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.
| |
Introduction to Stick Diagrams
Our aim in introducing stick diagrams is to show how to understand and
solve equations for unknown length or number x. In the first example
below we will go from a given value of x to a stick diagram constructed or
implied by that value of x. Then we will explore a few simple examples
where we find the value of x from a stick diagram. This is the simple
and slow start of a longer story. So follow the story or chains of reason here
because at the end of this story, I or we hope you will be able to solve
equations in one and more unknowns, and beyond that, you will have a path for
tutoring or leading others to this ability. So have patience.
Step 1. From value of x to one or more stick diagrams.
Step 2 below, going the other way, is more important.
Suppose x represents the length of a stick of length 15. We will
write x = 15. We draw two parallel sticks to represent this situation in a
geometric manner.
The equation x = 15 implies x +5 = 20. We can draw the following to
represent
this equation or situation x + 5 = 20.
The foregoing shows how we can go from a value of x to an equation in x
and a stick diagram which describes that equation. That is step 1 in
our journey.
Step 2. From a Stick Diagram to a value of x, or finding an unknown length
from a stick diagram.
Imagine we are given x + 4 = 16 for an unknown length x. We want to
find x. To that we describe the equation by the stick diagram.
This diagrams indicates a stick of length x + 4 has the same length as a
stick of length16. Now observe 16 -4 = 12 or 12 + 4 =
16.
x
|
4
|
16
|
| <---
12 ---> |
<--- 4 --> |
We are almost done. Now shorten the top and bottom sticks by 4
units. This cutting off or subtraction of 4 units gives the diagram
and suggests x = 12. Let us check this suggestion.
Check: When x = 12 (or when x is replaced by 12)
x + 4 = 12 + 4 = 16
as requested.
So x = 12 is a solution of x + 4 = 16.
Another Examples of Step 2. Finding an unknown length from a stick
diagram.
Suppose x is a length which satisfies solve x + 8 = 28.
Find the length x with the aid of stick diagrams.
| Stick Diagram |
Operation |
Equation |
|
|
Given |
x +9 = 29 |
|
|
Preparation
for a subtraction
|
observe
29 - 9 = 20
or
29 = 20 + 9
|
|
|
Subtract 9 |
x + 9 = 29
9 = 9 _
x = 20 |
|
Conclusion: Must have x = 20 |
If I was solving 2x + 5 = 20 in class, I would just fill in the
table and skip the work before it. Each table consists of a diagram in
the left column, a description of what is done or given in the middle column,
and the equivalent equations in the rightmost column. At the moment, you
are required to draw the stick diagram in the solution of the equation. That
is a crutch. Later on, only the equation column is required with a few
words to explain the operations.
Let us check this suggestion.
Check: When x = 20 (or when x is replaced by 20)
x +9 = 20 + 9 = 29
So the conclusion x = 20 works.
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www.whyslopes.com
Solving Linear Equations
|(Feb 14, 2005)
A reference for solving linear
equations and for recognizing word problems in essentially one variable.
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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