you don't own an elephant.

We all are slightly gullible. It is a matter of politeness not to challenge a speaker. On hearing a question, we like (or tend) to think each question posed is correct, honest and not misleading. But we need to continually watch for questions that are not realistic, especially if the speaker does allow us to challenge them. Their words may force upon us unchallenged assumptions or suggestions. Suggestive questions need to be recognized – if not stopped. They need to be challenged and corrected to prevent the reasoning from continuing in an absurd or deceptive direction.

A series of suggestive questions is intimidating and forceful. When the suggestions in them remain unchallenged, you may find yourself at the end of a long chain of suggestive reasoning, agreeing to or not challenging some repugnant ideas. So watch for misleading questions. The questions and possibly the speaker are false. Step by step, or question by question, such false reasoning needs to be exposed. The exposure could start with the very first question, and then the next, and the next, and so on.

When a speaker, in posing and answering suggestive questions, leads you to false or repugnant conclusions, such a speaker has lied and mislead you. Your intelligence has been deliberately or accidentally insulted. The speaker, a possible villain, has taken advantage of your politeness or silence. Faulty reason or lies may be hidden in suggestive questions.

Hype, Hype, Hype, Hooray

People try to persuade us in many ways. We need to recognize the fair and unfair ways, or the sensible and nonsensical ways. In persuading ourselves and others, we need to recognize and appreciate or reward careful logic. Efforts to persuade and lead us are met in advertising, public relations, political campaigns, religion, law, business, mathematics courses (yes), and even your family. Advertisements and sales pitches may give an excessively favorable impression of a product. That is, few people, parties or companies will point to the bad or weak parts in their service or product. Because of a favorable impression or promise, we may choose one service or product much to our later regret.

Words can be used not only to teach and inform but also to direct or misdirect others. Here different messages can be given to different people. For example in talking about an adult-only subject, a child may be given or understand one message, while the elders understand another (or both). This may give a simple, half-innocent, example of a creative, deceptive ambiguity. In time the child grows up. Delivery of two different messages at once becomes more difficult as the child learns. Double meanings are then seen by the child, and no longer useful. The child is less gullible. More blatantly, appearances and words can mislead us.

Ambiguity and inconsistency are tools of some politicians and some sales agents for whom only the result (selling a product, service or conclusion) counts. For example, a leader or salesperson may suggest different and contrary ideas to different people. Watch for this inconsistency. Does it reflect a maturing attitude or a deceptive tongue?

In honest debate between people, question and issues are addressed one by one as they appear and the course of debate is not changed to avoid answering awkward questions. Unfortunately, for the sake of persuasion, political speakers may respond to only part of a question and shift the topic of conversation, so that the original topic is neglected. It is a shallow and insulting kind of response that goes for the most part unchallenged in public debate.

Numbers, and not just words, can be used to mislead people. Numerical descriptions of situations need to be understood. Averages for instance may be computed using different ways. It is also deceptive to let people think that one calculation is used instead of another. It is also deceptive, more precisely meaningless, to use statistics without saying how they were computed. In mathematics, a statistic is just a number computed from collected data. Further examples of and warnings about numerical or statistical methods of deception can be found in the following two books.

  1. How to Lie with Statistics by D. Durf, 1954, Norton and Company, ISBN -0-393-31072-8
  2. Use and Abuse of Statistics by W. J. Reichmann, 1961, Pelican Books, ISBN 0-14- 02-0707-4

Ethics For Persuasion

When you want others to agree with an action or idea, how should you speak? The only way to convince others is to give them reasons acceptable to them. But in doing this, our reasons for the action or idea could be different from the ones acceptable to them. When this is the case, we should say so. In this, some diplomacy may be required. The honesty advocated here is awkward when you speak to people who do not allow reasons different from theirs for a common goal.


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


www.whyslopes.com >> - Volume 1A Pattern Based Reason >> Chapter 5 Deception Next: [Chapter 6 Chains of Reason.] Previous: [Chapter 4 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]

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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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