Section Topics
Fraction,
Fraction with Units, Fractions & Ratios; and Proportionality
forwards & backwards.
Section Pages
1. Addition of Units
2. Units and Equal Signs
3. Products with Units
2. Fractions with Units
4. Simplification of Fractions
5. Fraction Reciprocals & Division
6. Converting Units in Fractions
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Fractions With Units
(arithmetic & algebra with units)
In my school days, mathematics courses
were too pure to mention or show how to work with units in arithmetic and
algebraic calculations. So calculations with units in chemistry and physic course
appeared without sanction from my mathematics courses. But mathematics courses
should support the use of units, and calculations with units may appear in the
application of trigonometry and the development of calculus. This
subsection of the fraction folder provides a remedy.
| Arithmetic and algebra with units represents alternate
title for this subject of fractions with units. |
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An operational command of calculations
with units could be sufficient for further use in the representation of rates
and proportionality constants and for further use in calculation in chemistry,
physics and money matters. In this arithmetic with units of measurement,
products and quotients of monomials may be formed and simplified with
monomials that contain units to unlike powers. In contrast, sums and
difference of monomials to have a physical context may only be formed and simplified only
when the monomials are real multiples of each other. Inclusion of this topic will help later in examples of
exponent addition and subtraction with monomials in one to several variables
x, y, z, ... and in their products or quotients.
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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 Some 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]
| In general quantities in daily life
involve basic units of measure, that is of time, length, mass (or weight)
and of money too - an artificial concept. The units and their
numerical multiples may be multiplied together form monomials -
expressions equal to a real number times a product of units to nonnegative
powers. Writing these monomials as numerators and denominators in
fraction like expression (they look like fractions) gives fractions with
units. This section on arithmetic and algebra with units shows how form
and simplify expression for monomials form by products of units and
fractions formed by taking monomials for numerators and
denominators.
Advanced Students: The presentation here is
informal, but it could be codified in a formal way. See the first or
second chapter of Henri Cartan's work below for a model. . That
codification coupled with invariance ideas leads in the physical sciences
to similarity analysis and similarity requirements for solutions of
equations. |
End Notes for Teachers
- In applied
mathematics calculations with proportionality, multiples of simple
and compound units will serve as proportionality constants and rates with
physical or social meaning. An operational command of calculations
with units could be sufficient for further use in the representation
of proportionality constants and for further use in calculation in
chemistry, physics and money matters where units appear.
- Operations with monomials involving units and
their quotients are similar to operations on monomials in
variables x, y, z etc and their quotients. The latter too
may represent formal operations on expressions that have no meaning
for students other being marks on paper. In contrast, the
previous note implies a role for polynomial like operations on units
in calculations.
- In the modern axiomization of
mathematics as seen in school and colleges, the codification is limited to pure
numbers. Units are not discussed. But as a service to the physical and social
sciences, money matters included, the algebraic role of units in their
computations can be codified or modeled. That can be done
informally. Henri Cartan's work Elementary
theory of analytic functions of one or several complex variables,
translation of THEORIE
ELEMENTAIRE DES FONCTIONS ANALYTIC D;UNE OU PLUSIEURS VARIABLES COMPLEX, could
be extended to
provide a formal codification of operation with units.
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Math How-TOs
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Algebra 4. Geometry
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Online
Volumes (orders)
1, Elements of
Reason. 1996
1A. Pattern
Based Reason 1995
1B. Math
Curriculum Notes 1996
2. Three
Skills for Algebra 1995
3 .Why.Slopes.&.More.Math.1995
Skill
& Concept
Review or Development
1. Decimal
Arith - Video Based ]
2 Fractions
3. Fractions
with Units
3. Solving
Linear Equations -
making alg easier
4. Formulas
forwards & Backwards - unifying theme for Algebra
5. Proportionality,
Back- & For-wards - theme at work.
6. Logic
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7. Euclidean-Geometry
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Study Slopes - a context
10. Quadratics
11 Polynomials
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Polys - a context
13 Functions
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Theory, Richly
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Radicals & logs.
16 Calculus
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18
Electric
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19 Maps,
Similarity & Trig, (alt view)
20 Complex
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21
Logic with Symbols+truth tables
22 Consistent
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