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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 >> 10 LCM GCD and Euclid GCD Algorithm

### 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

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.