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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.
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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 6 Change of Language Next: [ Chapter 7 Prep for Calculus Arithmetic Exercises.] Previous: [Chapter 5 Islands-and-Divisions-of-Knowledge.]   [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 6, A (technical) Language Change

Implication rules can be stated in several ways. We need to recognize them.

One-Way Implication Rules

In the chapter Implication Rules , we met the rule

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

Rules like this can be said in different ways. This gives variety and choice in the way in which we write rules. The form of a rule does not matter, if we understand exactly what it says. The above one-way rule can also be rewritten (or restated, again without changing its meaning) using the words IF and THEN as follows.

IF Aunt Jane visits her nephew Tom's home THEN Tom goes out to play.

The word IF introduces a condition, namely Aunt Jane's visit to her nephew Tom's home. The word THEN introduces the consequence, what should occur, when the condition is satisfied. Here the consequence is Tom goes out to play. Since the original rule can be rewritten in the IF condition THEN consequence form, we say the original rule and the if-then form are conditional statements.

Note that a statement If A then B is only false when the situation or condition A occurs, but the anticipated consequence B does not.

Another way of writing the above one-way Aunt Jane and nephew Tom rule (with no change in meaning) is given by:

Aunt Jane's visit to her nephew Tom's home IMPLIES Tom goes out to play.

The words forces or makes may be used instead of the word implies. We could also use the word suggests, but in everyday use, a suggestion is optionally obeyed or followed while a rule (when it is correct) should or must be obeyed or followed. In talking about rules, we use the words implies, forces or makes for those rules we expect will be obeyed, or more precisely will never be disobeyed in the circumstances at hand. The explicit identification of such circumstances is exhaustive unless the circumstances in question are understood from a context, an obvious one, we hope.

Postscript : Instead of writing If A then B we may write B if A. The latter states that the situation B will happen if the situation A happens. That being said we cannot say that

B if and only if A

holds when there is a third situation C different from A, a situation which may occur when A does not, such that B if C also holds.

In the case

B if A
and also
B if C

the situation B may occur because of situation A or situation C, that is, due to A OR C. So when situation B occurs, the occurrence may be implied by A, C or another situation.

However, we can assert or state B if and only if A holds when B follows from the occurrence of A and whenever B occurs, so must A.

Two-Way Implication Rules

In the previous chapter Implication Rules, we met the rule

Tom goes out to play
when and only when
Aunt Jane visits his home.

This is an example of a two-way rule. Two-way rules can also be said or presented in different ways. Again the form of a rule does not matter, provided we recognize exactly what is meant. The above rule also can be rewritten (or restated, again without changing its meaning) in the if-and-only-if form:

Aunt Jane visits her nephew Tom's home
if and only if
Tom goes out to play.

This form suggests we call such rules biconditional statements. The prefix bi- here signals two ways. Whenever the condition (or situation) Aunt Jane visits her nephew Tom's home occurs, the other condition (or situation) Tom goes out to play must also occur, and vice-versa, if this rule is to be never-disobeyed.

You may prefer to say if and only if instead of when and only when. For instance, I might say or suggest to you: I will do that for you if and only if you do this for me. Alternatively, I might say or suggest to you: I will do that for you when and only when you do this for me. Tone provides the only difference between the two suggestions. Both of these suggestions represent a two-way obligation to which we might agree. Confusion or disappointment or false expectations may happen when suggestions such as these are not explicitly accepted or rejected.

Two-Way or Two One-Way Rules

The two-way Aunt Jane and nephew Tom rule above is rewritten (with no change in meaning) as

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

and also that

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

In this form, the two-way rule is seen to be the same as two one-way implication rules, each going in the opposite direction.

Equivalent Conditions (or Situations)

Two situations or conditions A and B, each of which must happen whenever the other does, are said to be equivalent to each other. So when a first situation is equivalent to a second, each situation implies and is implied by the other.

Conditional versus Biconditional

One-way and two-way implications are called conditional and biconditional statements (or rules), respectively.

The Abbreviation Iff

The terms and phrases

  • if and only if
  • when and only when
  • iff (shorthand for if and only if)

can all be used instead of each other. They are interchangeable. No matter what term or phrase is used to indicate a two-way implication, the difference between one-way and two-way needs to be remembered. Otherwise, statements, definitions and assertions will be read incorrectly.


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


www.whyslopes.com >> Volume 2 Three Skills For Algebra >> Chapter 6 Change of Language Next: [ Chapter 7 Prep for Calculus Arithmetic Exercises.] Previous: [Chapter 5 Islands-and-Divisions-of-Knowledge.]   [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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