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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.

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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whyslopes.com >> Secondary Mathematics - A Practical Approach >> Chapter 3 - Algebra Starter Lessons Next: [Chapter 4 - Logic for Reading Writing and Geometry etc.] Previous: [Chapter 2 - Why Sets.]   [1] [2] [3] [4][5] [6] [7] [8] [9] [10]

Chapter 3. Algebra Starter Lessons, Effective

The algebraic way of writing and reasoning with letters and further symbols is a great mystery and great source of discomfort for many students and teachers. The mystery invites rote learning and computational approach to secondary mathematics. Many students do not end their studies of mathematics follows from immersion. The textbooks of my youth employed the algebra was of writing and reasoning but did not introduce it. Mastery was left to chance. As a student, I was left to provide my own rationalization for the shorthand role of letters and symbols. With it I was able follow the logic in mathematics and science texts, the mastery made the latter possible, but I lacked the verbal ability to discuss and share my rationalization with others.

Tackling the algebra problem in secondary mathematics and calculus has been a long-term aim. Success can be reported in the form of methods, online here this, for introducing the algebra shorthand way of writing and reasoning using words and gemometry. To learn how, explore the following carefully:

Guide to Site Algebra Starter Lessons.

  • Skip those on 1 Working With Sets. The presentation as is is best left to the last years of secondary school or the first year of college. Correct, but needs to be made more accessible.

  • The section 2 Formula Forward Use - Evaluation is also valuable for the evaluation of arithmetic expressions. The section gives instruction and multiple examples of how evaluate formulas and arithmetic expressions, so that each step of the calculation can be seen and checked as done or later. In evaluation and in the written work needed for solving equations, using them forward and backward, the aim is to show students how present their work for the efficient confirmation or correction of skill mastery. The latter has to be seen to believed.

  • The section 3 Solving Linear Equations begins with a three column stick diagram method for solving linear equations ax+b = cx+d, a method intended to reinforce fraction skills while leading students from an easy manipulative approach with sticks, that is line segments, to the standard solutions of the latter equations, one that involves no sticks. After that this section introduces systems of equations in essentially one-unknown and trianglar system of equations to introduce students to the solution of systems. Most word problems in earlier secondary school are best formulated as systems in essentially one unknown, systems that are easily solved by transformation into one equation in one unknown. In the early secondary school treatment of word problems, students have to pass through mental gymastics to arrive one equation in one unknown. Formulating the word problem as system in essentially one unknown by passes the mental gymnastics - provides a mechnanical alternative, one in which algebraic skills are directly developed. Including systems easy to solve early secondary school aims to provide skill and confidence. In all, examples here show how to write and check solutions so that skill can be seen or corrected as needed. The example are accompanied by message: when a check fails, the mistake or mistakes will be found between the start of the solution and the end of a check. Not all checks are fault-free. Chapter 15 on Solving Linear Equation in site volume contains more examples. The latter include a neat example of the algebraic or literal solution of equations ax + b = c, all to illustrate and introducte the algebraic way of writing and reasoning.

  • The section 4 Computation Rules and Function Notation:

    covers the following.

    The subsequence dicussion of equilavent computation rules sets the stage for the explanation of axioms or rules for algebra as statements that two computation rules with different forms may give the same result. The latter in turn sets the stage for applying axioms or rules of algebra to changing the form of arithmetic and algebraic expressions in ways that leaves their values unchanged. Jump to lesson 7 below and return to the next two lessons as needed.

