- What is
a Fraction: Meaning of a Fraction - A whole number counts
how many ones, A fraction counts how many parts of equal
value. Algebraic Description included.
- Fraction Multiplication I. What is a Unit
Fraction of a Unit Fraction? What is a half of a third? What is a
unit fraction of a unit fraction? A unit fraction has one in the
numerator. Algebraic Description included.
- Fraction Multiplication II. Unit
Fraction of a Simple Fraction: what is a half of two thirds? What
is a quarters of seven tenths?
- Fraction Multiplication III. : What
is a Fraction of a Fraction: what is seven quarters of three
tenths?
- Equivalent
Fractions: What is the difference between two
quarters and one half? What is the difference between six eights
and three quarters? What is the difference between an apple and
four quarters of the apple? The thought that there is no
difference, that different fractions may describe the same amount,
quantity, leads to the idea of equivalent fractions. Examples of
fraction simplifying and equivalence included.
- Multiplication
Algebraic Development. The first, easy case treats the case
where the denominator of one fraction is a divisor of the numerator of
the second fraction The general case follows by raising terms in
the second factor to apply the easy case.
- Mixed
Numbers and Equivalent Fractions: We may describe a distance
as 3 half meters or we may describe the same length as 1½ meters.
Physically, there is no difference in the distance, only its
descriptions. The descriptions 3/2 meters and 1½ meters both
have the same value physically. So we declare the numerals 3/2 and 1½
to have the same value, or to be equivalent. Likewise 3
fifties (half-hundreds) is the same a 1½ hundreds. There is another
instance where the fraction 3/2 and the mixed numeral 1½ may be
identified as adjectives for the same count or measure.
- Comparison
of Fraction and comparison of Mixed Numbers: Algebraic
Description included of the first included.
- Fraction Addition I:
Easy case of like denominators - the easy case Algebraic
Description included.
- Fraction Addition II: General
Case of unlike denominators. the general case follows from
raising terms (as little as possible) to use the easy case.
- Examples to show "raising terms" similarities
between comparison, addition and subtraction of fractions.
- Fraction Addition III: Methods
for adding and subtracting Efficiently - Questions and
Problems: (a) How is the list method used to obtain a least common
denominator = the least common multiple of a pair of denominators? (b)
How can the prime number decomposition (also known as factorization)
be used to calculate the LCM and GCD of a pair of whole numbers? (c)
How can Euclid's algorithm be used forwards and backwards to calculate
the GCD (the normal result) and then the LCM of a pair of whole
numbers? (d) Employ the M-method to find the sum of
eighteen 21st and nine 14-ths? Say which of the latter is more than
the other, and find how much more.
- Fraction Multiplication IV: Efficient
Ways to Multiply Fractions: Learn how to calculate a few
products of fractions with and with the cancellation methods described
below for "efficient" multiplication, or more precisely
efficient or easier simplification after (cross) cancellation of
common factors.
- Fraction Division, Compound Fractions and
Reciprocals: A
Physical Introduction to Fraction Division: Explanations,
twice-over, are given in the next lesson.
- Fraction Division Methods Explained. See the previous page
for an introduction of the fraction division methods or formulas below.
- Two
Step Development Option - Explanation in two smaller steps, first with the
easy like denominators case and second with unlike denominators (Raising
terms in dividend and divisor fractions turns the second case into the easy case).
-
One
Step Development Option - Explanation in one big step.
- How to do
Arithmetic with Rational Numbers, that is signed fractions, signed whole
numbers and signed mixed numbers. Here a model for introducing arithmetic
with real numbers.
- Arithmetic Videos - Real Player Format
-
Primes,
How to Recognize Them. Extras include statement and
justification of rules for division by 2, 3, 5, 9 and 11, and the
calculation of remainders for division by 2, 3, 5, 9 and 11.
-
Fractions,
Operations With. Addition, Multiplication and Reduction
(Simplification) using primes, LCM, GCD. Euclid's Algorithm for
computing the GCD of a pair of whole numbers provides a method for
simplifying fractions, quickly without using prime decomposition
of numerators and denominators.
-
Greatest
Common Divisors, Calculation using Primes or Euclid Algorithm.
-
Least
Common Multiples, Calculation using Primes or Greatest Common
Divisor
Related Reading: Seeing
how to Solve
Linear Equations with fractional operations on Stick Diagrams may help
with fractions and develop the algebraic skills needed to understand the
algebraic description of fraction operations in this site. |
Unit of measurement are part of applied mathematics and external to pure
mathematics. Yet calculations involving units of weight, mass, length, time,
money (hand it over) and so on appear in the measurements and calculations of
daily life alone or as part of calculations with rates and proportionality
constants.
- Origin
and Addition of Units: Measurements system appear in daily
life, science and technology. They involve calculation with
units. This lesson introduce standalone units, and indicates how to
add subtract multiplies of them in a way that resembles the forward and
backward use of a distributive law for counting and measuring.
