Appetizers and Lessons for Mathematics and Reason 
www.whyslopes.com        (missing links in the exposition of both)   ( Français)  
What does it mean to use a formula forwards and backwards

 Logic mastery is key to easing or avoiding learning difficulties in work & studies. 

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

Links To Tutoring Services

Parents:-Help. your. child. or. teen.study

Site  Math Folders

1. Arithmetic Flash Video Lessons  11-2008
2.  Algebra Videos (to appear)
3.-Fractions-Rates-Proportns-Units-2006
4.  Algebra, Odds & Ends, HS level-2001
5. 
Solving Linear Equations  04-2005
6.-Euclidean-Geometry/Complex No.s 
7.  Analytic Geometry/Functions 2006
8.  Number Theory. 2006-7
9.  Complex Numbers More 2001. 
 
10  Exponents, Radicals & logs. 2008
11. Calculus  2005

12.Real  Analysis 1995
13. Electric Circuits Etc  2007
How-TOs/ Reference - 08- 2008
1. Arithmetic
2. Algebra 
3. More Algebra 
4. Geometry  
5. More Geometry
6. Calculus

Home Page

ParentsSite 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  Parents and teachers need to say no for little things, keep in practice, otherwise the authority to say no for big things will vanish.  Parental Authority: Use it or lose it.  

Students: Take charge of your learning.  Try to read like a lawyer for better work & study skills, but do not take everything literally.   In particular, two logic puzzles  are keys to site content, and to greater work and study skills.  

Welcome.  If you like chapters 1 to 12, 14, 16 & 17 in Volume 2,  Three Skills for Algebra, site exploration is recommended.  If not, more experience in work and study may help lead you to a greater liking. 

Teachers & Tutors:  If you are mathphobic, start with  Three Skills for Algebra.  Site pages stems from self-contained inductive principles and standards for training, coaching and the direct development of observable skills.  In support of the latter, site folders and how-TOs  offer smaller steps, more steps  and alternative steps to aid  instruction.   Skill development may require natural talent, but should do so in a minimal manner.   Form a group of  mathematics teachers & tutors for collecting and sharing methods for observable skill development. That is a professional duty. Invite speakers to offer those steps.  As a group talk about critical path analysis, about the ends and values of instruction.     If you have a great method for skill development, invite colleagues into your classroom to observe and refine it. If you a difficulty with one part of skill development, invite a colleague to discuss the matter.  Do not be shy. Think of the long-term benefit.   Difficulties in education are not private matters. Difficulties in mathematics education are  common. Exposure is needed for resolution.  Mathematics education/training should  include food for thought, but  it should also provide and emphasize observable and measurable skills and talents. Do not waste student time with the former when observable skills and talents are weak or absent. Where course design and curriculum looks like spaghetti, object. 

See sites mathsisfun.com purplemath.com and themathpage.com for more help  Multiple views are better than one. 

A Secret - Written Work is Every Thing: For  a pair of decimals with too many digits to keep in your mind, imagine the task of adding them without the aid of a calculator.  You would write the decimals, one above the another, with place values aligned,  and then add them digit by digit with carries as needed.  That provides a first example of how written methods aids and shows work.  Mathematics mastery is largely an on-paper activity with steps drawn or written to obtain results and to  record what was done, so that your mastery of mathematics is observable and correctable.  Mental skills are helpful, but for ease and better performance,  learn how to write or draw ideas and steps on paper to do and record mathematics.  Anything less will lead to difficulties.    Good luck. 

Where is Mathematics?  ask what would happen if numbers, geometry and algebra was to disappear.  Numbers and geometry are part of daily life. Decimals appear in telling and tracking time, distance, speed, mass or weights,  measures, costs and revenues.  Geometry appears in life like drawings, plans and maps.  Algebra too appears as formulas describing calculations that may be done forward first and backwards later. Mathematics education includes and may cover first practical and concrete skills and concepts before and then beside preparation for calculus.  That we hope, can be done in a way that serves other ends for mathematics. 

What is Mathematics Mastery? If you want to learn mathematics well you need to have a strong work ethic.  You need to watch for and master  methods and practices for writing and thus recording and doing work on paper in a step-by-step manner that can be seen and understood by others.  While you may get credit for right answers, answers that you arrived at in your head,  there is still a great need to show work on paper.  While you may think or hope that showing work is not needed,  the ability to record steps in arithmetic, geometry and algebra on paper allows us to develop and follow the onger and longer chains of reason that mathematics mastery requires. 

