|
Odds & Ends
Group I
1. Hints for Exams
2A. Exact Arithmetic
2B. Fractions Briefly
4.. Square Roots
5. Straight Lines
6. Problem Solving Methods
7. Trig and Complex No.
9. History of No.s
10. ln(x) and exp(x)
13. Rename the > Sign
14. Problems: Quadratics
15. Problems: Algebra Test
16. Problems: Linear Eqns I
17. Problems: Linear Eqns II
18. Problem Solving Hints
20. Independent Variables
21. Why Logic
22. Why Math
23. The 15 Times Table
24. The 20 Times Table
25. Algebra Formulas
26. On Learning Maths
28. Navigation +Time
29 Quibble-What is Algebra
30. Logic in Maths
31. Real Number Operations
Learn More
Group II
Constant Retirement Rate
Road Safety
3 Strikes Law in California.
Math HOW-TOs
9 Steps in Maths
[ Back ] [ Up ] [ Next ]
| |
Independent & Dependent Variables
Consider the rectangle area computation formula
Area A of Rectangle = [Width W of Rectangle] * [
Length L of Rectangle]
or A = WL. Here A, W and L are three quantities that may vary between
rectangles or for expanding rectangles - an unrolling carpet for instance covers
an expanding rectangle.
- If W and L are given or specified first, they are called the independent
variables, and A = WL is called the dependent variable.
- If you have one or several problems in which A and W are given and L is to
be found from the equation or condition A = WL, then A and W are called the
independent variables and L is called the dependent variable.
In general, if you have several quantities in a problem, the quantities that
are found or determined from the values of the others may be called dependent
variables, while the others are independent variables. There should be no
equation or relationship between the independent variables that can be used to
compute one of them given the rest. (I hope this clear enough for a first
introduction to the concept of dependent and independent variables. If not this
introduction should be rewritten.)
|
_______
| | |
// _ _ \\
/\
/\
<| (o) (o) |>
\ | | /
\___ _/
||
-/[]\-
||
/ \_
Professor Whyslopes:
-
Site value lies in the difference
between its ideas and yours.
-
If one site explanation is not to
your liking, try another. Each one is different.
|
Two gaps
- The Old Algebra Gap: Algebra
appears with too few words of explanation in high school and college
mathematics. Online Volumes 2 and 3 offer remedies.
Chapters
8 to 12 in Volume 2 put more words into the explanation and
comprehension of algebra. Chapter
14 in Volume 2 with its explicit discussion of the direct and
indirect use a formulas identifies a unifying theme for mathematics
and logic - all rules and patterns will be used forward and backwards.
Chapters
2 to 6 and 12 to 18 in Volume 3 may further ease or avoid the very
challenging use of algebra in the high level mathematics: calculus.
Calculus requires earlier high school mathematics at full strength: (i)
This logically complete but long lesson on complex
numbers shows how to simplify the senior high school
exposition of circular trig functions upto to formulas in the plane
for vectors dot and cross-products. The lesson provides the
route that would have been taken in course design if the key element
of the lesson, a December 2009 invention, had been available in
the 1950s. For further algebra skill development. See the site
coverage of fraction
with units, proportionality,
ratios and rates,
polynomials, quadratics
functions
and straight
line slopes and equations.
- The Arithmetic Gap: An exact and efficient
mastery of arithmetic with decimals and fractions is best (required)
for the high level study of mathematics alone and in science,
technology and business. Pages here on arithmetic
with decimals and integers, on fractions
and solving
linear equations with fractional
operations on stick diagrams may help fill the gap. That
exact and efficient command should be obtained in the last years of
primary school and the first years of secondary school.
|
Skill mastery in
mathematics has to be seen to believed. To that end,
learn or teach how-to write and draw the steps in mathematical
figuring or reasoning clearly. Do not try to save space
by doing a sequence of step in one place. Instead, do or record the
steps in sequence on a separate lines to make each step obvious and
verifiable.
|
|
| |
|
|
|
www.whyslopes.com
site
search
Parents: Help
your Child/Teen Learn covers Speaking
Skills, Reading
& Writing,
Preparing for Science &
Having Patience, etc
Math How-TOs
1. Arithmetic
2. Algebra
3. More
Algebra 4. Geometry
5 More
Geometry 6. Calculus
>> densely written
>> use as skill checklists
Online
Volumes (orders)
1, Elements of
Reason. 1996
1A. Pattern
Based Reason 1995
1B. Math
Curriculum Notes 1996
2. Three
Skills for Algebra 1995
3 .Why.Slopes.&.More.Math.1995
Skill
& Concept
Review or Development
1. Decimal
Arith - Video Based ]
2 Fractions
3. Fractions
with Units
3. Solving
Linear Equations -
making alg easier
4. Formulas
forwards & Backwards - unifying theme for Algebra
5. Proportionality,
Back- & For-wards - theme at work.
6. Logic
- Math Free, good for precision in work & studies
7. Euclidean-Geometry
(leanly)
8. Slopes
and Lines
9. Why
Study Slopes - a context
10. Quadratics
11 Polynomials
12 Factored
Polys - a context
13 Functions
- For-& Back -wards
14 Number
Theory, Richly
15. Exponents,
Radicals & logs.
16 Calculus
- Examples & Advice
17. Real
Analysis
18
Electric
Circuits Etc (So So)
19 Maps,
Similarity & Trig, (alt view)
20 Complex
numbers
21
Logic with Symbols+truth tables
22 Consistent
Story Telling
23. Even
More Logic
|
|