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Appetizers and Lessons for Mathematics and Reason
by A. Selby, Ph. D.   Feedback & Questions

20 pages in French: Algèbre  
 Définition d'une variable
  
La raison basée sur les  règles et modelés


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The four topics: Fraction Guide ] [ Fractions with Units Guide ] Ratios & Fractions Guide ] Proportionality Guide ] Links ]


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. 
 

Four Topics

Section Entrance

Fraction Guide
Fractions with Units Guide
Ratios & Fractions Guide
Proportionality Guide
Links

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 ]

 

For Senior High School  & Calculus Students

  <| (o)   (o)   |> 
 \     | |      / 
\___ _/

||
 -/[]\- 
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   / \_ 

Words  to clearly introduce algebra and variables have been missing in course design. For people who cannot do algebra, 
the missing words may explain or ease their difficulties.  Volume 2 ,Three Skills for Algebra,  in Chapters 8 to 14 & 18 etc, puts words before symbols to providing the missing words in a way that enrich the comprehension of all.  Those words form the middle part of a algebra (and logic) lessons aimed at helping or improving all of  high school mathematics and also calculus course design & delivery. 

For Avid Readers in School & Out - Online Books 
   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
Tour their 
forewords.   

Calculus Prep or Help: See Volumes 2 & 3, and this bigger Calculus Guide.  If your  calculus   questions is not answered here, submit it. Over time, that may complete the site development of calculus. 

For Parents: Speaking Skills, Reading & Writing Preparing for Scienceends, values and methods for work and study,  parent- friendly maths skill development booklets for ages 4-14.

Mostly For High School

Intro to Solving Linear Equations
 
- a different paths for junior and even senior high school students. Question for Tutors: When do you use and when you skip the stick diagram method here?

Fraction Skills,  thought-based  development, Ages 10 to 14 may need a tutor.  Students who have to understand in order to do may like the development in all or part. 

For Senior High School Mathematics & Calculus

5
wordy Logic Chapters
4 curious Algebra Chapters
Words before & besides symbols. A Key Algebra forward & backwards Chapter   
 

First Calculus Preview (1st intro)
Four Calculus Chapters  (2nd intro)
Intro to Complex Numbers (long)
Intro to Mathematical Induction (romantic & wordy at first)

Tutors & Instructors: These lessons introduce skills differently Would you recommend them? 

More Topics 

1. Decimal Arithmetic  Reference!
2. Integers - Intro to Signed No.s

3.  Fractions - fully explained.
4.  Fractions  with Units  
5.   Number Theory
6.    Solving Linear Equations  
Formulas for- & backwards -  
8.  Proportionality, Back- & For-wards.   
9. Logic Chapters:   
10.  Euclidean-Geometry  
11.  Slopes & Equations of Straight Lines.  (Take I. See take II below)
12.  Why Study Slopes
13. Maps, Plans,  Similarity & Trig,  
  (Take II included here)
14.  Quadratics: Starter lessons
15.  Polynomials: Starter lessons 
16 Why Factor Polynomials:  
17   Functions - Forwards & Backwards.  
18.  Exponents, Radicals & logs.  
19
Complex Numbers before trig (new advance/ starter lesson)
20.  DC Electric Circuits Etc 
21.
Real  Analysis 
22. The Olde Complex No, Trig
& Vector Section.
23. More Calculus Stuff
- written after Volumes 2 and 3.

Level I Material: New Stuff
Time and Date Matters
Level I Arithmetic. 
Money Matters
Measurement Matters
Matters of Chance (Risk Control)
Logic Chapters (leave what's not clear in Level I to Level II)
Using/Making Maps and Plans.
(A variant of
Maps, Plans,  Similarity & Trig,  to appear here).

For Instructors
-
Education Essays   (opinions, possibilities, references) 
- Free Advice and Directions for teaching primary & high school maths will be given in online meeting place with voice & whiteboard.   
- Math & Logic  How-TOs 
1. Arithmetic
2. Algebra
3. More Algebra
4.  Beginner Geometry
5.  More Geometry
6. Calculus 
7. Show Work or Logic 
These may be too dense for students.

Offering ideas to change education makes this site different.  Nothing ventured, nothing gained.  Site material is mathematically  correct, and where not, please report errors. The two level program POMME in the site entrance implies multiple paths for instruction. Supporting those paths in turn implies a clear destination  for site development and perhaps a new name.


 


 www.whyslopes.com > Four Topics & Links Entrance > Fractions with Units Guide     Back ] Next ]
The four topics: Fraction Guide ] [ Fractions with Units Guide ] Ratios & Fractions Guide ] Proportionality Guide ] Links ]


Road Safety Message   Walk on a side walk. If that is not possible, try  not to  walk on a road with your back to the traffic.
Try to see what  trucks, cars, buses or bicycles are coming, so that you may step out of their way.  Put safety first. .

Support for Technical Mathematics from Number Theory to Calculus Prep

A. More Arithmetic a must for algebra etc D. Logic In Mathematics G. Algebra with Take Home Value I. Vectors & Functions
Decimal Lesson - Reference  
Counting & Addition
   (8 lessons)
Comparison to Subtraction
  (9 lessons)
Multiplication
( 11 lessons)
Long Division  (12 lessons)
Decimals and Primes (8 lessons)
-Primes & Composites 
-Primes Factorization
-Greatest Common Divisors & Multiples.
 
