|
HIP,
HIP, HIP, Hooray
YOU are better than YOU think. Show yourself how:
|
// _ _ \\
/\ /\
<| (o) (o) |>
\ | | /
-/[]\-
||
/ \_
||||||||||||||||||||||||||||
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
liinear2007 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.
| |
Preparation for calculus, is demanding. It requires
the will to master fractions, algebra, logic, geometry, trig and
functions. Mastery needs to be developed and proven in writing.
Have another identify and correct all logic and notation errors (some cosmetic
variation allowed) in your written work as well as provide
encouragement. While calculators may help, mathematics is best learnt
the old-fashioned way with pencil and paper. The ability to record and
communicate solutions on paper is a must.
After logic, preparation for calculus offers a focus, a purpose
and base for most high school or college mathematics. Indeed,
preparation for calculus prepares for all subjects requiring some
mathematics.
Many profession far from mathematics require
mathematics mastery because its mastery shows an error in one step makes it
and everything after wrong. Knowing that makes a person more careful.
(1) Logic mastery
provides the first step in Logic
mastery is said to ease or avoid many learning difficulties
(2) After logic, site preparation for calculus continues
with algebra chapters reviewing or
introducing algebra or the shorthand roles of letters and symbols. Talking about
Three Skills for Algebra provides a new way to start or review the
subject.
The introduction to algebra begins with a
unique idea, namely we can talk about and describe numbers and quantities
before and then besides the use of letters and symbols. The site essay what
is a variable (a bit long) puts words before symbols. See if it
helps. The introduction to algebra continues with an old idea that we
can use letters and symbols, alone or in groups, to represent numbers and
quantities. That provide a shorthand way to refer to numbers and quantities by
themselves or in formulas. A formula itself represent a number, the number
that appears when each time the formula is used.
Site algebra
chapters 10 explore the shorthand art of representing numbers by letters
and symbols alone or in groups, and provides some rules and patterns.
Site algebra
chapter 14 and 15 also explore the difference between arithmetic and
algebraic solutions to questions - the aim being to show the power of algebra,
or the shorthand role of letters and symbols in describing solutions to
similar questions, all at once.
The introduction of algebra in chapters 8 to 14 can be
understood with the use of a calculator, and most likely can be read
alone. But preparation for calculus requires mastery of fractions as
well.
(3) The new site area Solving
Linear Equations with stick diagrams provides an introduction to algebra in
which fractions sense and operations can be seen in terms of line segments and
operations on them. See if you can master all this by yourself or with
help. In live instruction, I might put this step earlier, before the second or
first. But for most readers, placement third may be easiest.
The use of stick diagrams to solve some linear equations is a
disposal crutch. Students may solve linear equation in one unknown with these
diagrams in order to understand the equation only approach. While I advise
student to write out the stick diagrams fully and correctly, as a step to
solving equations via a sequence of equations, logically connected,
without the stick diagram crutch, I may not insist that a student who
makes the leap properly ahead of others, go back to write out solutions in the proper
stick diagram format.
Success in calculus requires fraction sense.
The hope here is that Solving Linear
Equations with Stick Diagrams will lead you to understand fractions - what
they are and how to work with them. If not, you will need to learn
or recall how to efficiently
add, efficiently
and efficient divide
fractions by hand. The Arithmetic
Videos (50+ in Realplayer format) may be viewed and Exercises on Mostly
Fractions (answers given) will also test and may develop
fraction know-how. More lessons, not needed we hope, appear the
site area Fractions, Ratios, Rates, Proportions and
Units
Problem solving skills are worth having. Steps 1 to 3
provide some tools for it in and out of mathematics. In problem solving
try to think in the box first - there may be a tried and tested recipe to save
you time and energy - before thinking out of the box. Solving
problems may involve a mix of trial and error, and strategies to lessen the
errors. For instance, in jigsaw puzzles face up or face down, working on the
edges first is a good method for solving the puzzle. In real life, if you can
not find a recipe to solve a problem, the existence of a solution may not be
guaranteed, a limitation of rule and pattern based reason. Did you hear
of the skilled worker, bright and intelligent, and well-trained, but the tools
in the worker possession were not right for job. That can happen to all
of us, but less often for the well-prepared.
(4) This site does offer enough examples in arithmetic and
algebra for students to see and try. The purple
math algebra lessons) and/or the math math
league help topics for grades 4 to 8 provide remedies. Good-bye.
