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Fractions with Units
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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.  

  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.

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.

  1. 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. 
  2. 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.
  3. 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.
  4. 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 x2y2

     =  

     3 y
    2  xz 

  5.  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

     
  6. 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. 
  7. 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 

  1. 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.  
  2. 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. 
  3. 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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2   Fractions  
3.  Fractions  with Units  
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5.  Proportionality, Back- & For-wards - theme at work.
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