Guide to Section Lessons on Proportionality

Proportionality Relations
forwards and backwards

The following pages summarize the main ideas. That sets the stage for a more detailed or elementary approach - another site to do.

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.

Proportional Reasoning, algebraic perspective

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 Equations introduce 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).

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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Proportionality Relations forwards and backwards

An Algebraic Prerequisite

Proportionality Relations
forwards and backwards