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Mathematics and Logic - Skill and Concept Development

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Online Volumes: 1 Elements of Reason || 2 Three Skills For Algebra || 3 Why Slopes Light Calculus Preview or Intro plus Hard Calculus Proofs, decimal-based.
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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.
- 1 versus 2-way implication rules - A different starting point - Writing or introducting the 1-way implication rule IF B THEN A as A IF B may emphasize the difference between it or the latter, and the 2-way implication A IF and ONLY IF B.
- Deductive Chains of Reason - See which implications can and cannot be used together to arrive at more implications or conclusions,
- Mathematical Induction - a light romantic view that becomes serious.
- Responsibility Arguments - his, hers or no one's
- Islands and Divisions of Knowledge - a model for many arts and disciplines including mathematics course design: Different entry points may make learning and teaching easier. Are you ready for them?

Early High School Arithmetic

Deciml Place Value - funny ways to read multidigit decimals forwards and backwards in groups of 3 or 6.
- Decimals for Tutors - lean how to explain or justify operations. Long division of polynomials is easier for student who master long division with decimals.
- Primes Factors - Efficient fraction skills and later studies of polynomials depend on this.
- Fractions + Ratios - See how raising terms to obtain equivalent fractions leads to methods for addition, comparison, subtraction, multiplication and division of fractions.
- Arithmetic with units - Skills of value in daily life and in the further study of rates, proportionality constants and computations in science & technology.

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?
- Formula Evaluation - Seeing and showing how to do and record steps or intermediate results of multistep methods allows the steps or results to be seen and checked as done or later; and will improve both marks and skill. The format here allows the domino effects of care and the domino effects of mistakes to be seen. It also emphasizes a proper use of the equal sign.
- Solve Linear Eqns with & then without fractional operations on line segments - meet an visual introduction and learn how to present do and record steps in a way that demonstrate skill; learn how to check answers, set the stage for solving word problems by by learning how to solve systems of equations in essentially one unknown, set the stage for solving triangular and general systems of equations algebraically.
- Function notation for Computation Rules - another way of looking at formulas. Does a computation rule, and any rule equivalent to it, define a function?
- Axioms [some] as equivalent Computation Rule view - another way for understanding and explaining axioms.
- Using Formulas Backwards - Most rules, formulas and relations may be used forwards and backwards. Talking about it should lead everyone to expect a backward use alone or plural, after mastery of forward use. Proportionality relations may be use backward first to find a proportionality constant before being used forwards and backwards to solve a problem.

Early High School Geometry

Maps + 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 human-made 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

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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 Rules

Introduction

Are 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 Puzzle

A One-Way Implication Rule

To 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 Questions

Try to answer the five questions below. Be careful. The questions may trip you. Answers follow. See if you agree with them.

Answers are given twice
  1. in pop-up answers above, and
  2. in text below
See if you agree with them.2
  1. When the rule is obeyed, what can you say happens for sure when Aunt Jane visits her nephew's home? This is easy. [pop up answer]
  2. When the rule is not disobeyed, what can you say happens for sure about Aunt Jane when Tom is out playing? Be careful. [pop-up answer]
  3. When the rule is not disobeyed, what can you say happens for sure about Tom when Aunt Jane is not visiting? Be careful, again. [pop-up answer]
  4. What must happen for the given rule to be disobeyed? This is another easy question. [pop-up answer]
  5. When the rule is not disobeyed, what can you say for sure about Aunt Jane when Tom does not go out to play? See the answer to the fourth question. [pop-up answer]

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 Answer

The 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 Answer

The 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 Answer

The 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 Answer

The 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 Answer

The 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.

From Volume 1A, Pattern Based Reason, version of this chapter

The CONTRAPOSITIVE - Optional Reading

The first situation

A AND not B

is inconsistent with the implication rule

IF A THEN B.

So in circumstance where the latter implication rule IF A THEN B. holds (is not disobeyed), we conclude or require the first situation

A AND not B

not to occur. The non-occurrence of A AND not B in turn implies the original implication

IF A THEN B

and the contra positive implication

IF not B THEN Not A

Since both imply not( A AND not B), the two implications are equivalent to each other and to the non-occurrence of A AND not B.

The Second Puzzle

A Two-Way Implication Rule

Try 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.
  1. When the rule is obeyed, what can you say happens for sure when Aunt Jane visits her nephew's home? This is easy. [Answer]
  2. When the rule is not disobeyed, what can you say happens for sure about Aunt Jane when Tom is out playing? Be careful. [Answer]
  3. When the rule is not disobeyed, what can you say happens for sure about Tom when Aunt Jane is not visiting? Be careful, again. [Answer]
  4. What must happen for the given rule to be disobeyed? This is another easy question. [Answer]
  5. When the rule is not disobeyed, what can you say for sure about Aunt Jane when Tom does not go out to play? See the answer to the fourth question. [Answer]
Answers are given twice
  • in popup boxes, and
  • in text below. (as in the printed version)
See if you agree with them.

