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Logic and Mathematics Skill & Concept Development with How-TOs Français: 26 pages
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Logic 5 Chapters Arithmetic 10 Steps Algebra 12 Starter Steps & 5 Advanced Steps
Work & Study 23 Tips Geometry 15 Steps Calculus 70 Lessons. See Site Map

Ages 15+: Why study slopes Polynomials Quadratics Why factor polynomials Logarithms Functions
What is similarity Euclidean geometry leanly Coordinates + complex no.s Vectors DC Electric Circuits
Ages 12+: Prime factorization Written work formats Decimal place value Extend arithmetic skills orally
What is a variable 5. Fraction Operations by Raising Terms Solving Linear Equations: Take I Take II


Online Volumes: 1 - Elements of Reason, 2 - 3 Skills For Algebra, 3 - Why Slopes and
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, 1A - Pattern Based Reason, 1B - Skill Development Principles + Troubles

Welcome: Site content may develop critical thinking, improve reading and writing, and build mathematics skills. See online chapters on on logic and pattern based reason.

Teachers: This December 2011, 5-phase framework offers a context for mathematics & logic instruction. Phases 1 to 3 focus on skills with actual or potential value for adult & daily life. College-oriented phases 5 & 4 focus on calculus & preparation for it. Phases 1 to 4 may also serve trades & professions not dependent on calculus.

Site Review: Math resources ... span ... arithmetic, logic, algebra, calculus, complex numbers, and Euclidean geometry. Lessons and how-tos .... provide a good foundation for high school and college ... mathematics. Read more.

Home < Work and Study Tips << S Adding words to algebra

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Adding Words to Algebra

While some calculations may be described with words - think of the perimeter calculation example, many calculations can be represented or given by an algebraic expression or formula that is best seen and grasped in a silences in a glance. Formulas like the compound interest or growth formula and like the quadratic formulas fall in this category of being seen and grasped in silence. The use and development of formulas for lengths, areas, volumes, speed, distance, time, simple interest, compound growth and even or odd numbers, introduces students to the shorthand role of letters and symbols in mathematics.

A. Naming and Identifying Formulas

Outside of algebra, a picture is worth a thousand words. Inside, formulas to awkward to read aloud term by term may also be worth a thousand words. However, both pictures and formulas may be named or identified with words. Outside of algebra, the phrase Mona Lisa identifies a painting of Leonardo da Vinci. Inside of algebra, formulas may be identified by name

  • the compound interest formula
  • the quadratic formula

Inside algebra, descriptive or identifying phrases also identify formulas.

  1. Triangle area calculation formula
  2. Square area calculation formula
  3. Circle Perimeter Formula
  4. Circle area formula
  5. Sphere or ball surface area formula.
  6. Sphere volume calculation
  7. Cube surface area expression
  8. Box volume formula

In these phrases, the words formulas, expression and calculation may be used interchangeable.

In dealing with fractions, we may speak of general and/or efficient methods or formulas for adding, subtracting, multiplying and dividing. At the higher level in the mathematical subject of calculus, the phrases product rules, quotient rule, chain rule, integration by parts identify operations with words.

Words to name or identify formulas and operations is a simple way to expand the role of words in mathematics.

B. Talking about Numbers, Amounts and Quantities

Following the appearance of algebra and logic in and apart from mathematics education, the oral element introduce above may expand. The first skill for algebra which I introduced in fall 1983 emphasized that we can talk about numbers, amounts and quantities without doing any arithmetic and even apart from or parallel to the use of letters and symbols to denote them. So we may talk about and describe numbers, amounts and quantities as being known or not, measure-able or not, private or not, confidential or not, forgotten or not, constant or not, and varying or not. Some of these descriptive terms are not usually part of mathematics, but their presence suggests how we may understand and explain the concepts of a numbers, amount or quantity being known or not, constant or not, or variable or not. I have written an essay What is a Variable that informally introduces and expands the concept before algebra begins. While pure mathematics and logic introduce technical definitions of what is a variable, those definitions are too complex for people just learning algebra. In the first instance, when a letter that denotes a number, amount or quantity that is unknown, constant or variable then the letter too will be called an unknown, constant or variable, respectively. And in the use of formulas, letters often denote numbers or measure whose value is to be given or found. What I am advocating here is a simpler use of language, a use closer to everyday use. While numbers, amounts and quantities may be described or talked about apart from algebra or the use of letters to denote them or their values, doing so while denoting them by letters or symbols expands the role of words in algebra and so in mathematics.

