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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.
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.
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Conic Sections
Here are a few details, an overview, that may help provide a context for
notes and explanations elsewhere. Conic sections are often studied in
late high school mathematics courses. See too the site
page on simplifying square without a calculator.
The intersection of a plane with a cone gives conic sections. In the
plane, those sections may take the form of a circle, ellipse, parabola,
hyperbola or a pair of straight lines. The selection of a coordinate
system in the plane leads to equations in "standard" or near standard
form. From the coefficients or parameters in the equations, the location of
geometric features can be computed.
Occurrence of Conics, http://britton.disted.camosun.bc.ca/jbconics.htm,
has drawings of conics (intersection of planes with cones) plus their
appearance in in daily or non-daily life.
The full study of conic sections could involve
- the description of cones apart from coordinates
- the description of cones with coordinates, that is through equation or
formulas for them.
- the description of a plane with coordinates
- the choice of orthogonal coordinate system (x, y,z) such that the plane
corresponds to Z = 0 (or another constant), and a description of the cone
within that coordinate system.
- the description of the intersection of the cone with the plane via a
quadratic equation Ax2 +Bxy +Cy2 + Dx + Ey + F =
0
- the transformation of the quadratic equation into standard form by a
combination of rotations to eliminate the Bxy term and translations to
eliminate the linear Dx + Ey terms.
- the use of the equation in standard form to classify the intersection as
an ellipse, parabola, hyperbola etc.
A short course on conic sections might mention 1 and possibly 2, leave
steps 3 and 4 for a course on linear algebra, and start with the transformation
of a quadratic equation Ax2 +Bxy +Cy2 + Dx + Ey + F = 0
into standard form. The latter requires (i) rotations and their description with
sines and cosines of the rotation angle, (ii) translations based on completing
the square. The special case B = 0 only requires the translations.
A short course on conic sections might describe the conic sections in terms
of set of points which satisfy geometric conditions involving distance to
so-called focal points and straight lines (directrix, major and minor axes,
conjugate and ????? axes). Those geometric condition distances provide
equations based on the Pythagorean distance formula and hence a few square
roots. The elimination of those square roots
involves their transformations of the equations to isolate them, so squaring
does the elimination. That transformation needs to be done once or
twice. By following in the steps of our predecessor to select a coordinate
system based on the given focal points and lines, the foregoing square root
eliminations transformations leads to equations without square which convention
declares to be the standard form. The aim of steps 5, 6 and 7 above is to
use rotations and translations to change the quadratic equation Ax2 +Bxy
+Cy2 + Dx + Ey + F = 0 into one of those standard forms.
Parameter in the standards forms allow one to identify the location of
focal points and axes and directrix, etc, in order to graph the conic section
in a plane. Going back and forth between the graph and the geometric
features would be one of the exercises in the short course. The standard form
are say equations in coordinates x and y (not necessarily the original
coordinates above) simplify enough to obtain formulas for x and y. The
analytic properties of those formulas and their numerical evaluation may help
in the more precise location of points in a conic section. The analytical
properties of the formulas, their domain and ranges in particular, echo,
reflect and imply properties of the conic sections.
Quadratic equations of the form y = ax2+bx+c lead to parabolas.
Indeed, in your mathematics classes, parabolas may be first defined as the
graph of quadratic y = ax2+bx+c. The discussion
of conic sections gives other equivalent ways to introduce parabolas.
See the chapter on islands and divisions of knowledge in site volume Pattern
Based Reason to reflect more on the possibility that a body of knowledge
may have several different entry points.
All the foregoing develops algebraic skills. Some applications may be
found in the following areas:
- astronomy where planets, comets and stars may follow elliptical, parabolic
or hyperbolic orbits (solution of two body systems).
- engineering where the classification of level sets of quadratics Ax2
+Bxy +Cy2 + Dx + Ey in two (or more variables) as ellipses,
parabolas or hyperbolas may have implications for the stability of
structures and chemical reactions.
- mathematics where the classification of level sets of quadratics Ax2
+Bxy +Cy2 + Dx + Ey in two (or more variables) as ellipses
or hyperbolas implies the existence of a minimum, maximum or saddle-point in
the graph of the quadratic or a function which it approximates.
Remember that high school mathematics curriculums expanded in the late 1950s
and early 1960s when rocket scientists (a dangerous profession for the world)
were in demand.
