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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.

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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## The Math Forum and Site Content

From 1995 to the present, the Math Forum Editors in Newsletters and one Hotspot of the month have been tracking my site's content and growth. Newsletters articles - site descriptions - are quoted below, latest first.

The descriptions in the Mathforum Newsletters and in other site reviews is not part of the formal peer review process met by articles appearing in mathematics and mathematics education publications. For university hiring and promotion committees, the informal reviews were not authoritative. Yet in the popularization of mathematics, or site content, they may still count.

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20 April, 2012 Vol. 17, No. 16

THE MATH FORUM @ DREXEL INTERNET NEWS

Grants, Scholarships, and Awards
Summer Workshops for Math Teachers | Whyslopes.Com Updated

WHYSLOPES.COM UPDATED
```

www.whyslopes.com/index.php /0400Secondary_Mathematics_-_A_Practical_Approach/

``` Alan Selby has added a new section to his site, entitled
"Secondary Mathematics for Ages 11+, a Practical Approach."

"Secondary Mathematics" provides a coherent guide to
technical innovations that simplify the learning and
teaching of arithmetic and number theory practices,
geometry. The new section ends with two calculus previews
with a description of  a light calculus course, more
descriptive than deductive, aimed  at introducing the key
ideas in differential and integral  calculus. The appendix
"Calculus with Proofs for the Keen or  Gifted" describes
how to reform calculus.

To better reflect his site's extent and intent, Selby has
also re-named his site -- originally called "Appetizers and
Lessons for Mathematics and Reason," and now "Logic and
Mathematics Skill and Concept Development."

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23 December, 2011 Vol. 16, No. 51

THE MATH FORUM @ DREXEL INTERNET NEWS

Effective Teaching with the Math Forum's Problems of the
Week eInstruction Contest Winners | WhySlopes.com Updated

WHYSLOPES.COM UPDATED
```
www.whyslopes.com/index.php/ 0399Mathematics_Skill_Development_Framework/
``` Alan Selby recently posted a five phase framework for
mathematics skill development at his site.

