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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 >> Chapter 21 Occurrence-Tables Next: [Chapter 22 Contrapositive and Vacuously True Implications.] Previous: [Chapter 20 Shorthand-Symbols-as-Pronouns.]   [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 21, Occurrence Tables

The Special Usage of Three Words: not, and & or

Given a situation A, we can talk about the negative situation not A. Given a situation A and another situation B, we may talk about two further situations

  1. A and B (conjunction), and
  2. A or B (inclusive or).

The meanings of the terms or phrases are explained next.

NOT A and NOT (NOT A)

Given a single situation A, we can speak of another situation NOT A. The situation NOT A is said to occur when the situation A does not occur. Further, the situation NOT A is said not to occur when the situation A occurs. This is summarized in the following table.

row A NOT (A)
1 occurs occurs not
2 occurs not occurs

Language note: a situation A is said to be true when it occurs and not true (false) when it does not occur.

The following table

row A NOT A NOT (NOT A)
1 occurs occurs not occurs
2 occurs not occurs occurs not

shows that the situation NOT (NOT A) occurs when A occurs and that the situation NOT (NOT A) does not occur when A does not occur. This suggests that the situation A is equivalent to the situation NOT (NOT A).

The Special Usage of Three Words (Continued)

The word AND

The situation A and B is said to occur if both situations A and B occur. Otherwise, it is said not to occur. See the table below.

row situation A situation B A and B
1 occurs occurs occurs
2 occurs occurs not occurs not
3 occurs not occurs occurs not
4 occurs not occurs not occurs not

The situation A and B occurs provided
rows 2, 3 and 4 in the above never occur.

In each row, a possible combination of the occurrence or nonoccurrence of the situations A and B is shown in the middle two columns. In the last column, we put a note to say whether or not, the situation A and B occurs or occurs not.

* Language Note. The phrase A and B is also labelled (called) the conjunction of the situations A and B. The situation A and B is said to be true when and only when both the situations A and B occur (= are true).

The Special Usage of Three Words

The At-Least-One-Usage of the word OR

In everyday speech when you use the word or in a phrase like John or Andrew will go to the store, the usual expectation is that only one will go, not both. But there is another use of the word or favored in logic. The word or is employed in the at least one sense (as is done in logic and mathematics). With this sense or usage, the previous phrase is understood in the inclusive sense: John or Andrew, or both, will go to the store. We now proceed and we will use the word or in the at least one sense.

The situation (A or B ) is said to occur if at least one of the two situations A and B occurs. Otherwise, it is said not to occur. This is summarized in the following table.

row situation A situation B A or B
1 occurs occurs occurs
2 occurs occurs not occurs
3 occurs not occurs occurs
4 occurs not occurs not occurs not

The situation A or B can be said to occur
provided the situation in row 4 does not occur.

Remember the at least one usage differs from the exactly one usage of A or B which means either A or B occurs, but not both. In contrast, in the at least one usage, A or B means either A or B occurs, or both.

We have to be careful with the word or. Its meaning depends on the speaker and possibly the listener. That is, confusion and ambiguity results when two people in question use the same words but give them different meanings. To eliminate this ambiguity in everyday speech, write and say one of the following:

  • A or B, or both,
  • A and/or B
  • A or B, but not both.

When listening, you will have to ask what is meant. Legal texts use the phrase A and/or B to signal that at least one of the two cases A and B can occur.

2. Occurrence Table for One-Way Implications

Any rule which can be stated in the form if a first situation A occurs, then a second situation B occurs, in brief, if A then B or A implies B, is called a one-way implication.

A one-way implication which is never disobeyed is said to hold and to be (always) true. For a one-way implication rule if A then B, we recall the following:

  1. The rule is obeyed when both situations occur.
  2. The rule is not disobeyed when the first situation A does not occur but the second B occurs.
  3. The rule is not disobeyed when the first situation A does not occur and also the second situation B does not occur.
  4. The rule is disobeyed if the first situation A occurs but the second situation B does not.

The last two items 3 and 4 can be summarized by saying that disobeying a one-way implication rule is impossible when the first situation A does not occur. When not disobeyed, the rule is said to be obeyed by default. The following table, an occurrence table for the one-way implication rule if A then B, summarizes what has been said.

row situation A situation B if A then B
1 occurs occurs obeyed
2 occurs occurs not disobeyed
3 occurs not occurs not
disobeyed
4 occurs not occurs not not
disobeyed

In each row, a possible combination of the occurrence or non-occurrence of the situations A and B is shown in the middle two columns. In the last column, we put a note to say whether or not the if-then rule is obeyed, disobeyed, or not disobeyed.

Row 2 represents the situation in which A occurs but B does not. Observe that in this situation, the rule is disobeyed. In the situations represented by the other three rows, the rule is not disobeyed. A one-way implication rule if A then B is said

  1. to be always true,

  2. to always hold

when it is never disobeyed. The one-way implication if A then B is always true when the situation described in row 2 in the above table never occurs.

Remark. If situation A never occurs, the implication rule if A then B is never disobeyed amd it is said to be vacuously true.

3 Occurrence Table for Two-Way Implication Rules

A rule which can be stated, or restated, in the form
The first situation A occurs when and only when the second situation B occurs
or in the form
The first situation A occurs if and only if the second situation B occurs
is called a two-way implication rule. For each two-way implication rule note that:

  1. The rule is obeyed when both situations occur.

  2. The rule is disobeyed when the first situation A occurs without the second situation B occurring.

  3. The rule is disobeyed when the second situation B occurs without the first situation A.

  4. In brief, the two situations in a two-way implication rule must both occur or both must not occur, for the rule to be not disobeyed.

The next table summarizes the above remarks for any two-way implication rule A if and only if B.

row situation A situation B A if and only if B
1 occurs occurs obeyed
2 occurs occurs not disobeyed
3 occurs not occurs disobeyed
4 occurs not occurs not not disobeyed


As said before, a two-way implication rule is said to be always true when it is never disobeyed. This requires that the situations in rows 2 and 3 of the above table do not occur. That is, the above two-way implication rule A iff B is true (never disobeyed) provided neither of the situations A and B occurs without the other.


Converses for One-Way Implications

A Definition

The converse to the implication rule if A then B is the rule if B then A. Note that interchanging the first and second situation A and B yields the converse to a rule. From this definition or perspective, we see that the converse of the converse is the original rule. Check this.

When we know a rule if A then B is never disobeyed, we have no guarantee that the converse rule if B then A is never disobeyed. The reason for this is as follows. The rule if A then B is true if the situation A never occurs without the situation B. The converse rule if B then A is true if the situation B cannot occur without the situation A.

Reminder. Now we can easily answer the following question: What can we say for sure about the event A when (i) the rule if A then B is never disobeyed, and (ii) the event B occurs? Your answer should be not much, or nothing, without further information.



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


www.whyslopes.com >> - Volume 1A Pattern Based Reason >> Chapter 21 Occurrence-Tables Next: [Chapter 22 Contrapositive and Vacuously True Implications.] Previous: [Chapter 20 Shorthand-Symbols-as-Pronouns.]   [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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