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Algebre Définition d'une variable La raison basée sur les règles et modelés  
Appetizers and Lessons for Mathematics and Reason
A T3 (Teach the Teacher) website

Site Reviews, Easy Reads for the not too young, Etc.

Easy Reads for the not too young

• Logic chapters 1 to 5 in Three Skills for Algebra may help in senior high school mathematics where rules and patterns are employed forwards and backwards,  and may also help  in daily life where understanding and writing agreements or contracts depends on understanding and writing rules and agreements.  Read Logic chapters 1 to 5 for self- defense. 
  
  In playing games, we need read it rules carefully. At work instruction have to be carefully read. In learning,  we need to read the rules and patterns of an art or discipline carefully.
  

• Apart from mathematics, Volume 1A, Pattern Based Reason, describes in general, the benefits, origins and limitations of  some rules and patterns in use everyday life and in politics. There-in lies a request for thoughtful and considerate leadership.
  

• The  algebra chapters 8 to 18   in Three Skills for Algebra use words to introduce or rationalize the algebraic shorthand role of letters and symbols. . Chapter 14 introduces the backward use of a formula in the arithmetic and algebraic solutions to problems. The forward and backward use of rules and patterns is everywhere in mathematics, chemistry, engineering, physics and business.   Volume 2 covers topics that students starting calculus should have mastered..

• Ease or avoid calculus fears & difficulties.  These arithmetic exercises with hints of algebra for students starting calculus may test and strengthen skills, or warn of weaknesses.  This geometric preview  and chapters 2 to 6 in Volume 3, Why Slopes and More Math,   give a context for the senior high school level study of slopes and of factored polynomials. The same material may be employed at the start of calculus to make it easier.  Calculus asks students to calculate derivatives (slopes for straight lines) and to do  sign analysis, that is,  to say identify interval where derivatives or slopes are zero, positive or negative.  While calculus upto the calculation of derivatives is algebraically challenging,  the sign analysis and interpretation as introduced in geometric preview  and chapters 2 to 6 is very simple. Moreover, it helps develop algebraic skills in a way that makes the calculation of derivatives and before that limits, much easier.  Chapters 11 and 13 provide a geometric introduction to the concepts of second derivatives in the context of graphing slope versus position and speed versus time.  Chapter 12 talks about units for slopes and rates of change.   

Skill Development Methods, small & big,  from fractions to calculus

Here is an alternative presentation - one with still more advice and directions. 

1. Fraction Operations (comparison, multiplication & division included). developed  by raising terms to transform into easy cases. 

2. Format for formula evaluation  ( a small idea useful for evaluation of  arithmetic & algebraic expressions in general) shows how to do and present work  more clearly

3. Solving  Linear Equations  via fractional operations on sticks (line segments) makes use of letters and stick-free solution understandable.  

4. Easily Solved Systems of equations in essentially one unknown gives a model for formulating word problems, as a way to ease or avoid  mental gymnastics with algebra. 

5. Integers  introduced Geometrically give a model for the geometric introduction of  most  operations on signed numbers. See Arithmetic How-TOs for more.  The full treatment may come later.

6. Early Geometry in the form of Plan and Map usage provide a practical, self-contained, early math (trig-free) approach to solving similarity problems by drawing diagrams to scale and then measuring on the diagram. Here is a self-standing prequel to trig.

7. Wordy Introductions to Logic may develop precision reading and writing needed in maths, all further studies, home life and work for better performance or self-protection  - Romeo and Juliet make mathematical induction easily to understand anad explain - Chains of reason provide a model for reason in Euclidean Geometry  outside mathematics

8. Three Skills for Algebra (Talking about Numbers, Describing Calculations, Describing when calculations are equal, what is a variable) may ease or avoid  fears & difficulties and clarify concepts that obvious to some, but not  ALL. The algebraic way of writing and reasoning needs to be introduced with words - rationalized.

9. Essay What is a variable  puts words before and beside symbols at a level the calculus or precalculus student will understand.  Mathematicians: Arguments against past verbal descriptions of variables do not apply here. 

