YOU are better than YOU think. Show yourself how:
|
// _ _ \\
/\ /\
<| (o) (o) |>
\ | | /
|
Learn to read notes and textbooks like a lawyer, so that no nuance, no subtlety and no clause escapes your attention. Read logic
chapters 1 to 5 in online volume Three
Skills for Algebra for greater skills & confidence in
work |
-/[]\-
||
/ \_
||||||||||||||||||||||||||||
Logic chapters 1 to 5 re- appear not in sequence, as is or longer, in Volume 1A, Pattern Based Reason, Bon Appetite.
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
Logic mastery makes the hard, easier. Logic mastery leads to better, stronger and richer comprehension. Logic mastery improves reading and writing. Logic mastery ease learning difficulties. Logic mastery gives a headstart. In sum, logic mastery will develops critical thinking, improve reading and writing, and give a firmer base for work and studies at many levels. Good luck.
After logic, (a) continue reading Three Skills for Algebra, chapters 8 to 14 and do so alongside site area on solving liinear Equations ; or (b) see this calculus starter lesson and Volume 3, Why Slopes & More Math, chapters 2 to 6;
|
// _ _ \\
/\ /\
<| (o) (o) |>
| |
| |
\
/
\ = /
|
Caution: Site advice is approximately correct, for some circumstances, not all. That leaves room for thought |
-/[]\-
||
_ / \
||||||||||||||||||||||||||||
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.
Try the Twiddla
Whiteboard. In principle, it allows
to people to draw and chat together online on a copy of this webpage or a clean
sheet. The chat may be via text or audio. Visit www.twiddla.com
to set up whiteboards to work with the webpage of your choice.
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.
The following essays are not all related to mathematics.
[ Constant Retirement Rate ] [ Road Safety ] [ 3 Strikes Law in California. ] [ Math HOW-TOs ] [ 9 Steps in Maths ]
The following odds and ends material may help. Links to area webpages are in the right margin. The discussions and links below give general advice and directions, and points to material outside this site area. Bon Appetite.
Key Skills and Concepts
- Logic mastery is the key to better study skills, the key to enriching skills and understanding, and the key easing or avoiding learning difficulties in many arts and disciplines. That advice cannot be repeated too often. See the first steps in Logic below
- Algebra is hard for many. Words have been missing or unclear in the description verbalization of skills and concepts. By learning to talk about number and quantities, describe them with words, before and then besides the shorthand roles of letters and symbols, we can more clearly understand and explain the algebraic or mathematical way of writing and reasoning. See the long essay What is a variable and the Algebra chapters 8 to 12..
- Formulas Forwards and Backwards: Chapter 14 employs the Compound Interest formula directly and indirectly (forwards and backwards), and compares arithmetic and algebraic ways for this. The pattern of first learning a formula or method directly and then using it indirectly or backwards is a key to algebra alone and in further subjects.
- Solving linear equations in one or two unknowns is another key part of junior and high school mathematics. Explore the site area Solving Linear Equations with Stick Diagrams for an introduction and then come back to Chapter 15 in this Volume
- Skill Consolidation: See Arithmetic
Videos on Primes, LCD, GCD, Fractions, Square Rootsand
do some Arithmetic
Review Problems to check or perfect your figuring and fraction sense and
skills. On paper, high school mathematic programs state arithmetic mastery
is a must for algebra. In practice, many students rely on their calculators
and so sabotage or delay their own education in mathematics. Every
student should know the sum and
Times table for all pairs of whole numbers 0 to 12. In a spare moment or two, practice filling in these tables by repeated addition.
More About Area Content:
Better study Skills, First Steps in Logic, Arithmetic, First Steps in Algebra, Finite Mathematics, More Logic. See links to postscripts in right margin.
| For Better Study Skills |
First Steps in Logic | Arithmetic & Fractions |
| A
Appendices B How to Learn C How to Read D What to do in School and Why E. How to Study Mathematics |
1.
Introduction 2. Implication Rules 3. Chains of Reason 4. Induction (Longer Chains) 5 Knowledge Islands Indirect Logic |
Arithmetic
WebVideos 7 Arithmetic Review Problems See Solving Equations with stick diagrams if your fraction skills and sense need improvement |
| If you plan to study calculus, do the Arithmetic Review Problems and have them corrected by another. Problems like these catch errors that slow or undo too many students entering calculus. Ouch. | ||
Altogether, the logic chapters provide a unique mathematics-free introduction to the direct and indirect definition and rule-based thinking that appeared in Euclid's work a long time ago (2300 years ago)
|
First Steps in Algebra |
||
| 8
The Three Skills 9 First Skill What is a Variables 10 Two More Skills 11 Why Shorthand 12 Shorthand Usage 13 What's Next 14 Compound Interest 15_Solving_Linear_Equations 16 Painless Proofs 17 Pythagoras |
Fractions and Solving Equations with stick diagrams (spring 2005 postscript) - | |
| A.
Letters & Lengths B. Stick Diagrams C. Beyond Sticks D..Almost One E. Binary Systems F. Larger Systems |
Stick
Diagram Examples 2x + 5 = 20 3x + 10 = 32 5a + 16 = 3a+ 24 (½)x + 8 = 24½ (¾)a + 16 = (¼)a+ 24 (¾)q + 17 = 32 |
|
| Stick diagrams give a visual introduction to algebra which develops fraction sense and skills. Try them before or besides chapter 15, Solving Linear Equations, whatever works | ||
| Teachers: Here some algebra lesson plans for the more effective development of the shorthand ways of writing and reasoning. | ||
|
Finite Mathematics |
||
| 21
What's Next 22. Geometric and Arithmetic Sums 23 Summation Notation 24 Investments, Loans, Pensions - Personal Money Calculations 25 Mathematical Induction and Recursion Proofs, Product Notation, & Factorial Notation |
26
What's Next 27 Pronouns in Logic 28 Occurrence Tables 29 Contrapositive 30 Truth Tables 31 Indirect Reason |
Some of this logic chapters may be used in the above treatment of analytic geometry. |
www.whyslopes.com
Algebra, Odds & Ends,
1. Hints for Exams 2A. Exact Arithmetic 2B. Fractions Briefly 3. What is a Variable? 4.. Square Roots 5. Straight Lines 6. Problem Solving Methods 7. Trig and Complex No. 8. Complex Applet 9. History of No.s 10. ln(x) and exp(x) 13. Rename the > Sign 14. Problems: Quadratics 15. Problems: Algebra Test 16. Problems: Linear Eqns I 17. Problems: Linear Eqns II 18. Problem Solving Hints 19. Functions & Sets 20. Independent Variables 21. Why Logic 22. Why Math 23. The 15 Times Table 24. The 20 Times Table 25. Algebra Formulas 26. On Learning Maths 27. Maths in Biology 28. Navigation +Time 29 Quibble-What is Algebra 30. Logic in Maths
Odd and Ends, Essays
Constant Retirement Rate Road Safety 3 Strikes Law in California. Math HOW-TOs 9 Steps in Maths
Study With Others: twiddla.com has developed a collaborative whiteboard with audio & text chat that allows students, tutors & teachers to explore & scribble on blank pages and copies of webpages together, If scribbling is awkward with one browser, try another.
In Volume 2, Three Skills for Algebra, Chapters 8 to 14 and postscript What is a Variable point to a greater & clear use of words in algebra. Chapter 14 introduces a 4th skill for algebra, an elaboration of the third: - The direct and indirect use of formulas, numerically and algebraically, is unifying theme that should be mentioned aloud, with words, in each and every use of formula.