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Mathematics and Logic - Skill and Concept Development

with lessons and lesson ideas at many levels. If one site element is not to your liking, try another. Each one is different.

30 pages en Francais || Parents - Help Your Child or Teen Learn
Online Volumes: 1 Elements of Reason || 2 Three Skills For Algebra || 3 Why Slopes Light Calculus Preview or Intro plus Hard Calculus Proofs, decimal-based.
More Lessons &Lesson Ideas: Arithmetic & No. Theory || Time & Date Matters || Algebra Starter Lessons || Geometry - maps, plans, diagrams, complex numbers, trig., & vectors || More Algebra || More Calculus || DC Electric Circuits || 1995-2011 Site Title: Appetizers and Lessons for Mathematics and Reason

Mathematics Concept & Skill Development Lecture Series: Webvideo consolidation of site lessons and lesson ideas in preparation. Price to be determined.

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Are you a careful reader, writer and thinker? Five logic chapters lead to greater precision and comprehension in reading and writing at home, in school, at work and in mathematics.
- 1 versus 2-way implication rules - A different starting point - Writing or introducting the 1-way implication rule IF B THEN A as A IF B may emphasize the difference between it or the latter, and the 2-way implication A IF and ONLY IF B.
- Deductive Chains of Reason - See which implications can and cannot be used together to arrive at more implications or conclusions,
- Mathematical Induction - a light romantic view that becomes serious.
- Responsibility Arguments - his, hers or no one's
- Islands and Divisions of Knowledge - a model for many arts and disciplines including mathematics course design: Different entry points may make learning and teaching easier. Are you ready for them?

Early High School Arithmetic

Deciml Place Value - funny ways to read multidigit decimals forwards and backwards in groups of 3 or 6.
- Decimals for Tutors - lean how to explain or justify operations. Long division of polynomials is easier for student who master long division with decimals.
- Primes Factors - Efficient fraction skills and later studies of polynomials depend on this.
- Fractions + Ratios - See how raising terms to obtain equivalent fractions leads to methods for addition, comparison, subtraction, multiplication and division of fractions.
- Arithmetic with units - Skills of value in daily life and in the further study of rates, proportionality constants and computations in science & technology.

Early High School Algebra

What is a Variable? - this entertaining oral & geometric view may be before and besides more formal definitions - is the view mathematically correct?
- Formula Evaluation - Seeing and showing how to do and record steps or intermediate results of multistep methods allows the steps or results to be seen and checked as done or later; and will improve both marks and skill. The format here allows the domino effects of care and the domino effects of mistakes to be seen. It also emphasizes a proper use of the equal sign.
- Solve Linear Eqns with & then without fractional operations on line segments - meet an visual introduction and learn how to present do and record steps in a way that demonstrate skill; learn how to check answers, set the stage for solving word problems by by learning how to solve systems of equations in essentially one unknown, set the stage for solving triangular and general systems of equations algebraically.
- Function notation for Computation Rules - another way of looking at formulas. Does a computation rule, and any rule equivalent to it, define a function?
- Axioms [some] as equivalent Computation Rule view - another way for understanding and explaining axioms.
- Using Formulas Backwards - Most rules, formulas and relations may be used forwards and backwards. Talking about it should lead everyone to expect a backward use alone or plural, after mastery of forward use. Proportionality relations may be use backward first to find a proportionality constant before being used forwards and backwards to solve a problem.

Early High School Geometry

Maps + Plans Use - Measurement use maps, plans and diagrams drawn to scale.
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- Coordinates - Use them not only for locating points but also for rotating and translating in the plane.
- What is Similarity - another view of using maps, plans and diagrams drawn to scale in the plane and space. Many human-made objects are similar by design.
- 7 Complex Numbers Appetizer. What is or where is the square root of -1. With rectangular and polar coordinates, see how to add, multiply and reflect points or arrows in the plane. The visual or geometric approach here known in various forms since the 1840s, demystifies the square root of -1 and the associated concept of "imaginary" numbers. Here complex number multiplication illustrates rotation and dilation operations in the plane.
- Geometric Notions with Ruler & Compass Constructions :
1 Initial Concepts & Terms
2 Angle, Vertex & Side Correspondence in Triangles
3 Triangle Isometry/Congruence
4 Side Side Side Method
5 Side Angle Side Method
6 Angle Bisection
7 Angle Side Angle Method
8 Isoceles Triangles
9 Line Segment Bisection
10 From point to line, Drop Perpendicular
11 How Side Side Side Fails
12 How Side Angle Side Fails
13 How Angle Side Angle Fails

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www.whyslopes.com >> Volume 2 Three Skills For Algebra >> Chapter 5 Islands-and-Divisions-of-Knowledge Next: [Chapter 6 Change of Language.] Previous: [Chapter 4 Longer Chains of Reason.]   [1] [2] [3] [4] [5] [6][7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42]

Chapter 5, Islands and Divisions of Knowledge

Recall the difference between one- and two-way implication rules:

A one-way implication rule says that when a first situation occurs, so must a second. It does not say that when the second occur, so must the first. (The second situation may occur without the first).

