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||

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

 (Optional Book Orders)
More Site Areas 
1. Help Your Child or Teen Learn 
2. Solving Linear Equations
3. Fractions Ratios Rates Proportions & Units
4. Euclidean Geometry
5. Analytic Geometry/Functions 
6. Number Theory
7. More Calculus
More Site Areas 
8. Complex Numbers 
9. Qc Maths  Education  
10. Secondary IV(?) maths
11. Real  Analysis 
12. LaTeX2HotEqn:
13. Electric Circuits Etc  
14.  Français
15. Algebra, Odds & Ends, Etc
More Site Areas 
16. Math Education Essays
17. Telling & Working with Time
18. Maps, Plans & Drawings
19. Quantitative Skills for  home, shopping and work 
20. Statistics Useful, or Not.

Test the
Twiddla Whiteboard


[Site Entrance & Hub]Back ] Area Entrance ] Next ][Site Exit]


YOU are better than YOU think. Show yourself  how:  

      |      
//  _   _ \\
/\             /\
  <|  (o)   (o)   |> 
 \     | |      / 

Read  logic chapters 1 to 5  in online volume Three Skills for Algebra  for greater skills & confidence in  work 
and study

 -/[]\- 
||
   / \_ 
 ||||||||||||||||||||||||||||

 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;

      |      
//  _   _ \\
/\             /\
<|   (o)   (o)  |> 
     | |     |
   \             /   
\    =   /

Caution: Site advice is approximately correct, for some circumstances, not all. That leaves room for thought

 -/[]\- 
||
  _ / \     
 ||||||||||||||||||||||||||||

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.

Abuse of Equal Sign

The solution of the equation

3
4
x
4

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

3
4
x =3
3     

in place of writing x = 3. While a person who writes 

x =3
3     

 

may mean x = 3, the expression 

x =3
3     

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.
  1. two different symbols (or expressions) have the same value
  2. two different calculations or expressions have the same value,
  3. 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.
4+5
=
7+2
true as both sides have the same value 9
r2
=
r·r
true as the right hand side says how to compute the left-handside.
3x+2
=
x+6
true when and only when x has the value 2.
x+4
=
x+6
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.

 

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?




www.whyslopes.com

[Top of this Page] [Site Exit] Back ] Area Entrance ] Next ]
[Comments, Reactions, Feedback][ Road Safety Message ]
: Favourite SitesBBC News  and mathematics portion of  English National Curriculum  

ll trademarks and copyrights on this page are owned by their respective owners.
Copyright to comments & contributions are owned by the Poster. 
The Rest © 1995 onward by site author,   Alan Selby,
a 1983 McGill. Ph. D. in mathematics
All Rights Reserved.