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# Mathematics and Logic - Skill and Concept Development

Questions: Will these ends and values motivate? Will smaller & more steps in site lessons and lesson ideas build skills and confidence?
Should we emphasize how ideas & methods depend on earlier ones? Does concept & skill mastery need to be checked to be believed? What is a Variable?

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.

Bright Students: Top universities want you. While many have high fees: many will lower them, many will provide funds, many have more scholarships than students. Postage is cheap. Apply and ask how much help is available. Caution: some programs are rewarding. Others lead nowhere. After acceptance, it may be easy or not to switch.

For students of reason in society, science and technology: Pattern Based Reason describes origins, benefits and limits of rule- and pattern-based thought and actions. Not all is certain. We may strive for objectivity, but not reach it. Postscripts offer a story-telling view of learning: [ A ] [ B ] [ C ] [ D ] to suggest how we share theories and practices.

#### Site's Best Lessons

##### For Logic

These online chapters may amuse while leading 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. Site Theme: Different entry points may be easier or harder for knowledge mastery.

##### For Arithmetic

Deciml Place Value - funny ways to read multidigit decimals forwards and backwards in groups of 3 or 6, US-CDN, UK-German and Metric SI style.

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.

##### For 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.

www.whyslopes.com >> Volume 3 Why Slopes - A Calculus Intro Etc >> Chapter 17. Area Approximation Next: [Chapter 18. Slopes Areas Integration.] Previous: [Chapter 16. Velocity Approximation.]   [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]

## Chapter 17. Area Approximation

Volume 3, Why Slopes and More Math.

Saying precisely 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.

1. An inner and lower/under 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 \em lower \rm or $\le A$.

2. An outer and upper/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 $\ge A$. A point in a square but not on an edge is said to be an interior point of the square.

3. 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 inbetween approximations are possible. Intermediate approximations yield an middle area estimate between the upper and lower estimates.

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 $S$.

One or more of the above approximations may be familiar to you from your elementary school days.

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 inbetween. Such halving results three sequences of numbers or quantities.

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 $A$ should be the common, finite, limiting value $L$ of the approximations as the sides of the covering squares become smaller (approach zero). This says how to compute the area $A$ with an unlimited accuracy if a common, finite limiting value $L$ exists for the approximations.

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.

Optional (Advanced Material: 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.

www.whyslopes.com >> Volume 3 Why Slopes - A Calculus Intro Etc >> Chapter 17. Area Approximation Next: [Chapter 18. Slopes Areas Integration.] Previous: [Chapter 16. Velocity Approximation.]   [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]

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?

### 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.
... 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. ...

##### For Geometry

Maps + Plans Use - Measurement use maps, plans and diagrams drawn to scale.

Euclidean Geometry - See how chains of reason appears in and besides geometric constructions.

Coordinates - Use them not only for locating points in the plane or space.

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 assumes no knowledge and some knowledge of the tangent function in trigonometry.

What is Similarity - another view of using maps, plans and diagrams drawn to scale in the plane and space. May buildings in space are similar by design.

##### For Calculus

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 - this 1995-96 lesson introduces calculus skills and concepts. It may also may be given to introduce further function maxima and minima both inside and at the ends of closed intervals.

Check Arith. Skills - too many calculus and precalculus students do not have strong arithmetic and computation skills. The exercises here check them while numerically hinting at equivalent computation rules.