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YOU are better than YOU think. Show yourself
how:
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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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A Fourth Skill For Algebra
Direct and Indirect Use of Formulas, or Forwards and Backward Use of
Equations
Every formula met in mathematics, accounting, science, technology etc may be
used directly and indirectly, that is forwards and backwards.
The simple message that the forward and backward use of formulas (direct
and indirect use) is part of high school mathematics and beyond
names a required skill and allows us to recognize, identify and thus emphasize
the most frequent pattern in high school mathematics and beyond.
This message needs to be given explicitly and early in secondary mathematics.
Otherwise the underlying skill become part of the hidden, or silent and unspoken,
agenda in mathematics courses.
Teachers: Consider combining the www.purplemath.com
a two page lesson on solving
literal equtions with the message above and the examples and exercises
indicated below. The page banner above was Forward and Backward use of
equations but it now reflects the purplemath lesson, Solving Literal
Equations.
First Site Example
Direct and Indirect Use of the Rectangle Area Computation Formula
Chapter 10
in Three
Skills for Algebra in discussing
Direct use of A =WL assumes W and L are given. Indirect use assumes A and one of
W and L is given, and leads to the calculation or formulas W = A/L or L =
A/W. The explanation of those formulas is a step towards algebraic
reasoning - the direct and indirect or forward and backward use of formulas.
More Examples: Formulas for perimeters and areas of
squares, circles, triangles, rectangles etc can be used forwards and
backwards. Finding the value of a proportionality constant k say in an
equation y = k x represents an indirect or backwards use of an equation,
a pre-requisite to further forward and backward use of the equation y = kx.
The calculation of parameters a and b in y = ax + b (or y = mx +b) represents
another backward use of a formula or equation. Quebec students in
secondary III have met the forward and backward use of the Pythogorean
equation c2=a2+b2 where c is the length of
the hypotenuse and the two numbers a and b are the lengths of the other two
sides (legs) of a right triangle.
To Do: : Post some details and exercises here
to further illustrate and emphasize the forward and backward use of common
formulas.
Going Further (More on Substitution)
The aforementioned Chapter
10 before the forward and backward use of a formula goes further in showing
how to describe a the calculation of a box V = H(WL) and show how to employ
substitution (a new concept for students) to go between this formula and V
= HA where A = WL. Details are given in the chapter. The
details may be easier to grasp if numerical examples are added to this
exposition.
Seeing how a box volume formula V = hA and V = h
(WL) can be transformed into each other illustrates and may introduce the
notion of equivalent expressions. The law applied here is A = WL is a
geometric law rather than an algebraic law (like the distributive law).
None, the idea that an expression represents a number or quantity and that
there may be more than one ways to compute the number or quantity is key to
the notion of equivalence. Students thus see how substitution in
formulas leads to new formulas, how arithmetic patterns may be used to
use formulas directly and indirectly, and how algebraic solutions may be more
general or powerful than arithmetic solutions.
Algebraic Exercises:
- Find a formula for the area of square in terms of its perimeter
(easy)
- Find a formula for the area of circle in terms of its perimeter
(easy)
- Find a formula for the perimeter of square in terms of its areas
(harder)
- Find a formula for the perimeter of circle in terms of its areas
(harder)
See www.purplemath.com
two page lesson on solving
literal equtions for hints or to learn more.
The exercises could be easier after reading the first sections of Chapter
15 and Chapter
14 in Three Skills for Algebra. The chapter 15 material may be
easier..
The first sections in Chapter 15, Solving
Linear Equations in online site Volume, 2. Three
Skills for Algebra, derives an algebraic formula for the solution of
equations of the form ax + b = c, and so emphasize the use of algebraic
shorthand reasoning to imply solutions for many problems of a given form at
once. All the foregoing emphasizes the power of algebra, or the shorthand way
of writing and reasoning with letters in place of numbers. That being said,
numerical experience is still required with formulas and their graphs,
otherwise the connection between numbers and algebra may too weak.
A Deeper Site Example
for now or later or never.
This Chapter
14 introduces the direct and indirect use of the
compound interest formula A = P(1+i)n.
Chapter
14 presents algebraic and arithmetic solutions that may be used to
check the calculator skills of students while developing the algebraic way of
writing and reasoning. In the compound interest formula A = P(1+i)n
three of the four amounts A, P and i and n are assumed known, and the problem
is calculate or find a formula for the missing fourth. The use of this formula
is indirect when the left hand side quantity A is given or known, and the task
is to find the value of the principal P, the interest rate i or the number of
compounding periods n. Add to chapter 14 coverage, numerical
confirmation that the algebraic solution works. The algebraic solutions
for the indirect use of formulas involve substitution and assumes the
pattern (AB)/B = A. Coverage of Chapter 14 is recommended as
part of the next topic: exponents and radicals.
Once as a too pure, applied mathematician, I did not investigate or become
familiar enough with the numerical behavior of my formulas since I was too
convinced of the power of algebra.. Numerical experience with formulas needs to
accompanying the development of algebraic reasoning skills. A balance is
required.
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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)
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