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 >> Volume 2 Three Skills For Algebra >> Chapter 6 Change of Language Next: [ Chapter 7 Prep for Calculus Arithmetic Exercises.] Previous: [Chapter 5 IslandsandDivisionsofKnowledge.] [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] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] Chapter 6, A (technical) Language ChangeImplication rules can be stated in several ways. We need to recognize them. OneWay Implication RulesIn the chapter Implication Rules , we met the rule When Aunt Jane visits her nephew Tom's home, Tom goes out to play Rules like this can be said in different ways. This gives variety and choice in the way in which we write rules. The form of a rule does not matter, if we understand exactly what it says. The above oneway rule can also be rewritten (or restated, again without changing its meaning) using the words IF and THEN as follows. IF Aunt Jane visits her nephew Tom's home THEN Tom goes out to play. The word IF introduces a condition, namely Aunt Jane's visit to her nephew Tom's home. The word THEN introduces the consequence, what should occur, when the condition is satisfied. Here the consequence is Tom goes out to play. Since the original rule can be rewritten in the IF condition THEN consequence form, we say the original rule and the ifthen form are conditional statements.
Another way of writing the above oneway Aunt Jane and nephew Tom rule (with no change in meaning) is given by: Aunt Jane's visit to her nephew Tom's home IMPLIES Tom goes out to play. The words forces or makes may be used instead of the word implies. We could also use the word suggests, but in everyday use, a suggestion is optionally obeyed or followed while a rule (when it is correct) should or must be obeyed or followed. In talking about rules, we use the words implies, forces or makes for those rules we expect will be obeyed, or more precisely will never be disobeyed in the circumstances at hand. The explicit identification of such circumstances is exhaustive unless the circumstances in question are understood from a context, an obvious one, we hope.
TwoWay Implication RulesIn the previous chapter Implication Rules, we met the rule
Tom goes out to play This is an example of a twoway rule. Twoway rules can also be said or presented in different ways. Again the form of a rule does not matter, provided we recognize exactly what is meant. The above rule also can be rewritten (or restated, again without changing its meaning) in the ifandonlyif form:
Aunt Jane visits her nephew Tom's home This form suggests we call such rules biconditional statements. The prefix bi here signals two ways. Whenever the condition (or situation) Aunt Jane visits her nephew Tom's home occurs, the other condition (or situation) Tom goes out to play must also occur, and viceversa, if this rule is to be neverdisobeyed. You may prefer to say if and only if instead of when and only when. For instance, I might say or suggest to you: I will do that for you if and only if you do this for me. Alternatively, I might say or suggest to you: I will do that for you when and only when you do this for me. Tone provides the only difference between the two suggestions. Both of these suggestions represent a twoway obligation to which we might agree. Confusion or disappointment or false expectations may happen when suggestions such as these are not explicitly accepted or rejected. TwoWay or Two OneWay RulesThe twoway Aunt Jane and nephew Tom rule above is rewritten (with no change in meaning) as Aunt Jane visits her nephew Tom's home implies Tom goes out to play, In this form, the twoway rule is seen to be the same as two oneway implication rules, each going in the opposite direction. Equivalent Conditions (or Situations)Two situations or conditions A and B, each of which must happen whenever the other does, are said to be equivalent to each other. So when a first situation is equivalent to a second, each situation implies and is implied by the other. Conditional versus BiconditionalOneway and twoway implications are called conditional and biconditional statements (or rules), respectively. The Abbreviation IffThe terms and phrases
can all be used instead of each other. They are interchangeable. No matter what term or phrase is used to indicate a twoway implication, the difference between oneway and twoway needs to be remembered. Otherwise, statements, definitions and assertions will be read incorrectly. Selby A, Volume 1A, Pattern Based Reason, 1996. www.whyslopes.com >> Volume 2 Three Skills For Algebra >> Chapter 6 Change of Language Next: [ Chapter 7 Prep for Calculus Arithmetic Exercises.] Previous: [Chapter 5 IslandsandDivisionsofKnowledge.] [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] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] 
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. 