(5) The new site area Solving
Linear Equations with stick diagrams also introduces systems of
equations that are triangular or essentially one unknown because they are easily
solved and because the first words problems in the first years of algebra
involve essentially one unknown.
With the introduction of decimal notation in
the 1500s, methods of decimal arithmetic methods turned figuring (addition,
subtraction, multiplication, division) into thoughts that could be recorded
and developed on paper. Before calculators, arithmetic after the 1500s
was easy done mechanically via on-paper operations with decimal notations for
whole numbers and the fractions which could be written exactly as
decimals.
With word alone we can describe
calculations we may want to do or give to another by speaking aloud or
writing. And to lessen the amount of writing, we may use abbreviations.
The use of letters and symbols to denote operations and numbers or quantities
takes the abbreviation step even further. Many but not all calculations
describable with words can be described by shorthand formulas with
abbreviation or letters and symbols standing in for operations (addtion +,
subtraction - , times x, division / ) and for numbers and quantities.
Often the shorthand description of a calculation is more compact, and in being
more compact can be seen and read quickly and precisely in writing on paper.
So many calculations are best seen and read silently on paper than read
aloud. So the shorthand description of calculations with expressions and
equations on paper is more efficient than the word description. It is perhaps
too efficient as students have to learn how to read a language that is
precisely written, but too awkward to read aloud precisely. That has
been a barrier to introducing algebra. But as with arithmetic,
operations and thoughts that are difficult to perform mentally or describe
with spoken words, can recorded and developed on paper in a written form that
be seen and understood at glance. Here the language is worth a thousand
words.
The ability to solve systems of equations in essentially one
unknown simplifies the solution of word problems in essentially one unknown. So
the ability should be put first. Otherwise, the solution of word problems
in essentially one unknown avoids forces us to do more work mentally than is
necessary. Asking a students to recognize the essential unknown in a word
problem bypasses the shorthand advantages of algebra.
The algebra
chapter 15 on solving linear equation, written before stick diagram were
conceived, to be read beside this stick diagram area is also emphasizes
the arithmetic and algebraic solutions to problems. The power of algebra
to solve many problems at once is illustrated again, but there is a
limitation. For large systems of equations, arithmetic or numerically methods
provide more practical approach to problem solving. So algebra provides a
powerful tool for solving many problems at once, but there are limits to it
use in that regards.
(6) In high school mathematics, there are two treatments of
geometry which coexist and supplement each other, the first without and the
second with. Logic mastery helps
here.
Euclid about 300 BC in his elements produced a codification
of geometry before the invention of coordinates by Renes Descartes 1800 year
later. Knowledge of Geometry before coordinates is employed in the
development of geometry with coordinates (analytic geometry, unit-circle trig,
complex numbers, calculus, and so on).
The new site area on Euclidean-Geometry
on geometry before coordinates offers a full thought-based explanation of
the following topics.
This part of the site preparation for calculus, a framework
for learning and teaching Euclidean Geometry deductively, may be
difficult to follow alone unless a student has seen most of the skills and
concepts elsewhere. That incompleteness suggest more details
should be included, a todo for later.
Try to read the following topics in sequence alone or with help because it
provides a very good base for Euclidean-Geometry
with some notes or observations to make the exposition more coherent.
- Correspondence
between triangles. Here is an explicit definition, not always seen in
class.
- Isometry
of Triangles - Here is a definition.
- Side-Side-Side
method
- Side
Angle Side method
- Angle-Side-Angle
method
- Isoceles
and Equilateral Triangles
- Side-Side-Side
triangle construction Failure
- SAS
triangle construction Failure or Near Failure
- ASA
triangle construction Failure - links with the parallel postulate
- Parallel
Lines - and angles associated with a transversal.
- Triangle
Angle Sum - from the parallel postulate
- Similarity
and Minimal Conditions for
- Right
Angle Trig., from Similarity
- Trig
& Similarity - More about the Connection
- Parallelograms
and their Properties
- Arrows
and Vectors in the Plane (skip)
A few notes - optional reading
- Correspondence between triangles are often used in
the early discussion of isometry and similarity without any
definition. So we begin with that.
- The issues of triangle duplication and isometry
via the triangle construction methods and isometry criteria (SSS,
SAS and ASA) is separated from whether or not the construction
method work. Lengths and angles must satisfy some inequalities
before the methods work. Those inequalities are automatically
satisfied by data coming from an existing triangle.