- Units and Equal Signs:
Mathematical practice requires the equality sign to have the "forward
and backward" reflective property that a = b when and only when b = a.
Here may read the equality sign to mean the same as or is equivalent
to. Yet in daily life, the statement 3 apples + 4 oranges
is, gives or equals 7 fruits, departs from this practice. We are not
going change the habits of daily life, but should be aware of the difference
here between the technical and common use of the word equals or the symbol =
for it.
- Products of
Units and Numbers: The first part of this
lesson shows how to add subtract numerical multiplies of them in a way
that resembles the forward and backward use of a distributive law for
counting and measuring. Carries and borrows in addition and
subtraction provide a unit-free form of conversion. In counting and
arithmetic, conversions are further present in expressing counts or numbers
in groups of ones, tens, hundreds and so on, and in adding, subtracting and
multiplying such counts. The second part of this lesson talks
about changing and converting the units used to measure or keep track
of quantities. Your fortune may be measured in pennies. or in dollars?
The third part show how to form products of quantities. The
operations here are similar to work with monomials where the product of 4xy
with 5 xy3z is 20 x2y4z.
Instead of using letters x, y and z, we use measurement units and counting
units as well, and could be more meaningful for students.
- Fractions With Units.
This lesson introduces fractions with units and extends the concept of
equivalent fractions to fractions with units. These fractions with
units are analogous to ratios of monomials and their simplification:
|
12 xy3
8 x2y2z |
= |
3 y
2 xz |
- Simplification
of Fractions with Units. Simplification of fractions
with units is analogous to to simplification of ratios of monomials
(fractions with monomials) in one to several variables.
|
24 yw8
9 w2y4z22 |
= |
8
3 |
w6
y3z22 |
- Reciprocals,
Division and Compound Fractions for Fractions with Units
This two part lesson shows how to (i) write the Reciprocals of
Fractions with Units, and how to (ii) divide Fractions with
Units, and how to (iii) evaluate the corresponding compound fractions where
numerators and denominators are fractions with units. Equivalent fractions
reappear here as part of the simplification of results. The
operations here analogous to calculating reciprocals of ratios of
monomials in variables x, y and z, etc; to dividing such ratios and to
evaluating compound fractions with those ratios as numerators and
denominators. three part lesson shows how to (i) write the
Reciprocals of Fractions with Units, and how to (ii) divide
Fractions with Units, and how to (iii) evaluate the corresponding compound
fractions where numerators and denominators are fractions with units.
Equivalent fractions reappear here as part of the simplification of
results. The operations here anonymous to calculating
reciprocals of ratios of monomials in variables x, y and z, etc; to dividing
such ratios and to evaluating compound fractions with those ratios as
numerators and denominators.
- Conversion of Units in
Fractions With Units: A physical quantity may be described in
different units. Here we show how speed (it is written as a fraction
with units) written in distance per hour may be expressed in distance per
minute. The trick of multiplying by a fraction with value 1 (a
fraction with units which is equivalent to one) appears here.
To Learn More about how about how fractions with units are treated and occur
in daily life and physics, see the Units in Calculations pages
in site Volume 3:
[7 Velocity]
[7 Varying Velocity Example]
[7. Velocity Calculation]
[7 Changing Units]
[7 Same Velocity Motions]
[10 Slopes without Units.]
[10 Units & Slopes]
[10 Units in Cost vs. Quantity]
[10 How Units Appear]
[10 Unit Elimination]
[10 Partial Elimination]
[10 Interest & Units]
[12 More on Units]
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The following lesson cover the
properties of two term and multiple term ratios. Fractions may be
identified with two term ratios. Two fraction are equal or
equivalent when and only when the corresponding two term ratios are equal
or equivalent. But fractions can be added, subtracted, multiplied
and divided while the same operations are not defined for two- and
multiple term ratios. While we may call a fraction, a ratio or a
rational number, ratios are different. Triple term ratios exist,
but triple term fractions do no exist.
- Fractions As (two term) Ratios and
Fractions Versus Ratios:
Fractions are often called ratios, and vice-versa. But the vice-versa
only holds for two term ratios. This lesson identifies fractions with
two-term ratios and contrasts the properties of fractions and two-term
ratios. (Ratios cannot be added, subtracted or compared, but
like fractions, the terms in ratios can be raised or lowered).
- Implied or Derived Ratios - New
Fractions and Ratios from Old:
If two fractions are equivalent then their reciprocals are also
equivalent. Likewise if a pair of two-term ratios are equivalent,
interchanging the first and second terms of each ratio in the pair leads to
a pair of equivalent ratios. Beyond that, more equivalent ratios can
also be generated from a pair of ratios. Food for
thought: How may equivalent fractions or ratios may be
formed from the relations ad = bc?
- Multiple Ratios: Multiple
Term Ratios - Three Term Ratios to be precise. We read the triple
ratio a : b :c as a to b to c. We further write
a : b: c :: A: B:
C
to say two triple ratios a : b: c
and A: B: C are equal or equivalent when
and only when
there are other ways to say when two triple ratios are equal or
equivalent.
