Guide to Section Fraction with Unit Lessons

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.

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.

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.

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 Algebra. Second, 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 Algebra. Where 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

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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Fractions With Units (arithmetic & algebra with units)

End Notes for Teachers

Fractions With Units
(arithmetic & algebra with units)