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 >> More Algebra >> 4 Functions >> 7 Functions with finite domains Next: [8 Set view of relations and functions.] Previous: [6 Set Existence Formation and Notation.] [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] Functions with finite domainsDomains and Ranges of functions y = f(x) may be defined as follows when y and x denote real numbers.
In the previous lessons, we saw how to give or describe a dependence of a number or quantity y on other numbers or quantities by a formula y = f(a,b,c). But we can also describe how dependence in other ways using forwards and backwards.
with arrow diagramsA function, dependency, map or assignment f may defined by arrow diagram.
The domain of f,
The range of f,
The range is a subset of the target set {a,b,c,d,6.} So the map is not surjective (onto) The map f is many to one as f(5) and f(2) are equal to c. Equivalent Ways with tablesWe could have defined the previous function with a horizontal table
or vertical table
as you like. Any letter may be used in place of x.
Another Table ExampleHere we use a table to define h(x).
The table says how to compute a function h. From the table, we may evaluate the mapping h at each element of its domain {0, 1, 2, 3, 4, 5, 6}
The domain of h is set of numbers
in the first row. The range of h is the set of numbers
in the second row. Yet another table exampleA table of values
in which there is no duplicate numbers or objects in the xrow gives a function f with
The range of f,
In this example
f(2) = 3 f(3) = 1 f(4) = 4 List MethodA function f may be described by specifying it values at points in a set. f(2) = 3, f(4) =11 f(8) = 2 The foreging gives a function f with domain {3, 4, 8} and range {3, 11, 2} List Method in GeneralA function f defined for a set of distinct values x_{1}, x_{2}, ... x_{n}. by specifying its values y_{1}, y_{2}, ... y_{n} at those numbers, so that
Here the domain of definition of f,
is a finite set. The range of f
is a finite set. (Remember to eliminate duplicate values of y so that elements of the range are not listed twice.) Using ordered pairsA function f in mathematics may be specified by a set of ordered pairs. For example f = { (1,3.4), (2.5, 4), (2.1, 5), (1, 8) } represents the function f with
The function domain, the set of items for which is defined, is
Plotting the ordered pairs gives the graph of f. The set of points
provides the graph of f. So we may write
and identify the function with its graph. The graph is a set of points in the coordinate plane. So the study of functions y =f(x) where y and x are real numbers becomes part of analytic geometry. The stage is now set for the following. Analytic Geometry View of Functions in the planeHere the set of points in the plane is denoted by IR^{2} = {(x,y) such x and y are real numbers} A finite set S of points (x, y) in the coordinate plane IR^{2} which satisfies the vertical line property, namely each vertical line intersect S at most one point. In this case, when the line x = a intersects the set S at a point (a,b), the computation associated rule f puts f(a) = b. Site to do: Put an illustration here ] The set S may be given by a list of order pairs or by their plot (graph) in the plane. www.whyslopes.com >> More Algebra >> 4 Functions >> 7 Functions with finite domains Next: [8 Set view of relations and functions.] Previous: [6 Set Existence Formation and Notation.] [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] 
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. 