Calculus is the college or senior high school mathematics subject required for college or university studies in accounting, business, money matters, science, engineering and health.

Calculus is  very demanding.  Half the students who take college calculus fail. Try to be in the other half. 

Calculus  employs at full strength  earlier elements of high school mathematics, namely  functions, trig, algebra (including polynomials), mathematical induction, more logic,  geometry and exact arithmetic.   But preparation for calculus should not be the only end for secondary school mathematics. other shorter term ends need to be included to provide a motivation and a context.  Preparation for calculus could be buried in quantitative skill development for daily life and for numerical and geometric methods there-in.

Advanced Level Help:   Volume 3, Why Slopes and More Math offers starter and preview lessons for calculus to make the hard easier.  These arithmetic review problem with hints of algebra may catch a few weaknesses at the start of calculus or in earlier instruction.

More Mid-Level Help: Talking about three skills for algebra and what is a variable  introduces more words into understanding     and explaining algebra.  In mathematics, the question of what does it mean to use a formula or rule  forwards and backwards (link for students) identifies with words a recurring pattern or unifying theme (link for tutors-teachers).   

Mid-Level Help   Seeing the difference between one and two-way implication rules may provide lawyer like, precision reading skills needed for success and self-defense in studies, in work and  in making agreements.  Besides that, put mathematics aside and see if you can enjoy the leading  math-free logic chapters in Volume 2.  You know not studying leads to poor marks.  If you decide to study, take some time to master logic. That exercise and logic mastery together will lead to better study skills.  Good luck.

Base Level Help:   Arithmetic Flash Video Lessons  review and explain operations with decimals and integers in great depth - perhaps too deeply.  Solving Linear Equations lightlyconnects (i)  algebra with geometry and fractions  - fractional operations on line segments), connects (ii) word problems with systems of equations in essentially one unknown - let algebra make the hard easier. For a further   review and explanation of arithmetic with fractions see the site folder on  Fractions etc.    If you are adult or older teen,  the lesson here should be a review and should make what was hard in the past, easier - and the review should unify and connect the ideas and skills, half of which or more, you have seen before.

Site Reviews 
kindest first

  1. Magellan, the McKinley Internet Directory, 1996: 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

    Current Location:  Volume 2, Three Skills for Algebra - for the most part. 
  2. The World-Wide Web Virtual Library Education by Country - Canada, Nov. 2005: This online classroom offers appetizers and lessons for math from arithmetic to calculus or why slopes; for deductive reason (logic) and critical thinking; and for learning in general. Included here are opinions on the communication of skills and mathematics instruction. The logic appetizers are math free. Each appetizer is different. If one is not to your liking try another. ...

  3. The NSDL Scout Report for Mathematics Engineering and Technology-- Vo. 4, No.  4 (Feb.  2005)  [A new]  section of the website,  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. Links to related [online] chapters ...  are also provided.

    Location: Site Folder. Solving Linear Equations  04-2005.  The stick diagram approach was invented during a third high school teaching practice in that program and posted online just before the fourth.

  4. The NSDL Scout Report for Mathematics, Engineering, & Technology -- Vol. 1, No.  8 (May 2002)  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, [ouch] the site is not well organized. This does not take away from the quality of the information, though.

    Remark: Six year later, November 2008, the site has been reorganized. Site pages are now (we hope) clearly and properly organized into books or folders with chapters or lesson groups. 

  5. Education Planet Newsletter, top math sites, 2001: ... The commentary and online books available at this site provide a very rich guide to mathematical reasoning and high school math to calculus. The emphasis here is on the thinking part of math rather than the actual manipulations themselves. There is also information here for parents as well as teachers. Now you can help your students gain insight into mathematics rather than just helping them memorize formulas.

    Current Locations:  Volume 2, Three Skills for Algebra: and Volume 3, Why Slopes and More Mathematics,  Help. your. child. or. teen.study and Volume 1B, Math Curriculum Notes. The entrance page to site folder  Mathematics Education Essays includes a long article on the thinking part of mathematics and how it may have to co-exist with rote learning (or postponed explanations).

  6. Math Forum News Letter: Aug, 2001. ...

     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

    Current Location:  In  Euclidean Geometry & Complex No, and Complex No. Folders

    Original & Current Locations:
      Volume 3, Why Slopes and More Mathematics for complex numbers & Site Folder : Complex Numbers More on the short way to reach the latter using complex numbers. Since 1996 introductions of complex numbers have appeared in Volume 3,  alone in a single 2001 page on complex numbers, then  in the Analytic Geometry/Functions folder and, saving the best for last, in Euclidean-Geometry folder.  

