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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 >> 1 Decimal Place Value

### Notes

Food for thought. The SI lesson below would be sufficient for upper secondary schools in many countries , and independent of the use of local naming systems for big and small numbers and measures.

The exercise of reading decimal aloud with several places before and after the decimal point is not a practical skill. It is a skill which can provide comic relief in a mathematics class while developing and reinforcing place value comprehension. A possible modification of the following lessons to favour UK-terms meaning of billions and trillions is indicated below.

1. Place Value in Three Digit Whole Numbers describes the ones, tens and hundreds place value of the digits and, more importantly, says how 3 digit number aloud, digit by digit.

2. Groups of Three Place Value for Multidigit Decimals gives a single example to introduce or review the ones, thousand, millions, billions, trillions and quadrillions place value of groups of three in multidigit decimals. Here place value is determined by reading a number backwards (from right to left) while the normal way of reading identifies the groups of three and their place value from left to right.

3. More on Groups of 3 Multi-Digit Place Value gives more examples to example to introduce or review the ones, thousand, millions, billions, trillions, quadrillions and quintillions place value of groups of three in multidigit decimals. Here again place value is determined by reading a number backwards (from right to left) while the normal way of reading identifies the groups of three and their place value from left to right. Recognizing place in groups of three backwards has to be done before the number is read aloud.

4. Groups of 3 Place Value in Decimal Fractions gives a single example to introduce or review the thousandths, millionths, billionths, trillionths and quadrillionthss place value of groups of three in multidigit decimal fractions. Here place value is determined by reading a number forwards(from left to right and can be done while the decimal fraction is read aloud.

5. More on Groups of 3 Place Value in Decimal Fractions gives a more examples to introduce or review the thousandths, millionths, billionths, trillionths, quadrillionths and quintrilionth place value of groups of three in multidigit decimal fractions. As just said, place value is determined by reading a number forwards(from left to right and can be done while the decimal fraction is read aloud.

6. Groups of 3 Place Value in Mixed Decimal Fractions gives a single example in which place value of groups of three in a mixed decimal number or fraction is based on place value of groups of three determined backwards, that is, contrary to the normal direction of reading in English from a decimal point while place value of groups of three after the decimal point is determined in accordance with the normal reading direction.

7. More on Groups of 3 Place Value in Mixed Decimal Fractions gives more examples of the appearance of place value from quintillions to quintillionths in a multidigit decimals that included a decimal point.

8. Review Lesson 1 2 4 and 6 - All in One present four examples, one from each lesson.

9. Place Value Review - Decimal form of Avogrados number included. Students in science courses may appreciate the order of magnitude of Avogrado's number after this stand-alone extension of the previous lessons introduces them to 602 septrillions.

10. Names for Big Numbers and Powers of 1000 Expansion goes from the power of ten expansion of multidigt decimals to powers of 1000 = 103 groups of three expansion of multidigit decimals with digits before and after a decimal point. A reference is given.

11. Place Value SI Standard International way. The SI (System International) systems of units and measures provides "standard names" for big and small numbers from yota to yocto for powers of 103 and it reciprocal 10-3 that can occur in discussing group of three place value in multidigit decimals. The alternative names are given along side dare-we-say common US-style names for numbers, big and small. A reference is given. for

### Modification for UK big and small number naming.

In the UK, a billion is not a thousand million, is it is a million million. Further, a trillion is not a thousand million, it is a million billions. That suggests a variation of the foregoing in which multidigit decimals may be read aloud in full or partial groups of six, where each group of six consists of two groups of three. Thus

56 789 345 230 456 678 . 560 678 999 12

56 thousand and 789 billions
345 thousand and 230 millions
456 thousand and 678 ones or units,
560 thousand and 678 millionths, and
999 thousand and 120 billionths

That being said, some variation of this ideas may work better. Again, the skill of reading multidigit decimals aloud in groups of three or six is not a practical. It is a skill aimed at developing and re-enforcing place value.

Is this better? For reading in full or partial groups of three digits, UK instructors might emphasize the use of SI terms yota to yocto for powers of 103 and it reciprocal 10-3 because groups of three are easier to digest than groups of six, and groups of three may lead to better comprehension of the SI naming system for large and small numbers and measure.

www.whyslopes.com >> Arithmetic and Number Theory Skills >> 1 Decimal Place Value

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.