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 >> Work and Study Tips >> H Jigsaw puzzles and problem solving Next: [H more  Routine to nonroutine problem solving.] Previous: [G. Written work formats for developing and showing skill.] [1] [2] [3] [4] [5] [6] [7] [8] [9] [10][11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] Problem Solving Tips and MethodsSolving problems in mathematics and other subjects sounds practical. There are two kinds of problems, routine and not. The solution of routine problems may be given in class for students to apply. When a problem is routine, routine solutions should be employed. So routine solution methods for routine problems should be met and memorized to avoid the extra work required for solving problems whose solution is not given. The rule of thumb is a follows.
The Jigsaw Puzzle ApproachProblem solving may be like putting together a jigsaw puzzle. In solving a jigsaw puzzle, we may begin with the sides as pieces with straight edeges are fewer in number and must be aligned, after that the more difficult to place inside pieces may be fitted together. Jigsaw puzzles may be made more challenging by hiding the picture they are suppose to form, or by assembling the pieces upside down. That being said, with the pieces picture side up, we may put try to put them together with trial and error as needed, but with continuity and drawn shape limiting the trial and error. This trial and error combination of pieces that go together may be ad hoc, opportunistic and in general combinatorial. The trial and error requires persistence. With that, over time, more and more of the puzzle will be solved until, if all the pieces are present, the problem is fully solved. Unfortunately, jigsaw puzzle pieces may walk away over time, so there no guarantee that all the effort made will lead to complete picture to solve the puzzle. More generally, when we are tackling a nonrouting problem or puzzle, the existence of solutions is not always certain. Text Book Problems and ExercisesFor most textbook problems and exercises, all the pieces or elements needed to solve the problem are likely to be present in the current or previous chapters. They may just need to be fitted together in ways similar to the worked problems or examples in the text or course notes. The similarity will be close for the easiest problems and further for the more complicated ones. Skill in following textbook patterns may become routine or almost so with practice, with careful reading of the text or notes, with care not to forget earlier skills and methods. In senior high school and college mathematics and mathematical subjects, problem solving may remain routine and feasible with time and effort to see and master all the problem solving methods present in notes or a textbook. Thinking Outside the Box when needbeWhat is routine for one is not for another. Experience counts. Where a student may have to think hard to solve a problem, an older student or an instructor may tackle a problem based on past experience. Problem solving may think out of the box or the confines of earlier problem solving practices, when none of the latter practices apply, Thinking out of the box means look for new angles or different perspectives for tackling or addressing the problem. Or, it may involve tackling what appears to be a related, similar or easier problem in the hope that experience with the latter will make the original problem addressable. Not all certain. And for problems from real life, solutions may be routine, solutions may be difficult to find, or the existence of solutions may be not be known. Some trial and error may be required with success not always certain for the original formulation of a problem. Real World ProblemsIn real world problems and questions unlike most problems and exercises in a book, there may be no given pattern to follow. Not all is certain. Here may be missing pieces or extra pieces, and no guarantee that the solution can be done. Preparing to Solve ProblemsMaster Logic: Again, poblem solving is like putting together a jigsaw puzzle. In the case of textbook problems, all the pieces are present and just need to be fitted together following the clues, and an possible a picture showing the desired result. In the case of real world problems, there may be missing pieces or extra pieces, and no guarantee that the solution can be done. Problem solving besides thinking out of the box and being opportunistic an combinatorial in looking for clues to use alone or with others requires precision in reading, writing and figuring. Imprecise logic and language abilities will lead to difficulties. Precision reading and writing, and opportunistic trial and error skills for problem solving may be refined and developed (we hope) by reading the following chapters in site Volumes 1A and 2.
These appetizers and lessons show how rules and patterns may fit together to arrive at conclusions or solve SOME problems. Other problems are just too hard. We can't prevent that. Master FractionsMany applied mathematics problems involving chopping and combining lengths, areas and volumes. So you need to know how to take a proper or improper fraction of a length, area or volume. You need to understand that one length may be 2.5 times or 2½ times or (5/2) times another. Any if you do calculation, you need to do it with care or at least do it with the knowledge that an error in one step makes all that follows wrong. The ability to figure well and precisely, so that you answer is correct, shows or suggests the ability to follow methods, one step at a time and one step after another in any subject, and in problem solving as well. Algebra Word ProblemsIf your interest is in solving algebra word problems at the high school level, I would recommend learning how to solve linear equations in several unknowns in an effortless fashion. High school students who can solve linear equations in one unknown are often given word problems where extra variables have to be eliminated to formulate a single equation in one unknown quantity to solve. The trick here is to draw or extract a single equation from the given information. But in most such words problems, it is easier to extract or draw from the given information several linear equations in several unknowns to solve. Each sentence in the word problem gives an equation in one or more unknowns or quantities. Now the algebraic way of writing and thinking can be used to eliminate variables and to solve for the one or more quantities of interest in an effortless fashion. The algebraic solution of linear equations involves the elimination of variables to obtain say one equation in one unknown. This elimination process may be better done and recorded with algebraic notation. Going directly to one equation in one unknown to solve a problem requires more work to be done with words. www.whyslopes.com >> Work and Study Tips >> H Jigsaw puzzles and problem solving Next: [H more  Routine to nonroutine problem solving.] Previous: [G. Written work formats for developing and showing skill.] [1] [2] [3] [4] [5] [6] [7] [8] [9] [10][11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] 
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. 