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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.

Bright Students: Top universities want you. While many have high fees: many will lower them, many will provide funds, many have more scholarships than students. Postage is cheap. Apply and ask how much help is available. Caution: some programs are rewarding. Others lead nowhere. After acceptance, it may be easy or not to switch.

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 >> Arithmetic and Number Theory Skills >> 5 Integers

     1 Integers as Coordinates.
     2 Integers Multiplies of a Unit Moverment.
     3 Adding Movements with same direction.
     4 Adding Movements wiht opposite directions.
     5 Zero Movement and Additive Inverses.
     6 Multiplication by Natural Numbers.
     7 Multiplication by Signs.
     8 Multiplication by Signed Numbers - Integers.
     9 Multiplying Integers.
     10 Integer Multiplication Formulas.
     11 Adding Integers - Formulas and Examples.
     12 Adding Integers - More Examples.
     13 Subtraction with Additive Inverse [swf file]
     B Integer Long Division - Multiple Choices.
     C Divisibility by 11 - Integer Recognition Method.
     D Remainders Modulo 11 Pair Rule [swf file]

Folder Content: 16 pages.


These lessons and three appendices include exercises to consolidate and extend understanding of integers.

The lessons provide a hands-on, thought based development. Lessons assign three geometric roles to integers to develop and explain rules for integer arithmetic.

  • Role I: Integer are first introduced as coordinates for points on a line, where adjacent points are a unit distance apart.
  • Role II. Integers then serve as multipliers in the definition of integer multiples of a unit movement, integer multiples that can be added and multiplied by whole numbers and then integers.
  • Role III. Integers themselves may describe movements, how many steps to the left or right, along a straight line, and so can be identified with movement, integer multiples of a unit movement, now called a step. That third role or identification leads allows integers to be added and multiplied.

Exercises are included in most lessons.

Three appendices cover an associative law, division quotient and remainder options for integers, and how remainder arithmetic explains the alternating sum of digits test for divisibility of decimals by 11.

Extension: This three role, geometric development and explanation of integers starting with unsigned whole numbers provides an example to follow for the development and explanation of (i) rational numbers starting with unsigned fractions; and (ii) real numbers starting from unsigned (positive) real numbers or their decimal representation. The site exposition of complex numbers continues the geometric development of number theory.

The Main Lessons

Saying how to do an operation defines it.

  • Lesson 1: 1 introduces the first role of integers a signed whole or natural numbers serving as coordinates for points a unit apart along an infinite straight line.
  • Lesson 2 introduces the second role of integers as multipliers in providing integer multiples of a movement, a unit movement along a line.
  • Lesson 3 shows how to add integer multiples of a unit movement when those multiplies have the same (like) direction.
  • Lesson 4 shows how to add opposite and same direction integer multiples of a unit movement.
  • Lesson 5 introduces the idea of additive inverses for integer multiples of a unit movement. That answer the question what movement added to another will result in a zero net movement.
  • Lesson 6 says how to multiple integer multiples of a unit movement by a natural number, zero or a whole number.
  • Lesson 7 shows how to apply the positive and negative signs + and - to a vector. The application of the plus sign + to a movement is the identify operation while the application of the negative sign - to a movement yields its additive inverse, that is, a movement with the same direction and opposite direction.
  • Lesson 8 shows how to multiply an integer multiple of a unit movement by an integer. Since each nonzero integer may be written as a plus or minus sign prefixed to a whole number, we take multiplication by a nonzero integer to be the operation of multiplying by its whole number part (also known as a length or magnitude) followed by an application of its sign to the latter product. There is a further refinement or extension of the second role of integers as multipliers of movements. The first role is to provide signed coordinates for points a unit distance apart along a line.
  • Lesson 9 identifies integers with movements - so many steps in one direction or another. There-in lies a third role for integers - a movement role. That leads to an extension of lesson 8 in which multiplying integer multiples of a unit movement turns into multiplying integers by integers.
  • Lesson 10 continues lesson 8 and 9 to obtain rules for products of signs and products of integers that be easily applied and learnt by rote, or seen as a consequence of lessons 1 to 9.
  • Lesson 11 applies the lessons 3, 4 and 5 on the addition of integer multiples of a unit movement to the third movement role of integers. That leads to methods for addition of integers.
  • Lesson 12 echoes lesson 11 in providing rules for adding integers, rules that are easily understood and repeated, with or without explanation of why they work.

The Appendices

  • Appendix A (optional reading) continues lesson 11in an algebraic manner (a departure from the presentation in the other lesson) states an associative law for multiplication of integer multiples of unit movements by integers. That may - proof to be written - imply an associative law for integers.
  • Appendix B (more optional reading) looks at the long division method for pairs of whole numbers for pairs of nonzero integers - long division of their whole number parts (lengths, magnitudes) implies several dividend = quotient times divisor plus remainder relations, where the sign of the remainder may be like or unlike the sign of the dividend and/or divisor.
  • Appendix C links to the alternating sum of digit test for divisibility by 11 to remainder arithmetic, modulo 11, with the aid of integers. The link here is given without proof. Indeed the whole treatment of remainder calculations provides rules to apply and follow to repeatable and reproducible results, without full (algebraic) explanation of why these practices work. The explanation why is or or is to be included in the site coverage of algebra.

www.whyslopes.com >> Arithmetic and Number Theory Skills >> 5 Integers

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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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