 |
OnlineVolumes
1, Elements of Reason.
-with foreword for all volumes
1A. Pattern Based Reason
- striving for objectivity, etc
1B. Math Curriculum
Notes
inductive principles etc
2. Three Skills for
Algebra
- unifying themes + study skills
3. Why
Slopes & More Math - previews & starter lessons for
elementary & advanced calculus.
See Volume 2 and 3 if you are preparing kids for calculus.
More Site Areas
1. Help Your Child or Teen
Learn.
2. Linear Equations
& Fraction Skills - Sec I to V level
3. Fractions Ratios
Rates Proportions Units - Sec I & II
4. Euclidean
Geometry - Sec IV
5. Analytic Geometry -Sec
IV & V
6. Number Theory. Sec V
&VI
7. More Calculus Sec V
& VI
8. Complex Numbers Sec
II to VI
9. Qc Maths Education
10. Secondary IV(?) math
11.Real
Analysis College level
12. LaTeX2HotEqn College level
13. Electric Circuits Etc
Sec IV+
14 Français - Sec III +
15. www.whyslopes.com Entrance
Level Pages:
This Calculus Preview and Chapters 2 to 6 in Volume 3 offer lessons to make the hard easier at the start of calculus, or to provide a context for the study of slopes and factored polynomials before calculus.
Your IP
Address &
how to use it
Three Links for Teachers:
(i) First
Year High School Math - Lesson Plans with Fraction Focus
(ii) Second
Year High School Math - Lesson Plans with an algebra focus
(iii) Algebra Lesson
Plans
Parents: Site Area Helping Your Child or Teen Learn covers 1. Speaking Skills, 2. Reading & Writing, 3. Preparing for Science, 4. Math Work Books, 5.Books for Parents, 6. Mathematics for ages 6 to 14, 7. Having Patience -you'll need it. Chaperone your sons and daughters through jumpMath workbooks for grades 3 to 8 along side site lessons for grades 7 to college and material elsewhere. Parents and teachers need to say no for small things of little consequence to build and maintain authority to say no for larger matters. Parental authority: use it or lose it, but do not abuse it.
Lesson Plans and lessons
Secondary I - fractions
& allied concepts (decimals, percentages)
Secondary II - Algebra (arithmetic versus algebraic methods, backward
use of formulas and proportionality equations)
Secondary III - to come(?)
Secondary IV - Functions to Trig & Statistics
Algebra Lesson Notes & Ideas for All levels
Site Eurekas
Page Contents: [top] [Online Books] [Advice & Directions] [Study Tips] [Site Eurekas] [Online References] [16+ Site Areas]
-
Algebra and Fraction Skills Combined. Thanks go to Linda P. for inventing a three column format for Solving Linear Equation with stick diagrams - Teachers take note: - fractional operations on line segments, the stick diagrams, introduces algebra visually while strengthening arithmetic sense and skills. Emphasizing solution checks allows students identify and undo their own mistakes. Here is material for junior to senior high school students and even college students learning or in difficulty with fractions and algebra.
-
Words before or besides symbols: The non-verbal nature of mathematics, that is, the use and appearance arithmetic and algebraic expressions or formulas better written and seen silently than read aloud element by element, has made learning and teaching harder than need-be. While letters introduced as pro-numerals, pronouns or placeholders for numbers and quantities in formulas or algebraic expressions may be called variables, the non-verbal nature of mathematics and its modern written development as marks & symbols on paper has neglected or overlooked the use of words before and besides the shorthand roles of letters, marks, symbols and expression in developing and recording and codifying mathematical calculations and concepts. In other words, we can use spoken words before and beside letters and symbols to the nature and introduction of mathematics clearer and more verbal. In particular, we can describe numbers and quantities, talk about them, without doing arithmetic and before or besides the use of letters and symbols. See the first skill for algebra and the long essay what is a variable to learn more - to put more words in the introduction of algebra. Here is material easily read by avid readers in junior high school and above, adult mathphobics included. Teacher & Tutors: See too Algebra Lesson Plans for more ideas, likely to be effective, in developing algebraic skills at the junior high school to college level.
-
Calculus: The non-verbal element of mathematics appears further in the ed decimal-free view of real numbers, limits, continuity and convergence in calculus and beyond. But a decimal-based view is sufficient for most and it provides a starting point for the decimal-free view. While pure modern mathematics can be developed without diagrams and decimals, pure mathematics is not for beginners nor for many who apply mathematics. Mathematics education needs to depend on diagrams and decimals to provide all outside of pure mathematics, a concrete view. The site introduction to calculus begins with two previews, one geometric and the second more algebraic, which together provide students with an easier path to follow - a re-invention perhaps of a 1960's approach to defining slope functions (a.k.a. derivatives) for polynomials. Fresh or not, the site introduction to calculus shows how to develop algebraic skills gradually to ease or avoid sudden full strength requirements for them in calculus. That is to say, a rearrangement of the order of topics in calculus, or simply an inclusion of a preview beforehand, may make skills and concepts easier to learn & teach. A few well-placed ideas makes a difference.
