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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Changing Units (Digression) The measure
or unit of slope, speed and velocity in the last example is given
by the ratio [ km/min] or km per
minute, or km per one sixtieth of an hour. But other units
(or ratio of units) are possible. The question becomes: what units
do you like for the expression of these quantities. It is possible
to change units of time and distance.
Multiplying by the number 1 does not affect the
value of an expression, but the number 1 can be written in
many ways. Some, not all, are helpful.
These different ways can sometimes help in changing the units
used for the expression of a quantity. A few slope or velocity
based examples follow. Note that
[60min/1hr] = 1 implies the
following
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4
3 |
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km
min |
= - |
4
3 |
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km
min |
×1 |
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1
2 |
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km
min |
× |
60min
hr |
= -30 |
km
hr |
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3
2 |
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km
min |
× |
60min
hr |
= -90 |
km
hr |
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In summary, multiplying by 1 = [60min/hr] does not change the speed or velocity m, but it does help
to change the units from minutes to hours. Note to do the reverse
change, multiply by
instead.
Note that care must be
taken in selecting the ratio of units whose value is 1.
A faulty choice will introduce more units instead of
permitting some to cancel.
Remark. An alternate method is to substitute
1 min = [1/60] hr into an expression. For
example
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8
3 |
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60
1 |
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km
hr |
= 160 |
km
hr |
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as before. Substitution replaces one unit by an expression
for it in terms of another. With this method, there is
little or no hazard of introducing units that don't cancel,
but the algebra or arithmetic requires a little more thought.
The choice of unit conversion method depends on where you
would like to do the work or reasoning.
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www.whyslopes.com
Fractions, Ratios, Units, Rates
& Proportionality
Fraction
Starter Lesson
(simplify, multiply, divide & then add or subtract)
An Alternative Starter
Lesson
(take your pick, or try both)
Area Map & Intro
Fraction Starter Lesson A
Fraction Starter Lesson B
1 What is a Fraction
2 Multiplication I
3 Multiplication II
4 Multiplication III
5 Equivalent Fractions
6. Mixed Numbers
7 Comparison
8 Addition I
9 Addition II
10 Addition III
11 Multiplication IV
12 Division
13 Two Term Ratios
14 Implied Ratios
15 Multiple Ratios
16 Units in Arithmetic
16 Longer Explanation
16 Change Units
16 Products of Quantities
16. Fractions with Units
16. Division+Reciprocals
17 Proportionality
17 Examples
18 Rates & Slopes EGs
18 Constant Rate
18 Varying Rate
18 Velocity Calc., EGs
18 Changing Units
18 Slopes and Units
18 Slopes, No Units
19 RealPlayer Videos
Links
Arithmetic Videos - Real Player Format
Decimal Addition
Methods
Decimal
Subtraction Methods
Decimal
Multiplication Methods
Decimal Division
Methods
Fractions
Primes
Greatest Common
Divisors
Least Common Multiples
Square Root
Simplification
Area Content Summary
- Fraction Starter Lesson
- Real Player Videos on Operations with Primes and
Fractions
- Continuous Ruler & Line Segment
model for fractions and operations on fractions - Number Theory Area
points to the general model.
- Distinction between Ratios and Fractions, a nuance:
While binary ratios a:b may be identified with a fraction, triple
ratios a:b:c and further multiple ratios cannot.
- Saying how to add and subtract like monomials in
units and their powers, and saying how multiply and divide like and
unlike monomials leads to fraction like expressions involving units
and a framework for discussion rates - ratios of quantities - a
framework for handling proportionality constants, and framework for
carrying units through calculation in quantitative disciplines
Hint: See site area on solving linear equations to strengthen
fraction sense and algebra skills together. Good luck.
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