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# Mathematics and Logic - Skill and Concept Development

Questions: Will these ends and values motivate? Will smaller & more steps in site lessons and lesson ideas build skills and confidence?
Should we emphasize how ideas & methods depend on earlier ones? Does concept & skill mastery need to be checked to be believed? What is a Variable?

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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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 Lessons

##### For Logic

These online chapters may amuse while leading 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. Site Theme: Different entry points may be easier or harder for knowledge mastery.

##### For Arithmetic

Deciml Place Value - funny ways to read multidigit decimals forwards and backwards in groups of 3 or 6, US-CDN, UK-German and Metric SI style.

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.

##### For 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.

www.whyslopes.com >> Mathematics Skill Development Framework >> ------- More Algebra and Slope-based Calculus Preview Next: [ ------- Implementation Notes.] Previous: [ ------- Euclidean and Analytic Geometry with Complex Numbers and Trigonometry.]   [1] [2] [3] [4] [5] [6] [7] [8] [9][10] [11] [12] [13] [14] [15] [16] [17]

## More Algebra Islands

### Mathematical Recursion and Induction

The discussion of mathematical induction and recursion represents a standalone island of know-how combining logic and algebra. Mathematical induction and inductive definitions may be covered to explain or derive formulas for geometric sums and binomial coefficients. Mathematical induction may be woven into the further development of skills and concepts to add to the logical structure of mathematics as seen in this phase and the following ones.

Mathematical induction and recursion, starting with site logic chapters, is not a difficult topic to develop. Its coverage sets the stage for factorial notation, summation notation, product notation, binomial coefficients and summation formulas for sums: arithmetic, geometric and other. In any light development of calculus, that is calculus limited to polynomials and their ratios, mathematical induction would also play a role in obtaining rules for differentiation of polynomials directly and in composite functions where the last applied or outer function is polynomial.

### Algebra Island: Natural Logarithm and Powers, etc.

The forward and backward use of the compound interest formula and encounters with radical and powers there-in provides a context or hook for the algebraic description of the properties of the natural logarithm, its inverse the exponential functions, and the expression of radicals and powers in terms of the latter inverse and the exponential powers. Deriving and providing those expression shows how and when radicals and powers can be computed.

Site algebra steps shows how the natural logarithm and its inverse or antilog, the natural exponential functions may be introduced numerically with an algebraic description of their fundamental properties. Following that, their properties may be used to show how to compute radical, powers and more exponentials using the natural logarithm and its inverse, the exponential function. Showing how to compute the latter with the aid of logarithms and its inverse indicate when they can computed as well. In essence, the site treatment of the foregoing provides unifies the otherwise disconnected treatments that I have seen in secondary mathematics.

### Algebra Islands: Sets Concepts and Notation - A Balanced Approach

In mathematics for practical use, set concepts and operations of membership, union, intersection, and complements, an may help in the description of sets of numbers, in counting objects by dividing them among disjoint or overlapping set - think of Venn Diagrams; in discussing and illustraing logic - think again of Venn Diagrams; in the algebraic codification of probability theory in this phases, and in the algebraic description and codification of functions and their properties in the next phase. Before mastery of the algebraic statement of conditions to define subsets - see the safe set theory below, sets may be defined by a roster method - listing their members; and sets may introduced or depicted Geometrical as a set of points in the plane in as a set of points in a Venn Diagram.

Note: When sets are introduced by the roster method, we may allow elements to be repeated in the list, and then introduce a roster shrinkage method to make the list shorter. As a first example,

$\{\frac{25}{5}, 15, 20, \frac{50}{10}\} = \{5, 15, 20 \}$

Then in talking about rational numbers as being the set of unsigned fraction $\frac{p}{q}$ where p and q denote whole numbers, we do not have to talk about duplicate values or equivalent fractions in the list because roster shrinkage will replace one by a single representative, say the one in which p and q are relatively prime.

#### Safe Set Theory

Everyday language uses the terms set, group and collection interchangeable. More formally, set concepts and operations in mathematics are codified in via the concepts and operations involving membership, union, intersection, complements, and what has been called set builder notation. The latter in fact would be better described as subset formation notation. In the eighteen hundreds, the too free or wild creation of sets led to self-reference paradoxes. To make set formation safer in the discussion of numbers, sets of numbers or their set theorectic equivalents were assumed. Then set formation was limited to forming subsets of a given set A, to set products sets

$A \times B = \{ [a,b] | a \in A, b \in B\}$

that is sets of ordered pairs from two existing sets A and B; and to forming the power set 2A- the set of all subsets of an existing set A. The set operations of intersection, union and complement by taking placing a larger set may be cast as safe.

### Algebra Island: Probability Theory and Counting Methods

The coverage of probability is reserved to the end of phase 3. It requires coverage of function notation and review or coverage of set concepts and operations: membersbip, union, complement, intersection. The counting principles employed to the find the number of ways events or outcomes can be occur entail mastery of recursion in computation and in mathematical induction.

From home and elementary instruction, students may be familar with likelihood and chance. That is, Earlier use of likelihood in deciding whether or not to take a chance will be algebraically extended here into set-based probability theory. Mastery of probability and allied concepts of expected value may be presented as a base for taking or avoiding risks. The question of when to buy a lottery ticket or to visit a casino may be addressed in class.

### Inverse Functions in Calculus or Before

Given the graph of a function y = f(x), one may use a vertical line method to find the value of f(x) from the graph, given the value of x in its domain. In that traditional, the graph of y versus x takes the y-axis to be vertical and the x-axis to be horizontal. To find or study the inverse of the function f, we may graph x versus y for the equation x = f(y). The table of values and graph of the latter is given by transpose the table of values and graph for y = f(x), provided we still keep the values of y along the vertical axis and those of x along the y-axis. On this graph of x =f(y), we may still try to apply the vertical line method to find y, given x. That works if the graph as a whole or a restricted portion of it, satisfies the vertical line rule. The foregoing provides a simple "dual" way to introduce and define a left inverse of a function. If we assume the transpose of a graph also transposes it tangent line, a formula for the derivative of the inverse is also suggested if not rigourous implied. For rigour, see the site treatment of this matter. The fact that the transpose [b,a] of a point [a,b] are reflections of each other may be mentioned in the foregoing, but is not critical. And in this development, the modern mathematics definion of a real-valued function y -f(x) as set of points in the plane need not appear. We simply take a formula for f(x) and it graph as two different computation rules for the same function.

www.whyslopes.com >> Mathematics Skill Development Framework >> ------- More Algebra and Slope-based Calculus Preview Next: [ ------- Implementation Notes.] Previous: [ ------- Euclidean and Analytic Geometry with Complex Numbers and Trigonometry.]   [1] [2] [3] [4] [5] [6] [7] [8] [9][10] [11] [12] [13] [14] [15] [16] [17]

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 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. ...

##### For Geometry

Maps + Plans Use - Measurement use maps, plans and diagrams drawn to scale.

Euclidean Geometry - See how chains of reason appears in and besides geometric constructions.

Coordinates - Use them not only for locating points in the plane or space.

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 assumes no knowledge and some knowledge of the tangent function in trigonometry.

What is Similarity - another view of using maps, plans and diagrams drawn to scale in the plane and space. May buildings in space are similar by design.

##### 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.

Why factor polynomials - this 1995-96 lesson introduces calculus skills and concepts. It may also may be given to introduce further function maxima and minima both inside and at the ends of closed intervals.

Check Arith. Skills - too many calculus and precalculus students do not have strong arithmetic and computation skills. The exercises here check them while numerically hinting at equivalent computation rules.