A two-way implication rule says that:

  1. when a first situation occurs, so must a second, and
  2. when the second situation occurs, so must the first.

A two-way implication says that when each situation occurs, so must the other. (Therefore if the two-way rule is to be obeyed, when one situation does not occur, neither can the other.)

The examples in the chapter Chains of Reason involved one-way implication rules. They showed that one-way implication rules can sometimes be put together to get further implication rules. You may remember we had one implication rule about Charles that was not used to get any conclusion.

Two analogies (Isolated Islands or Ignorable Rooms)

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.


Selby A, Volume 1A, Pattern Based Reason, 1996.


www.whyslopes.com >> Volume 2 Three Skills For Algebra >> Chapter 5 Islands-and-Divisions-of-Knowledge Next: [Chapter 6 Change of Language.] Previous: [Chapter 4 Longer Chains of Reason.]   [1] [2] [3] [4] [5] [6][7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42]

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Road Safety Messages for All: When walking on a road, when is it safer to be on the side allowing one to see oncoming traffic?

Play with this [unsigned] Complex Number Java Applet to visually do complex number arithmetic with polar and Cartesian coordinates and with the head-to-tail addition of arrows in the plane. Click and drag complex numbers A and B to change their locations.

Pattern Based Reason

Online Volume 1A, Pattern Based Reason, describes origins, benefits and limits of rule- and pattern-based reason and decisions in society, science, technology, engineering and mathematics. Not all is certain. We may strive for objectivity, but not reach it. Online postscripts offer a story-telling view of learning: [ A ] [ B ] [ C ] [ D ] to suggest how we share theory and practice in many fields of knowledge.

Site Reviews

1996 - Magellan, the McKinley Internet Directory:

Mathphobics, this site may ease your fears of the subject, perhaps even help you enjoy it. The tone of the little lessons and "appetizers" on math and logic is unintimidating, sometimes funny and very clear. There are a number of different angles offered, and you do not need to follow any linear lesson plan. Just pick and peck. The site also offers some reflections on teaching, so that teachers can not only use the site as part of their lesson, but also learn from it.

2000 - Waterboro Public Library, home schooling section:

CRITICAL THINKING AND LOGIC ... Articles and sections on topics such as how (and why) to learn mathematics in school; pattern-based reason; finding a number; solving linear equations; painless theorem proving; algebra and beyond; and complex numbers, trigonometry, and vectors. Also section on helping your child learn ... . Lots more!

2001 - Math Forum News Letter 14,

... new sections on Complex Numbers and the Distributive Law for Complex Numbers offer a short way to reach and explain: trigonometry, the Pythagorean theorem,trig formulas for dot- and cross-products, the cosine law,a converse to the Pythagorean Theorem

2002 - NSDL Scout Report for Mathematics, Engineering, Technology -- Volume 1, Number 8

Math resources for both students and teachers are given on this site, spanning the general topics of arithmetic, logic, algebra, calculus, complex numbers, and Euclidean geometry. Lessons and how-tos with clear descriptions of many important concepts provide a good foundation for high school and college level mathematics. There are sample problems that can help students prepare for exams, or teachers can make their own assignments based on the problems. Everything presented on the site is not only educational, but interesting as well. There is certainly plenty of material; however, it is somewhat poorly organized. This does not take away from the quality of the information, though.

2005 - The NSDL Scout Report for Mathematics Engineering and Technology -- Volume 4, Number 4

... section Solving Linear Equations ... offers lesson ideas for teaching linear equations in high school or college. The approach uses stick diagrams to solve linear equations because they "provide a concrete or visual context for many of the rules or patterns for solving equations, a context that may develop equation solving skills and confidence." The idea is to build up student confidence in problem solving before presenting any formal algebraic statement of the rule and patterns for solving equations. ...

Senior High School Geometry

- Euclidean Geometry - See how chains of reason appears in and besides geometric constructions.
- Complex Numbers - Learn how rectangular and polar coordinates may be used for adding, multiplying and reflecting points in the plane, in a manner known since the 1840s for representing and demystifying "imaginary" numbers, and in a manner that provides a quicker, mathematically correct, path for defining "circular" trigonometric functions for all angles, not just acute ones, and easily obtaining their properties. Students of vectors in the plane may appreciate the complex number development of trig-formulas for dot- and cross-products.
Lines-Slopes [I] - Take I & take II respectively assume no knowledge and some knowledge of the tangent function in trigonometry.

Calculus Starter Lessons

Why study slopes - this fall 1983 calculus appetizer shone in many classes at the start of calculus. It could also be given after the intro of slopes to introduce function maxima and minima at the ends of closed intervals.
- Why Factor Polynomials - Online Chapter 2 to 7 offer a light introduction function maxima and minima while indicating why we calculate derivatives or slopes to linear and nonlinear curves y =f(x)
- Arithmetic Exercises with hints of algebra. - Answers are given. If there are many differences between your answers and those online, hire a tutor, one has done very well in a full year of calculus to correct your work. You may be worse than you think.


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