www.whyslopes.com:  Parents:  Help Your Child/ Teen Learn
Appetizers and Lessons for Mathematics and Reason  (Français)
Online tutor-teacher seminars available

Online Volumes (Book Orders)
1,  Elements of Reason. (1996)
1A. Pattern Based Reason  (1995)
1B. Math Curriculum Notes (1996)
2. Three Skills for Algebra  (1995)
3.-Why Slopes & More Math  - (1995)

Read chapters 1-5, 8-12, 14, 16 & 17 
in  Three Skills for Algebra 

More Site Areas 
1.  Solving Linear Equations  (04-2005)
2.  Fractns, Rates Proportns, Units  (2006)
3.  Algebra, Odds & Ends, HS level-(2001)
4.  Euclidean Geometry  & Cmplx NOs-new 
5.  Analytic Geometry/Functions (2006)
6.  Number Theory. (2006-7)
7.  Complex Numbers (2001)
8.  Calculus Introduction (2005)
More Site Areas 
99   Real  Analysis (1995)
10. Secondary IV(?) maths (2006-7)
11. Math Education Essays  (2006-7)
12. LaTeX2HotEqn: (2004)
13. Electric Circuits Etc  (2007)
14. Quebec Math Ed (2004 -8))
15  Prequel to the How-TOs (06-2008)
16-Virtual-Classroom (Take a peek)
Tutor -Teacher-How TOs-(08- 2008)
1. Arithmetic Reference
2. Algebra 
3. More Algebra 
4. Geometry  
5. More Geometry
6. Calculus
7. Logics in Maths
Site Tour:  Click here,  here,  herehere,
 here,   herehere and  here.
On leaving, if your next step would be a 
www  search, do that here.


Employ an online or offline tutor at your own risk from 

AU:  tutorfinder.com.au
CDN :  findatutor.ca 
CDN: .i-tutor.ca
CDN: Montreal Tutors
NZ:   findatutor.co.nz
UK:   tutorhunt.com 
UK:  tutors4me.co.uk
USA:  wiziq.com
USA: ziizoo.com

YOU are better than YOU think. Show yourself  how:

      |      
//  _   _ \\
/\             /\
  <|  (o)   (o)   |> 
 \     | |      / 

 For better work & study skills, read logic chapters 1 to 5  in  Three Skills for Algebra. Sooner is better. Good luck.

 -/[]\- 
||
   / \_ 
 ||||||||||||||||||||||||||||


 Logic Mastery
 Amazing, Amusing, Amorous,  Delicious, Delightful, Edifying, Strengthening Elixir. 
It eases work & learning difficulties Makes the hard easier. Opens eyes. Leads to greater precision.
in reading and writing

Do not leave here without it -  Logic mastery  will develops critical thinking, improve reading and writing, and give a firmer base for work and studies at many levels. Good luck.

   |      
//  _   _ \\
/\             /\
<|   (o)   (o)  |> 
     | |     |
   \             /   
\    =   /

Caution: Site advice is approximately correct, for some circumstances, not all. Site How-TOs are logically developed, but not tried and tested. That leaves room for thought and refinement..

 -/[]\- 
||
  _ / \     
 ||||||||||||||||||||||||||||
 

After logic  (a) continue reading Three Skills for Algebra, chapters 8 to 14  and do so alongside site area on solving linear2007 Equations ; or (b) see this calculus starter lesson and Volume 3, Why Slopes  & More Math, chapters 2 to 6;


For online automated help in senior high school maths & calculus, visit  quickmath.com  For Automatic Calculus and Algebra Help with derivatives, integrals, graphs, linear equations, matrix algebra, visit calc101.com  With  overlap, each site quickmath & calc101offers a different range of services, some free, some not, all based on webmathematica. Good luck.


Explore collaborative whiteboards from groupboardtwiddla  or scriblink.


Chapter 4

Repeatable & Reproducible Results

Previous: One- Versus Two-Way Implications

When we do arithmetic, we follow rules. If each of us does not make a mistake, we will get the same result as each other. Each calculation is reproducible. Other people know how it was done. They can repeat it, and see if they agree with our results. When we get a result by following rules or instructions, other people can check our results. All they have to do is follow the same rules.

Just as we have methods for doing arithmetic carefully and precisely, we also have suggested methods for thinking carefully about what to accept, to do or to decide. The ability to read and understand rules or suggestions precisely is needed not only in persuading ourselves or others but also in following recipes, instructions and rule-based reason.

The suggestions or rules which give repeatable and reproducible results are the most certain and possibly the most correct. Reliable rule-based processes give repeatable and reproducible results. These results do not depend on who gets them. For instance, a good recipe can be followed by any cook - provided the cook can find all the ingredients. Results should depend on the recipe, but be independent of the cook.


Chapter Subsections: First Puzzle ] Second Puzzle ] One- Versus Two-Way ] Talking About Logic ] Implications vs Suggestions ] One Versus Two Way Committments ] [ Repeatable & Reproducible ] Limits and Benefits ] Accidental Rules ] Steps for Better Reason ]

Next: Limits and Benefits

 

www.whyslopes.com
Volume 1A, Pattern Based Reason

 Chapters 1 to 24

FOREWORD
Three Remarks

1 Introduction
2 Communication
3. Elements of Reason
4 Implication Rules
5. Deception
6 Chains of Reason
7 Longer Chains
For & From Consistency
8. Language Change
9 Next Chapters
10 Responsibility
11 Accidental Patterns
12 Knowledge Islands
13 Euclidean Logic
14 Deductive & Empirical 
     Views of Mathematics

15 Objectivity
16 Origin of Rules
and Patterns
17 Objective Ways

18. Waking up
19. Symbols  & Logic
20. Pronouns or Symbols
21. Truth Tables I.
22. Truth Tables II
22. Biconditional
22. Contrapositive
23. IF-THEN table
24. Indirect Reason Again

To reason often means to persuade someone of the need for an idea or action. That someone could be yourself. So be careful.

1A Logic Postscripts
- online only

+Proof by Absurdity alias proof by contradiction
+How the demand for consistency supports the law of the excluded middle
+Reality versus or with the aid of Imagination
+Links for reason, logic and crtical thinking
+Three Remarks
+History Lost or Missing

There is a difference between
knowing how to spend money,
and having money to spend.

There is likewise a difference
between mastering a skill
and having meeting a situation in which it applies.

 



 


 .


Site Tour:  Click here,  here,  here,  
here
,   hereherehere and here.

On leaving, if your next stop would
 be an internet search, do it here.

Back ] Area Intro ] Up ] Next ] [Top of this Page]  
Road Safety Message  Do not walk on a road with your back to the traffic. 
  Favourite SitesBBC News  and mathematics portion of  English National Curriculum  
[Site Author Contact Form]  If you find an error, please cut and paste its 
web page location (URL) into its report.

All trademarks and copyrights on this page are owned by their respective owners.
Copyright to comments & contributions are owned by the Poster. 
The Rest © 1995 onward by site author,   Alan Selby (
email) All Rights Reserved. 

www.whyslopes.com