- The failure and near failure of the ASA angle
points to and provides a context for the parallel line
postulate. The latter represents here an extrapolation of
experience with the ASA triangle construction method.
- Properties of parallel lines, in particular the
angles formed by transversals are developed next. The latter imply
the sum of angles in a triangle is 180 degrees or two right angles.
- The classical development of right triangle
trigonometry then follows from similarity. We see how
trigonometry hides similarity considerations and gives an
alternative to them solution of missing side and angles problems for
triangles. Similarity is implicit in trigonometric computations.
- Properties of parallelograms follow and combine
earlier properties of triangle construction or isometry criteria and
the properties of parallel lines. Finally, arrows and
their addition are introduced in the plane to show what can be
done before the use of coordinates. In analytic geometry, the
use of numerical coordinates, allows for the easier and further
development of skills and concepts in geometry etc.
|
(6) The new site area on Analytic-Geometry
introduces the coordinate based treatment of geometry. In it, ordered pairs of
real numbers represent points in the plane. Sets of ordered pairs represent
lines, circles and further geometric figures in the plane. By using ordered
pairs and sets of order pairs to model or codify Euclidean geometry, there
is no need to rely on drawing or sketches to arrive at conclusion in geometry.
And in modern pure mathematics after calculus (optional studies for most), that
is done. But preparation for calculus and calculus itself are best understood by
mixing the coordinate and pre-coordinate views of geometry and coordinate and
pre-coordinate views of trigonometry and functions.
The developers of the modern mathematics school
and college curriculums in the late 1950s and the 1960 decade tried a too pure
approach to the exposition of mathematics. The two treatments of geometry,
coordinate and pre-coordinate existed, were used, but care was taken in theory
not to mix the patterns or axioms of one with the another while the practice
required. Moreover axioms (assumed patterns) for real numbers and their
consequences were not mixed with the common knowledge and common assumptions
about the decimal representation of coordinates. The refusal to mix led
to a separation and even an inconsistency between the theory and
practice. For consistency in exposition, if not with pure mathematics,
the preparation for calculus and the calculus
intro at this site mixes coordinate and pre- coodinate geometry (favouring
coordinate views where possible). It identifies real numbers and defines real
numbers by their decimal representation(s) in order to make the discussion of
limits, convergence and continuity simpler to learn and teach. The
mixing here may provide the algebraic-deductive maturity and context that
students of pure mathematics almost surely require for the comprehension of
context-independent modern mathematics
| Analytic
Geometry (summer 2005) Geometry and
trigonometry with coordinates builds on geometry
without. Functions not part of geometry required can be described with
ordered pairs, or be sets of points drawn in the plane. So they
are be included here with or in analytic geometry. |
- Real Numbers
- Real No.s More
- Linear Inequalities
- Say More Positive
- Rectangular Coords
- Distance Formulas
- Add & Multiply Points
- Polar Coordinates
- Radians
- (A) Vectors
(A) Coordinate Arithmetic
(A) Navigation on Maps
(A) Addition Geometrically
(A) Rotation
- (C) Complex Numbers
(C) Distributive Law
(C) Properties
(C) Complex Conjugates
(C) Pythagoras New Proof
- (T) Unit Circle Trig
(T) Complex Numbers &Trig
(T) cis or exponential FNS
(T) Dot & Cross Products
(T) Cosine Law
(T) Pythagoras Converse
|
- (L) Lines Slopes Equations
(L) Deriving Eq'ns
(L) Perpendicular
(L) Numerically
(L) 3 Eqn Forms
(L) Algebraic View
(L) System of Equations
(L) Odds & Ends
- (PT) Dilatations
(PT) Translations
(PT) Rotations
- Quadratics
- Conic Sections
- (FN) Functions
(FN) With Formulas
(FN) With Sets
(FN) Vertical Line Rule
(FN) Horizontal Line Rule
(FN) Inverse Functions
(FN) Why Sets
(FN) Odds and Ends
|
| The
assumption of coordinates covering lines, planes and space connects
geometry and algebra in the thought-based development of distance
formula, vectors, complex numbers, unit-circle trigonometry. etc. In
steps 1 to 12, The Analytic
Geometry site mixes assumptions about real numbers and Euclidean
Geometry in the plane to provide a simple development of vectors and
complex numbers in a way that aids the development of trig identities,
the proofs of cosine law, and the trigonometric representation of dot
and cross-products. The treatment of inverse functions
emphasizes that a curve, set or relation in the plane which satisfies
both vertical and "horizontal" line rules defines two
functions or computation rules clearly inverse to each other, and
that the graph in standard position of the function obtained from the
horizontal line rule is obtained by the reflection [a,b] ===> [b,a]
in the plane. For the inclusion of polynomial in functions see Visual
Aids for Column Multiplication Methods at this site. The site Number_Theory
areas contain ideas useful for operation on polynomials. |
(7) The site calculus
introduction starts with a geometric
and algebraic preview of calculus.
