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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?

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 >> Arithmetic and Number Theory Skills >> 3 Prime Factorization Skills >> 5 Prime Factorization and a Square Rule Next: [6 Sieve-of-Eratosthenes and Square Rule.] Previous: [4 video Prime Factorization Introduction.]   [1] [2 swf] [3] [4 swf] [5 swf] [6][7] [8] [9 swf] [10 swf] [11 swf] [12] [13] [14 swf] [15 swf] [16 swf] [17 swf] [18] [19 swf] [20 swf] [21]

## Prime Factorization and a square rule

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#### Quick Prime Identification and Composite Number Factorization Method

If a whole number N less than 169 = 132 is not divisible by 2, 3, 5, 7 nor 11 [the primes smaller than 13] then the number is prime. Otherwise, it - the number N - is a product of 2, 3, 5, 7 or 11 with another factor N' less than 169.

Knowledge of times tables, divisibility rules or a calculator - one displaying results to two plus decimals, may be used to recognize multiples of 2, 3, 5, 7 and 11. Two decimals are sufficient because the fractions one half to one eleventh are all more than 0.01 = one hundredth. See Efficient Square Rule Use to quickly learn more.

In learning and applying algebra exactly, one usually needs to compute with fractions or ratios of whole numbers less than 169 or so. Learning how to efficiently use the above method for recognizing primes and obtaining prime factorization of of whole numbers less than 169 is sufficient for most ends and purposes purposes in high school and college mathematics and science courses. The above method is based on the square method below.

If a whole number N less than the square of a given prime is not divisible by all the the primes smaller than given one then the number is prime. Otherwise, it - the number N - is a product of one smaller prime and another factor N' less than the square of the given prime.

This lesson focuses on obtaining prime factors and prime decomposition of whole numbers. Prime decompositions are also known as prime factorizations.

The the first primes 2, 3, 5, 7 or 11 are the smallest primes. A theorem of Euclid (300 B.C.) shows there the number of primes is unlimited. Given any finite sequence of prime numbers, their product plus one is not a multiplie of any in the sequence. So it is prime or has another prime as a factor. The latter observation implies one plus the product of the first m primes has a prime factor, the m+1st or later prime. So the list of prime numbers is un-ending - in other words is not finite or infinite.

Theorem A. If N = AB is product of two whole numbers A and B where A > B > 1  then A2 > N and N2 > B.

Proof:  N = AB < AA = A2 .Therefore A2 > N. Similarly N = AB > BB = B2 .Therefore B2 < N.

The above theorem implies if N is a product of two whole numbers, both greater than one,  than the smallest squared will be less than or equal to N and the largest squared will be greater than or equal to N. Now the smallest factor is a prime or it a multiple of a prime. In either case there is prime whose square is less than or equal to the smallest factor squared and hence less than or equal to the original number N. Whence the next theorem holds.

Theorem B. If N is composite then N has a prime factor p with square p2 < N.

### A Square Rule Method for Identifying Primes

When an implication rule (i) If A then B holds, the contrapositive of that rule, that is, (ii) if NOT B then NOT A, must also hold. Anything else would contradict or be inconsistent with the original rule (i). See the logic chapters in site Volumes 1A or 2 to learn more.

Theorem C - Contrapositive of Theorem B. If N has is not divisible by each primes p with square p2 < N then N is prime.

Theorem C provides the first part of square rule or method for recognizing prime and composite numbers

The the first six primes 2, 3, 5, 7 or 11 and 13 have squares 4, 9, 25, 49, 121 and 169. That should be by your lnowledge of the times table or decimal multiplication implies this.

Theorem C has the following consequence:

If N < 169 = 132 is not divisible by any of the primes 2, 3, 5, 7 or 11 then N is a prime number.

To check if a whole number N < 169 is a prime, is enough to check for divisibility by 2, 3, 5, 7 or 11. The latter can be done with the help of divisibility rules for recognizing multiples of these primes, with the help of the 10 or 12 times table, or with the aid of a calculator.

### A Square Rule Method for Prime Factorization

Theorem: If a whole number N is composite then N has a prime factor p < then the whole number N has a prime factor p with square p2 < N.

Contrapositive: If a whole number N is not divisible by each primes p with square p2 > then N is prime.

In the prime decomposition of a whole number N with the check for divisibility of N with the smallest primes first. That will require less arithmetic work - fewer divisions - than starting with larger primes.

A rule of thumb, the probability that a given whole number N is divisible by prime p is greater than the probability that N is divisible by another prime q when p < q. Exercise - explain why.

In the prime decomposition of a whole number N, if you find q is the smallest prime among those with square $\le$ N that divides the whole number N, then $N = q \times M$ where M is a whole number with the property all primes less than q do not divide M. But q will divide M if $q^2$ divides N. Otherwise, q will not divide M. Moreover the set of primes $p \ge q$ that have have square $\le$ M are a subset of the set primes $p \ge q$ with with square $\le$ N. That implies fewer primes have to be checked in the prime factorization of M. The prime factorization of N will be given by q times the prime factorization of M.

Remark: The square rule itself is well-known. The site innovation consist of providing a name for it and emphasizing it as a tool to make master of primes and prime factorization easier to learn and teach.

Exercise: Identify all multiples of 7 less than 121 that are not also multiples of smaller primes 2, 3 and 5.  The first three are indicated above.

www.whyslopes.com >> Arithmetic and Number Theory Skills >> 3 Prime Factorization Skills >> 5 Prime Factorization and a Square Rule Next: [6 Sieve-of-Eratosthenes and Square Rule.] Previous: [4 video Prime Factorization Introduction.]   [1] [2 swf] [3] [4 swf] [5 swf] [6][7] [8] [9 swf] [10 swf] [11 swf] [12] [13] [14 swf] [15 swf] [16 swf] [17 swf] [18] [19 swf] [20 swf] [21]

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.