  • The subsection 5. Real Numbers lesson span the properties of real numbers and their employment as coordinates along a line and in the plane. The lesson on multiplication of real numbers echoes earlier lessons on sign numbers. In practice, computations with real numbers are approximately. The use of calculators allows computations to be done to 3+ decimals, the more the better, for the sake of accuracy. Calculators should be used to show students the decimal approximation of the PI - the ratio of circle perimeter to its diameter, to 5+ decimal places. To many students in primary school do not learn that PI is 3.14 approximately, to 2 decimal places - they and some of their teacher may think the approximation is exact. The use of calculators to display approximations to PI to 4 and more decimal places may correct that mistaken belief. The value of PI can be physically approximated by measuring the perimeter of a circle with a given diameter, and then calculating the ratio. At this level, the latter can be taken as its definition - an ideal for empirical but not mathematical verification - oops! The mathematical fact that PI has an infinite, non-repeating decimal expansion has to be given. At this level, the proof of the latter is not for all. The proof is reserved for undergraduate or graduate students in advanced mathematics courses.

    The axioms for Real Numbers etc lesson describes the arithmetic properties of addition and multiplication only. Subtraction and Division are not mentioned. However, subtraction of number can be recast as adding the negative or additive inverse of the number, while division by a number can recast as multiplying by it multiplicative inverse or reciprocal. Once the recasting is done in an arithmetic expression or computation rule, here an algebraic expression, the algebraically described computation properties in the axioms and earlier oral rules for adding and multiplying multiple addends and terms can be applied to the recast form of the expression or computation rule. Chapter 18 in the site Volume 2, Three Skills for Algebra, tries to describes and illustrate the underlying concepts. The chapter was written in 1995-6 when I was struggling with how to make algebra skills and concepts clearer through the use of words and numerical examples. Chapter 18 in the site Volume 2, Three Skills for Algebra, also describes and illustrate the properties of real numbers given here in a step by step, complementary manner. Two views are better than one, we hope.

  • The section 6. More, Less and Greater Than Inequalities - Comparison includes several lessons on comparing real numbers not by their lengths or magnitudes, but essentially by their position when used as 1D coordinates along a "real number" line. Here I try to bend previous treatments by introducing the notion of 10 be more than -5 by 15, or -1 be less than 6 by 7. I would keep the name of the less than sign as is, but in the context of Real Numbers [as coordinates along a line] rename the greater than sign and call it the more than sign in order to step away from the comparison of size, magnitude or length implicit in the comparison of unsigned numbers. This section may be covered ligthly to introduce the less than and more than signs in the context of signed numbers, that is Real Numbers.

  • The section 7. Axioms, Logic and Equivalent Equations includes five lessons.

    • 1 Equivalent Computation Rules: This first amd key lesson views the distributive law a(b+c) = ab+ac as the equivalence of two different computation rules, one provided by h(a,b,c) = a(b+c) and the other by h(a,b,c) = ab+ac. In the context of area computations, the equivalence of computation rules h and g is implied by the presume equality of two different ways of computing the area of a rectangle with sides of length a and b+c. The notion that h and g denote two different forms of the same function can be discussed later. The geometric context allows the letters a, b and c to denote the length of line segments. The equivalence can also be verified by hand-computation, by calculator-computation and by writing short computer programs to evaluate each side. A counting context is also offered. Examples of other computation rules are included in the lesson.

    • 2 Addition and Multiplication Axioms: The second lesson casts Addition and Multiplication Axioms for Real Numbers etc in terms of equivalent computation rules. The distributive law again appears here.

    • 3 Product Axioms - Two Forms: The third lesson present two forms of the product rule. First, we observe from the consideration of decimals that the product of two non-zero numbers or lengths is nonzero. The word lengths here is used to indicate a possible extension or variant of this consideration: The area of a rectangle with two nonzero sides, the lengths, is "clearly" non-zero. The contrapositive form of this product rule, it can be written as an implication rule IF A THEN B, is the statement or axiom that if a product of two real numbers is zero than at least one of the factors must be zero.

    • 4 Subtraction and Division Axioms: The fourth lesson - more a lesson idea than a lesson, adds subtraction and division axioms. The aim here is to imply how the addition and multiplication axioms may be applied by converting subtraction and division operations into addition and multiplication operations through the use of additive inverses - the negative of a number, and through the multiplicative inverses - the reciprocal of number, respectively.