C. Using Formulas etc Forwards and Backwards.

A theme that transcend algebra. A Fourth Skill For Algebra.

Every formula met in mathematics, accounting, science, technology etc may be used directly and indirectly, that is forwards and backwards.

The simple message that the forward and backward use of formulas (direct and indirect use) is part of high school mathematics and beyond names a required skill and allows us to recognize, identify and thus emphasize the most frequent pattern in high school mathematics and beyond.

This message needs to be given explicitly and early in secondary mathematics. Otherwise the underlying skill become part of the hidden, or silent and unspoken, agenda in mathematics courses.

Teachers: Consider combining the www.purplemath.com a two page lesson on solving literal equtions with the message above and the examples and exercises indicated below. The page banner above was Forward and Backward use of equations but it now reflects the purplemath lesson, Solving Literal Equations.

First Site Example

Direct and Indirect Use of the Rectangle Area Computation Formula

Volume 2, Chapter 10, in discussing Direct use of A =WL assumes W and L are given. Indirect use assumes A and one of W and L is given, and leads to the calculation or formulas W = A/L or L = A/W. The explanation of those formulas is a step towards algebraic reasoning - the direct and indirect or forward and backward use of formulas.

More Examples: Formulas for perimeters and areas of squares, circles, triangles, rectangles etc can be used forwards and backwards. Finding the value of a proportionality constant k say in an equation y = k x represents an indirect or backwards use of an equation, a pre-requisite to further forward and backward use of the equation y = kx. The calculation of parameters a and b in y = ax + b (or y = mx +b) represents another backward use of a formula or equation. Quebec students in secondary III have met the forward and backward use of the Pythogorean equation c2=a2+b2 where c is the length of the hypotenuse and the two numbers a and b are the lengths of the other two sides (legs) of a right triangle.

To Do: : Post some details and exercises here to further illustrate and emphasize the forward and backward use of common formulas.

Chapter 10 before the forward and backward use of a formula goes further in showing how to describe a the calculation of a box V = H(WL) and show how to employ substitution (a new concept for students) to go between this formula and V = HA where A = WL. Details are given in the chapter. The details may be easier to grasp if numerical examples are added to this exposition.

Seeing how a box volume formula V = hA and V = h (WL) can be transformed into each other illustrates and may introduce the notion of equivalent expressions. The law applied here is A = WL is a geometric law rather than an algebraic law (like the distributive law). None, the idea that an expression represents a number or quantity and that there may be more than one ways to compute the number or quantity is key to the notion of equivalence. Students thus see how substitution in formulas leads to new formulas, how arithmetic patterns may be used to use formulas directly and indirectly, and how algebraic solutions may be more general or powerful than arithmetic solutions.

Algebraic Exercises:

  1. Find a formula for the area of square in terms of its perimeter (easy)
  2. Find a formula for the area of circle in terms of its perimeter (easy)
  3. Find a formula for the perimeter of square in terms of its areas (harder)
  4. Find a formula for the perimeter of circle in terms of its areas (harder)

The exercises will be easier after reading the first sections of Chapter 15 and Chapter 14 in Volume 2, Three Skills for Algebra.

Remark. In arithmetic to calculus and beyond in mathematics and the physical sciences, and in the use of implication rules in or from logic, all rules and formulas will be used forwards and backwards, some time bothways in the same problem, especially those involving proportionality constants. In the latter, finding one represents a backward use of a formula, after which the formula may used forward or backwards. use.

Teachers & Tutors: Site pages offer better or best practices for providing skills - simpler than expected & comprehensive but for exercises. For your charges, your duty is to study them alone or in groups and develop skill building exercises & activities to share. Start now. The effort here is the best I can do. Others are welcome to refine or exceed it. Please do.