The first can be understood with a knowledge of slopes to straight lines. The
second employs factor polynomials in sign analysis. The two preview
together develop gradually and further the algebraic skills needed at full
strength in calculus. The calculus intro, the online book Why
Slopes and More Math (chapter 14), and its appendices online in the Real-Analysis-Decimal-View
site area develop almost as far as possible, the simplifying decimal view of
limits, convergence and continuity, a view sufficient for most who will not
study pure mathematics, a view that will those who study a context for the
decimal-free view of pure mathematics. There-in lies a reasonable compromise
between the needs of the public and the requirements of modern mathematics
(set-based, decimal-free, diagram-free, context-independent or -free).
| |
whyslopes.com
Entrance Level
Montreal Area Tutors
Pages For Teachers
Site Entrance & Hub
Permissions for Instructors
Lesson Plans - Sec I
Lesson Plan, Sec II
Lesson Plans - Sec III
Secondary Maths, Core Elements
Site History/Content
Site Reviews
Vol 1. Elements of Reason
Maps Plans Drawings
Quantitative_Skills/index.html
Order Site Books
McGill-Quebec Nonsense & BS
HIP, HIP, HIP, Hooray for
site
content & history. Hype, Hype,
Hype, Hoorary, for deception.
Your IP Address & how to use
it
Pages for Students
Site Entrance & Hub
25 hours per course
Site Areas by Age and Subject
Montreal Tutors
Entrance Continued
Still More Advice
Head Start Page
More Advice & Directions
Aims to adopt to aid
Arithmetic Check List
Fraction Skill and Concept Check List
Site History and Content
Books to Read
Complex No.s Intro.,.
Calculus Motivation
Calculus. Guide Short
Calculus. Guide-Long
Calculus Guide - Longest
Links - Many Subjects
Links - Games/Activities
Long Site Intro
Logos Cafe
Logic Check List
Mathematics Cafe
Math CheckList
A Site Map
Advice for Secondary I Students
Three Ways to be a Better Student
Reason for HS Mathematics
Three Links for Teachers:
(i) First
Year High School Math - Lesson Plans with Fraction Focus
(ii) Second
Year High School Math - Lesson Plans with an algebra focus
(iii) Algebra
Lesson Plans
Help U Learn/ Teach
- Algebra
words before symbols
- direct & indirect
use of formula, numerical versus algebraic solutions - what
is a variable (more words)
- Arithmetic
- exercises
- with fractions
-
videos on primes, lcm, gcm,lcd, square roots etc
- Calculus - geometric
preview, algebraic
preview,
3 study guides,
much more
- Complex numbers
-starter lesson with java applet - easy
consequences for trig & vectors in the plane
- Education
- Empirical Course Design
& Delivery
- Fractions
- alone
- by rote
- with
algebra
- videos
- Functions - introduction
hindsight
- composition aka
substitution -
- Geometry, Euclidean - Correspondence
of triangles, Triangle
construciton, duplication & Isometry - Failure
of ASA & the // line postulate - angle
sum in triangles -//
grams - Triangle
Similarity
- Geometry-
Analytic - functions, polynomials, complex numbers, unit circle
trigonometry
- Logic
- First Steps -
Symbols in Logic
-
Occurrence &
Truth Tables - Indirect
Reason -Indirect
Reason More
- Proportionality
- Definition -
Direct & Indirect Use - Numerical versus Algebraic Solutions
- Real Analysis
- Decimal View of concepts
and of proofs
- Rules &Patterns in Science, Technology & Society
- Pattern Based Reason
- Mathematical Reasoning, empirical, inductive or deductive
- Units
- in rates & slopes &
(?) derivatives
- in ratios
& proportions - slopes & rates included
- Complex Numbers & Vectors & Trig
- trig expression for dot
& cross - cosine law
|