10. The Forward and Backward use of formulas (also rules and patterns) is  a unifying theme for senior high school and college mathematics and science. Beyond that, note in the mathematics of  proportionality  this theme appears with backward use (finding the proportionality constant) put first.  

11. Complex numbers & properties introduced geometrically & rigorously before the development of periodic trig functions will simplify  simplify the high school level 2D geometric development of   trig and  vectors.  The simple geometric proof here of the distributive law is the key. The advantages of using complex numbers in the exposition of trig was well-known in the 1940s or earlier.   Why complex numbers were not geometrically developed before trig in the course designs of the 1950s or 60s is a bit of mystery.  Some inquiry or research may explain why.  Since 1976, this site author looked for a simple proof, found or re-invented several, only to learn in February 2010 that giving a geometric proof was an exercise in Secondary Mathematics, A Functional Approach for Teachers, H. F. Fehr,  D. C Heath and Company Boston 1951.  

12. How to quickly  introduce multiplication, addition, subtraction and long division of polynomials geometrically.  Justification is another matter.

13. Function Theory for Senior High School and Calculus Students - Multiple Viewpoints explained and reconciled. 

14. Geometric  and Algebraic (Chapters 2 to 6  in Volume 3) Calculus Previews:  These offer an end earlier studies or a start for calculus in a manner that strengthens algebra skills and eases or postpones calculus difficulties. 

Welcome.  If you are a high school or college level student,  explore site pages to develop or check skills you have just seen in class or explore site material if you would like to become a tutor or a teacher, or to decide whether or not to inform your school or college math department about this site. 

For starters, if you have time,  read  Logic Chapters 1 to 5 and Algebra Chapters 8 to 14 in online Volume 2, Three Skills for Algebra. Logic mastery will help skills and performance at work, in study and YES, in playing games. In each area, rules and instruction  understood exactly may give an advantage.  Chapter 4 gives a Romeo and Juliet view of mathematical induction. There is a unifying theme for algebra in Chapters 14. 

Most of your instructors have not seen the skill development methods, fresh and recycled, in site pages. Invite them to explore this site.  Mention site reviews.  Say what you like or dislike. 

Instructors:  First, this T3 (Teach the Teacher)  offers skill development methods  from arithmetic to the start of calculus.  With regrets, the  mathematics curricula put forth by English educational authorities in North America and the UK, those I have seen,  are not satisfactory.  Technical gaps persist, are accepted or go unnoticed. See below. Such gaps  compound difficulties.   Moreover the underlying premise in old course design (a premise inherited in presented course designs) of preparing for calculus and advance mathematics does not serve common  needs.  We could put first the  common and likely needs of daily life in the streets before preparation for advanced mathematics (calculus) begins in earnest.  In that, skill development should be observable and verifiable. Third,  teachers need to observe and discuss each others methods -  what works, what does not, and when.  Schools and colleges need the foregoing discussion for improvement of and continuity in course design and delivery.    Calling this a  T3 (Teach the Teacher) site is not a claim to perfection. Much can be improved.  It is a invitation to take site methods as a lower bound for instruction, methods to exceed or equal.  

Help Elsewhere for students - Do not put all your eggs in one basket: Three text-based sites  mathsisfun,  purplemath and themathpage are well-done.   The BBC also provides help (examples) in: mathematics and many other subjects for students.   The Bright storm Flash Video Site:  (membership required) for secondary  mathematics US style and some calculus lessons with an emphasis on the mechanics (the how, not the why), Brightstorm flash videos are neat and usually well-done except for notational lapses - doing calculations in place instead of doing one step per line, one step after another.  

Three Gaps - Why Mathematics is harder and less liked than need-be.

What would you give to be an Algebra Power User?  

Would you give time and effort to read  (i)  the site folder on  Solving Linear Equations; (ii)   algebra chapters 8 to 18 and the essay what is a variable. in the book, Three Skills for Algebra.  That is a good start. Look for what is different is in the site intro to algebra.