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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10. Addition with unlike Denominators, efficiency matters
Two fractions may be added together using any common denominator. For
example, the use of common denominator 12 = 2*6 = 3*4 leads to
15
6 |
+ |
7
4 |
= |
30
12 |
+ |
21
12 |
= |
51
12 |
= |
4 |
3
12 |
= |
4 |
1
4 |
the use of common denominator 24 = 4*6 = 6*4 leads to
15
6 |
+ |
7
4 |
= |
60
24 |
+ |
42
24 |
= |
102
24 |
= |
4 |
6
24 |
= |
4 |
1
4 |
and use of common denominator 36 = 6*6 = 9*4 leads to
15
6 |
+ |
7
4 |
= |
90
36 |
+ |
63
36 |
= |
153
36 |
= |
4 |
9
36 |
= |
4 |
1
4 |
For all three choices of common denominators, the least and other,
conversion to a like denominator, addition and simplification all lead to the
result 4¼ . But the use of smaller common denominators involves smaller numbers
in the computation and hence less simplification work in the end. The use
of the least common denominators usually gives the most efficient way to add and
subtract fractions with unlike denominators. So try to use the least common
denominator.
There is one exception that comes to mind, that occurs when the product of
the original denominators in the addends (the fractions being added)
gives a power of ten, for example 10, 100, 1000, 10000, and so on. In the
latter case, divisibility rules for division by 2, 5 and 10 may lead to easier
simplification despite the presence of larger numbers.
- [Play
Video] 5 minutes. How to add fractions
using common denominators. Here the common
dominators is the lowest or least common
denominator (LCD) and its given by the least
common multiple (LCM) of the denominators in
the fractions added together. Here the listing
multiples method is used to compute the
LCM. The alternative of not using the LCD for
the fractions is explored to show what happens
when the LCD is not used.
- [Play
Video] 3 minutes Another example
of how to add fractions with and without
the least common denominators with an
explanation that not using the LCD (least
common denominator) leads to ratios that
can be simplified. So use of LCDs is promoted.
- [Play
Video] 3 minutes - Another example of
the listing multiples method to find the
LCM and thus the LCD for the sum of two
fractions.
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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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