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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

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# 20 by 20 Multiplication Table

During the school year, practice filling in the 10 or 12 times table at least three times correctly without a calculator. Observe how the numbers increase by a constant amount in each row and in each column.

* 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
2 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40
3 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60
4 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80
5 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
6 6 12 18 24 30 36 42 48 54 60 66 72 78 84 90 96 102 108 114 120
7 7 14 21 28 35 42 49 56 63 70 77 84 91 98 105 112 119 126 133 140
8 8 16 24 32 40 48 56 64 72 80 88 96 104 112 120 128 136 144 152 160
9 9 18 27 36 45 54 63 72 81 90 99 108 117 126 135 144 153 162 171 180
10 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200
11 11 22 33 44 55 66 77 88 99 110 121 132 143 154 165 176 187 198 209 220
12 12 24 36 48 60 72 84 96 108 120 132 144 156 168 180 192 204 216 228 240
13 13 26 39 52 65 78 91 104 117 130 143 156 169 182 195 208 221 234 247 260
14 14 28 42 56 70 84 98 112 126 140 154 168 182 196 210 224 238 252 266 280
15 15 30 45 60 75 90 105 120 135 150 165 180 195 210 225 240 255 270 285 300
16 16 32 48 64 80 96 112 128 144 160 176 192 208 224 240 256 272 288 304 320
17 17 34 51 68 85 102 119 136 153 170 187 204 221 238 255 272 289 306 323 340
18 18 36 54 72 90 108 126 144 162 180 198 216 234 252 270 288 306 324 342 360
19 19 38 57 76 95 114 133 152 171 190 209 228 247 266 285 304 323 342 361 380
20 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 360 380 400

• Do you know about the domino effect of mistakes and errors in arithmetic? If not, ask parents, teachers or fellow students to explain what that might be. If yes, if you know about it, then you know about the need to be careful in each arithmetic step you take. The domino effect of errors appears both in and outside of arithmetic. In it, an error in one step likely makes all following steps and any result wrong. Taking care to avoid the domino effects is a must for building skills and confidence in all skill based subjects at home, at work and in school. Good luck.

• Understanding exactly or precisely what is said or written will avoid confusion when you are following instructions at home, at school and at work. Learning to write and speak exactly or precisely will avoid confusion of others when you explain or give instructions as well. When you are old enough (or now), read the site math free chapter on two logic puzzles. The chapter may help you read, write and speak with precision. That in turn may help you avoid the domino effects in many subjects.

Good Luck.

### Arithmetic Steps

Some good practices for skill development in arithmetic appear in site arithmetic and number theory steps. Where these good practices are not best, say so.

1. Definition of Primes, Simplified : A simpler definition of prime numbers which takes advantage of the 12 times tables to identify small primes upto 13. In particular, a whole number is said to be prime if it is not the product of two smaller whole numbers. With the word smaller in the definition, the whole numbers 11 and 13 are prime because it is not given by a product inside the 10 and 11 times table. [This was the small example given above]

2. Quick Prime Factorization of Small Whole Numbers: Emphasis of a square or square root rule to provide QUICK prime factorization skills for whole numbers less than 169 = 132. In particular, a whole number less 169 is prime if and only is it is not a multiple of the primes 2, 3, 5, 7 and 11 less than 13. Simple divisibility rules and calculators (an overkill) here may be used to recognize multiples of 2, 3, 5 and 11. Quick prime factorization of whole numbers is a key to exact and efficient fraction practices employed in mathematics from algebra to calculus. There is no escape.

3. Fraction Operations Explained: A thought-based development of addition, comparison, subtraction, multiplication and division operations starting with simpler cases where operations are easily explained, and continuing on to general cases where all operations are justified by raising terms. In higher mathematics, if not elementary mathematics, comprehension of why methods work is highly valued, it is part of the spirt of mathematics mastery. Understanding how and why operations are justified should move you away from learning by rote. Reference: fraction operations by raising terms

4. Arithmetic and Fractions With Units: Figuring with denominate numbers, that is multiples of units of measure for physical quantities and units of value for monetary quantities. This practical value for calculations involving speed, rates in general and associated proportionality constants in daily life and also in practical and applied arts and sciences. (In algebra taught by rote, you may see similar figuring with multiples and powers of variables in products and quotients. The path here has more meaning and is very practical)

5. Oral Dimension of Arithmetic: Verbal description and extension of common practices for finding counts, totals and products by forming and adding or multiplying subcounts, subtotals and subproducts. Here calculation practices are introduced and described orally instead of symbollically, the latter being harder for many to grasp. For many, how to calculate averages and how to calculate perimeters of polygon are best described with words, the use of letter or symbols being to complicated to understand in the first instance. Mastery of common practices for counting, totaling and multiplying do not have to wait for their algebraic description. Instead, the verbal forms can be given. [These arithmetic notes expand on part of the big example given above.]

6. Place Value Revisited: An exposition of place value in decimals with places before an after the decimal point in groups of three may amuse and inform. In it, students in North America may learn how to read aloud and write on paper the decimal

6,571,045,375,905,333,034,412.450,033,870

as 6 sextrillions, 571 quintrillons, 45 quadrillionths, 375 trillionths, 905 billionths, 333 millions, 34 thousands, 421 ones, 450 thousandths, 33 millionths and 870 billionths. In contrast, students elsewhere may use the following "SI" (system international) method how to read aloud and write on paper the decimal form of 6 zettaunits, 571 exaunits, 45 petaunits, 375 teraunits, 905 giga-units, 333 megaunit, 34 kilounits 421 ones, 450 milliunits, 33 microunits and 870 nanounits.

7. Addition, comparison, subtraction, multiplication and division of decimals: The site development may covered more lightly than presented. The development of place value methods for all but long division is thought-based. Why methods work are both indicated. Long division method is given without justification, but with a method to check results. In all methods, students will meet the domino effects of care and mistakes. Avoiding the latter provides an end, value and tool for skill mastery, an echo of the old fashion idea that figuring well is a sign of practical intelligence.

8. Signed Numbers: The site description of arithmetic with integers and arithmetic with signed numbers is not bad. The site objective so far has not been to cover everything in mathematics, but to develop and express ideas on how mathematics should be learnt or taught. That being done, a clearer account of arithmetic with signed numbers is due.

9. More Steps To Elaborate - not in site material: Talk about scientific notation, arithmetic with, and arithmetic with mixed decimals - that is, decimals with multiple places before and after the decimal point. Relate foregoing to fraction skills and practices. Explain the comparison, addition and subtraction of scientific in terms of of finding a common factor or denominator.

www.whyslopes.com >> Arithmetic and Number Theory Skills >> The 20 Times Table Next: [Expression Evaluation how to show work.] Previous: [The 12 Times Table Visually.]   [1] [2] [3] [4] [5][6 swf] [7 swf] [8] [9] [10]

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.