    • 5 Equality in Algebra: The fifth lessons discusses equality axioms or practices. The same number may have several different representations. Substitution Practice: All can be used interchangeable in computations. Indeed, one representation may be replaced by a more convenient one for the sake of aiding a computation, while results remain repeatable and reproducible. Equality axioms and practices show how one equation may imply another, and vice-versa. That leads to the concept of equivalent equations or equivalent systems of equations.
    • 6 Equations and Systems - Equivalent or Implied: The sixth lessons slowly introduces concept of equivalent equations or equivalent systems of equations just mentioned.

  • 8. Unifying Theme For Algebra: In primary and secondary level mathematics, and calculus too, many tables, rules, patterns and formulas may be employed not only directly - in the forward manner, but also indirectly in a backward manner. Addition and times tables may be used backwards to answer subtraction and division questions, some - not all. In logic, the contrapositive of an implication rule represents an indirect or backward use or formulation. In calculus, differentiation rules will be reversed to provide anti-differentiation or integration methods. In secondary algebra, the study of formulas entails not only their forward or direct use, but also their indirect or backward use. In the UK, a quantity given by a formula is termed the subject of the formula. The formula itself may involve one or several other quantities or variables. Using the formula for the original quantity to find an algebraic expression for one of the other quantities is termed changing the subject. It entails mastery of the algebraic or literal [with letters] way of writing and reasoning. The aim of section lessons is gradually and systematically develope this algebraic way of writing and reasoning. The lessons here are unique. They endeavour to fill a gap I have seen in the development of algebra since my own school days in the 1960s and 1970s where I saw or sensed the algebraic ways of writing and reasoning was required but not systematically introduced. Here is a remedy. It may be easier and simpler than you expect. Good luck.

    This section consists of the following lessons:

    Some lessons duplicate material from site Volume 2, Three Skills for Algebra. Upper level secondary mathematics and science is based on the backward use of formulas. Talking about the latter recognizes and vocalizes a commonality and so gives a unifying theme for the learning and teaching of algebra.

  • The section 9. Proportionality Backwards and Forwards This section includes the following webpages. Some represent lessons. Other represent lesson ideas that tutors or instructors will have to expand and clarify for learners. The coverage is rather rich. It may be spread over mid- and upper secondary school level instruction in mathematics and science.

    1. Proportionality Concepts and Practices- Three plus Kinds of Proportionality Relations, Forwards and Backwards: The lesson says what is (defines) Direct, Joint, Inverse Proportionality and describes how to shift or generate proportionality relations from each others. In a proportionality relation (or equations), algebraically interchanging the dependent quantity with an independent one via a backward use of the relation leads to further proportionality relations of the same or different type. The use of proportionality relations begins with the backward use problem of finding the value of a proportionality constant. Once its value is known, the proportionality relation can use in the forward direction to find values of the dependent variable, or in the backward direction to find values of a so called independent variable.

    2. Proportional Reasoning, algebraic perspective
      . This lesson overlaps the others.

    3. Twenty or so Examples of Proportionality and Multiple Ratios or Proportions: Many examples of proportionality relations appear in high school mathematics and physics. Here is a list of some (most if not all) that may be met. Remember each proportionality relation will be used forward and backwards in multiple ways.

    4. Two and Multiple-Term Ratios, a proportionality constant viewpoint. Fraction and ratios are overlapping concept and have overlapping roles in arithmetic, but they are not identical even though fractions a/b where a and b are whole numbers may be called ratios. In mathematics ordered pairs of whole numbers a and b may appear in coordinate form (a,b) or [a,b]; in ratio form a:b and in fraction form a/b.

    5. Proportionality Constants for Equivalent Fractions: The numerator is proportional to denominators in any fractions equivalent to a given one - a simple matter.

  • 10. Five Examples of Algebraic Reasoning. The material here is optional. That being said, keen or gifted students may test their algebraic reasoning skills by reading here the fraction lesson: 2 Fraction Operations Physical Development. It provides an algebraic perspective of how raising terms can lead and justify methods for addition, subtraction, comparison, multiplication and division of fractions.