The essay "Which Way to Go" introduces it in plain language:

```
www.whyslopes.com/index.php /0399Mathematics_Skill_Development_Framework /0050Which_Way_To_Go.html
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18 February, 2011 Vol. 16, No. 7

THE MATH FORUM @ DREXEL INTERNET NEWS

NCTM Annual Meeting: Early Registration Discount
Whyslopes.com Revamped | Wolfram Fun Facts

WHYSLOPES.COM REVAMPED

www.whyslopes.com

Alan Selby has overhauled his website of "appetizers and
lessons for mathematics and reason." Look for the math
formulas neatly typeset with LaTeX, Flash videos, and other
rich media supported by his new content management system.

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September 24, 2010 Vol. 15, No. 39

THE MATH FORUM @ DREXEL INTERNET NEWS

Macalester Problem of the Week | Whyslopes.com Updated

WHYSLOPES.COM UPDATED

Since we last featured the complex numbers section of
Alan Selby's site "Appetizers and Lessons for Mathematics
and Reason" in March this year, he has added a new section,
Mathematics Education Revisited:

www.whyslopes.com/mathematics_education_essays/

Selby has also included a section called POMME: Progressive
Observable Motivated Mathematics Education, which he
describes   as "a two-level program for practical

www.whyslopes.com/#POMME

Postscript: Now in Archives Directory

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March 26, 2010 Vol. 15, No. 13

THE MATH FORUM @ DREXEL INTERNET NEWS

Math Awareness Month
Capturing Teacher Knowledge Study | Complex Numbers

COMPLEX NUMBERS

www.whyslopes.com/complex.html

Since we ... featured the fractions section of Alan Selby's
site "Appetizers and Lessons for Mathematics and Reason" in
January this year, he has revised and expanded the section
on complex numbers, organizing it under these topics:

- Addition and Multiplication [of vectors]
- What Are Complex Numbers?
- Second Way to Compute Products
- Consequences of Two Ways to Compute Products

These pages provide a way to introduce complex numbers
before the study of periodic (unit-circle based)
trigonometric functions.

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January 8, 2010 Vol. 15, No. 2

THE MATH FORUM @ DREXEL INTERNET NEWS

Math Forum Online Workshop: Algebraic Reasoning
Math Forum Online PD Courses
2010 Census | Fractions, Ratios, Proportions
May 30 2005 Vol. 10, No. 22

FRACTIONS, RATIOS, PROPORTIONS

www.whyslopes.com/etc/fractions/

Since we last featured Alan Selby's site "Appetizers and
Lessons for Mathematics and Reason" in 2005, he has revised
and greatly expanded the section on fractions and ratios,
organizing it under these topics:

- Fractions
- Fractions with Units (arithmetic and algebra with units)
- Ratios and Fractions (or ratios versus fractions)
- Proportionality Relations Forwards and Backwards

For all of Selby's pages on high school mathematics and
calculus topics, see:

www.whyslopes.com/

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May 30 2005                            Vol. 10, No. 22

THE MATH FORUM @ DREXEL INTERNET NEWS

MathMem | Fractions and Ratios | Math-A-Thon

FRACTIONS AND RATIOS

www.whyslopes.com/etc/fractions/

Alan Selby has added a new section called Fractions and
Ratios to his site "Appetizers and Lessons for Mathematics
and  Reason." This new area "offers a thought-based
development of  fractions, what they are, how to compare,

Selby organized his pages under these topics:

1. What is a Fraction
2. Multiplication I: Unit Fraction of Unit Fraction
3. Multiplication II: Unit Fraction of a Simple Fraction
4. Multiplication III: Simple Fraction of a Simple
Fraction
5. Equivalent Fractions
6. Mixed Numbers
7. Comparison of Fractions
8. Addition and Subtraction I: Case of Like Denominators
9. Addition and Subtraction II: Case of Unlike Denominators
10. Addition and Subtraction III: Efficiency Matters - Use
of LCDs
11. Multiplication IV: Efficient Ways to Multiply Fractions
12. Division of Fractions
13. Two Term Ratios
14. Implied Ratios
15. Multiple Ratios

The earlier site area on solving linear equations visually
reinforces Selby's concept of fractions and arithmetic
operations on them, while geometrically introducing algebra.

www.whyslopes.com/etc/solvingLinearEquations/

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21 February 2005                            Vol. 10, No. 8

THE MATH FORUM @ DREXEL INTERNET NEWS

What the US Can Learn from Singapore's Mathematics System
Geometry Module | Solving Linear Equations

SOLVING LINEAR EQUATIONS

www.whyslopes.com/etc/solvingLinearEquations/

Alan Selby has added a new section called Solving Linear
Equations to his site "Appetizers and Lessons for
Mathematics and Reason." The section introduces stick
diagrams as a way to "provide a concrete context for many
of the rules or patterns for solving equations -- a context
that may develop equation solving skills and confidence
informally before the  algebraic statement of the rule and
patterns for solving equations."

Selby organized his pages under these topics:

- Letters and Lengths
- Stick Diagrams
- Beyond Sticks
- Almost One
- Binary Systems
- Larger Systems

The earlier online chapter 15, Linear Equations, offers
further information and examples for the development of
algebraic reasoning skills in the solution of linear
equations.

www.whyslopes.com/etc/ThreeSkillsForAlgebra/ch15.html

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13 August 2001                           Vol. 6, No. 33

THE MATH FORUM INTERNET NEWS

Complex Numbers and Trig - Alan Selby | NASA CONNECT
Dare to Fly with Class

COMPLEX NUMBERS and TRIG -  Alan Selby

www.whyslopes.com/complex.html

In an expansion of "Appetizers and Lessons for Mathematics
and Reason," Alan Selby has added new sections on Complex
Numbers and the Distributive Law for Complex Numbers. They
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

www.whyslopes.com/freeAccess/distributiveLaw.html

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19 March 2001                             Vol. 6, No. 12

THE MATH FORUM INTERNET NEWS

String Figures, Mathematics, Origin - Probert
Appetizers and Lessons for Math and Reason - Selby
2001 Regional Conference - Mid-Atlantic Eisenhower
Consortium @ RBS

APPETIZERS AND LESSONS FOR MATH AND REASON - Alan Selby

www.whyslopes.com/

Alan Selby's mathematics appetizers range over arithmetic
review problems, notions of what variables are, skills
leading to algebra, painless theorem proving, complex numbers
with some trig, the importance of slope (some calculus), a
decimal perspective of error control and continuity (more
calculus), and renaming the "greater than" sign (back to
algebra).  Intended for the learning and teaching of
mathematics at the high school and college level, the site
also offers some reflections that support teachers' use of
the materials for their own learning.

Sections include: ...

- Pattern Based Reason (Vol 1A)
www.whyslopes.com/volume1a/index.html
A book in eighteen chapters that describes the origins,
benefits and limits of rule- and pattern-based thought
and processes in daily life, society, science and
mathematics.

- Applet
www.whyslopes.com/freeAccess/applet.html
Shows how to multiply and add complex numbers (vectors
in the plane).

...

www.whyslopes.com/ParentCenter/index.html
Suggestions on how to help children aged 7-16 learn in
school.

A version in French is also available:

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25 November 1996                     Vol. 1, No. 8

Alan Selby's Appetizers | Dynamic Algebra | NCTM-L Archive

THE MATH FORUM INTERNET NEWS

www.mathforum.org/

FEATURED INTERNET RESOURCE: ALAN SELBY'S 'MATH APPETIZERS'

www.cam.org/~aselby/lesson.html#math

ALAN SELBY's mathematics appetizers range over arithmetic
review problems, notions of what variables are, skills
leading to algebra, painless theorem proving, complex
numbers with some trig, the importance of slope (some
calculus), a decimal perspective of error control and
continuity (more calculus), and renaming the greater
than sign (back to algebra).

Advice on how to read, how to learn, why go to school,
etc. is included. The tone is sometimes funny, and the
writing is dense, rich, and intriguing. There are
reflections on teaching, so these materials can be used
in the classroom and as a place for teachers to learn.

Alan's explanations of mathematical concepts using words
and stories are particularly strong. Don't miss his
Appetizers and Lessons for Logic, also on this site.

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Math Forum June Hot Spot of the Month in
www.mathforum.org/library/hotspots/95.html

Alan Selby's Lessons and Appetizers for Math and Logic

Three Skills For Algebra (a first image of mathematics
after arithmetic -  why letters or symbols are favored in
algebra in place of numbers); Two logic puzzles to show the
difference between a one-way and a two-way rule; Painless
Theorem Proving; Longer Chains of Reason: What is Mathematical
Induction?;   Complex numbers Etc. - A geometrical story
based on the addition and multiplication of arrows in the
plane; Chains of Reason - math-free examples of rule-based
reasoning; How Logic or Rule-Based Reason Appears in Math;
VNR Concise Encyclopedia of Mathematics. Also Appetizers
on Other WWW Pages.

```

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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.
... 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.