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Appetizers and Lessons for Mathematics and Reason  
for calculus, preparation for calculus & math ed. reform, etc
Online Volumes ( Book Orders)
1,  Elements of Reason.
1A. Pattern Based Reason 
1B. Math Curriculum Notes
2. Three Skills for Algebra
3. Why Slopes & More Math

 Mathematics Course Designers: LAMP offers food for thought.
 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

 ||Définition d'une variable || Algèbre || Arithmetique || Logique ||La raison basée sur les règles et modelés||
[ Back ] [ Advice & Directions ] [ Next ]

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.

Learn to read notes and textbooks like a lawyer, so that no nuance, no subtlety and no clause escapes your attention.

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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.

Trigonometry or Similarity
a question of convenience

In solving for missing sides or angles in triangles you may be told  to use similarity methods directly or trig functions.

And in answering questions, students should apply the requested method to show their mastery of it. But if the requested method has not been mastered, using another method may give part marks or not.

Below shows how the use of trigonometry represents an indirect use of similarity.

Direct Use of Similarity

Assume we cannot measure the height h directly.  To find the height h of the triangle which casts a shadow 100 metres long when the sun's angle of elevation in the sky is 50 degrees, we may draw a similar right triangle to scale with the ASA triangle construction method and base length of 2 centimetres (2 cm)

Now

Therefore

Direct Use of Trig
(Indirect Use of Similarity)

By using trigonometry, we have another way to compute the height H. That is, we may use the tangent function

to conclude that

H = tan (50o)(100 metres).

Here we may compute  tan (50o) with the aid of a table or calculator.  That calculation hides the presence of similarity. But we may find tan( ) from the right triangle

or any other similar triangle (drawn carefully)  using

 

and our ability measure the opposite side, here h. Then

Then H = tan (50o)(100 metres) gives

The latter implies

Conclusion or Observation.

In solving problems involving missing sides of  a triangle, the use of tables or calculators to look-up or find the values of a trig function is but an alternative to computing those values from a triangle similar to one with the missing sides. Using a calculator or table is more convenient.

Remark. Before the use of calculator,  the values of trigonometric functions might be tabulated for use in calculations.   For instance, we could by measurement (not a very precise method), estimate the values of cosine, sine and tangent at  5, 10, 15, 20, ..., 85 degrees by doing measurement and computing the required ratios.  The latter would require many right triangles to be drawn.  After that we could solve for missing sides etc in given right triangles with acute angles 5, 10, 15, 20, ..., 85 by consulting the table or the triangles we used to fill the table.  We could also interpolate between the values in the table to obtain estimated values of sine, cosine and tangent for use in further computations.   However, that work has been done for us (by more refined methods), and the results tabulated in tables and built-in the computational abilities of calculators.

 

 
 

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4. Euclidean Geometry

Advice & Directions
Correspondence
Isometry
Side-Side-Side
Side Angle Side
Angle-Side-Angle
Isoceles
Right Bisector Construction, Etc.
Perpendicular - Point to Line
SSS Failure
SAS Failure
ASA Failure
Parallel Lines
Angle Sum
Similarity
Right Triangle Similarity
Trig  or Similarity
Parallelograms
Kites From Triangles Duplication
Parallelogram from Triangle Duplication
Addition of points in the plane
Multiplication of Points in the Plane
Distributive Law, Step I
Distributive Law, Step II
Distributive Law, Step III

Above Average Students in Geometry may enjoy  the site geometric introduction of complex numbers and the wordy volume 1A,  Pattern Based Reason.

For algebra, logic starter lessons,  see Volume 2, chapters 1 to 12, plus 14, 16 and 17.

Analytic Geometry, Functions & Trig

(FN) What are Functions?
(FN) Functions - More
SZM: Sign, Zero, Monoticity
(L) Lines Summary
(P) Polynomials (*,+,-)
(Q) Quadratics
(D) Simplify Square Roots
(T) Unit Circle Trig
Conic Sections

More For Analytic Geometry:

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

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

Lesson Plans and lessons

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

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

 


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