Column Multiplication Methods for Arithmetic and Algebra

Duplicate Material, and Deliberately So

Column multiplication methods appear in primary school with decimals and take advantage of place value. The appear in middle to senior secondary school in the multiplication of polynomials. Such methods can be introduced geometrically. See Column Multiplication Methods in General. The latter ideas using letters to denote lengths, subsegments, along the side of a rectangle. The letters limited to two per side could also represents the integral and fractional part of a mixed number. The result is a column multiplication methods for the product of sums, one justified geometrical in the case of positive or unsigned numbers. But operationally the method works in general - we assume that in place of proving for the sake of accessiblity. The multiplication method provides a simple, mechanical replacement for the FOIL method taught in algebra. For polynomials, see Column Multiplication Method.

Remark:Mastery and sanction of column multiplication methods extends the distributive law given in algebra. The recommendation here for the sake of an operational command of mathematics is to give the methods and the law, and allow studies in pure mathematics if taken, to explain the redundancy, or how the law implies the methods. The practical aim of secondary mathematics here is not to give a lean axiomatic base for mathematics, but an operational command based on consistent rules and practices, axioms included. Lean may follow later in specialized courses taken by students mastering the more theorectical aspects of university mathematics, engineering or science. Lean too early is a burden.


The algebra starter lesson end with enriched material - optional readings

A Origins of Counting and Figuring Methods: It includes the following

B Real Numbers Extrinsic Development:
It includes the following masterial:

1 Fractions with Finite Decimal Expansions
2 Counting Digits in Decimal Multiplication
3 Location of Point in Decimal Multiplication
4 Location of Point in Decimal Addition
5 Fractions with Infinite Decimal Expansions
6 Infinite Decimals Ending in 9 repeating
7 Arithmetic with Infinite Decimal Expansions
8 Division and Mulplication of Compound Fractions
9 Division with Digits after Decimal Point
10 Numbers given by Infinite Aperiodic Decimal Expansions
11 Signed Number Addition and Addition Properties
12 Real Numbers Line - Signed Coordinates
13 Arrows and Vectors in a Plane
14 Vector Head to Tail Sums and Resultants
15 Head to Tails in-place Addition - Associative
16 Collinear Horizontal Arrows-Vectors
17 Arrows Rotate to Reverse with Length Unchanged
18 Geometrically Why Vector Addition Commutes
19 Signed Multiples of Vectors
20 Length and Direction of Collinear Vector Sums How to Add Definition
21 Addition of Multiples of a Single Vector
22 Multiplication of Signed Numbers
23 Distributive Law - Two Derivations
24 Signed Numbers - Arithmmetic Properties
25 Mid-way Convergence to Axiomatic Approach
26 More Less Greater Than Comparison
A Signed Number Arithmetic Review
A Modular and Remainder Arithmetic

Learn More - Readings for Now or Later

  1. Volume 2 Three Skills For Algebra. See Chapters 8 to 17 and optional 18. Tutors, teachers and learners should read if enjoyable.
  2. Volume 3 Why Slopes and More Mathematics. The algebra shorthand way of writing and reasoning is employed at full-strength in calculus. Chapters 2 to 7 of this work provide an algebraic light calculus preview, one that stems from the observation that the middle part of a calculus course is algebraically less challenging that the leading parts. This preview may be employed at start of a first course in calculus. It may also be used in secondary mathematics before calculus to a context for the study of slopes and polynomial factorization, all in a way that should advances algebra skills.

whyslopes.com >> Secondary Mathematics - A Practical Approach >> Chapter 3 - Algebra Starter Lessons Next: [Chapter 4 - Logic for Reading Writing and Geometry etc.] Previous: [Chapter 2 - Why Sets.]   [1] [2] [3] [4][5] [6] [7] [8] [9] [10]

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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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