Answers to the Second Puzzle

The 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.

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. The questions and answers follow.

  1. When the rule is obeyed, what can you say happens for sure when Aunt Jane visits her nephew's home? Answer: Tom must be out playing (no change).
  2. When the rule is not disobeyed, what can you say happens for sure about Aunt Jane when Tom is out playing? Answer: Aunt Jane must be visiting (the answer has changed).
  3. When the rule is not disobeyed, what can you say happens for sure about Tom when Aunt Jane is not visiting? Answer: Tom is not outside playing (the answer has changed).
  4. What must happen for the given rule to be disobeyed? Answer: Either Aunt Jane must be visiting and Tom does not go out to play or Tom must be out playing without Aunt Jane visiting (the answer has changed).
  5. When the rule is not disobeyed, what can you say happens for sure about Aunt Jane when Tom does not go out to play? Answer: Aunt Jane is not visiting (no change).

One Versus Two Way Implications

The 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.

  1. A one-way implication rule says that when a first situation occurs, so must a second. It does not say that when the second occurs so must the first. The second situation may occur without the first.
  2. A two-way implication rule says that
    1. when a first situation occurs, so must a second, and
    2. when the second situation occurs, so must the first.
    A two-way rule says that when each situation occurs, so must the other. Therefore if the two-way rule is to be obeyed, when one situation does not occur, neither can the other.

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?

From Volume 1A, Pattern Based Reason, version of this chapter

One- Versus Two-Way Commitments

In speaking to someone, you may promise I will do you a favor if you do one for me. Now this promise is a one-way commitment. If the other person does you a favor first, your promise obliges you to do the other a favor. But if you do the other person a favor, your promise does not oblige him or her to do you a favor unless the person has made a similar promise. When you want a two-way obligation, you have to be careful and precise with your words and promises. In particular, you need to reach an agreement with the other person or party in which, besides your promise to do a favor in return for one, the other also promises you the favor you want in return.

Repeatable & Reproducible Results

When we do arithmetic, we follow rules. If each of us does not make a mistake, we will get the same result as each other. Each calculation is reproducible. Other people know how it was done. They can repeat it, and see if they agree with our results. When we get a result by following rules or instructions, other people can check our results. All they have to do is follow the same rules. Just as we have methods for doing arithmetic carefully and precisely, we also have suggested methods for thinking carefully about what to accept, to do or to decide. The ability to read and understand rules or suggestions precisely is needed not only in persuading ourselves or others but also in following recipes, instructions and rule-based reason. The suggestions or rules which give repeatable and reproducible results are the most certain and possibly the most correct. Reliable rule-based processes give repeatable and reproducible results. These results do not depend on who gets them. For instance, a good recipe can be followed by any cook - provided the cook can find all the ingredients. Results should depend on the recipe, but be independent of the cook.

Limitations and Benefits

To see the benefits and limitations of logic and rule-based thought is important. We should not say or judge or conclude too much when facts and evidence are missing, hidden, withheld or uncertain. Faulty, misleading suggestions need to be recognized. More can be suggested than proven. The methods of arithmetic and logic are but tools for reaching conclusions when reliable information and rules are available to be put together. The question of what rules to apply is always present. Untested rules say what might occur. They should be used only with suspicion. The more reliable and tested rules are to be preferred. Rule-based reason gives certainty or agreement in some situations, not all. We need to identify the level of uncertainty in all the rules and information we use. Such awareness allows the recognition of the weak or weakest spots in our reasoning or judgment processes.

Accidental Rules

The initial one-way implication rule said:

When Aunt Jane visits her nephew Tom's home, Tom goes outside to play.

This rule describes a pattern. This rule is said to be true if it is never disobeyed. This rule is said to be false if it is disobeyed at least once. We can talk about the truth and falseness of a rule in the past, present, future or in some special situation. Given a rule or a possible pattern, we would like to know in which circumstances it is never disobeyed. The five questions show us how to use this rule when we know it is not disobeyed. A sixth question is

What, if anything, can we do to check or guarantee that a given rule is never disobeyed in the circumstances of interest?

We could perhaps observe all the visits of Aunt Jane to see that Tom goes out to play each and every time. If he does not once, the rule is false. It has been disobeyed. [3]

[3] Note that this rule will never be disobeyed if Aunt Jane never visits. In the latter case, the rule is said to be vacuously true.