-Prime Factorization Aids 
(Learn how to find factors quickly)
-Prime Factorization Examples
 
-Counting & Generating. Factors

-Divisibility Rules and Remainders for Division by 2, 3, 5, 9 and 11.
Integers (12 lessons) Intro to Signed Numbers
Fractions (< 20 lessons)  Essential Skills & Concepts 
Ratios & Fractions (3 lessons):  Similarities & Differences
  
Units in calculations
Fractions  with Units
B.  Basic Algebra
Solving Linear Equations  
- in one unknown. Intro  with stick diagrams?
the normal way
 & with good nttn.
(the nttn that reappears in Gaussian Elimination. |
-in more unknowns: simultaneous equations essentially one unknown. the let algebra do the work view of  word problems.
  - still in more unknowns:  Gaussian Elimination via substitution, by equality or comparison, by operations on equations
C. More Algebra
Words before symbols: See if U like the lengthy chapters 8 to 12 in Volume 2, Three Skills for Algebra  
What is a Variable.  The answer here  is a simple prequel to the modern mathematics viewpoint.
First, every rule & pattern U meet in math, logic & science will be used forwards and backwards.  Get a head start with this theme by reading  Chapter 14 in Three Skills for AlgebraSecond, in the study of Proportionality Relations (3 dense lessons here) finding the proportionality constant gives an initial  backward  use of the proportionality formula.
 Talking about words before symbols and the forward and backward use of formulas gives words to make algebra simpler & clearer.  
If you can not read or write precisely, you will have difficulty in following instructions.  One wordy remedy  is given by chapters 2 to 5  in Three Skills for AlgebraWhere does Logic or a geometric model for reason Appear in Mathematics? The answer lies in  Euclidean-Geometry    In North America, Euclidean Geometry disappeared from high school mathematics as it was too hard. The light treatment here is a possible remedy.
E.  More Geometry
The Pythagorean Theorem. Chapter 17 from  in Three Skills for Algebra uses algebra and geometry   to show why the  Pythagorean equation  for right triangles holds. Its forward and backward use  is common exercise..  At a more theoretical level, the Pythagorean theorem leads the discovery that not all lengths can be  fractional multiples of a unit length. That geometrically implies a  need for and even existence of irrational numbers.
Analytic Geometry:
Common Practices with  Maps and Plans drawn to scale  give coordinate-dependent base  for senior high school development of similarity, trig, vectors and straight lines.   
Complex Numbers: This lesson on
Complex Numbers  draws on Euclidean and Analytic geometry. Sbortcuts simplifiy  trig identities, the cosine law; and   trig formulas for 2D dot- and cross-products. 

F. Logarithms, Exponentials,
Roots & Powers

Logarithms, exponentials, rational and real powers for secondary students. This  complete Operational Viewpoint. (Sufficient for the precalculus forward and backward use of compound growth and decay formulas in biology, physics, chemistry,  personal finance, and calculus. To learn more, if you study calculus,  see chapter 19 of Volume 3, Why Slopes and More.Math

In Volume 2, Three Skills for Algebra, chapters
  1. Geometric Sums Etc,
  2. Notation For Sums,
  3. Personal Money Maths and
  4. Some Finite Mathematics
identify methods useful in money computations, methods needed for calculus. Your teachers or other writer may present the same ideas with greater clarity and detail - A site to do.

H. Polynomial & Quadratics

Analytic Geometry:   -  Slopes and Lines - Take 1.   Take 2 appears in site section Maps and Plans.   Two views are better than one.  I may combine them later.  -In my school days, slopes appeared year after year.   This Why  Slopes calculus preview on graphs of functions y = f(x) explains why.  Enjoy.
Quadratics and Polynomials: Operations on Polynomials:
Meet a light and ultraquick geometric introduction to  multiplication, addition and subtraction of polynomials. Then see how the foregoing combine to permit long division of polynomials.    Compare Fractions  with Units. Enrichment: A Plus:  The Geometric introduction here gives or is almost identical to a justification for column methods in decimal arithmetic. 
Geometric Derivation of the Quadratic Formula  The account here gives a starter lesson for the more algebraically harder geometric-free derivation. If you study physics, chemistry or trigonometry, you will need to know about quadratics, their factorization and the quadratic formula.
Technical Value: The study of polynomials  high school mathematics has technical value as part of the senior high school mathematics preparation for calculus.  This simple account of Why Factor Polynomials   (Chapters 2 to 6 in Volume 3 .Why.Slopes.&.More.Math.) will give a context for the study of polynomials,  their factorization, and sign analysis of functions, all in a way that should improve your algebraic thinking and reasoning skills. 
Vectors in the Plane (2 simple lessons)
- Navigation with vectors or arrows
- Sum of Motions
- more lessons to be added later.
Operations on movement or vectors along the line and in the plane have value in mathematics in defining and implying the properties of real and complex numbers before the assumption of those properties as axioms.  Vectors and their properties appear in physics, its mathematical description and formulation. 
Functions - Forwards & Backwards.  Here is a full technical reference (24 lessons) for use in a calculus or precalculus course as needed. In it, the set viewpoint of functions expression of modern pure mathematics.  comes from the set-based codification and
In the mathematics education reforms of the 1960s in North America, primary and secondary school mathematics were expressed in terms of sets. That expression has now retreated from primary and secondary school texts. But it still lingers on, and can be very useful, a source of clarity and precision, in the situations where it should be retained: Counting with the aid of sets and functions; the description of functions; the high school account of probability theory; and in the discussion or illustration of ideas in logic. 

J. Pre-Calculus Skill Check

Arithmetic Skill Check.  In the calculus courses I taught 1983-89, too many students had weak skills in arithmetic. I would give and carefully correct these exercises to tell students what they needed to review and master.  
-  All the skills and concepts in 
Chapters 1 to 24 or Volume 2, Three Skills for Algebra: Look for those you do not understand and fill the gaps. Do so quickly while balancing this advice with  your other duties.  Good luck.

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