YOU are better than YOU think. Show yourself how:
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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.
Example 4: Solve 3x - 4 =-11 (animated presentation)
Observe Error Recovery & Advice in Solution
 |
Example 5: Solve x + 12 = 10x - 30
| Operation | Equation | ||||||||
| given | x + 12 = 10x - 30 | ||||||||
| add -x to both sides | 12 = 9x - 30 | ||||||||
| add 30 to both sides | 42 = 9x | ||||||||
| Switch sides | 9x = 42 | ||||||||
| Divide both sides by 9 (or multiply by 1/3) |
|
Note:
- We could have added -x + 30 to both sides in one step, not two.
- We switched sides for cosmetic reasons. That is we prefer to write b =15 in place of 15 = b.
- The switch would not have been needed if we had started with the equation 10x - 30 = x + 12 instead of x + 12 = 10x - 30. If x is a solution of one, it is also a solution of the other. This switch would have put the bigger coefficient on the left hand side and result in 9x = 42.
| Operation | Equation | ||||||||
| given | x + 12 = 10x - 30. | ||||||||
| switch sides to get bigger coefficient for x on left. |
10x - 30 = x + 12 | ||||||||
| add -x to both sides | 9x - 30 = 12 | ||||||||
| add 30 to both sides | 9x = 42 | ||||||||
| Divide both sides by 9 (or multiply by 1/3) |
|
You see the solution does not change.
Check: For
| x = | 4 | 2 3 |
we have
| LHS = x+ 12 |
RHS = 10x-30 |
||||||||||||
| = | 4 | 2 3 |
+ | 12 | |||||||||
| = | 10 | (4 | 2 3 |
) | -30 | ||||||||
| = | 16 | 2 3 |
|||||||||||
| = | 40 | + | 20 3 |
- | 30 | ||||||||
| = | 10 | + | 6 | 2 3 |
|||||||||
| = | 16 | 2 3 |
|||||||||||
Observe the left hand side (LHS) has the same value as the right hand side (RHS). So the value
| x = | 4 | 2 3 |
works.
Example 6: Solve -3x + 4 = 10
 |
Example 7: Solve 6x + 4 = 4x + 14
Solution Steps:
| Operation | Equation |
| given | 6x + 4 = 4x + 14. |
| add -4x to both sides | 2x + 4 = 14 |
| add -4 to both sides | 2x=10 |
| multiply both sides by ½ | x = 5 |
We could have added -4x + -4 to both sides in one step instead of two.
Check: For x =5
| Left Hand Side | = 6 x + 4 = 6*5+4 = 30+4 = 34 |
Right Hand Side | = 4x + 14 = 4*5+14 = 20 +14 = 34 |
So for x =5, the Left Hand Side and Right Hand Side have the same value.
www.whyslopes.com
Solving Linear Equations
|(Feb 14, 2005)A reference for solving linear equations and for recognizing word problems in essentially one variable. Skill in arithmetic with fractions is a must for algebra. .
Area Entrance Proper Use of Equal Sign A. Letters and Lengths B.. Solving Linear Eq'ns. C. Solving Linear Eq'ns D.Almost One E: 2D Systems - Sub Method. E: Continued E: Still More F. Larger Systems
Area Entrance (i) x + 20 = 29 (ii) 2x + 5 = 20 (iii) 3x + 10 = 32 (iv) 5a + 16 = 3a+ 24 (v) (½)x + 8 = 24½ (vI) (¾)a + 16 = (¼)a+ 24 (vii) (¾)q + 17 = 32 (viii) 13 =[2/3]x +7 twice (x) Animated Examples (i) Integral Coefficients (A) (ii) Integral Coefficients (B) (iii) Fractional Coefficients (iv) With parameters
Arithmetic Videos
Decimal Addition Methods
Decimal Subtraction Methods
Decimal Multiplication Methods
Decimal Division Methods
Fractions
Primes
Greatest Common Divisors
Least Common Multiples
Square Root Simplification
Site books and further webpages on learning and teaching mathematics and pattern based reason may develop critical thinking, improve reading and writing, and give a base for learning or teaching high school and college mathematics.
Great_Expectations: If you can learn to follow a multi-step methods in any subject precisely, you can do so in other subjects, as well.
Good news: Site pages identify what you need to study.
Bad news: Site pages do not explain everything
Worse news: Learning takes time, yours
Lesson Plans and Ideas for Teachers & Tutors:
Secondary I - fractions & allied concepts (decimals, percentages)