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YOU are better than YOU think. Show
yourself how:
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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.
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Abuse of Equal Sign
The solution of the equation
is x =3. But is an error, a mistake, a major misuse of the equal sign to
insert an = 3 besides the x in the above equation to obtain or write
in place of writing x = 3. While a person who writes
may mean x = 3, the expression
actually means a third of x is 3.
Mathematics and English teachers should mark what is
written, not what might of been meant, so their students learn to write
precisely.
Precision is important. A person who does not write exactly
what he or she means does not know how to read precisely what is written in
their notes and textbooks, and so is easily confused.
Moreover, in mathematics, confusion about notation, what is proper or
not, leads to errors in all calculations and in problem solving.
Ouch!
2.1 Proper Use of the Equal Sign
Here are a few words about the equal sign. The equal sign is used to say two
different items have the same value. So the equal sign = may be used to say or
suggest the following.
- two different symbols (or expressions) have the same value
- two different calculations or expressions have the same value,
- the value of a number or quantity is the same as the value of another
expression.
The suggestion of identical values can be true or false depending on
circumstances. See the next examples.
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true as both sides have the
same value 9 |
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true as the right hand side
says how to compute the left-handside. |
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true when and only when x has
the value 2. |
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always false as x + 6 = (x+4)
+2 is two units more than x+4 |
Here the first equation or equality holds (meaning is true) since both 4+5
and 7+2 are expressions with the same value 9.
The second equation r2
= r·r always holds, no matter what value you give to r. It
tells us how to compute the number or quantity described by the expression r2.
The third equation 3x+2 = x+6 holds (is true) when and only when x
= 2.
When x has a value other than 2, the statement (suggestion or
assertion) that 3x+2 has the value as x+6 is false.
The
fourth statement x+4 = x+6 is always false.
No value given to (or
substituted for) x will make x+4 and x+ 6 have the same value.
Adding 4 and adding 6 to a number gives different results, different
values, no matter what the number is.
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www.whyslopes.com
Analytic Geometry
Area Entrance
Entrance + Pages Below this page
Pages at Current Level
Area Entrance
Pages at Above this Page
Extras: Not all perfect.
Equal Sign Use/Abuse
Real Numbers
Say More Positive
Linear Inequalities
Triangle Inequality
Absolute Value |x|
|x| Eq'ns & Inequalities
Rectangular Coords
Shortest Path
Distance Formulas
Add & Multiply Points
Polar Coordinates
Radians
(A) Vectors
(A) Coordinate Arithmetic
(A) Navigation on Maps
(A) Addition Geometrically
(A) Rotation
(PT) Translations
(PT) Dilatations
PT: Rotations
Links to Site Pages outside this site area follow - co- and pre-
requisites.
Road
Safety Message
Easy Consequences of this (newest) Complex
Number. Starter Lesson follow below to provide an alternate
development of HS or college maths.
Vec & Cmplx No Applet
B2 C. Conjugates
B3 Pythagoras
B4 Distance
B5 Rt Triangle Similarity
B6 Trig., Functions
B7 Dot & Cross Products
B8 Cosine Law
B9 Exponential & cis fns
B10 Easy Trig Identities
B11 Set Viewpoint
Arithmetic Videos - Real Player Format
Decimal Addition Methods
Decimal
Subtraction Methods
Decimal
Multiplication Methods
Decimal Division Methods
Fractions
Primes
Greatest Common Divisor
Divisors
Least Common Multiples
Square Root
Simplification
Using formulas forwards
& Backwards - A unifying theme for algebra skill development - the 4th
skill in Volume 2, Three
Skills for Algebra!
What
is a Variable?
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