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.
Many Equations with Essentially One Variable
Example: Suppose the letters A, B, C and T satisfy the four equations
A = 2T
B = 3T
C = 4T and
T + A + B + C = 60
That means A has the same value in the first and fourth equation, and so on. Now we use the first three equations
A = 2T
B = 3T
C = 4T
to replace the A, B and C in the fourth equation
T + A + B + C = 60
That gives
T + 2T + 3T +4T = 60
In other words
10 T = 60
from which we get T = 6. The first three equations by substitution now give
A = 2T = 2*6 = 12
B = 3T = 3*6 = 18
C = 4T = 4*6 = 24
Check: we expect A =12, B = 18, C = 18 and T=6 to satisfy the original equations
A = 2T
B = 3T
C = 4T and
T + A + B + C = 60
The first three are satisfied by the way in which computed A, B and C from T while T + A + B + C = 6 + 12 + 18 + 24 = 60 by direct addition in your head, or if need-be, shame on you, with a calculator.
Remark: The fourth equation T + A + B + C = 60 is essentially an equation only in T because the other amounts A, B and C can be expressed in terms of T and thus eliminated by replacement or substitution in the fourth equation. Because the system
A = 2T
B = 3T
C = 4T and
T + A + B + C = 60
gives or reduces almost immediately a single equation in the one unknown T to solve, we say this is an essentially one variable system. We can go further. See the next example.
First Animated Example
Exercise (1) : Solve the essentially one variable system
A = 3T+ 8
B = T+ 4
C = ½T
D = 2 T - 12 and
T + A + B + 4C + D = 100
and check your answers.
The values of the unknown T may be a positive or negative proper or improper fraction, or zero. ( I do not know. I have not solved this system. I have only written it.)
Second Animated Example
Third Animated Example
www.whyslopes.com
Solving Linear Equations
|(Feb 14, 2005)a secondary I to V reference for solving linear equations and for recognizing word problems in essentially one variable whether you like it or not, 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
Up 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
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)