17 What is Area - Saying how to compute a quantity defines it

Covering A Region by Squares

The areas of squares and rectangles may be calculated by calculating the product of the lengths of their sides. In the plane, the area of a bounded region S rectangular or not, may be approximated by covering the region S concerned with small squares, all of the same size, overlapping, if at all, only at their edges.

The example of an elliptic shaped region S is shown. Each covering by small squares gives three methods for approximating what the area A of the region should be.

An inner (or lower) approximation to the area A of the region S can be obtained by summing up the areas of all the squares contained completely in the region S. This inner approximation is expected to yield an estimate lower or £ A.

An outer (upper or over) approximation of the area can be obtained by summing up the areas of the squares which have an interior point in common with the region. This outer approximation is expected to yield an estimate higher or ³ A. (A point in a square but not on an edge is said to be an interior point of the square.)

A middle approximation might be obtained by adding to the inner approximation, the areas of those square which are completely in or more than half-in the region S. Other in-between approximations are possible. Intermediate approximations yield an middle area estimate between the upper and lower estimates.

FOOTNOTE: From a computational perspective, more than half-in but not completely in is not easy to define. This could be a matter of visual judgment - a step outside of the domain of rule-based mathematics. To give a mathematical algorithm, the toss of a coin might be sufficient, or a judgment could be made on how many of the four triangles formed by the diagonals are included completely in the region S.

Each of the above approximations is expected to improve as the squares are quartered (their sides halved) repeatedly and indefinitely. The latter would cause the lower estimate to increase, the upper estimate to decrease while the middle estimate together with the area A presumably approximate, remaining in between. Such halving results three sequences of numbers or quantities.

Note: The lower estimates yield the increasing sequence, the upper estimate yield the decreasing sequence, and the middle estimates yield a sequence between the previous two.

The area of a region is defined by the methods for approximating it. That is, the region has an area A = L if and only if the three numerical approximations described above all approach a single finite limiting value L. This limiting L is then called the area of the region. Otherwise, with some disappointment perhaps, we may say that the area is not defined. (Alternatively, we might define inner and outer areas using the limiting values of the inner and outer approximations and identify circumstances in which they are equal.

FOOTNOTE: For instance, for some odd (pathological) region, inner and outer area approximations may approach an inner limit L_inner and an outer limit L_outer which are not equal. Regardless, these limiting values may define what is meant by inner or outer concepts of area. There has been a concern with identifying conditions in which inner and outer approximations etc approach the same limits. For more information, a very specialized (that is, not for everyone) book or course on area computation, more precisely, integration theory, is indicated.

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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Chapter 17
Area Approximation

Covering A Region by Squares

Skill Development Tips
For All

Chapter 17Area Approximation

Covering A Region by Squares

Skill Development Tips For All

Chapter 17
Area Approximation

Skill Development Tips
For All