Secondary Mathematics for Ages 11+, A Practical Approach for home-tutoring or -schooling, or for schools & colleges with local curriculum control. Study how to include site content - its skill development how-TOs and innovations into present or future lesson plans - some reading required.

Road Safety Messages and Questions: When and why should you face traffic when walking along a road or cycle path? Is it a good idea to hang limbs outside of cars etc? What gives more protection in a crash: a car, motorbike or bicycle? See too, the BBC-Belgium story Texting and Driving - texting & the impossible test - the article links to a gruesome utube video on the subject

The Logic of Injustice: How Texas sent an innocent man to his death - The wrong Carlos. Some judgments are irreversible. Procescution: Where and when prosectors play to win rather than for justice, guilt beyond a reasonable doubt goes unrespected due to prosecutors who putting winning first, those innocence before the law may be convicted. Some procescutors offices in continuing to accuse after a pardon due to reasonable doubt or innocent being shown, may sucessfully oppose compensaton for false convictions by asserting a pardon individual is still under suspicion. Then the pardoned individual or the latter's estate is not compensation for years or decade of improper or false imprisonment, or for execution. Site chapters on Logic
and some in Pattern Based Reason may slowly lead to greater precision in reading, applying and writing laws.

May 2012, Composition Starting: Pre-School and Primary Mathematics - Quantitative Skills, An Intellectual View, Feedback Welcome:

The 8 Most Popular Site Inlinks

20 Times Table - the most popular site page - popular pages - unexpected.
Fractions & Ratios - with lesson on raising terms to introduce & justify times, division & comparison as well addition & subtraction
Parent Center - See below
Volume 1, Elements of Reason - Intro to all site books.
What is a Variable - best for ages 13+
Written work formats for Arithmetic and Algebra - a skill method and standard!
Complex Numbers Visually - best for ages 13+
Natural Logs, Exponentials, Powers, Roots

Division of Labour: This site offers advice and directions with pointers to resources elsewhere, if known, when they help or lessen the need to write more.

Parent Center: Help your child or teen learn:

Parent-friendly Work Booklets for ages 3+ to 13 Use these or others to check or build skills. Other booklets are available but these booklets allow parents unsure of themselves in mathematics to help their children. The selection acquired in Canada is published in the USA. So it has a US orientation. In retrospect, the selection shows parents what to check with the booklets or by other ways, the choice is theirs. But in retrospect, the selection does not cover integral and fractions liquid weights and measures - ask the publishers to correct that! For ages 9 to 12 say, parents may compensate by showing boys and girls how to use weights or mass, and further measures in food preparation. Beyond that children may be shown how to measure and calculate angles, lengths and areas [proportional amounts too] directly or by using maps and plans drawns to scale. Learning how to gather and measure all the ingredients, pots and pans for a dish or a meal, along with cleaning up sets the stage for like activities or experiments in science courses, and in developing organizational skills, gives boys and girls a head start. Good luck. At the other extreme, more comprehensive than light, if your motto is McCainian: drill, drill, drill then Toronto mathematician and actor John Mighton's jump math organization has jump math workbooks for at least grades 3 to 8 for at-home and in-school use - training sessions for teachers available. Jump math has been expanding to cover older students. Jump Math Samples: plus Fractions for Grades 3-4 & Grades 5-6 [Read] Free Resources grades 1 to 8 [unread - likely to be good]. and

Mathematics Skills For Ages 3 to 14 - technical!

Skills with take home value - A few ideas

Basic skills include time-date-calendar Matters; money matters; map, plan and scale diagram matters;counting, measuring and figuring; decision making with logic and likelyhood; being careful and being aware of the domino effect of mistakes; reading and writing with precision.

Is your child able to add, subtract and multiply amounts of money, work with fractions, work with clocks and calendars, work with maps and plans, and measure length, weight-mass and volume? Schools may promote your son or daughter without providing basic skills in reading, writing and arithmetic.

