Fractions and Units
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Ratios and Proportions
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0. Preview/Summary
1 What is a Fraction
2 Fraction Multiplication I
3 Fraction Multiplication II
4 Fraction Multiplication III
5 Equivalent Fractions
6. Mixed Numbers
7. Fraction Comparison
8 Fraction Addition I
9. Fraction Addition II
10 Fraction Addition III
11 Fraction Multiplication IV
12 Fraction Division & Reciprocals
13 Compound Fractions
14 Operations with Units I
15 Operations with Units II
16 Operations with Units III
17 Operations Unit IV
18. Operations with Units V
19. Operations with Units VI.
20. Operations on Units VII
Webpages 1 to 13
provide examples to explain what is a fraction, and to show how to add,
subtract, compare, multiply and divide fractions efficiently and exactly. In the process, algebraic
descriptions of operations are indicated not as requirement for fraction
mastery but as enrichment option. A mastery of compound fractions will be employed
later in the
explanation of long division, decimal place value methods for mixed numbers
involving tenths and hundredths.
Pages 14 to 20 extend arithmetic with fractions to
include arithmetic with units of measure that may appear in daily life or
science. Rates of change and proportionality constants
involving quantities without unlike measures may be expressed as
"fractions" in which numerators and denominators are multiples
of units of measure to first or higher powers. Operations with
units, their products and quotients, is needed for calculations with
numbers, amounts and measures in daily life and in senior high
mathematics and science. Operations on units are very similar to operations with
polynomials and so may serve as preparation or a prequel to the
latter. |
A. Two Term Ratios
B. New Fractions/Ratios from Old
C. Triple Term Ratios
(i) Proportionality - 4 kinds + EGS
(ii) Fractions & Proportionality
(iii) Slopes & Proportionality I
(iv) Slopes & Proportionality II
(v) Distance-Time Proportionality
(vi) Distance-Time Proportionality
(vii) Rates-units-proportionality
Pages A, B and C very carefully review the link and
separation between ratios and fractions. Proper fractions may be
identified with the two term ratio of a part to a whole.
Page (i) describes 4 kinds of proportionality, hints at
the forward and backward relations between them, and gives 20+
examples.
Pages (ii) to (vii) give several examples of
proportionality and express proportionality constants with the help of
units, their products and quotients. There-in lies an application of
operations with units that extends arithmetic with fractions in the left
column.
While ratios a:b involving a pair of
numbers can be identified with a fraction a/b and even a
proportionality constant k = a/b = a fraction equivalent to a/b,
ratios a:b:c may appear in the discussion of proportions, but they
cannot be identified with fractions of the form A/B. So there is
a difference between fractions and the concept of what a ratio or
proportion when more than two numbers or quantities appear.
Note: Operations with units I to VII provide a framework and
context for working with rates of change and proportional constants -
quantities that involve units alone or compounded via products alone or in
fractions. |
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Web Videos: RealPlayer Videos illustrate
calculations with Primes, Fractons, LCD, GCD that help in fraction skill
development. Many more webvideos in flash format are in preparation.
The starter
lesson (fraction summary page) points to fraction know-how. Mastery
of simplification, cross-cancellation in multiplication (an exercise in
simplification), division of fractions (another exercise in efficient
multiplication and simplification), and then addition and subtraction with
least common denominators and more simplification. Simplification may employ
rules for recognizing multiples of 2, 3, 5 and 10, and exploit or emphasize 10
or 12 times table. Simplification and more simplification (lowering terms) is
the theme. However, raising terms appears in the addition and subtraction of
fractions with unlike denominators as an aid to these operations and via the
choice of least common denominators, to simplification.
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Recap
Operations with Units: Pages 14 to 19
introduce arithmetic with units alone or in fractions. In applied mathematics, a
quantity or measure is described by a number times a unit of measures.
Calculations with those quantities may be done with the numerical coefficients -
the whole numbers or fractions etc that say how many times a unit appears, or
calculations may be done by carrying units in the calculation themselves. That
has its advantages and disadvantages. But in chemistry and physics at the senior
high school levels, multiples of units appear in arithmetic operations.
Those arithmetic operations are introduced mostly by example to pages 14 to
19. Units alone or in fractions. So Fraction like operation appear.
Polynomial Prequel: Operations with units of measure are very similar to
operations with variables x, y and z, etc. in the algebraic description of
calculations with monomials, polynomials and rational functions. So
operations with units of measure may serve as a prequel to operations to the
latter.
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[ Next ]
Fractions, Rates, Units &
Proportions
A. Fractions without and then with Units
of Measurement
0. Preview/Summary
1 What is a Fraction
2 Fraction Multiplication I
3 Fraction Multiplication II
4 Fraction Multiplication III
5 Equivalent Fractions
6. Mixed Numbers
7. Fraction Comparison
8 Fraction Addition I
9. Fraction Addition II
10 Fraction Addition III
11 Fraction Multiplication IV
12 Fraction Division & Reciprocals
13 Compound Fractions
14 Operations with Units I
15 Operations with Units II
16 Operations with Units III
17 Operations Unit IV
18. Operations with Units V
19. Operations with Units VI.
20. Operations on Units VII
B. Two and Multiple Term Ratios
Forward and Backward Use of Proportionality. More on Units in
calculations or graphing.
A. Two Term Ratios
B. New Fractions/Ratios from Old
C. Triple Term Ratios
(i) Proportionality - 4 kinds + EGS
(ii) Fractions & Proportionality
(iii) Slopes & Proportionality I
(iv) Slopes & Proportionality II
(v) Distance-Time Proportionality
(vi) Distance-Time Proportionality
(vii) Rates-units-proportionality
Oops: If you see an error in this site area, please report
it.
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
Flash webvideo replacements for the above videos are
or will appear in the Arithmetic
Videos folder, asap.
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Caution: Site advice
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Site How-TOs
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