Ellipses from drawing to equation
(Elimination of Square Roots)
Answer to a Problem (written December 1997)
From A Student
I will also show you some examples of each section in my math book, and their
instructions and let you see can you explain to be how to work those problems
out.
My math book gives the definition of a few of these terms, they are: An
ellipse is the set of all point P in the plane such that the sum of the
distances from P to two points is a given constant.
Each of the fixed points is called a focus (plural: foci) of the ellipse.
One problem in my book goes like this: Find an equation of the ellipse having
foci F1(-4,0) and F2(4,0) and sum of focal radii 10, use the distance formula. A
point P (x,y) is on the ellipse if and only if the following statement is true:
PF1+ PF2 = 10. See can you help me step by step to solve this problems.
Response
Draw a coordinate system or graph in which you can locate F1 and F2. Next
take a thread with length 10. Attach its ends to F1 and F2 with pins or by
passing the thread through the paper and attaching the ends on the backside with
some tape. Adjust the attachment so that when the thread is taut (fully
extended) the visible length is 10
P
/\ ---------- Hold taught with the end of a pencil.
/ \ This string should have length 10
/ \
---------/------\----------------------------------------------
F1 F2
(-4,0) (4,0)
Now move the pencil with string taut (fully extended) so that it draws a
curve. That curve tightly drawn will be an ellipse. Say, allow or assume
that the coordinate of the pencil point P is (x,y).
Now as you move P or change (x,y), the following equation should be
satisfied.
(distance of P to F1) + (distance of P to F2) = 10 = L
The distance of P to F1 is given by a square root of
(x-(-4))**2 + (y-0)**2 =(x+4)**2+y**2 = first quadratic
The distance of P to F1 is given by a square root of
(x-4)**2 + (y-0)**2 =(x-4)**2+y**2 = second quadratic
The equation
(distance of P to F1) + (distance of P to F2) = 10 = L
has the form
sqrt( first quadratic) + sqrt (second quadratic) = L
where L = 10 = length of string. We eliminate the square roots by observing
that
sqrt( first quadratic) = L - sqrt (second quadratic)
This isolates one quadratic. We can eliminate by squaring both sides. This
yields
first quadratic = L**2 - 2L*sqrt(second quadratic) + second
quadratic
Now isolate the remaining root by observing
2L*sqrt(second quadratic) = L**2 - second quadratic - first
quadratic
The right hand side simplifies to a linear expression in x and y. So squaring
again gives
4L**2(second quadratic) = (a linear expression)**2
The result is an equation quadratic in x and y to be tidied up and put in
your favourite standard form. In general, you could try this with F1 having
coordinates (a,b), F2 having coordinates (c,d) and the length of the string
being L. L should be greater than the distance between F1 and F2.
More
With a string of fixed length L and two focal points F1 and F2, you can draw
half or all of ellipse with the aid of a pencil.
If F2 and F1 move towards each other, the ellipse become more and more
circular. Half the distance between F1 and F2 is called the eccentricity e of
the ellipse. This eccentricity is zero for a circle.
If F2 and F1 are moved away from each other, the ellipse drawn because more
elongated in the direction between them these points, and it becomes narrower in
the perpendicular direction. The maximum distance between them is the length L
of the thread. In this case, the ellipse has become a line segment.
If the distance between F1 and F2 is greater than L, than for every point P,
the triangle inequality implies
(distance of P to F1) + (distance of P to F2) >
(distance of F1 to F2) = L
We assume that L is greater than the distance between F1 and F2.
The ellipse, if it not a circle, has two axes of symmetry.
The major axis passes through the two points F1 and F2. This ellipse
is widest in this direction.
The minor axis is given by the perpendicular bisector of the line segment
joining F1 and F2. The ellipse is narrowest in this direction.
I hope this late reply helps. It may become another appetizer or lesson at my
website. Thanks. The other problems are left for you solve. it very important
that you understand any solution you find or obtain step by step. The details
count.
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www.whyslopes.com
Analytic Geometry
& Functions, etc
Area Entrance
Entrance + Pages Below this page
Pages at Current Level (C) Complex Numbers
(FN) What are Functions?
(FN) Functions - More
SZM: Sign Analysis
(L) Lines Summary
(P) Polynomials (*,+,-)
(Q) Quadratics
(D) Simplify Square Roots
(T) Unit Circle Trig
Conic Sections
Links
More Links
Area Entrance
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
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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