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.
Learn to 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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Chapter 18
Arithmetic Rules and Patterns (algebraically described)
Rule-based reasoning is used in the changing of formulas
and equations. Somewhat flexible rules say how or what is permitted. The
flexible rules in algebra can be applied one at a time or one after another to
arrive at formulas and equations or to draw conclusions on or from them (the
formulas and equations, that is). But understanding the rules requires the
algebraic way of writing and thinking to be well understood beforehand, else
they, the rules, will not make sense.
Chapter Sections: [Order of Operations] [ 18 Changing Formulas ] [ 18. Proper Use of Equal Sign ] [ 18. Replacement & Substitution ] [ 18 Real Numbers & Quantities ] [ 18 Rules for Algebra ] [ 18 Sums as Factors I ] [ 18 Sums as Factors II ] [ 18 Addition Properties ] [ 18 Sum Associative Property ] [ 18 Sums and Number 0 ] [ 18 Replacing Subtraction by Addition ] [ 18 Times Properties ] [ 18 Sum Grouping and Ordering ] [ 18 Product Associative Property ] [ 18 Products with the Number 1 ] [ 18 Product Grouping and Ordering ] [ 18 Power Rules ] [ 18 To Divide, Multiply ] [ PS: Rules for Fractions and Division ] [ 18 Inconsistent Nttn ]
1 Order of Operations
Parentheses are (often) used to show the order in which arithmetic (+, -,
¸ and ×) is done in a calculation. The order can
sometimes be changed without changing the result. Rules or properties of
arithmetic say when. These rules and properties say how to move the parentheses
about, and how to omit them, without changing the result obtained. Here the
arithmetic may change, but the result does not.
In this section, we talk about the use of parentheses in the description of
calculations that are or could be done. The rules of arithmetic for shifting or
omitting parentheses state when two would-be calculations should give the same
result are described in the following sections.
In arithmetic, the order in which the arithmetic is done may change the
result. So some caution is required. In describing calculations we also need to
give the order in which the additions, subtractions, multiplications and
divisions can be correctly done. The order is based on the following
conventions:
- expressions within a pair of parentheses (¼)
are to be computed before those outside. So the stuff ¼,
whatever it is, within a pair of innermost parenthesis are done before those
outside.
- without parentheses to show what calculation is to be done,
multiplications and divisions are to be done before additions and
subtractions. Multiplication and division are said to have a higher
priority.
Departing or changing the order in which arithmetic is done could give an
incorrect answer. Here are some more examples which show that the order of
operations sometimes affects results. Your problem is to know when.
- The expression 17-(10-3)
gives 17-7 = 10 but (17-10)-3
gives 7-3 = 4.
- The expression ([4/5] ¸[5/16])×[2/3] gives
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64
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128
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This is different from
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- The expression (5¸6)¸2
= ([5/6])¸2 = [5/12] but 5 ¸(6¸2)
= 5¸3 = [5/3]. The parentheses cannot be
omitted.
- (8-5)-2 = 3-2
= 1 while 8-(5-2) = 8-3
= 5. So the parentheses are important.
- But (5·4) ·3 and 5 ·(4 ·3) both give the same result.
Sometimes the order in which arithmetic is done affects the result. In this
case, parentheses and conventions are needed to say what is done first. So
unless you know a rule which says the order indicated by parentheses and the
priorities assigned to arithmetic operations (+, -,
×, ¸) can be changed, you should be very careful.
When in doubt, don't.
In teaching, I had respect for the student who would identify
in his or her arithmetic (or reasoning) what was uncertain. That was a sign of
careful thinking. I tried not to reward students who tried to hide their
guesses. In learning, once a student has identified the limits and
uncertainties in his or her knowledge, that student is ready and able to learn
more.
Chapter Sections: [ 18 Changing Formulas ] [ 18. Proper Use of Equal Sign ] [ 18. Replacement & Substitution ] [ 18 Real Numbers & Quantities ] [ 18 Rules for Algebra ] [ 18 Sums as Factors I ] [ 18 Sums as Factors II ] [ 18 Addition Properties ] [ 18 Sum Associative Property ] [ 18 Sums and Number 0 ] [ 18 Replacing Subtraction by Addition ] [ 18 Times Properties ] [ 18 Sum Grouping and Ordering ] [ 18 Product Associative Property ] [ 18 Products with the Number 1 ] [ 18 Product Grouping and Ordering ] [ 18 Power Rules ] [ 18 To Divide, Multiply ] [ PS: Rules for Fractions and Division ] [ 18 Inconsistent Nttn ]
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www.whyslopes.com
2. Three Skills for Algebra
Foreword, Chapters
& Appendices
Foreword
1. Introduction
2. Implication Rules
3. Chains of Reason
4. Romeo and Juliet
4. Induction Mathematical
5 Knowledge Islands
6 Old Language
7 Arith Skill Check
7. The Next Chapters
8 The Three Skills
8 VNR-Concise-Encyclopedia
PS. What is a Variable
9. Algebra Talk
10 Two More Skills
11 Why Shorthand
12 Shorthand Usage
13 What's Next
14 Compound Interest
15 Linear Equations
PS I. Distributive Law
PS II. Polynomials
16 Painless Proofs
17 Pythagoras
18 Rules of Algebra
19 Functions & Sets
20 Degrees & Radians
21 What's Next
22. Arith & Geometric Sums
23 Summation Notation
24 Your Money
25 Induction & Recursion
26 What's Next
27 Pronouns in Logic
28 Occurrence Tables
29 Contrapositive
30 Truth Tables
31 Indirect Reason
A. Advice For Learning
Words Before Symbols:
What is a Variable?
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
Solving Linear Equations:
A. Letters and Lengths
B. & C. Solving Linear Eq'ns
with stick diagrams.
(i) x + 20 = 29
(ii) 2x + 5 = 20
(iii) 3x + 10 = 32
(iv) 5a + 16 = 3a+ 24
(v) (½)x + 8 = 24½
(vI) (¾)a + 16 = (¼)a+ 24
(vii) (¾)q + 17 = 32
(viii) 13 =[2/3]x +7 twice
(x) Animated Examples
(i) Integral Coefficients (A)
(ii) Integral Coefficients (B)
(iii) Fractional Coefficients
(iv) With
Parameters
Problem Solving with Linear
Equations in one or many
unknowns, and in essentially
one unknown - Symbols before
words.
C. Solving Linear Eq'ns
without
Stick Diagrams
D.
Problems in
essentially one unknown
E: 2D Systems - Sub Methods.
F. Larger Systems
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