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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.

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

www.whyslopes.com >> Arithmetic and Number Theory Skills >> 3 Prime Factorization Skills

### More Notes

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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.

Recognition of prime factors and the prime decomposition of whole numbers speeds calculations of LCM, GCD and LCD in exact arithmetic with whole numbers and fractions. They also lead to cosmetic simplification of square roots. The recognition of common factors in numerators and denominators of fractions alone or in products helps with the reduction of fractions and via cancellation leads to efficient methods for multiplying fractions.

##### Description of Folder Lessons
1. [video] how Products are bigger than factors. This observation simplifies the identification of primes and composite numbers.

2. Prime or Composite less than 16.. Instead of saying a whole number is prime when and only when it is not a product of whole numbers larger than one, we say a whole number is prime when and only when it is not the product of two or more smaller whole numbers larger than one. The addition of the word smaller, redundant and technically not required due to lesson 1, none the less makes prime number identification easier to learn and teach. This webpage employs the definition of primes and the 12 times table to recognize that the whole numbers 2, 3, 5, 7, 11 and 13 as primes.

3. [video] Primes and Composites from 9 times table. Saying a whole number is prime when and only when it is not the product of two or more smaller whole numbers larger than one allow small primes in the 9 times table to be quickly identified. Here is a short form video version or variant of the previous lesson.

4. [video] Prime Factorization Introduction. This video lesson introduces and illustrates prime factorization for a few whole numbers. In the last example, the equivalence between tree notation and the use of equal signs in the development of prime factorization is emphasized.

5. Prime Factorization and a Square Rule. This lesson provides a more detailed discussion of prime factorization and introduces a well-known, but I suspect hitherto nameless rules, for identifying primes and obtaining prime factorization. The name square rule is coined. This rule or method provides students with a quick manual method for obtaining the prime factorization of whole numbers. This quick method can be employed with the aid of the divisibility rules for the small primes 2, 3, 5 and 11, when or if mastered, or with the aid of calculators that display sufficiently many digits after the decimal point. See the calculator usage notes and cautions below.

6. Sieve-of-Eratosthenes upto 100. Discussion of this Sieve (or filter) shows how striking out of multiples of 2, 3, 5 and 7 is sufficient by the square rule to identify all primes in a list or table of whole numbers 1 to 100.

7. Calculator Usage Notes and Cautions. The square rule for identifying primes and obtaining prime factorizations, two side of the same coin, can be employed with the aid of calculators that display sufficiently many digits after the decimal point. Here are notes on how with some cautions.

8. [video] Prime Factorization upto 19. Given the identification of all primes less than 100, this video is redundant. However, it is retained to illustrate the use of the square rule. An alternative approach would be to use the 18 times table alone or a subtable of a larger times table.

9. [video] Prime Factorization upto 19 squared. The square of 19 is 361. provides examples of prime factorizaton with the square rule for a few numbers less 361.

10. [video] Prime Factorization upto 23 squared. The square of the prime 23 is 5629. provides examples of prime factorizaton with the square rule for a few numbers less 529.

11. Efficient Square Rule Use. The last part of the lesson Prime Factorization and a Square Rule provides a few hints or directions for the efficient use of the square rule. This lesson on efficient use provides examples.

12. LCD GCD and LCM calculations using Primes. Finding least ommon ddominators, least ommon multiples and greatest common divisors are of sets of whole numbers, two or more, represent number theory practices of service in simplifying, adding, comparing, subtracting, multiplying and dividing fractions, and of service in cosmetic conventions for the exact representation of square, cube and higher roots of whole numbers. This webpage gives some examples of LCD GCD and LCM calculations using Primes, or more precisely prime factorizations.

Lessons 13 to 18 present and illustrate methods to count and find all whole number factors of a given whole number using tables, products and trees. In the study of quadratics, factoring by inspection requires the identification of all integer factors of a given integer. The methods here or variant of them turn that identification part of this factoring a quadratic by inspection into a systematic art.
13. [video] Factors of 24 using primes

14. [video] Factors of 24 Take II

15. [video] Factors of 20 using Prime Factorization

16. [video] Factors of 980 using primes

17. Identify and Count Factors using Primes

18. [video] Count Factors given Prime Factorization

19. [video] Prime Factorization Unique. This video lesson introduces the notion that the prime factorization of a whole number will be independent of the route that gives the factorization. Knowing about the uniqueness and how it implies the latter is nuance that may included for completeness, or not.

20. Uniqueness of Prime Factorization. This lesson describes the uniqueness in general - offers reason for it.

www.whyslopes.com >> Arithmetic and Number Theory Skills >> 3 Prime Factorization Skills

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.
... 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.