In observing some but not all of her past visits, we may see the pattern that when she visits he goes out to play. These observations only describe the past. Patterns observed in the past can or might change in the future. We have to judge how likely this is. In contrast, seeing a rule is not obeyed at least once, or just once, is enough to say the rule is false - not always obeyed. Vocabulary: A situation in which a rule is disobeyed is said to provide a counter-example to the rule. In summary, seeing a rule is obeyed a few times is enough to suggest a pattern. Seeing a rule is obeyed a few times is not enough to imply with complete confidence that it is never disobeyed. Observations may only suggest a pattern is developing. They may lead us to conjecture or guess that the rule will always be obeyed or at least never be disobeyed. A difference between being suspicious and being certain exists. Patterns seen may suggest rules, but not prove them absolutely. A rule which suggests that every time an event occurs, another event will occur cannot be checked or proven absolutely. Such a rule can be assumed for the sake of getting conclusions. When is the rule reliable? What can be done to test our assumptions? Our confidence in the resulting conclusions depends on the reliability of the rules and implications used. The reliability, origin and testing of rules, instructions, recipes, suggestions and implications need more inspection. Where is the proof? Sometimes proof is not available. So we may pretend (assume) a rule is never disobeyed to reach conclusions or to make suggestions from it. Each pretense or assumption represents a weak spot - a possible gamble or source of error, in our reasoning. [4]

[4] In arithmetic, an error or wrong number early in our calculation casts doubts on the rest of the calculation. Similarly in reason, a false step or assumption casts doubts on the rest of the reasoning and the conclusions drawn from it.

More will be said on this subject of what rules are reliable. The chapter Accidental Patterns will echo many of the ideas introduced here.

Steps for Better Reason

A first step in rule- and pattern-based reason is to see and understand the difference between one-way and two-way implication rules. People too often think a one-way implication rule is a two-way implication rule. That can be confusing and misleading. It leads to false expectations and arguments. The ability to read and understand one- and two-way implication rules precisely further helps in following instructions and recipes and also in deciding which rules to apply.

A second step is to be aware of and cautious about suggestive and misleading questions. When asked a question, we politely try to answer without challenging the suggestions or assumptions made in it. Some questions take advantage of our politeness. Pause when encountering such questions. Don't always answer immediately. Rather, think if the question asked assumes too much or makes assumptions with which you are ill at ease. Those that do should be avoided or challenged. The chapter Deception, Suggestive or Misleading Questions speaks further about this issue and this second step.

A third step is to chain, link or connect implication rules together to create more implication rules for getting conclusions. (The verbs to link, to chain and to connect all have the same or similar meaning here. They are used interchangeably. Each can be used instead of any other for the sake of variety.) The chapters Chains of Reason and Longer Chains of Reason show how implication rules can be used one at a time and one after another.

The remaining chapters on reason in this book describe how patterns and implication are written and found, and how their reliability can be judged. A fourth step in logic or reason is to talk about how patterns and implication rules are found, invented and employed in daily life, technology, science and mathematics. The world is full of patterns, implications and suggestions. Some are more certain, more reliable and more correct than others, while others are completely false. We must try to identify which are which. Uncertainty is not welcome, yet not knowing what is unsure is worse. Locating weak spots in reasoning permits a search for replacements.



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]

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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 head-to-tail addition of arrows in the plane. Click and drag complex numbers A and B to change their locations.

Pattern Based Reason

Online Volume 1A, Pattern Based Reason, describes origins, benefits and limits of rule- and pattern-based 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 story-telling view of learning: [ A ] [ B ] [ C ] [ D ] to suggest how we share theory and practice in many fields of knowledge.

Site Reviews

1996 - 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; pattern-based 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 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, spanning the general topics of arithmetic, logic, algebra, calculus, complex numbers, and Euclidean geometry. Lessons and how-tos 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.
- Complex Numbers - Learn how rectangular and polar coordinates may be used for adding, multiplying and reflecting points in the plane, in a manner known since the 1840s for representing and demystifying "imaginary" numbers, and in a manner that provides a quicker, mathematically correct, path for defining "circular" trigonometric functions for all angles, not just acute ones, and easily obtaining their properties. Students of vectors in the plane may appreciate the complex number development of trig-formulas for dot- and cross-products.
Lines-Slopes [I] - Take I & take II respectively assume no knowledge and some knowledge of the tangent function in trigonometry.

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.
- Why Factor Polynomials - Online Chapter 2 to 7 offer a light introduction function maxima and minima while indicating why we calculate derivatives or slopes to linear and nonlinear curves y =f(x)
- Arithmetic Exercises with hints of algebra. - Answers are given. If there are many differences between your answers and those online, hire a tutor, one has done very well in a full year of calculus to correct your work. You may be worse than you think.


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