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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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Stick Diagrams is a new invention for
visually providing a context for the solution of equations in one unknown.
Worked examples follow with and then without 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
See if you can follow the solutions and the solution checks in them.
Each Example comes with one or more silent movie (animated
gif) solutions of further examples.
If you do not follow the use of parameters do not worry.
Take peak at Chapters
15 in Three Skills
for Algebra. It will give you a second perspective on solving linear
equations starting with examples simpler than you have mastered in this site
area, but that second perspective also may also help you understand the
following example. It is an example from the end of Chapter 15.
Notes for Teachers
- Most students understand the use of stick diagrams in solving linear
equations. One recent student could not see the equivalent between the
stick diagrams and equations, but he could use the diagrams to solve.
More examples might have helped. Not all is certain in mathematics
education.
- A vertical pair of equal-length sticks (line segments) with the second
below the first is use to represent an equation in which the length of a
line segment is the unknown. A sequence of operations easily seen and
understood may shorten or lengthen or multiply each stick by the same
amount, so that a sequence of stick pairs or equation results. The aim
is obtain a pair of equal length sticks with one has the sought for length
and the other have a length given by a number. That solves the equation. The
solution method here employ subtraction, multiplication and division to
shift from one pair of equi-length sticks to another pair in order to solve
for or isolate the unknown. Some students will not catch on to the idea that
operations should lead from one pair of equi-length sticks to another.
- A three column table summarizing the operations appears in each
example and is followed a check of the solution obtained. By checking,
students know at the end of their calculation whether or not a mistake has
been made in obtaining a solution. The first column in the table
presents a sequence of stick diagrams. The third column gives the
corresponding equation. The middle describes the operation in going
from one pair of sticks to another (or one equation to another). Ideally the
description is written in a way that it describe the operation on a pair of
sticks and the corresponding equations well. The filling in of the table
introduces the notion that what is done to one stick or one side of an
equation has to be done to the other side as well to maintain equality of
the lengths each stick or equation represents.
- The equations here and in the exercises may be solved without using the
stick diagrams. Some students may see the stick diagrams and decide not to
use them. But their use introduces the notion that what is done to one
stick or one side of an equation has to be done to the other side as well to
maintain equality of the lengths each stick or equation represents.
Ask students who see the stick diagrams and decide not to use them to have
patience and to take the time to demonstrate they mastered stick diagram
usage. Implicit in their usage are all the rules of algebra for
solving equations. Those rules will be formally given later with
reference and illustration by the stick diagram method of solving
equations.
- Students should be required to check that the solution they obtained
satisfies the original equation, and be told explicitly if the right hand
side does not equal the left hand side for your solution that they have to
look for the error (or if time is short, acknowledge their solution is
wrong). Finding that the the right hand side does not equal the left hand
side and saying nothing, or worse claiming to have done the problem points
to a lack of comprehension.
- In class or a solution given by a student, the three column table may be
filled in one row at a time and one row after another with no work before
it. However in these webpages, the solution is provided in paragraph form
step by step before the three column table is written to summarize the
proceeding.
- The examples here involve only unit and simple fractions ¼ ½
¾ mainly because they easily inserted typed on the keyboard. Other
fraction, proper and improper, and mixed numbers appear in the
exercises.
- The stick diagram method here employs only subtraction, division and
replication of segment lengths. Magnification and reduction of
diagrams is also useful to fit them in the width of a column. Example
equations are chosen so that all coefficient and terms in the stick diagram
method remain non-negative. The objective of the stick diagram method is not
to solve all linear equation, but to lead students to solving linear
equations by operations on equations by themselves without any geometric
representation by stick diagrams. Using parallel arrows would be a
method to extend the representation to include positive and negative
coefficients in equations, preferably selected to have non-negative
solutions.
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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
Up
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
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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