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Mathematics and Logic - Skill and Concept Development

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30 pages en Francais || Parents - Help Your Child or Teen Learn
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.
- 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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www.whyslopes.com >> Skills with take home value

Mathematics with Take-Home Value

Essay, Draft form.

Before a child goes to school, many mathematics, logic and language skills will appear in home life. Which ones have the most take-home value may vary. Different communities may have different needs. Some may serve practical ends. Many skills are required in buying and selling goods, property and services, in making informed decision in the presence or absence of certainty. Practical applications done and recorded in an observable and verifiable or correctable manner set an empirical standard for rigour, one in which the domino effects of wishful thinking and mistakes have immediate consequences.

We need to remember that the time in school of a child or teenager may be limited because of family circumstances or because of a voluntary termination of schooling. Thus Mathematics skill development should focus in the first instance on what TCPITs, the common person in the street, may need sooner or later in daily or adult life. There-in lies a place for a broad development of mathematics and logic skill with take-home value. That development should begin before the preparation for college studies, but may overlap because the latter preparation in part may make skills with take-home easier to learn and teach.

In daily life, many skills are mastered and used without a full comprehension of how and why they work. In preparing a meal, very few if any of us full understand the digestive system. Machines and mechanisms may be also be employed without comprehension of how they work and why. That being said, in counting, figuring and measuring, the ability to do is required while comprehension is optional. In the case of methods with take home value, the first task is to provide skill with methods without overwhelming students with explanations of why, but explanations included to the extent that they aid skill development, but do not overwhelm it. Skill mastery is the first aim of instruction in methods with take-home value.

Where primary instruction in methods with clear take-home value overlaps secondary instruction preparing students for college and also prepares students for further skills that have take-home value, Primary instruction has skills with take home value to development explanations of how and why rules and patterns give results is must to the point that it aids skill development without overwhelming a student. Beyond that, where secondary instruction preparation for college studies in business, engineering, science, technology and mathematics education should gradually emphasize comprehension of both how and why, with the explanation that skill without comprehension becomes limited. That being said, the theory or explanation of how and why should be lean. Where topics do not have immediate take-home value, reasons for them should be given. See site account of secondary ends and values - those to emphasize in college oriented skill development.

A. Time and Date Skills

People have a sense of time, that before, after and simultaneous. That sense of time and sequence holds not only in the right now, but also in the telling of stories of past and future events, and also in fiction.

Modern family life may be arranged in accordance with the time of day, day of the week, the month, the season and the year. Every one has a birthday. Year round, daily life is governed by seasons and the day of the week, what time to rise or eat, what time to sleep, when to work and study and pray. School and work life are run or organized around time or schedules. Events occur before, after or the same time. Civic and religious calendars provide schedules, celebrations and holidays. The teen or adult may have study and work schedule as well.

B. Money Skills

People buy and sell goods and services. People work for a living, There are budgets to balance in private life and in business. Money is counted, added, compared, subtracted, multiplied and divided. Life in families and communities often involves money matters. Talking about ends, values and methods for handling money at home, and in buying goods and services, or balancing a personal or business budget would provide a context and motivation for care and diligence in arithmetic with unsigned and even signed numbers - the use of the latter is optional. Students may shown how calculate net worth by adding subtotals. See corresponding remark below.

In the case of money matters, students may master the forward and backward use of compound interest/growth formulas and geometric summation formulas by rote initially, simply because of the possible take-home value. For example, numerical examples may be employed to empirically suggest or confirm the geometric sum formula. To ease or avoid raising the level of algebraic complexity, formulas may given and illustrated instead of being derived independently or from each other. In the first instance, the ability to apply methods with take-home value in a repeatable and reproducible manner is still more important than know-why.

C. Spatial Skills

Child may learn to recognize the spatial positions of before, after, behind, ahead, above, below and besides. That learning may come from experience or from others at home and in class. The tutor or teacher may check or develop this knowledge through simple questions and activities.

