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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 >> 2 Arithmetic with Decimals >> D Decimal Long Division Methods

### Notes

#### Long Division (LD) Examples and Theory

Teaching and Tutoring Tip: (1) The long division method and checks for it requires at full strength, multiplication, subtraction and addition methods for decimals. So long division represents a check of addition, subtraction and multiplication skills as well. Difficulties with the latter operations need to caught and remedied before preferably or while students do long division calculations. (2) Preparation for calculus requires students to add, multiply, subtract and divide polynomials. In that preparation, long division of polynomials resembles long division of decimals . So mastery of the latter will help with mastery of the former.

• Lesson 1. Where Does Division Appear? The division question of asking how many times does one quantity, number or unit go into another appears in counting, grouping and measurement.

1. For Distances and Length Measurement: How many times does a shorter length (a unit length perhaps) go into (go into) a longer length. The longer length is view a whole number multiple of the shorter length with a remainder.

2. For Dot Counting and Grouping: How many same size groups of say three dots can be formed from sixteen dots provide a first example.

3. For Volume Measurement: How many times does a small unit of volume go into a larger volume.

• Lessons 2 to 4 - Worked Examples of Long Division with 1 and 2 digit divisors.

(2) 6847 = 2282 × 3 + 1 (Dividend: 6847, Divisor: 3, Quotient: 2282, Remainder: 1)
(3) 7834 = 1119 × 7 + 1 (Dividend: 7834, Divisor: 7, Quotient: 1119, Remainder: 1)
(4) 89463 = 6881 × 13 + 10 (Dividend: 7834, Divisor: 7, Quotient: 1119, Remainder: 1)

These examples illustrate the long division method and format met in the site author's school days. These examples introduce the helpful practice of listing the first 10 multiples of the divisor. Single digit multiples are needed in the method. These examples also show how to check (test) that a computed or given quotient and remainder are correct.

Quotable quotes: Even if we don't know why long division works, we can check its results by testing whether or not

Dividend = Quotient × Divisor + Remainder.

Remember: If a check fails, there is an error between the start of the long division itself and the end of the check. And if the check fails, you need double check the check and the long division.

For single digit divisors, there is a short division format that is more compact (uses less space) than the long division format. Learn it if you wish.

• Lesson 5: Examples in which LD (long division) is done with & without extra zeroes to further indicate how, if not why, LD works.

• Lesson 6: Why the long division methods works. Here is a closer inspection illustrated with the long division (with extra zeroes) of 7275 by 2 to 7275 = 3637 x 2 + 1. This first explanation and lesson 8 are for people who like to understand long division fully as well as learning to do it and check results.

• Lesson 7. Long Division With a Mistake - How to handle mistakes in mathematics solutions

Long Division Method here suggests

5626 = 308 × 15 + 6

Observe how the error is spotted and ignore the extra zeroes that appear in the solution. Their presence will be explained below.

Tip 1: In class, when a check spots an error, the error itself will between the start of your solution and the end of your check. The error could be in your check.

Tip 2: In a test, do not erase your solution if you a spot an error or think you have made one. Do not erase your solution until you have written and checked a replacement. Better yet, cross out the solution instead of erasing it when you have the replacement. And if on a test, you do not have time to provide a replacement, give the solution and check, and write Oops! do not have time to correct in large print or boldly besides the check. The student who submits wrong a clearly written solution with a clearly written check, right or wrong, gets more respect and show a greater command of subject matter than the students who does not present work clearly. The advice do not erase your solution may help as well with instructors who are looking for reasons to reward your work. Good luck.

Tip 3: In homework, if you have time to correct your work, do so. If not follow the advice in tip 2, and change your error declaration to : Oops! Did not make time to correct.

• Lesson 8. Correcting the Error. Doing long division properly (with extra zeroes) to get

5626 = 375 × 15 + 1

and then checking the results. It works.

• Lesson 9. Yet another insight into why long division method works. This lesson inspects the numbers and subtractions that appear in long division of the dividend 5626 by a divisor 15 to show why 5626 = 375 × 15 + 1

• Lessons 10 and 11: More examples of long division.

• Lesson 12. The long division method in expanded form gives yet another explanation of why the LD method works.

Not Done Here or Yet - Later is anticipated: The treatment of long division for decimal fractions (mixed numbers plus a proper fraction equivalent to a fraction whose denominator is a power of ten) belongs after the coverage of fractions and in that coverage, a discussion of decimal fractions.

www.whyslopes.com >> Arithmetic and Number Theory Skills >> 2 Arithmetic with Decimals >> D Decimal Long Division Methods

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.