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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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Students: Start with the stick diagrams examples. Then explore the
rest of this site area.
Notes
The following pages offer a new path for learning and teaching how to
solve linear equations in high school or college.
[ Proper Use of Equal Sign ] [ A. Letters and Lengths ] [ B.. Solving Linear Eq'ns. ] [ C. Solving Linear Eq'ns ] [ D.Almost One ] [ E: 2D Systems - Sub Method. ] [ E: Continued ] [ E: Still More ] [ F. Larger Systems ]
The path reinforces arithmetic and algebra skills and concepts.
The optional start, lesson A.
Letters and Lengths, material drawn from a chapter 9 in site volume
Three Skills for Algebra, shows or recalls how letters may denote
lengths in the computation of areas of rectangles, triangles and circles. the
notion of describing or denoting a length by a letter is key to mastery of
stick diagrams, a notion that may be obvious to some and not need further
explanation.
Item B, Stick Diagrams, leads to
sequence of pages explaining how to use stick diagrams to solve some linear
equations.
(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
Stick diagrams by themselves provide a concrete or visual context for many of
the rules or patterns for solving equations, a context that may develop equation
solving skills and confidence before any formal algebraic statement of the rule
and patterns for solving equations.
Using geometry to develop algebra and fraction skills and
sense. For students starting algebra, it far easier to allow a
letter to stand for length of a given or drawn length that has yet to measured
or than is to allow a letter to denote a number yet to be given. That makes
stick diagrams more amenable to the development of algebraic skills and sense
through a three column presentation methods which equates stick diagrams and
operations on them with linear equations and operations on them.
Teachers: Some students may follow the stick
diagram reasoning without immediately seeing how the equations correspond to the
the stick diagrams. For such students, accept mastery of the stick
diagrams while pushing the algebraic equation viewpoint and give exercises
in translating the stick diagrams into equations to remedy this difficulty or as
a preventive measure at the very start of the coverage of stick
diagrams. (Addition May 17th): Put an example of the form x +
a = c first to provide a simpler introduction to the use of stick diagrams. Some
students need it. Others will find that too simple.
Cryptic Summary: Stick
Diagrams provide a new invention for the visual solution of a single
equations in one unknown, say x, starting with a pair of equi-length
sticks to represent the left and right side of the equation and then
proceeding through a sequence of operations which yield further pairs of equi-length
stick corresponding to simpler and simpler equations until the last pair gives
the value of the unknown x. Stick diagrams provide a visual guide and
introductory crutch for solving special linear equations which students will
eventually abandon in order to solve the general linear equation in one
unknown while learning the necessary skills on route. The sequence of
operation on stick diagrams illustrates and strengthens arithmetic with whole
numbers and fractions in a visual manner. Examples here use letters
other than x to as the unknown length in an equation. The sticks in the
diagrams might be replaced by vectors when negative constant terms are
permitted - an extension that might be useful, yet may not be needed due to
intended transient nature of the stick diagrams.
The use of stick diagrams is or should be a temporary phase in learning to
solve linear equations. The further pages show how to solve linear equation in
one or more unknowns without the use of stick diagrams.
-
Site pages on Fractions,
Ratios, Rates, Proportions & Units are for
students who want know-why besides their know-how. They are also for
teachers and tutors. Explore the pages one at a time and one after
another. Skip those not to your liking. Pages 13 to
15 cover ratios, simple or multiple. Pages 16 to 18 introduce units in
calculations and provide a setting for the discussion and definition of
rates and proportionality constants.
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The Arithmetic
Videos (Realplayer format) may be viewed apart from or besides fraction
lessons 1 to 12.
-
Exercises on Mostly
Fractions will test fraction know-how. The site area Solving
Linear Equations may help students visualize fractions while you meet a
geometric approach to algebra. I hope you can follow and enjoy the
underlying ideas.
-
For good material elsewhere, visit www.purplemath.com
for lessons on arithmetic, algebra and geometry.
Learn More
See the following chapters from the site book Three Skills for
Algebra:
8
The Three Skills For Algebras
9 The
First Skill
10 Two
More Skills
11 Why
Shorthand
14
Compound Interest
15
Linear Equations
16
Painless Proofs
17
Pythagoras
This site area on linear equations, Chapters 14
Compound Interest and 15
Linear Equations all provide routes to introduce and extend algebraic way of
writing and reasoning. Which one to follow first is a matter of taste. Nibbling
at all in parallel is an option until their digestion is complete.
Chapter 14
Compound Interest could be rewritten in terms of compound growth and decay
of populations and radioactive material, and and/or connected to exponential
growth and decay without any mention of compound interest.
Teachers
The new site page Teaching
Algebra describes a program to follow.
