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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.

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www.whyslopes.com >> Arithmetic and Number Theory Skills >> 10 LCM GCD and Euclid GCD Algorithm

     12 GCD 2700 288 via Primes [swf file]
     1 Least Common Multiples LCM Introduction [swf file]
     2 Least Common Multiple LCM intro via list method [swf file]
     LCM 60 45 Avoid List Method Use Primes [swf file]
     4 LCM of 8 and 10 via Primes [swf file]
     5 Common Divisors 60 45 via Primes [swf file]
     6 GCD from Primes [swf file]
     7 GCD and LCM from prime factorization [swf file]
     8 GCD from Euclidean Algorithm [swf file]
     9 GCD of 360 110 via Primes and Euclid Algorithm [swf file]
     10 Euclid Algorithm with 129 125 and with 45 14 [swf file]
     11 GCD 2700 288 via Euclid Algorithm [swf file]
     13 GCD from given Prime Factorizations [swf file]
     14 GCD of 650 110 via Primes LCM via Product Rule [swf file]
     15 GCD of 650 225 via Euclid Alg LCM via Product Rule [swf file]
     16 GCD and LCM of 650 225 via Primes [swf file]
     17 GCD LCM of 85 and 60 via Primes [swf file]

Folder Content: 17 pages.

Notes

  1. Least Common Multiples [LCM] Introduction. This video lists the first 14 multiples of 6, and the first 6 multiples of 14 to see if there is a smaller common multiple that 6 × 14 = 14 × 6. The video provides a hint of the role of primes in find the LCM of the two numbers. ??? KILL

  2. Least Common Multiple LCM intro via list method. This video answers the question what is a LCM, explains the motivation for LCM calculation, and introduces the list method for finding the LCM of a pair of small whole numbers, here 6 and 8. For these two numbers, the list method begins by writing or listing the first 6 multiples of 8 and the first 8 multiples of 6 to be list

  3. LCM 60 45 Avoid List Method Use Primes. This video explains why the use of prime factorization requires less work than the list method to find the LCM for two numbers, namely 60 and 45. Includes a clear introduction of the prime factorization based method for finding LCMs.

  4. LCM of 8 and 10 via Primes. This video shows how to find the least common multiple of 8 and 10 using their prime factorizations. The video explains the method. The video includes the list method as well for confirmation.

  5. Common Divisors 60 45 via Primes. This video employs the prime factorizations of 60 and 45 - obtained in the previous lesson - may be used to generate common divisor and to identify the greatest common divisor.

    Optional Question: How many common divisors are their. Master section on Combinatorics to answer.

  6. GCDs from Primes. This video shows how prime factorization of whole numbers may be used to find the greatest common divisors of the whole numbers.

  7. GCD and LCM from prime factorization. This video gives examples of how to compute Greatest Common Divisor and Least Common Multiples of a pair of numbers, each equal to product of primes - their prime factorizations.

  8. GCD from Euclid's Algorithm. This video gives a first example of Euclid Algorithm for find the greatest common divisor of two numbers, here 875 and 300. It then simplifies the fraction 875 over 300. Finally, it shows how to construct a small - in fact the least - common multiple of them for use in addition of two fractions with denominators 875 nad 300.

  9. GCD of 360 110 via Primes and Euclidean Algorithm. This video calculates the GCD of 360 and 110 with Euclid Algorithm and then verifies the same result can be obtained from prime factorization. Euclid Algorithm may be quickest - proof of that or discovery of that is left to further studies in mathematics.

  10. Euclid Algorithm for 129 125 and for 45 14. This video provides two more examples of greatest common divisor calculation with Euclid's algorithm. The GCD in both examples is 1. Thus implies that in each pair of numbers, the pairs are relatively prime - their prime factorization share no common primes.

  11. GCD 2700 288 via Euclid's Algorithm. This video calculates the greatest common divisor of 2700 and 288 via Euclidean Algorithm. Then it employs the GCD to simplify a fraction where one is the numerator and the other is denominator. Lastly, it employs number obtained from the algorithm to obtain a Least Common Multiple - LCM

  12. GCD 2700 288 via Primes.This video calculates the greatest common divisor of 2700 and 288 using their prime factorizations

  13. GCD from given Prime Factorizations. This video shows how to calculate GCD for numbers given as products of primes. Three products are given. The products are consider in pairs. Question: What the GCD of all three numbers?

  14. GCD of 650 110 via Primes. Then LCM via Product Rule. The product of two numbers equals the product of their GCD and LCM. We call that relation, a product rule. If the product GCD × LCM is known along with one of the factors, then the other factor can be calculated. That represents a backward use of this product rule.

  15. GCD of 650 225 via Euclid Alg. Then LCM via Product Rule. This video calculates the GCD of the two numbers, and then uses the product rule introduced in the previous lesson to obtain the LCM. The next video confirms the GCD and LCM computed here by deriving them from prime factorizations.

  16. GCD and LCM of 650 225 via Primes. This video confirms the GCD and LCM computedin the previous video using prime factorizations.

  17. GCD LCM of 85 and 60 via Primes. This video calculates the GCD and LCM of the two numbers 85 and 60 with the aid of their prime factorizations.


www.whyslopes.com >> Arithmetic and Number Theory Skills >> 10 LCM GCD and Euclid GCD Algorithm

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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. ...
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.

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