YOU are better than YOU think. Show yourself how:
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Logic
Mastery Do not leave here without it - Logic mastery will develops critical thinking, improve reading and writing, and give a firmer base for work and studies at many levels. Good luck.
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-/[]\- After logic, (a) continue reading Three Skills for Algebra, chapters 8 to 14 and do so alongside site area on solving linear2007 Equations ; or (b) see this calculus starter lesson and Volume 3, Why Slopes & More Math, chapters 2 to 6; 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. |
5. Equivalent Fractions
This lesson introduces and provides motivation for equivalent fractions.
Example:
|
Three Quarters |
3 |
|
||||||
| Each box is a quarter |
1 |
1 |
1 |
1 |
||||
| Each box is a half of a quarter | ||||||||
Each quarter is two eighths.. Hence 3 quarters is 6 = 3*2 eighths. Hence
3 |
= | 2* 3 2* 4 |
= |
6 |
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 |
of an object | ) = |
B |
of the object |
In shorthand we see
| N | * ( | 1 N |
of |
B |
) | = |
B |
or equivalently
| N | * |
B |
= |
B |
The latter in turn gives the common factor cancellation property
N* B |
= |
B |
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
| N | * | _1_ N |
= | 1 |
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)
An Alternative Starter
Lesson
(take your pick, or try both)
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.