-
Logic: indirect reason begins with contrapositive form of an implication. Indirect reason continues with proof by contradiction or absurdity. For example, the suspicions of a detective about who did the crime may be allayed by an alibi. With people normally being in two places at once, action at distance is not suspected in most crimes. That being said, in mathematics, the consistency of a system of axioms may not be known, but for a statement that may only be true or false, the inconsistency of a statement with the system may be a reason to add its negation as a requirement for the consistency of the system.
-
Senior High School Mathematics Revisited: An alternate High School Trig & Geometry Program: In the traditional development of trigonometry, six trig functions (sine, cosine, tangent, cosecant, secant and cosecant) are first defined for acute angles using right triangles and similarity principles. Then the same functions are extended using a unit circle in a rectangular coordinate system so that they are defined for all angles. The rewritten [complex numbers] page, December 2005, introduces a new, lean, logical development of senior high school mathematics based on the properties of real numbers and the "covariance" assumption that the sum of vectors is independent of the choice of coordinate systems. The development gives short way to reach and explain trigonometry for all angles & prove the Pythagorean theorem, trig formulas for vector dot- and cross-products, the cosine law and a converse to the Pythagorean Theorem. The foregoing combined with the new methods below offers a lean, alternative program for a full, logical and more accessible development of secondary mathematics, the part needed for calculus & technical or business trades. Missing details appear in the Number Theory site area discussion of the distributive law for real and complex numbers - details whose exposition may be improved - writing is an iterative affair.
-
Fractions, Ratios, Rates, Proportions & Units. Calculus demands fraction sense and also written work with "efficient" operations on fractions without a calculator. Ratios of two numbers a:b and proportional (?) between a pair of numbers may identified with a fraction a/b and all fractions equivalent to it. But binary and longer ratios a:b:c, and binary or multiple proportions may identified with a point in projective space with or without units. Products and quotients of units, addition of like units, and change of units need to be defined for the sake of (i) carrying units in calculations involving rates and proportions, and for the sake of (ii) illustrating addition and subtraction of exponents in products and quotients of monomials. Area content here revisits upper primary or junior high school material, but the presentation, a first draft perhaps, is for students or teachers at a higher level.
After writing site lessons on fractions, thinking about what is important or not, the site author has a greater appreciation for similar & earlier work in introducing and reviewing fraction skills and sense in the last years of primary school or the first year of high school.
-
Number Theory - (Sept 10th, 2005) Explore this development of numbers from tally sticks to the properties of real numbers with digressions into justifying decimal methods for comparison, addition, subtraction, multiplication and modular or remainder arithmetic methods for recognizing multiples of 2, 3, 4, 5, 6, 7, 8, 9, 10 and 11. Some technical parts need further explanations.
Remark The physical (or linear manifold) principle that a sum of displacements in the line or plane should not depend on the choice of unit length and direction implies the distributive law for real and complex numbers or coordinates. The latter principle implies a shorter development of trigonometry which bypasses most of the need for coordinate-free Euclidean Geometry is given or indicated in the site page: Complex Numbers & Trig, outside the site area on complex numbers.
Teachers & Gifted Students: High school mathematics programs in the past have explored multiple paths for the development of skills and concepts. Here is another one. A shorter development of trigonometry which bypasses most of the need for coordinate-free Euclidean Geometry is given or indicated in the site page: Complex Numbers & Trig,
1B, Mathematics Curriculum Notes, Chapters 1 to 12
Chapter 11: Primary School Mathematics
-Inductive principles for course design & delivery require a clear description of where and how skills and concepts may rest on earlier ones, so that difficulties may be explained and remedied by looking for what was missed or forgotten in earlier studies.
Mathematics is a demanding subject. All errors in notation and comprehension need to be identified and corrected. In reading, spelling and writing, students have to learn all the letters in the alphabet, not just some. and memorize spelling. Anything less implies difficulty.
Likewise in mathematics, students have to master key skills and concepts, one at a time and one after another. Anything less implies difficulty.
| for online sessions by chance when I am
online or appointment when I am off. The first session (saying hello) is
free. While talking online, we may scribble on Yahoo, MSN,
Skype or
whiteboards. The twiddla whiteboards has a built-in browser for students,
teachers and tutors in general to import webpages and explore/scribble on
them together. It also has audio in theory. [Session
length depends on supply and demand. Call during off-peak periods
for better service. ] |
|