Arithmetic and Number Theory Skills

Algebra Starter Lessons

1 Working With Sets
2 Formula Forward Use - Evaluation
3 Solving Linear Equations - Skip first step with students able to solve 1 eqn in 1 unknown.
4 Computation Rules and Function Notation
5 Real Numbers
6 More Less Greater Than Inequalities and Comparison
7 Axioms Logic and Equivalent Equations
8 Unifying Theme For Algebra
9 Proportionality Backwards and Forwards
10 Examples of Algebraic Reasoning
A Origins of Counting and Figuring Methods
B Real Numbers Extrinsic Development


Site coverage of formuala evaluation format, of computation rules and axioms, and of the forward and backward use of formulas and proportionality relations lessens the amount of natural talent needed to understand and explain algebra.

Geometry - maps plans trigonometry vectors

1 Maps Plans Measurement
2 Euclidean Geometry - Constructions + extras
3 Cartesian and Polar Coordinates
4 Lines and Slopes Take 1
5 What is Similarity
6 Trigonometry first steps
7 Complex Numbers
8 Unit-Circle Trigonometry
9 Lines and Slopes Take 2 with tangent function
10 Intersecting Straight Lines and Transversals
11 Parallel Straight Lines and Transversals
12 Function Translating and Rescaling
13 Vectors
14 Degrees to Radians and Radians to Degrees
15 Arc or Inverse Trigonometric Function

Pre-Teen and young teen mastery of skills and practices which should be common with map-plans-diagrams drawn to scale, contour interpretation included, has actual or potential take-home value for daily- and adult-life in solving routine problems. Elevating some practices to principles, axioms or postualates, provides a base for analytic and Euclidean geometry, an analytic view of similarity, and an efficient mastery of trigonometry and complex numbers. Right triangle trigonometry provide an analytic alternative to solving geometric problems by drawing diagrams to scale.

More Algebra

Natural-Logarithms Exponentials Powers Roots
Five Polynomial Operations
Quadratics Geometrically
Functions
5 Factored Polynomial Sign Analysis Examples
Rewriting algebraic substitution as function substitutions

The first topic leads to a full high school level theory for the forward and backward mastery of growth and decay models and for definition, range and domains of radicals, roots and powers. The next two topics make quadratics and polynomials easier to learn and teach. Site coverage of functions turns vertical and horizontal line rules into computation methods for evaluating functions.

70 Calculus Starter Lessons

Calculus Lessons Elsewhere:

  1. How to Ace Calculus: Street Wise Guide - Mostly Text.

  2. Flash Video for Calculus Phobics

They cover basic topics in ways likely to complement your notes, your textbooks and site material. When Goldilocks trespassed in the house of the three bears, she found three bowls of porridge, two not to her liking, and one just right. Different bears have different tastes. As invited guest here and elsewhere, if one or more explanations is not to liking, try another. It may be better or just right.

Unsolicited Advice

Learning to do and high marks if it comes to easy is often deceptive - light rather than deep. For that reason, students with learning difficulties determined not to let it get in their way may go deeper and farther than those with none. High marks, if the come easy, may be deceptive - provide a too light and not a deep mastery. That could have been your problem in secondary school, one that leads to comprehension shock or difficulties in calculus and more generally in the first year of college. Bon Appetite.


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Home < Work and Study Tips << S Adding words to algebra

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Logic-Reason for all
Careful Thinking
Chains of Reason
Mathematical Induction
Responsibility
Bodies-of-Knowledge

Arithmetic - Ages 10+
1. Deciml Place Value - fun
2. Decimals for Tutors
3. Prime Factors - quickly
4. Fractions + Ratios
5. Arith with units - science

Geometry
1 Maps + Plans Use
2 Euclidean Geometry
3 Rct +Polr Coordinates
4 Lines-Slopes [I]
5. What is Similarity
Algebra Starters - the base
1. Better Work Format
2. Solve Linear Eqns
3. Computation Rules
4. Axioms, Item 3 Viewpnt
5. Formulas Backwards
More Algebra
Logarithms-ax & m/nth roots
Five Polynomial Operations
Quadratics Geometrically
Functions || Vectors too
Arith. Skill Check+Answers
Calculus Prep/Preview
What is a Variable
Why study slopes
Why factor polynomials
Complex Numbers
Limits + Continuity

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