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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. For students of reason in society, science and technology: Pattern Based Reason describes origins, benefits and limits of rule- and pattern-based thought and actions. Not all is certain. We may strive for objectivity, but not reach it. Postscripts offer a story-telling view of learning: [ A ] [ B ] [ C ] [ D ] to suggest how we share theories and practices. Site's Best LessonsFor Logic
These online chapters may amuse while leading to greater precision and comprehension in reading and
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Deciml Place Value - funny ways to read multidigit decimals forwards and
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What is
a Variable? - this entertaining oral & geometric view
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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 two-way implication rules. Mastering the difference is a simple, first step, in rule and pattern-based 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 one-sided 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 One-Way 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 two-way 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 one-way. 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 one-way implication rule. The first aim of this chapter is to show you the difference between one- and two-way 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 one-way and a two-way implication rule. Seeing this difference signals that you can interpret precisely what a rule means.
The Second PuzzleA Two-Way Implication RuleTry answering the five questions again, using this two-way (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 two-way 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 one-way implication rule, while the second gives a two-way implication rule. The following words should further help you to see the difference between one- and two-way 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 two-way implication rule.
Seeing or recognizing the difference between one- and two-way implication rules makes you a more careful thinker. One- and two-way rules, recognized or not, are what we use to reach conclusions or make judgments. One and two-way 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?
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? 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
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2001 - Math Forum News Letter 14,
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for Complex Numbers offer a short way to reach and explain:
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cross-products, 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,
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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
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patterns for solving equations. ...
For Geometry
Maps + Plans Use - Measurement use maps, plans and diagrams drawn
to scale. For Calculus
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. |