YOU are better than YOU think. Show yourself how:
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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 |
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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;
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Caution: Site advice is approximately correct, for some circumstances, not all. That leaves room for thought |
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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.
Algebra Hint and Formula Sheet
(Crib Notes)
Straight Lines
| slope m = | Dy
Dx |
= | y2-y1
x2-x1 |
= | rise
run |
The point-slope form of equation for a line y-y1 = m·(x-x1)
The slope intercept form of equation for a line y = b +m·x or y = m·x + b. In answering questions, rewrite any equation you obtain for a non-vertical line into a slope intercept equation.
If L has nonzero slope m=m1 and a line K perpendicular to L has slope m2 then -1= m1·m2 . Thus m2 = -1/m1 = negative reciprocal of m1. When slope m1 of L is known, it can be used to compute the slope m2 of K without being given two points on the line K.
Quadratics
Formulas for
Factoring by inspection: (x+A)(x+B) = x2+(A+B)x + AB.
Completing the square: x2+2Qx = (x+Q)2 - Q2
Difference of two squares. (C+A)(C-A) = C2 - A2
By completing the square, each quadratic y = ax2+bx+c has an axis of symmetry with equation x = -b/2a.
If the discriminant b2-4ac > 0, then one or two solutions of ax2+bx+c = 0 are given by the quadratic formula.
| x = |
2a |
The plus sign gives one solution x+ and the negative sign gives another solution x-
The quadratic formula with the plus-minus is shorthand for two formulas:
| x = |
2a |
= x+ |
and
| x = |
2a |
=x- |
Natural Logarithms and Exponentials
Uniqueness Property: If a > 0, b> 0 and ln(a) = ln(b) then a = b.
Inversion Properties
-
ln(exp(x)) = x for all real x
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exp(ln(x)) = x if x > 0
Fundamental property of logarithms ln(ab) = ln(b) +ln(a) (proof available in calculus)
Fundamental property of exponentials: exp(x1) · exp(x2) = exp(x1+x2)
The fundamental property of logarithms implies
-
ln( 1/a) = (-1) ln(a)
-
ln(a m) = m ln (a)
Logarithms to Base c > 0.
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The button log(x) on a calculator computes log10(x).
Roots and rational powers of positive numbers
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The m-th root of a positive number
_
mÖb = a 1/m = exp( (1/m) ln(a) )
Exponentials of Real Numbers
a x = exp( x ln(a)) for all real x for a > 0 and not only for rational numbers m/n
Properties
(a x)y = a xy (Exponential of an exponential)
a xay = a x+y for all real numbers x and y when a > 0.
Now for the natural number e = exp(1) = 2.718281828... (irrational).
The natural logarithm of e, ln (e) = 1 Therefore
e x = exp( x) for all real x when a > 0
as a x = exp( x ln(a)). Calculators often have a button marked e x for the evaluation of the exponential function exp( x)
Caution: the capital EXP on some calculators will not help you with the calculation of exp(x). Use the button marked e x instead.
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.