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 2 Implication Rules  Forwards and Backwards Next: [Chapter 3 Chains of Reason.] Previous: [Chapter 1 Introduction to Chapters 2 to 6.] [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 2. Implication RulesIntroductionAre you a careful thinker? Can you understand exactly the meaning of a rule or pattern? Instructions for building or creating provide rules and patterns which say and suggest that when this is done, that should happen. Every cook and dressmaker knows the importance of following instructions carefully. Instructions and suggestions which are not repeatable and results which are not reproducible are not of interest to cooks and dressmakers. In this chapter, you will meet two puzzles. They show the difference between one and twoway implication rules. Mastering the difference is a simple, first step, in rule and patternbased thought. This first step is needed to precisely read rules, definitions and statements in all disciplines, including mathematics. To read carefully, do not imagine too much. To decide or choose among opinions and actions, you must understand the exact meaning of written and spoken words. You need this skill to understand, to follow, to write and to change rules, guidelines, instructions and laws, etc. Use your imagination in language courses. Use your imagination when you are reading novels (and newspaper opinion columns). When reading newspapers or listening to radio and television ask: Is the story presented in a onesided way? Headlines may suggest conclusions which are not in the stories or the text. Look at the details. Here imagination allows you to guess what the full story might be. But imagination provides only suggestion, not proof. Confidence in suggestions must come after proof is given, not before. Also use your imagination for poorly written rules to guess their meanings. Guesses and speculations give possible meanings. These may or may not be correct. Proof and evidence, or tests, may decide which among various possibilities, if any, are correct. Each of us needs to understand fully or as much as is possible, whatever we might be doing or learning. In reasoning, some rules and patterns are reliable. Others are guidelines. Each of us needs to know which is which. The First PuzzleA OneWay Implication RuleTo help you think and possibly cook more carefully, we look at a very simple puzzle. The puzzle consists of a rule and five questions. The questions test your ability to think carefully and to read exactly what is written. Once you have understood the answers and why they are true, your ability to think carefully and clearly will have advanced. The rule for the puzzle is as follows: When Aunt Jane visits her nephew Tom's home, Tom goes out to play. Five QuestionsTry to answer the five questions below. Be careful. The questions may trip you. Answers follow. See if you agree with them.
Hint: The rule provides no information and no reason explaining why Tom goes out to play whenever his Aunt Jane visits. The rule only describes what happens when Aunt Jane visits. We cannot say if he goes out to play to avoid Aunt Jane. We cannot say if he looks forward to her visits. The answers to the above questions only depend on the wording of the question and the given information or rule(s). So control your imagination. Do not assume or imagine too much. Suggestion: Discuss the questions with your family and friends. Some people will get correct answers immediately. Others require persuasion. Still others will not understand. Talking with people about the questions shows how well they think. The First AnswerThe first question is When the rule is obeyed, what can you say happens for sure when Aunt Jane visits her nephew's home? It`s answer is easy: Tom goes out to play. The Second AnswerThe second question is When the rule is not disobeyed, what can you say happens for sure about Aunt Jane when Tom is out playing? The answer is nothing. The rule only tells what happens when Aunt Jane visits. It does not say what must happen when Tom goes out to play. Tom could go out to play without Aunt Jane visiting. The rule does not say, nor does it suggest that Tom may only play outside when Aunt Jane visits. The rule does not say Aunt Jane must visit when Tom goes out to play. When the rule is not disobeyed, we cannot say much for sure or certain about Aunt Jane when Tom goes out to play. All we can say for sure is that she may be visiting or she may not be visiting. When she is not visiting, the rule cannot be disobeyed. When she is visiting, the rule is obeyed and so not disobeyed. In either case, the rule is not disobeyed. The above rule is one way. It says what should happen when Aunt Jane visits without saying that she must be visiting when Tom goes out to play. When Tom goes out to play, the rule is not disobeyed when Aunt Jane is not visiting. It gives no information on her whereabouts. An example of a twoway rule is given later. See the second puzzle. The Third AnswerThe answer to the third question When the rule is not disobeyed, what can you say happens for sure about Tom when Aunt Jane is not visiting? is like that of the second. When Aunt Jane is not visiting, the rule is not disobeyed if Tom goes out, and the rule is not disobeyed if Tom does not go out. When the rule is not disobeyed we can say nothing for certain about Tom when Aunt Jane is not visiting. The rule does not say that the only time Tom can go out to play is when his Aunt Jane visits. Again, the rule is only oneway. When Aunt Jane is not visiting, no information can be extracted from the rule. It says nothing about Tom. The Fourth AnswerThe fourth question is What must happen for the given rule to be disobeyed? The rule is disobeyed if Aunt Jane visits and Tom does not go out to play. That is, the situation where Aunt Jane visits and Tom does not go out to play must happen for the rule to be disobeyed. The Fifth AnswerThe fifth question is When the rule is not disobeyed, what can you say happens for sure about Aunt Jane when Tom does not go out to play? The rule will be disobeyed when Aunt Jane visits and Tom does not go out to play. To avoid the rule being disobeyed when Tom does not go out to play, Aunt Jane must not be visiting. The fifth answer is Aunt Jane is not visiting. The contrapositive way of writing the above rule is When Tom not go out to play, Aunt Jane not visit. For this contrapositive rule to be never disobeyed, what can you say for sure when Aunt Jane visits? Answer: Not (Tom Not go out to play), that is, Tom goes out to play. The contrapositive of the contrapositive is the original rule. Later chapters on logic give more information, just a little more, about the contrapositive. Some Vocabulary. The above rule is called an oneway implication rule. The first aim of this chapter is to show you the difference between one and twoway implication rules. The meaning and use of the word implication will be talked about later. The five questions should help you see the difference between a oneway and a twoway implication rule. Seeing this difference signals that you can interpret precisely what a rule means.
