12. Two Analogies

Islands and Divisions of Knowledge
Chapter 12

Two analogies and Ignorable Rooms
Duplicated Paragraph

One and two-way implications can also be joined. The ways in which this can be done are described below by analogies with one- and two-way streets, and one- and two-way doors. These analogies indirectly describe how rule-based knowledge is put together. In particular, rule-based knowledge is divided into separate segments. Each segment cannot be reached from another by chains of reason. The two analogies describing this situation further are presented next.

Islands Without Roads Between

Implications are like streets or roads. They may be traveled one-way or both ways. Streets (or implications) may lead nowhere. Others may lead to interesting and sometimes unexpected places.

Each road may touch several others. Each of these others may touch several more. But by foot or car, from one road, there is no guarantee that all roads can be reached. Moreover, when some one-way roads are present, poor planning may imply no return route for every possible starting point.

Maps make the exploration of any road system easy. All we have to do is read the map. Without a map, we have to explore the neighborhood in which we live, and hope we can find a path back. One-way streets are a danger here, unless another path back is available. Without a good map, we cannot say in advance, when we explore the streets, if we will get to an interesting or boring destination. To find out what is interesting, our only choice is to explore or to ask whether any one has made a map. We would like to learn from the experience of others, perhaps.

By road, not all destinations are accessible or reachable. We may for example have roads on several islands with no boats, ferries, planes, bridges or ships to take us between them. Without boats, ferries, planes, bridges, or a very low-tide, we have no route or connection between one island and the next. Without these extra routes, the roads (or implications) of one island are not linked to the roads of another. The streets on even a single island need not all be connected to each other. For example, imagine on one island that a mischievous or artless road planner has provided one-way roads all leading from one end of the island to the other. On such a road system, a return to the starting point is not possible. We can imagine another island in which the planner, mischievous or not, has placed a mixture of one- and two-way roads. From some starting points you can leave but not return. From some parts or destinations, you cannot leave. Between other starting points and destinations, you can go back and forth. And after going back and forth several times, you may forget which place was the destination or the starting point.

All the situations just described with one- and two-way streets can happen similarly in logic with one- and two-way implication rules. In other words, knowledge is linked by one- and two-way implication roads, spread over several islands. The map of this area is not complete. As we explore and forget, roads and routes new to us or our neighbors are uncovered or rediscovered.

Rooms Without Doors Between

Implication rules are also like doors or gates between sections of a building or estate. (Implication rules like doors join the rooms of a large palace, castle, house or prison. ) Some allow two-way passage. Others permit only one-way passage. All this can be a deliberate design or it could be due to a poor design.

When we restrict our paths to two-way doors, we can always retrace our steps exactly and get back to where we started. But one-way doors are different. To get back after going through a one-way door, we need to find another route back through some other door or doors. Otherwise, we are shut out of our starting room. That is, we suppose a one-way door can only be opened from one side, and that after use it snaps shut. When we go through a one-way door, we can get back to our initial side of the door only if there is a route back. But by passing through one-way doors, we may find ourselves locked out of the initial room we were in. We may further find ourselves locked in another room or section of the building.

Ignored Rooms

Whenever the building we are exploring has sections closed off or unreachable, we can ignore all maps of those sections. Making a map of the unreachable sections is not possible, except by guessing. Guessing is suggestive, yet not reliable.

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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Islands and Divisions of Knowledge Chapter 12

Two analogies and Ignorable Rooms Duplicated Paragraph

Islands Without Roads Between

Rooms Without Doors Between

Ignored Rooms

Islands and Divisions of Knowledge
Chapter 12

Two analogies and Ignorable Rooms
Duplicated Paragraph