The spatial sense of up and down, or above and below, may be easier to learn and remember than the sense of left and right.

In my child-hood window on life, reading of words and decimals from left to right was no problem. But I still had difficulty identifying left and right. I have no experience with literal and numeric dislectsia, I will ask a question. Would it be easier for some students with dislectsia in reading and writing English, or in reading and writing decimals horizontally have less difficulties in reading and writing words and decimals if the latter were written vertically. Column place value methods for counting and figuring with decimals could easily be transposed to provide row place methods. Calculators displays could also be transposed.

D. Geometry, Navigation and Construction Skills with maps and plans

Maps and plans drawn to scale (or not) are everywhere. In going to school and in traveling children may see maps and plans. Larger schools, colleges and workplaces may use building plans or maps to provide directions.

Geometry in part consists of direct measurements with units of length, on rulers and tape measures, with angle measures in terms of degrees and fractions of a circle or revolution. Students may identify a right angle with a quarter of circle before thinking about quarter revolutions. Scaling of Direct Measurement of lengths and angles on maps and plans drawn to scale then gives another method, not direct, for finding and using lengths and angles not measured directly. Triangle construction algorithms ASA, SAS, SSS and even AA(A) will be included among methods for drawing triangles, rectangles and the circles to scale or full size.

In travel and in construction, we may use maps drawn to scale to estimate or calculate lengths and even areas, and thus solve problems without or before any knowledge of higher mathematics in the form of trigonometry. Solving problems with maps drawn to scale has take-home value, and may be emphasized and illustrated in geography and science lessons. Reading maps and plans, understanding contour levels, are all parts of quantitative skill development.

For geometry the foregoing students may learn on paper to evaluate a formula for an area or volume given the necessary lengths and measures. They should also be able determine those lengths or measures from hands-on experience with the actual figure or scale drawings of it. Instruction should point out applications robustly and fully.

E. Counting, Measuring and Figuring Skill

Besides counts and measures of time, dollars and length, master measures of area and volume, and of mass and weight. In buying and selling goods and services, and in making things - that includes cooking and construction - people use and combine measures alone and in proportions. Examples include speed and the cost of a unit (a prequel to the discussion of per unit rates) of a good or service.

  1. Express Measure & Counts in terms of given and alternate units of each, respectively.

  2. Add, Compare and Subtract Counts and Measures - when possible.

  3. Multiply: Form the product of both with a number.

  4. Divide Measures & Counts by another measure or number.

  5. Each rate has a reciprocal. For example, the rate of 5 apples per 10 pennies is equivalent to the reciprocal rate of 10 pennies per 5 applies. The transformation of one rate into reciprocal may be taught along side the use of rates in arithmetic. Mastery here may be verbal. Skill here can be extended with the arithmetic with multiples of units and fractions with such mutliples in numerators and denominators.

  6. Example of the chain rule for rates is given by observation that 5 apples for each 10 oranges, and 10 oranges per 20 pears implies 5 applies per 20 pears as well. With the mastery of arithmetic with fractions, the chain rules can associated with a product of fractions with units.

Exposure to measurement matters in the home and in shopping will depend on the family and community life of students. Due to the variation, skill development in school will have provide the missing experience and a context for it. But all has take-home value.

F. Chance and Risk Skills

In decision making, not all is certain. Risks are present even for people who avoid games. Learning about chance and probability may help avoid situations or decision where the risk are high, or help in making decisions that lower risk and make the chances of success greater. Again, due to the variation, skill development will have provide the missing experience and a context for it.

G. Logic and Decision Making Skills

Logic mastery in mathematics or apart - say in a reading and writing course - may lead to greater care or precision in reading and writing, and so avoid or lessen difficulties in studies and in work. Precision in reading and writing may follow here from study of logic and from an awareness of the domino effect of errors.

