Appetizers and Lessons for Mathematics and Reason (www.whyslopes.com)
||Définition d'une variable || Algèbre || Arithmetique || Logique ||La raison basée sur les règles et modelés||

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1,  Elements of Reason.
1A. Pattern Based Reason 
1B. Math Curriculum Notes
2. Three Skills for Algebra
3. Why Slopes & More Math

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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 
and study

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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.


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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)

Secondary II - Algebra  (arithmetic versus algebraic methods, backward use of formulas and proportionality equations)

Secondary IV - Functions to Trig & Statistics

Calculus Intro 

Algebra Lesson Notes - All levels


 

 



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