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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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5. Equivalent Fractions
This lesson introduces and provides motivation for equivalent
fractions.
Example:
|
Three Quarters |
3
4 |
|
| Each box is a quarter |
1
4 |
1
4 |
1
4 |
1
4 |
| Each box is a half of a quarter |
|
|
|
|
|
|
|
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Each quarter is two eighths.. Hence 3 quarters is 6 = 3*2 eighths.
Hence
The foregoing equation may be read forwards or backwards. The
fractions
3 and 6
4 8
are equivalent.
A general discussion, an Algebraic Shorthand Description of
ideas, follows.
- Instructors: Give numbers in place of letter below.
- Learners: Assume N = 4 and M = 5 and B = 3 on first reading
below.
We may use the property of divisible objects (fractions included)
| N |
* ( |
1
N |
of an object )
= the object
|
Thus if we have a fraction
B
M
of an object then
| N |
* ( |
1
N |
of
|
B
M
|
of an object |
) = |
B
M
|
of the object |
In shorthand we see
or equivalently
The latter in turn gives the common factor cancellation property
The left and right hand side in foregoing equation are said to be equivalent
fractions. Replacing the left hand side by the right hand side in a
calculation is called a simplification, a reduction, a cancellation or a
lowering of terms. On the other hand, replacing the right hand side by the
left hand side is called raising terms. Raising of terms is useful in the
addition and multiplication of fractions.
In the computation of fractions, we may also use
View the following RealPlayer videos after this lesson
- [Play
Video] 3-4 minutes. Equivalent
fractions - Lowering and raising terms
(the values of numerators and denominators) to
obtain equivalent fractions. Simplification
involves lowering terms - cancelling common
factors or divisors on top and bottom.
Addition & subtraction of fractions may
involve raising terms to obtain a common
denominators. See below.
- [Play
Video] 2-3 minutes A few examples of
Simplifying Fractions - lowering terms by
canceling common factors until there are no
more common factors, so that the numerator and
denominator are relatively prime, that is
there prime decompositions have no primes in
common.
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www.whyslopes.com
Fractions, Ratios, Units, Rates
& Proportionality
Fraction
Starter Lesson
(simplify, multiply, divide & then add or subtract)
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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