|
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
liinear 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.
| |
4. Functions and Relations
Site Objective: Clarify and Provide a context for the set-based
description and codification of functions and relations.
The set based notion that a function can be represented or codified by a set
of ordered pairs is useful, but the notion that writing y = f(x) indicates that
y depends on x should come first.
Functions and Relations
In modern pure mathematics, functions and the allied concept of
relations are identified with sets of ordered pairs. That provides the
eventually useful, set-theoretic viewpoint. Yet before it you should meet and
understand the previous, broader and impure dependency viewpoint.
Logical or Pedagogical Preparation (Pre-requisites)
-
Word have been missing or used unclearly in
mathematics. The introduction of the notion of what is a variable and
a quick review of three skills for algebra, the use of notation in
mathematics, and the forward and backward use of formulas in chapters 8 to
14 in Volume 2, Three Skills for
Algebra, might fill gaps in the comprehension of algebra, and
develop the algebraic maturity needed pedagogically if not logically for the
current study of functions and relations.
-
While we may advocate the greater use of
calculators, courses in calculus and senior secondary school courses
still require students to master and understand exact arithmetic with
fractions without a calculator and the use of prime numbers and
factorization, material that appears in earlier courses. The site
area. Solving
Linear Equations introduction of stick diagrams can be reviewed
(despite opposition) so that students may visualize and consolidate some
fraction skills and concepts. The site area in full provides students and
teachers a model, a lower bound, for the solution of linear equations from
one equation in one unknown to systems of n equation in n unknowns where n =
2, 3 or 4. The site area Solving
Linear Equations in covering a simpler topic also develops a greater
algebraic maturirty,needed pedagogically if not logically for the current
study of functions and relations.
The following two columns point to a thought-based introduction
of functions suitable I hope for math 436 in Quebec.
Column I
Functions Before Sets
(Cover First)
|
Column II
Functions with a
Set-Theoretic Focus
|
Methods to Define Computation and Assignment Rules :
- Using
Formulas (with use of function notation to indicate dependence of
one number or quantity on several others. (math 436)
- Using
Arrow Diagrams, Tables and Sets of Ordered Pairs (listed or plotted)
- functions with finite domains. (math 436)
- Using
Curves and Infinite Sets of Points in the Plane - When the
vertical rule holds, a set of points or curve in the plane can be used
to define a function f(x) via the vertical line method.
Note: Graphing a function f gives a set of points or curve in
the plane for which the vertical line method for computing a function
yields the same function f. (math 436)
- Functions
with Infinite Domains - a few exercises (math 436)
For students who have met slopes and/or
polynomials before the discussion of functions, the Geometric
and Algebraic previews of
calculus will provide motivation for the study of slopes (why slopes) and
for the factorization of polynomials. The algebraic previews will
develop more algebraic skills and concepts, and still greater algebraic
maturity needed pedagogically if not logically for the current study of
functions and relations. These examples may be woven into the
monoticity analysis discussion of on what intervals, real-valued
function y = f(x) of a single real variable x are increasing or
decreasing. and what intervals those functions are positive, negative or
zero. A point is given by a very short interval.
Item I on the right hand side provides
related mateirial
|
Curve or Set Viewpoint of Functions and Relations
In the foregoing examples, you have seen sets appear in the description
of the domains and ranges of functions, and in the definition of function
using sets of ordered pairs. The latter implies or suggest the Set
Based View and Codification of what is a function in site pages with
the following ideas. (Here are more ideas for math 436).
- Set
Existence and Construction (technical starting point)
- Interval
Notation. Next (?) see Domains
and ranges for a zoo of functions using interval notation.
- Assignment
and Computation Rules without & then with ordered pairs.
- Concept
of a Relation, a Set-Based Codification and Generalization.
- Why
call a set of ordered pairs a relation? Numerical Exercise
Included.
- Source,
Target, Domain and Range Set for functions and relations - plus
Definition of subjection, injections and bijections - set
viewpoint
- Injectivity
of Real Valued Functions - injectivity, one-to-one, two-to-one,
many-to- one, or not one- to-one.
- Sign
Analysis, Zero Analysis, Where are functions positive, negative or
zero?
- Monotonicity
Analysis: Where are functions increasing, decreasing etc.
(Includes Optional: Why strictly increasing and strictly decreasing
functions are one to one, that is, injective. )
.
- Extrema
or Max-Min Analysis Where do they have their greatest and least
values. What are minima and maxima.
- Exercises
with Formulas and Graphs - Numerical Experience (!)
- Domains
and ranges for a zoo of functions using interval notation.
- The
absolute Value Function (Qc math 536)
- Functions
Revisited (for teachers, if not students)
|
Lessons Elsewhere:
The website Purplemath offers
the following lessons
Functions & Relations Continued
More Lessons here
-
(FN)
Absolute Value
-
(FN)
Step, Sawtooth & Abs. Value
-
(FN) Horizontal Line Rule
Examples
-
(FN) Inverse Functions
Examples
-
(FN) More
Ways to Define Examples
The last five Lessons 12 to 15 are for students in secondary V mathematics
536 Those lessons require mastery of the earlier lessons. The
calculus introduction pages
- Function Domains
- Polynomials - Domain & Range
may help as well.
| |
www.whyslopes.com
Lesson & Lesson Plans for
Sec IV (Maths 436)
a reference for learning and teaching functions, polynomials, solving linear
systems,
powers + exponents + bases + radicals (roots) , quadratic formulas, equations
of straight lines
1A. Master Logic
1B. Problems Solving Method
2A Solve Linear Equations i
2B.Solve Linear Equation II
2C Use Equal Sign Properly
2D. Perfect Arithmetic Skills
3 Words & Symbols
3 Goals to Set for Students
4 Use Equations Backwardly
5. Master Functions & Relations
6. Exponents & Radicals I
6 Exponents & Radicals II
7. Straight Lines
8. Polynomials (x,/,+/-)
9. Quadratics
10 Prove it
13 Similarity Scale Factors
12 Trig & Triangles
14 Statistics
MEQ Intermediate Objectives
Remarks for Teachers
Sit down and study - no one else can do that for you.
Advice and Directions
What to do in School & Why
How
to Study Maths & Why
Preparing
for Science
Good News: If you can learn to follow a multi-step
methods in any subject precisely, you should be able to do so in other
subjects, as well. Hint: Start with arithmetic
Words Before Symbols:
What is a Variable?
Level: Secondary II to VI, or Grades 7 to 12)
Introduction
Variation between Examples
Variation of Letters
A letter denotes a variable
Cases of Double Variation
Three Notions of a Variable
Constants, Parameters
& Variables
Talking about numbers
Dependent
or Independent
Variable, a Matter of Choice
Complex number starter lesson
Arithmetic Videos
Fractions
Primes
Greatest Common Divisors
Least Common Multiples
Square Root Simplification
Arithmetic Videos
Decimal Addition Methods
Decimal
Subtraction Methods
Decimal
Multiplication Methods
Decimal Division Methods
Fraction
Starter Lesson
(simplify, multiply, divide &
then add or subtract)
|