The Second PuzzleA TwoWay Implication RuleTry answering the five questions again, using this twoway (implication) rule Tom goes out to play when and only when Aunt Jane visits his home. instead of the original rule. How will the answers change? Rather, which answers change? This second rule can be restated as follows. Tom goes out to play when Aunt Jane visits his home.and also Tom goes out to play only when Aunt Jane visits his home.The first when part of this rule is disobeyed in the situation where Aunt Jane visits and Tom does not go out to play. The only when part of this rule is disobeyed in the situation when Tom goes out to play without his Aunt Jane visiting. Here are the five questions again.
Answers to the Second PuzzleThe twoway implication rule for the second puzzle is: Tom goes out to play when and only when Aunt Jane visits his home instead of the original rule. How will the answers change? Rather, which answers change? This second rule can be restated as follows. The first when part of this rule is disobeyed in the situation where Aunt Jane visits and Tom does not go out to play. The only when part of this rule is disobeyed in the situation when Tom goes out to play without his Aunt Jane visiting. The questions and answers follow.
One Versus Two Way ImplicationsThe two puzzles give examples of implication rules. The first puzzle gives a oneway implication rule, while the second gives a twoway implication rule. The following words should further help you to see the difference between one and twoway implication rules. Seeing this difference may help you understand better the answers to the above questions. They may also help you answer the five questions again using the twoway implication rule.
Seeing or recognizing the difference between one and twoway implication rules makes you a more careful thinker. One and twoway rules, recognized or not, are what we use to reach conclusions or make judgments. One and twoway rules can be used to suggest or persuade us of what needs to be done or avoided. Talking About Logic
As suggested above, you can give people the above rules or similar ones
before asking five questions. Before you do this, you should wait for a
receptive mood, especially if you are not in a classroom. For the sake of
an argument and some fun, you may ask after getting an answer, are you
sure? Or you may pretend a correct answer is wrong. Of course, you
will admit this ruse later, and explain why you really agree (or
disagree) with the answers. The aim is to see how people reason and more
importantly to strengthen their thinking skills. Logic inside and outside
of mathematics is supposed to give rules for thought, that is rules for
arriving at conclusions. Yet the only rule needed in the reasoning shown
above is as follows: Read exactly what is written and don't assume nor
imagine too much. Implications Versus Suggestions
In a dictionary you may find that the verb to imply also means
to suggest. Words which say when one event occurs so does or
will a second are called suggestions or implications. Suggestions and
implications can be true. True here means obeyed or at least not
disobeyed. Suggestions and implications can be false. False here means
disobeyed. In our reasoning process, we want to say with certainty that
when this occurs so will that. In practice, we may have to
be content with saying when this occurs, so may that.
Knowing which of our rules are sure or which are uncertain identifies the
weaknesses in our reasoning processes. The implication rules that are
never disobeyed provide the most certain suggestions in reason. In logic,
when we speak of implication rules, we speak of rules which we hope are
never disobeyed. Rules which might be disobeyed are called conjectures,
suggestions or guesses. Evidence (persuasion) should be required to
convince us that a conjecture or suggestion is a reliable implication. We
can imagine or suggest more than we can prove. Caution is advised on
hearing a rule. Before applying a rule, you need to know how certain it
is. Is it a reliable implication or merely an uncertain suggestion?
www.whyslopes.com >> Volume 2 Three Skills For Algebra >> Chapter 2 Implication Rules  Forwards and Backwards Next: [Chapter 3 Chains of Reason.] Previous: [Chapter 1 Introduction to Chapters 2 to 6.] [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. 