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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.
| |
Stick Diagram Solutions: Sixth Example
(brought to you by the number q, unknown on one side,
simple fraction for a coefficient)
Now in the equation (¾)q + 17 = 32 we imagine that q represents an unknown
length. In the stick below, the top stick has length (¾)q + 17 = 32 while the bottom stick
has length 32. The equation say both sticks have the same length, here 32.
Cutting off or subtracting 17 from both sticks (adding -17) gives
a stick of length (¾)q on top and a stick of length 15 = 32-17 on bottom. This
second stick diagram represents the equation (¾)q =15.
But (¾)q = (¼)q + (¼)q + (¼)q. So a third of the length of (¾)q is +
(¼)q. To find (¼)q, we will take cut
each stick into thirds:
The stick diagram suggests that (¼)q =5
Replicating the above diagram (¼)q four times (multiplying by 4 the
reciprocal of ¼) gives
(¼)q
|
(¼)q
|
(¼)q
|
(¼)q
|
5
|
5
|
5
|
5
|
and simplifying suggest q = 20
We could have gone directly to the latter conclusion by multiplying the
diagram
for equation (¾)q =15, or equivalently
by the reciprocal (4/3) of ¾ to find
| q = |
4
3 |
( |
3
4 |
q) |
= |
4
3 |
(15) |
= |
4
3 |
(3 x 5) |
= |
4 x5 |
= |
20 |
|
A Solution Table for (¾)q + 17 = 32 follows.
If I was solving (¾)q + 17 = 32 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.
|
Solution Table for (¾)q + 17 = 32 |
| Stick Diagram |
Operation |
Equation |
|
|
Initial Situation
Given |
(¾)q + 17 = 32 |
|
|
Subtract 10
(a.k.a Add -10) |
(¾)q =15 |
|
|
A third of (¾)q is (¼)q. So q is also a third of 15. (Multiply by
to go from 2nd row to 3rd.)
|
(¼)q = 5
|
(¼)q
|
(¼)q
|
(¼)q
|
(¼)q
|
5
|
5
|
5
|
5
|
| optional diagram - keep if it helps.
|
|
Replicate four times or multiply by 4
to go from 3rd row to 4th and then simplify
|
(¼)q +
(¼)q+(¼)q+(¼)q
= 5 +5 +5 +5 = 4 * 5
Or q = 20
|
Conclusion:
|
Check if q = 20 satisfies (¾)q + 17 = 32?
|
Left Hand Side |
=(¾)q + 17
=(¾)20 + 17
= 15 + 17
= 32
|
Right Hand Side |
= 32 |
Note: We can always check whether a number is a solution of an
equation or not by computing the left and right sides of an equation for or at
that number. If the two sides differ, the number is not a solution.
- If you are asked to show that a number satisfies an equation you do the
check.
- If you are asked to find a solution to an equation algebraically you
should show some work (besides trial and error) that leads to the solution.
Then you should check the solution.
Solutions of equations can always be checked. So before you hand-in an
answer, you can always check whether it is correct or not. And if it is not
correct, you should say so if you do not have time to find the correct
answer. Instructors want to see how you obtain the solution. If your
arithmetic without a calculator is usually good, the odd error in your work is
not as important as you showing that you have master an algebraic method for
solving problems.
| |
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
|