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 1A Pattern Based Reason >> Chapter 21 OccurrenceTables Next: [Chapter 22 Contrapositive and Vacuously True Implications.] Previous: [Chapter 20 ShorthandSymbolsasPronouns.] [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] Chapter 21, Occurrence TablesThe Special Usage of Three Words: not, and & orGiven a situation A, we can talk about the negative situation not A. Given a situation A and another situation B, we may talk about two further situations
The meanings of the terms or phrases are explained next. NOT A and NOT (NOT A)Given a single situation A, we can speak of another situation NOT A. The situation NOT A is said to occur when the situation A does not occur. Further, the situation NOT A is said not to occur when the situation A occurs. This is summarized in the following table.
Language note: a situation A is said to be true when it occurs and not true (false) when it does not occur. The following table
shows that the situation NOT (NOT A) occurs when A occurs and that the situation NOT (NOT A) does not occur when A does not occur. This suggests that the situation A is equivalent to the situation NOT (NOT A). The Special Usage of Three Words (Continued)The word ANDThe situation A and B is said to occur if both situations A and B occur. Otherwise, it is said not to occur. See the table below.
The situation A and B occurs provided In each row, a possible combination of the occurrence or nonoccurrence of the situations A and B is shown in the middle two columns. In the last column, we put a note to say whether or not, the situation A and B occurs or occurs not. * Language Note. The phrase A and B is also labelled (called) the conjunction of the situations A and B. The situation A and B is said to be true when and only when both the situations A and B occur (= are true). The Special Usage of Three WordsThe AtLeastOneUsage of the word ORIn everyday speech when you use the word or in a phrase like John or Andrew will go to the store, the usual expectation is that only one will go, not both. But there is another use of the word or favored in logic. The word or is employed in the at least one sense (as is done in logic and mathematics). With this sense or usage, the previous phrase is understood in the inclusive sense: John or Andrew, or both, will go to the store. We now proceed and we will use the word or in the at least one sense. The situation (A or B ) is said to occur if at least one of the two situations A and B occurs. Otherwise, it is said not to occur. This is summarized in the following table.
The situation A or B can be said to occur Remember the at least one usage differs from the exactly one usage of A or B which means either A or B occurs, but not both. In contrast, in the at least one usage, A or B means either A or B occurs, or both. We have to be careful with the word or. Its meaning depends on the speaker and possibly the listener. That is, confusion and ambiguity results when two people in question use the same words but give them different meanings. To eliminate this ambiguity in everyday speech, write and say one of the following:
When listening, you will have to ask what is meant. Legal texts use the
phrase A and/or B to signal that at least one of the two
cases A and B can occur. 2. Occurrence Table for OneWay ImplicationsAny rule which can be stated in the form if a first situation A occurs, then a second situation B occurs, in brief, if A then B or A implies B, is called a oneway implication. A oneway implication which is never disobeyed is said to hold and to be (always) true. For a oneway implication rule if A then B, we recall the following:
The last two items 3 and 4 can be summarized by saying that disobeying a
oneway implication rule is impossible when the first situation A does
not occur. When not disobeyed, the rule is said to be obeyed by
default. The following table, an occurrence table for the
oneway implication rule if A then B, summarizes what
has been said.
In each row, a possible combination of the occurrence or nonoccurrence of the situations A and B is shown in the middle two columns. In the last column, we put a note to say whether or not the ifthen rule is obeyed, disobeyed, or not disobeyed. Row 2 represents the situation in which A occurs but B does not. Observe that in this situation, the rule is disobeyed. In the situations represented by the other three rows, the rule is not disobeyed. A oneway implication rule if A then B is said
when it is never disobeyed. The oneway implication if A then B is always true when the situation described in row 2 in the above table never occurs.
Remark. If situation A never occurs,
the implication rule if A then B is never disobeyed amd
it is said to be vacuously true. 3 Occurrence Table for TwoWay Implication Rules
A rule which can be stated, or restated, in the form
The next table summarizes the above remarks for any twoway implication rule A if and only if B.
Converses for OneWay ImplicationsA DefinitionThe converse to the implication rule if A then B is the rule if B then A. Note that interchanging the first and second situation A and B yields the converse to a rule. From this definition or perspective, we see that the converse of the converse is the original rule. Check this. When we know a rule if A then B is never disobeyed, we have no guarantee that the converse rule if B then A is never disobeyed. The reason for this is as follows. The rule if A then B is true if the situation A never occurs without the situation B. The converse rule if B then A is true if the situation B cannot occur without the situation A. Reminder. Now we can easily answer the following question: What can we say for sure about the event A when (i) the rule if A then B is never disobeyed, and (ii) the event B occurs? Your answer should be not much, or nothing, without further information. Selby A, Volume 1A, Pattern Based Reason, 1996. www.whyslopes.com >>  Volume 1A Pattern Based Reason >> Chapter 21 OccurrenceTables Next: [Chapter 22 Contrapositive and Vacuously True Implications.] Previous: [Chapter 20 ShorthandSymbolsasPronouns.] [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] 
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. 