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 >> Arithmetic and Number Theory Skills >> 6 Fractions and Ratios Fraction Operations by Raising Terms  A Simple Innovation. 1 What is a fraction. 1 What is a fraction Take II. 2 Unit Fraction Multiplication. 3 Unit fraction of a fraction. 4 Fraction Multiplication. 5 Equivalent Fractions. 6 Multiplication Algebraically Take II. 7 Numerals Fractionals Quantals. 8 Numerals Fractionals Quantals  Take II. 9 Improper Fractions and Mixed Numbers. 10 Simplification of Fractions and Mixed Numerals. 11 Simplification  an Algebraic View. 12 Fraction Comparison. 13 Fraction Comparison  Algebraic View. 14 Adding and Subtracting with Like Denominators. 15 Adding and Subtracting with Unlike Denominators. 16 Addition Subtraction Comparision Compared. 17 Efficient Ways to Add and Subtract. 18 Efficient Ways to Multiply. 19 Dividing Fractions  How TO. 20 Dividing Fractions  the Why. 21 Reciprocals for Fractions and Wholes. 21 Working With Signs. 22 Complex  Compound Fractions. A Similarities between Fractions and TwoTerm Ratios. B Fractions and TwoTerm Ratios. C Equality for Fractions and TwoTerm Ratios and Fractions. D Three Term Ratios. Folder Content: 29 pages. NotesLessons below may revise or consolidate fraction skills and concepts with operations on lengths. In the process, algebraic descriptions of operations are indicated not as requirement for fraction mastery but as enrichment option. Methods for comparison, addition, subtraction, multiplication and division are all developed or justfied below by raising terms to transform a general case into simpler cases. Algebraic view of those operations, click here. StepbyStep Skill DevelopmentWhile all methods are likely to be wellknown in the mathematical sense, the methods themselves may be new to many students and teachers. These raising term methods may overwhelm some students but help others. Instructors and tutors should emphasize fraction sense and skill mastery by rote when and where explanations of why methods work do not help. Partial or full comprehension may be left to older or gifted students, and to students who need to understand the development or origins of methods before using them.
A. Ratios And Fractions

a b 
= 
A B 
and 
b c 
= 
B C 
there are other ways to say when two triple ratios are equal or equivalent.
Note: Triple ratios or triple proportionalities occur between the sides of similar triangles. More generally, multiple ratios or proportionalities occur between the sides of similar triangles.
The discussion of ratios or multiple ratios is best understood besides a discussion of proportionality.
Inner Versus Outer Terms  small point: In the discussion of equality of ratios a : b = A: B written in that order, the inner terms are small b and big A while the outer terms are small a and big B. In contrast, if we rewrite the equality as A: B = a : b, we find the inner and outer terms are interchanged. However, the equality requires the product of the inner and outer terms be equal, that is aB = Ab. That equality is not affected by rewriting a : b = A: B as A: B = a : b, and the resulting swap of inner and outer terms
www.whyslopes.com >> Arithmetic and Number Theory Skills >> 6 Fractions and Ratios
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?
Pattern Based Reason
Online 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 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:
2001  Math Forum News Letter 14,
2002  NSDL Scout Report for Mathematics, Engineering, Technology  Volume 1, Number 8
2005  The NSDL Scout Report for Mathematics Engineering and Technology  Volume 4, Number 4
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 trigformulas for dot and
crossproducts.
LinesSlopes [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.