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 >> Mathematics Skill Development Framework >>  More Algebra and Slopebased Calculus Preview Next: [  Implementation Notes.] Previous: [  Euclidean and Analytic Geometry with Complex Numbers and Trigonometry.] [1] [2] [3] [4] [5] [6] [7] [8] [9][10] [11] [12] [13] [14] [15] [16] [17] More Algebra IslandsMathematical Recursion and InductionThe discussion of mathematical induction and recursion represents a standalone island of knowhow combining logic and algebra. Mathematical induction and inductive definitions may be covered to explain or derive formulas for geometric sums and binomial coefficients. Mathematical induction may be woven into the further development of skills and concepts to add to the logical structure of mathematics as seen in this phase and the following ones. Mathematical induction and recursion, starting with site logic chapters, is not a difficult topic to develop. Its coverage sets the stage for factorial notation, summation notation, product notation, binomial coefficients and summation formulas for sums: arithmetic, geometric and other. In any light development of calculus, that is calculus limited to polynomials and their ratios, mathematical induction would also play a role in obtaining rules for differentiation of polynomials directly and in composite functions where the last applied or outer function is polynomial. Algebra Island: Natural Logarithm and Powers, etc.The forward and backward use of the compound interest formula and encounters with radical and powers therein provides a context or hook for the algebraic description of the properties of the natural logarithm, its inverse the exponential functions, and the expression of radicals and powers in terms of the latter inverse and the exponential powers. Deriving and providing those expression shows how and when radicals and powers can be computed. Site algebra steps shows how the natural logarithm and its inverse or antilog, the natural exponential functions may be introduced numerically with an algebraic description of their fundamental properties. Following that, their properties may be used to show how to compute radical, powers and more exponentials using the natural logarithm and its inverse, the exponential function. Showing how to compute the latter with the aid of logarithms and its inverse indicate when they can computed as well. In essence, the site treatment of the foregoing provides unifies the otherwise disconnected treatments that I have seen in secondary mathematics. Algebra Islands: Sets Concepts and Notation  A Balanced ApproachIn mathematics for practical use, set concepts and operations of membership, union, intersection, and complements, an may help in the description of sets of numbers, in counting objects by dividing them among disjoint or overlapping set  think of Venn Diagrams; in discussing and illustraing logic  think again of Venn Diagrams; in the algebraic codification of probability theory in this phases, and in the algebraic description and codification of functions and their properties in the next phase. Before mastery of the algebraic statement of conditions to define subsets  see the safe set theory below, sets may be defined by a roster method  listing their members; and sets may introduced or depicted Geometrical as a set of points in the plane in as a set of points in a Venn Diagram. Note: When sets are introduced by the roster method, we may allow elements to be repeated in the list, and then introduce a roster shrinkage method to make the list shorter. As a first example, \[\{\frac{25}{5}, 15, 20, \frac{50}{10}\} = \{5, 15, 20 \}\] Then in talking about rational numbers as being the set of unsigned fraction $\frac{p}{q}$ where p and q denote whole numbers, we do not have to talk about duplicate values or equivalent fractions in the list because roster shrinkage will replace one by a single representative, say the one in which p and q are relatively prime. Safe Set TheoryEveryday language uses the terms set, group and collection interchangeable. More formally, set concepts and operations in mathematics are codified in via the concepts and operations involving membership, union, intersection, complements, and what has been called set builder notation. The latter in fact would be better described as subset formation notation. In the eighteen hundreds, the too free or wild creation of sets led to selfreference paradoxes. To make set formation safer in the discussion of numbers, sets of numbers or their set theorectic equivalents were assumed. Then set formation was limited to forming subsets of a given set A, to set products sets \[ A \times B = \{ [a,b]  a \in A, b \in B\} \] that is sets of ordered pairs from two existing sets A and B; and to forming the power set 2^{A the set of all subsets of an existing set A. The set operations of intersection, union and complement by taking placing a larger set may be cast as safe.} Algebra Island: Probability Theory and Counting MethodsThe coverage of probability is reserved to the end of phase 3. It requires coverage of function notation and review or coverage of set concepts and operations: membersbip, union, complement, intersection. The counting principles employed to the find the number of ways events or outcomes can be occur entail mastery of recursion in computation and in mathematical induction. From home and elementary instruction, students may be familar with likelihood and chance. That is, Earlier use of likelihood in deciding whether or not to take a chance will be algebraically extended here into setbased probability theory. Mastery of probability and allied concepts of expected value may be presented as a base for taking or avoiding risks. The question of when to buy a lottery ticket or to visit a casino may be addressed in class. Inverse Functions in Calculus or BeforeGiven the graph of a function y = f(x), one may use a vertical line method to find the value of f(x) from the graph, given the value of x in its domain. In that traditional, the graph of y versus x takes the yaxis to be vertical and the xaxis to be horizontal. To find or study the inverse of the function f, we may graph x versus y for the equation x = f(y). The table of values and graph of the latter is given by transpose the table of values and graph for y = f(x), provided we still keep the values of y along the vertical axis and those of x along the yaxis. On this graph of x =f(y), we may still try to apply the vertical line method to find y, given x. That works if the graph as a whole or a restricted portion of it, satisfies the vertical line rule. The foregoing provides a simple "dual" way to introduce and define a left inverse of a function. If we assume the transpose of a graph also transposes it tangent line, a formula for the derivative of the inverse is also suggested if not rigourous implied. For rigour, see the site treatment of this matter. The fact that the transpose [b,a] of a point [a,b] are reflections of each other may be mentioned in the foregoing, but is not critical. And in this development, the modern mathematics definion of a realvalued function y f(x) as set of points in the plane need not appear. We simply take a formula for f(x) and it graph as two different computation rules for the same function. www.whyslopes.com >> Mathematics Skill Development Framework >>  More Algebra and Slopebased Calculus Preview Next: [  Implementation Notes.] Previous: [  Euclidean and Analytic Geometry with Complex Numbers and Trigonometry.] [1] [2] [3] [4] [5] [6] [7] [8] [9][10] [11] [12] [13] [14] [15] [16] [17] 
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. 