Mathematics Concept & Skill Development Lecture Series: Webvideo consolidation of site lessons and lesson ideas in preparation. Price to be determined. Bright Students: Top universities want you. While many have high fees: many will lower them, many will provide funds, many have more scholarships than students. Postage is cheap. Apply and ask how much help is available. Caution: some programs are rewarding. Others lead nowhere. After acceptance, it may be easy or not to switch. 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. Early High School Arithmetic
Deciml Place Value  funny ways to read multidigit decimals forwards and
backwards in groups of 3 or 6. 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? Early High School GeometryMaps + Plans Use  Measurement use maps, plans and diagrams drawn to scale.  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 humanmade 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 
www.whyslopes.com >> The Math Forum and Site Content Next: [Road Safety Questions.] Previous: [Site Reviews.] [1] [2] [3] [4] [5][6] [7] The Math Forum and Site ContentFrom 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. \/\/\/\/\/\/\/\/ 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, elementary and advanced algebra, elementary and advanced 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 renamed his site  originally called "Appetizers and Lessons for Mathematics and Reason," and now "Logic and Mathematics Skill and Concept Development." \/\/\/\/\/\/\/\/ 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 UPDATEDwww.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 \/\/\/\/\/\/\/\/ 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. \/\/\/\/\/\/ September 24, 2010 Vol. 15, No. 39 THE MATH FORUM @ DREXEL INTERNET NEWS Macalester Problem of the Week  Whyslopes.com Updated Mathematical Creativity and Giftedness 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 twolevel program for practical mathematics instruction." Read more about it here: www.whyslopes.com/#POMME Postscript: Now in Archives Directory \/\/\/\/\/\/ 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?  Addition and Multiplication Properties  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 (unitcircle based) trigonometric functions. \/\/\/\/\/\/\/ 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  Links For all of Selby's pages on high school mathematics and calculus topics, see: www.whyslopes.com/ \/\/\/\/\/\/\/ May 30 2005 Vol. 10, No. 22 THE MATH FORUM @ DREXEL INTERNET NEWS MathMem  Fractions and Ratios  MathAThon 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 thoughtbased development of fractions, what they are, how to compare, add, multiply and divide them." 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/ \/\/\/\/\/\/\/ 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 \/\/\/\/\/\/\/\/ 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 crossproducts  the cosine law  a converse to the Pythagorean Theorem www.whyslopes.com/freeAccess/distributiveLaw.html \/\/\/\/\/\/\/\ 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  MidAtlantic 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 patternbased 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). ...  Helping Your Child Learn www.whyslopes.com/ParentCenter/index.html Suggestions on how to help children aged 716 learn in school. A version in French is also available: \/\/\/\/\/\/\/ 25 November 1996 Vol. 1, No. 8 Alan Selby's Appetizers  Dynamic Algebra  NCTML 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. \/\/\/\/\/\/\/ 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 oneway and a twoway 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  mathfree examples of rulebased reasoning; How Logic or RuleBased Reason Appears in Math; VNR Concise Encyclopedia of Mathematics. Also Appetizers on Other WWW Pages. www.whyslopes.com >> The Math Forum and Site Content Next: [Road Safety Questions.] Previous: [Site Reviews.] [1] [2] [3] [4] [5][6] [7] 
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 headtotail
addition of arrows in the plane. Click and drag complex numbers A and B to change their locations.
Pattern Based ReasonOnline Volume 1A, Pattern Based Reason, describes origins, benefits and limits of rule and patternbased 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 storytelling view of learning: [ A ] [ B ] [ C ] [ D ] to suggest how we share theory and practice in many fields of knowledge. Site Reviews1996  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; patternbased 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
crossproducts, 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 howtos 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. 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. 