The leading chapters of Volume 2, Three Skills for Algebra, develop deductive reason (logic mastery) in a math-free way. Altogether, those chapters hint at the partial Euclidean organization and codification of rule and pattern based arts and disciplines.

Awareness of
the difference between say A if B and saying A if and only if B (or equivalent expressions) will sharpen reading and writing. And seeing how to chain implication rules together will help reasoning in general. These two elements of logic and further elements may be introduced when students are ready for them - the age level for that may depend on the student. The net result should fewer difficulties in work and study, and better results in general.


People in health care and instruction may sense an obligation to help others. But in buying and selling goods, properties and services, and in working for a living, decision may require money skills and logic or language skills to negotiate, to recognize the good and bad aspects of deals and contracts, and to recognize agreements that should not be made.

H. Proof

The aim of instruction in the above areas and in the parts of arithmetic they require is to develop observable and verifiable skills and know-how. As part of that, we will show learners how to do and record numerical and geometric steps in learn but sufficient show work formats that allow steps and results, from start to end, to seen and confirmed or corrected. Emphasizing that care and patience, and precision too, are needed to avoid the domino effect of errors and approximations in short and long chains of reason will be emphasized. The ability to figure well, and to read and write with precision, is an observable sign of intelligence of the practical kind for work and studies in general.

The word proof is usually associated with higher mathematics. But figuring with decimals, fractions and geometric reason on paper is based on doing and recording written and drawn steps, one at a time, one after another, in an observable and thus confirmable or correctable manner.

Errors are possible in arithmetic and geometry. When a student or another checks the students work, that work is serving as visible proof of the correct application of a skill or method, a proof that can be checked for fullness and correctness. The work done and recorded should show and demonstrate that figuring steps have been used properly. Results should be repeatable and reproducible in routine applications of mathematics methods, even before any acquaintanance with implication rules A if B. The call to show work in school or in a business, account keeping included, represents a common form of proof.

The students who does and records the steps of a calculation, one at a time, one after another, carefully, without errors, is not only proving skill but also providing a proof of the result. The recorded, error-free work is proof if it provided by correct method even if the composer of proof does not have a full comprehension of why the method is correct. That is situation is likely to arise in arithmetic where skill with arithmetic methods has more take home value than full comprehension of why the methods work. Rigour and proof in skill development is based on the observable mastery of skills, with steps done and recorded in a way that gives and implies a repeatable, reproducible and verifiable presentation and results. In that comprehension of why methods give results is optional.

I. Routing and Non-Routine Problem Solving

Rigour in observable skill development and problem solving does not refer to deductive logic, see the next item, it refers to observable and thus hard results, done and recorded for the doer and others to confirm or correct. In this the record or trace of figuring and drawing steps implies correctness, modulo that of the underlying methodsd. Steps done and recorded mechanically represent the underling rule and pattern based reasoning - for better or worse.

Rigourous skill development begins with an awareness of the domino effect of mistakes in following steps or instructions. That should imply more careful work and study effort. Proof and problem solving skill further includes approaching problems or puzzles with a systematic or deliberate trial and error exploration of what might fit or work. The combinatorial trial and error solution of jigsaws puzzles with edges first provides an example.

Routine problem solving is largely mechanical in that one requires or looks for existing rules and patterns directly and carefully. When a problem does not appear to be routine, or when routine solution methods are not satisfactory, the habit of always looking for pieces that fit or methods that may work represents a creative, combinatorial and opportunistic approach. Problem solving may ends with an awareness of what is is not yet routine or solutions not found. That leaves room for further thought. But problem solving may also end the careful or diligence application of rules and patterns, the accomplishment and record of steps that the solver or others may follow now to later to check and confirm results. With work done and recorded in an observable or repeatable and reproducible manner, solutions becomes rigourous. Moreover, the paths they follow may make further problems of a like kind, routine.

www.whyslopes.com >> Skills with take home value

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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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