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# Mathematics and Logic - Skill and Concept Development

with lessons and lesson ideas at many levels. If one site element is not to your liking, try another. Each one is different.

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.
More Lessons &Lesson Ideas: Arithmetic & No. Theory || Time & Date Matters || Algebra Starter Lessons || Geometry - maps, plans, diagrams, complex numbers, trig., & vectors || More Algebra || More Calculus || DC Electric Circuits || 1995-2011 Site Title: Appetizers and Lessons for Mathematics and Reason

Mathematics Concept & Skill Development Lecture Series: Webvideo consolidation of site lessons and lesson ideas in preparation. Price to be determined.

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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.
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- 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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## Problem Solving Tips and Methods

Solving problems in mathematics and other subjects sounds practical. There are two kinds of problems, routine and not. The solution of routine problems may be given in class for students to apply. When a problem is routine, routine solutions should be employed. So routine solution methods for routine problems should be met and memorized to avoid the extra work required for solving problems whose solution is not given. The rule of thumb is a follows.

• For Routine Problems, learn and use Routine Methods.

• For non-routine problems, be combinatorial, follow strageties, and try whatever may work, near or far.

There is a need to master or at least identify what problems are routine. Otherwise, you will spend time in looking for and inventing solutions for problems whose solutions should routine or automatic.

### The Jigsaw Puzzle Approach

Problem solving may be like putting together a jigsaw puzzle. In solving a jigsaw puzzle, we may begin with the sides as pieces with straight edeges are fewer in number and must be aligned, after that the more difficult to place inside pieces may be fitted together. Jigsaw puzzles may be made more challenging by hiding the picture they are suppose to form, or by assembling the pieces upside down. That being said, with the pieces picture side up, we may put try to put them together with trial and error as needed, but with continuity and drawn shape limiting the trial and error. This trial and error combination of pieces that go together may be ad hoc, opportunistic and in general combinatorial. The trial and error requires persistence. With that, over time, more and more of the puzzle will be solved until, if all the pieces are present, the problem is fully solved. Unfortunately, jigsaw puzzle pieces may walk away over time, so there no guarantee that all the effort made will lead to complete picture to solve the puzzle. More generally, when we are tackling a nonrouting problem or puzzle, the existence of solutions is not always certain.

### Text Book Problems and Exercises

For most textbook problems and exercises, all the pieces or elements needed to solve the problem are likely to be present in the current or previous chapters. They may just need to be fitted together in ways similar to the worked problems or examples in the text or course notes. The similarity will be close for the easiest problems and further for the more complicated ones. Skill in following textbook patterns may become routine or almost so with practice, with careful reading of the text or notes, with care not to forget earlier skills and methods. In senior high school and college mathematics and mathematical subjects, problem solving may remain routine and feasible with time and effort to see and master all the problem solving methods present in notes or a textbook.

### Thinking Outside the Box when need-be

What is routine for one is not for another. Experience counts. Where a student may have to think hard to solve a problem, an older student or an instructor may tackle a problem based on past experience. Problem solving may think out of the box or the confines of earlier problem solving practices, when none of the latter practices apply, Thinking out of the box means look for new angles or different perspectives for tackling or addressing the problem. Or, it may involve tackling what appears to be a related, similar or easier problem in the hope that experience with the latter will make the original problem addressable. Not all certain. And for problems from real life, solutions may be routine, solutions may be difficult to find, or the existence of solutions may be not be known. Some trial and error may be required with success not always certain for the original formulation of a problem.

### Real World Problems

In real world problems and questions unlike most problems and exercises in a book, there may be no given pattern to follow. Not all is certain. Here may be missing pieces or extra pieces, and no guarantee that the solution can be done.

### Preparing to Solve Problems

Master Logic: Again, poblem solving is like putting together a jigsaw puzzle. In the case of textbook problems, all the pieces are present and just need to be fitted together following the clues, and an possible a picture showing the desired result. In the case of real world problems, there may be missing pieces or extra pieces, and no guarantee that the solution can be done.

Problem solving besides thinking out of the box and being opportunistic an combinatorial in looking for clues to use alone or with others requires precision in reading, writing and figuring. Imprecise logic and language abilities will lead to difficulties. Precision reading and writing, and opportunistic trial and error skills for problem solving may be refined and developed (we hope) by reading the following chapters in site Volumes 1A and 2.

1. Implication Rules (Volume 1, Part I, Pattern Based Reason)
2. chains of reason (Volume 1, Part I, Pattern Based Reason)
3. longer chains of reason (Volume 1, Part I, Pattern Based Reason)
4. islands and divisions of knowledge (Volume 1, Part I, Pattern Based Reason)
5. painless theorem proving (Volume 2, Three Skills for Algebra)

These appetizers and lessons show how rules and patterns may fit together to arrive at conclusions or solve SOME problems. Other problems are just too hard. We can't prevent that.

### Master Fractions

Many applied mathematics problems involving chopping and combining lengths, areas and volumes.  So you need to know how to take a proper or improper fraction of a length, area or volume. You need to understand  that one length may be 2.5 times or 2½ times  or (5/2) times another. Any if you do calculation, you need to do it with care or at least do it with the knowledge that an error in one step makes all that follows wrong. The ability to figure well and precisely, so that you answer is correct, shows or suggests the ability to follow methods, one step at a time and one step after another in any subject, and in problem solving as well.

### Algebra Word Problems

If your interest is in solving algebra word problems at the high school level, I would recommend learning how to solve linear equations in several unknowns in an effortless fashion. High school students who can solve linear equations in one unknown are often given word problems where extra variables have to be eliminated to formulate a single equation in one unknown quantity to solve. The trick here is to draw or extract a single equation from the given information. But in most such words problems, it is easier to extract or draw from the given information several linear equations in several unknowns to solve. Each sentence in the word problem gives an equation in one or more unknowns or quantities. Now the algebraic way of writing and thinking can be used to eliminate variables and to solve for the one or more quantities of interest in an effortless fashion.

The algebraic solution of linear equations involves the elimination of variables to obtain say one equation in one unknown. This elimination process may be better done and recorded with algebraic notation. Going directly to one equation in one unknown to solve a problem requires more work to be done with words.

www.whyslopes.com >> Work and Study Tips >> H Jigsaw puzzles and problem solving Next: [H more - Routine to non-routine problem solving.] Previous: [G. Written work formats for developing and showing skill.]   [1] [2] [3] [4] [5] [6] [7] [8] [9] [10][11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]

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.
... 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.