The ideas here could be woven in early high school or late
primary school class for students ages 10 to 15 say. Past mathematics
curriculums called for an efficient mastery and comprehension of on paper
methods for arithmetic with whole numbers and fractions to serve as a basis for
algebra. Recent practice in classroom leaves many students without a
fraction sense or comprehension and without the ability to do and understand
arithmetic with fractions. The lack of fraction sense after the first year
of high school implies failure or a waste of time and energy in further
mathematics courses. I kid you not.
Note(1) : The discussion of two term ratios a:b (read a
to b) and mutliple-term ratios a:b:c (read a to b to c) historically (?) may
have come before the discussion and physical interpretation of fractions
a/b. Fractions themselves can be identified with twp-term ratios (and
may be called ratios) but a three or more term ratio cannot be
identified with a single fraction. There-in lies a
difference. Some ratios are not fractions.
Note (2): The discussion of ratios here is link to
proportionality - In equivalent fractions, the numerators are
proportional to the denominators with proportionality constant equal to the
common value of the fractions. In equivalent two-term ratios, the the
first term is proportional to the second term with proportionality
constant equal to common value of the associated equivalent
fractions.
Note (3): In the evaluation of formulas for perimeters
and areas, etc, students may see letters replaced by numerical values. Seeing
such substitutions could be part of the development of the algebraic way of
writing and reasoning. The shorthand description of how to add, multiply
and divide fractions provide further opportunities to describe or summarize
computations that could be done, and a further chance for students to see
letters as place holders for numbers, place holders than may be identified
with or replaced by numbers in actual computations.
If you are teacher or tutor, I hope you will see how to generate
more examples and illustrations. Those here give the main ideas but
more examples would help. The examples below are based on division of lengths or
rectangles due to the convenience or inconveniences of html in web page
production.
The puzzle of how to introduce the algebraic way of writing and reasoning
clearly and directly was first met by in high school days
1965-70. Difficulties of fellow students and instructor in understanding
and explaining algebra slowed the site author's education. The first algebra
chapters in the 1995-6 Volume 2, Three
Skills for Algebra, point to a solution - a greater verbalization in
mathematics in which the overlooked ability of describing or talking about
numbers and quantities is recognized and emphasized. That is before and then
besides the introduction of letters and symbols in algebra as placeholders
for numbers and quantities in calculations or their description. This spring
2005 site area Solving Linear Equations with fractional operations on
stick diagrams also introduces algebra in a parallel approach to the
foregoing, which comes first is a matter of taste, while
consolidating fraction sense and skills. The two approaches together
provide a solid base for algebra for students starting their teenage years, or
later remedial instruction. Thus earlier, clearer & likely to be
effective introduction of algebra should lead to shifts in course
content and design at the all levels in high school and college mathematics,
enriched to remedial.
Word processors and spell checkers to help us write. Yet to read and write,
we still need to learn the alphabet and how to use words. Likewise,
calculators and spreadsheets may help us with arithmetic, but to mathematics,
we still need to learn methods to represent, compare add, multiply, subtract
and divide whole numbers <101 by themselves or in fractions, and how to use
or describe arithmetic. Decimals approximation are fine until we need to do
arithmetic & algebra exactly in ways others can follow. Do not let
calculators remove the intellectual component of fraction sense and skills.
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www.whyslopes.com
Solving Linear Equations
|(Feb 14, 2005)
a secondary I to V reference for solving linear
equations and for recognizing word problems in essentially one variable
whether you like it or not, skill in arithmetic with fractions is a must for
algebra. .
Area Entrance
Proper Use of Equal Sign
A. Letters and Lengths
B.. Solving Linear Eq'ns.
C. Solving Linear Eq'ns
D.Almost One
E: 2D Systems - Sub Method.
E: Continued
E: Still More
F. Larger Systems
Area Entrance
(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
Arithmetic Videos
Decimal Addition Methods
Decimal
Subtraction Methods
Decimal
Multiplication Methods
Decimal Division Methods
Fractions
Primes
Greatest Common Divisors
Least Common Multiples
Square Root Simplification
Site books and further webpages on learning and
teaching mathematics and pattern based reason may develop critical thinking,
improve reading and writing, and give a base for learning or teaching high
school and college mathematics.
Great_Expectations:
If you can learn to follow a multi-step methods in any subject precisely,
you can do so in other subjects, as well.
Good news: Site pages identify
what you need to study.
Bad news: Site pages do not explain
everything
Worse news: Learning takes time, yours
Lesson Plans and Ideas for Teachers &
Tutors:
Secondary I -
fractions & allied concepts (decimals, percentages)
Secondary
II - Algebra (arithmetic versus algebraic methods, backward use of
formulas and proportionality equations)
Secondary
IV - Functions to Trig & Statistics
Calculus
Intro
Algebra
Lesson Notes - All levels
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