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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
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power users?
| | More Algebra Hints
Description: This lesson summaries the properties of straight lines, their
slope, equations, and the slopes of perpendicular or parallel lines
5. Straight Lines
| slope m = |
Dy
Dx |
= |
y2-y1
x2-x1 |
= |
rise
run |
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Two points are usually needed to compute the slope. For a straight line
segment, the slope m is a constant of proportionality between Dy
= y-y1 and Dx
= x-x1. The change Dy
in y is proportional to the change in Dx
in x.
The point-slope form of equation for a line y-y1
= m·(x-x1) implies y
= y1+m·(x-x1).
In the case [x1, y1] = [0, b] is the
y-intercept, equation y = y1+m·(x-x1)
becomes the slope intercept form of equation for a line y = b +m·x
or y = m·x + b. In answering questions, rewrite
any equation you obtain for a non-vertical line into a slope intercept equation.
After an equation of a line is written or given in form y = m·x
+ b, the coefficient of x gives m and the constant term b is the y-intercept,
that is the value of y when x = 0.
Graphing: Two points are usually needed to draw a straight line.
Use the x- and y- intercepts if the line does not pass through the origin.
For best results (greatest accuracy) in drawing a line, take two points far
apart. One point is enough is the line is horizontal or vertical. Label
the horizontal and vertical axises with their names and coordinates.
If L has nonzero slope m=m1 and a line K perpendicular to
L has slope m2 then -1= m1·m2 . Thus
m2 = -1/m1 = negative reciprocal of m1.
When slope m1 of L is known, it can be used to
compute the slope m2 of K without being given two points on
the line K.
To find the intersection point of a line y = m1x + b1 and
y = m2x + b2 , solve the equation m1x + b1
= m2x + b2 for x and then compute y.
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Professor Whyslopes:
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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.
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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.
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| Schools/Colleges:
Hire the site
author, as an online instructor, as technical support
for teachers, or advisor for curriculum review.
Site
Reviews may serve as references. See how online
whiteboards with voice and real-time writing make
online help possible with board content printable.
Text or written work scanned or saved to a pdf file
may be uploaded for discussion in the
whiteboard. |
www.whyslopes.com
Parents: Help
your Child/Teen Learn
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
Math
How-TOs etc 2008
1. Arithmetic
2. Algebra
3. More
Algebra
4. Geometry
5. More
Geometry
6. Calculus
Site
Description/Reviews by 3rd parties
Site
Math Lessons
1. Arithmetic
Flash Videos 11-2008
2. Algebra Videos (to appear)
3. Fractions
and More
4.. Solving
Linear Equations 04-2005
5. Euclidean-Geometry
To Complex No.s
6. Analytic
Geometry/Functions 2006
7. Number
Theory. 2006-7
8. Exponents,
Radicals & logs. 2008
9 Calculus
2005
10..Real
Analysis 1995
11 Electric
Circuits Etc 2007
12. .Algebra,
Odds & Ends, HS level-2001
13.Maps,
Plans, Similarity &Trig, with
Complex Numbers, 12-2009.
For Math
Instructors/Tutors/
Curriculum Committees
1. K0-11Applied Math Program Outline
2. Mathematics
education essays
3. LAMP
- an earlier applied math program.
4. (150 pages)
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