Note: Triple ratios or triple proportionalities occur between the
sides of similar triangles. More generally, multiple ratios or
proportionalities occur between the sides of similar triangles.
The discussion of ratios or multiple ratios is best understood besides
a discussion of proportionality.
Inner Versus Outer Terms - small point: In the discussion of
equality of ratios a : b = A: B written in that
order, the inner terms are small b and big A while the outer terms are small a
and big B. In contrast, if we rewrite the equality as A: B
= a : b, we find the inner and outer terms are interchanged. However,
the equality requires the product of the inner and outer terms be equal, that
is aB = Ab. That equality is not affected by rewriting a : b
= A: B as A: B = a : b, and the resulting
swap of inner and outer terms
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The following pages summarize the main ideas.
Chapter
14 of Three
Skills for Algebra with its explanation of the forward & backward
a formula, the compound interest formula should be regarded as a prerequisite to
the coverage here.
- Proportionality Concepts
and Practices- Three plus Kinds of Proportionality Relations, Forwards and
Backwards: The lesson says what is (defines) Direct, Joint,
Inverse Proportionality and describes how to shift or generate
proportionality relations from each others. . In a proportionality relation (or equations),
algebraically interchanging the dependent quantity with an independent one
via a backward use of the relation leads to further proportionality
relations of the same or different type. The use of proportionality
relations begins with the backward use problem of finding the value of
a proportionality constant. Once its value is known, the
proportionality relation can use in the forward direction to find values of
the dependent variable, or in the backward direction to find values of a so
called independent variable.
- Twenty
or so Examples of Proportionality and Multiple Ratios or Proportions:
Many examples of proportionality relations appear in high school mathematics
and physics. Here is a list of some (most if not all) that may
be met. Remember each proportionality relation will be used
forward and backwards in multiple ways.
- Two
and Multiple-Term Ratios, a proportionality constant viewpoint. Fraction
and ratios are overlapping concept and have overlapping roles in arithmetic,
but they are not identical even though fractions a/b where a and b are whole
numbers may be called ratios. In mathematics ordered pairs of whole numbers
a and b may appear in coordinate form (a,b) or [a,b]; in ratio form a:b and
in fraction form. The following treatment emphasizes the difference.
- Proportionality
Constants for Equivalent Fractions: The numerator is proportional
to denominators in any fractions equivalent to a given one - a simple
matter.
An Algebraic Prerequisite
The
Forward and Backward Use of Formulas and Equationsintroduce a universal & unifying theme in the mathematics
and science. This theme appear in Volume 2, Three
Skills for Algebra. See chapters 10 and 14.
The two equivalent phrases Forward and Backward Use (or Direct and
indirect use) voice, identifies and emphasized what has hitherto been a
silent theme in the teen and adult mathematics education. The phrases spoken
repeatedly in the classroom will alert students to this common thread and the
need to understand and master it.
Chapter 10 considers the
backward use of the rectangular area formula A = WL where
W denotes the width and L denotes the length of a rectangle
Direct or forward use of the rectangle area formula A = WL calls for the
value of A to be calculated from given value of W and L. A first backward use of this
formulas will find the value of the width W from the values of area A and length
L. Finding the length L from the values of A and W would be another backward or
indirect use of this formula. Chapter 10 does that exercise algebraically in
the hope that readers will follow.
Chapter 14 in Volume 2 employs the more complicated Compound
Interest formula in the form A = P(1+i)n directly and indirectly (forwards and backwards), and
gives both arithmetic (numerical) and algebraic (literal) solutions to solve
backward use problems. Every formula met in high school and college mathematics and science is
likely to be used backwards and forwards. The arithmetic approach to this may be
easiest or most natural for students in the first instance, but the algebraic approach and it
ability to solve many problems at once points to a power of algebra. Mastery of
the algebraic approach with that power is the objective. The algebraic
approach is essential, not all powerful.
For Right triangles, the
Pythagorean identity c2 = a2+b2
between leg lengths a and b, and hypotenuse length c
is never used directly. The near forward use would obtain c from the principal square root of a2+b2
before or after substitution of values for a and b. The arithmetic solution
would involve substitution first, while algebraic solution would involve
substitution after. A backward use find a, given b and c values,
would obtain a from the principal square root of c2- b2
before or after substitution of values for a and b in the identity.
Between the forward and backward use of formulas for area of rectangles
and compound growth A = P(1+i)n, formulas for area of triangles, squares,
r circles, trapezoids,
parallelograms and polygons; for volumes of spheres, cylinders, cones,
pyramids, and boxes (parallelepipeds); and for perimeters of triangles,
rectangles, circles and so on, provide opportunities to illustrate and
reinforce the backward use of equations using arithmetic and algebraic
solution methods.
Algebraic expressions for systems of linear equations in 3 or more
unknowns, can be derived, but the derivation and their expression is so
complicated numerical methods for solutions are preferred (except in special
cases).
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