  7. Math Forum News Letter,  November 25, 1996 ....  mathematics appetizers range over arithmetic review problems, notions of what variables are, skills leading to algebra, painless theorem proving, complex numbers with some trig, the importance of slope (some calculus), a decimal perspective of error control and continuity (more calculus), and renaming the greater than sign (back to algebra). // Advice on how to read, how to learn, why go to school, etc. ...  included. The tone is sometimes funny, and the writing is dense, rich, and intriguing. There are reflections on teaching, so these materials can be used in the classroom and as a place for teachers to learn.  // ... explanations of mathematical concepts using words and stories are particularly strong. ...

    Current Locations:  Volume 2, Three Skills for Algebra: and Volume 3, Why Slopes and More Mathematics. What does it mean to use a formula forwards and backwards

  8. Cornell Theory Center, Fall1995. Several puzzles and short discussions to help students understand algebra and proofs. If you don't like the first one you look at, try another; they're all quite different. 

    The Cornell Theory Center and the Math Forum provided the first two links to this site.

  9.  

  10. Math Forum June 1995 Hot Spot: Three Skills For Algebra (a first image of mathematics after arithmetic - why letters or symbols are favored in algebra in place of numbers); Two logic puzzles to show the difference between a one-way and a two-way rule; Painless Theorem Proving; Longer Chains of Reason: What is Mathematical Induction?; Complex numbers - A geometrical story based on the addition and multiplication of arrows in the plane; Chains of Reason - math-free examples of rule-based reasoning; How Logic or Rule-Based Reason Appears in Math;  ...

    Current Location:  Volume 2, Three Skills for Algebra:

 

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Pages for Students

Home
How to Improve Marks
Grades 7 to 12 Guide
1. Arithmetic Lessons
2. Algebra How-TOs
3. More Algebra How-TOs
4.  Geometry Steps Into
5. More Geometry
6. Calculus How-TOs
7. Logic in Math

Proper notation & format 
makes the hard easier.

Pages For Teachers

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Notes for Instructors
Vol 1, Elements of Reason
Sec I Objectives
Sec II Objectives
Sec III Possibilities
Permissions
A Site Map

Show a student how to learn and that helps one. Show a teacher or  tutor how to make skills and concepts easier for students and that helps many. 

Miscellaneous

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

What may be learnt and when depends on how skills and concepts are developed. Making the hard easier and clearer will allow earlier & richer development of skills and concepts.

YOU are better than YOU think. Show yourself  how:

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Take greater  charge of your work or studies: Read like a lawyer for better work & study skills, but do not take everything literally.

In particular, two logic puzzles  are keys to site content, and to greater work and study skills.  See if  you agree.

 -/[]\- 
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   / \_ 
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Help Me Learn/Teach;

  1. Algebra
    words before symbols - direct & indirect use of formula, numerical versus algebraic solutions - what is a variable (more words)
  2. Arithmetic
    -new Arithmetic Folder
    - exercises
    - with fractions
    - videos on primes, lcm, gcm,lcd, square roots etc
  3. Calculus - geometric preview, algebraic preview,
    3 study guides,
    much more
  4. Complex numbers
    -starter lesson with java applet - easy consequences for trig & vectors in the plane
  5. Education
    - Empirical Course Design & Delivery
  6. Fractions
    - alone
    - by rote
    - with algebra
    - videos
  1. Functions - introduction
    hindsight - composition aka
    substitution
    -
  2. Geometry, Euclidean - Correspondence of trianglesTriangle construction,  duplication & Isometry - Failure of ASA & the // line postulate - angle sum in triangles -// grams - Triangle Similarity
  3. Geometry- Analytic - functions, polynomials, complex numbers, unit circle trigonometry
  4. Logic
    - First Steps -
    Symbols in Logic -
     Occurrence & Truth Tables - Indirect Reason -Indirect Reason More
  5. Proportionality
    - Definition - Direct & Indirect Use - Numerical versus Algebraic Solutions
  6. Real Analysis
    - Decimal View of concepts and of proofs
  7. Rules &Patterns in Science, Technology & Society - Pattern Based Reason
  8. Mathematical Reasoning, empirical, inductive or deductive
  9. Units
    - in rates & slopes & (?) derivatives
    - in ratios & proportions - slopes & rates included
  10. Complex Numbers & Vectors & Trig
    trig expression for dot & cross - cosine law

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