www.whyslopes.com || Fit Browser Window

# 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.

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.

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.
-
- 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 >> 4 Remainder Arithmetic and Divisibility

### Notes

The following lessons provide a very slow, essentially prealgebraic introduction to remainder arithmetic for modulo 10, 5, 9, 3 and 2 to to explain the origin of divisibility rules for division by 2, 3, 6, 9 and 10. The lessons start with remainder arithmetic modulo 10 as decimals are based on counting in groups or powers of 10.

1. Remainder Arithmetic Modulo 10 calculates the remainders on division by 10 for two whole numbers 5430 and 253, and then asks what the remaindeer on division by 10 will be for their sums and products.

2. Remainder Arithmetic Modulo 10 more calculates the remainders on division by 10 for whole numbers 846134 and 25683, and then shows what the remainder on division by 10 will be for their sum.

3. Remainder Arithmetic Modulo 10 still more calculates the remainders on division by 10 for whole numbers 846134 and 25683, and then shows what the remainder on division by 10 will be for their product

4. Remainder Arithmetic Modulo 10 in general uses the patterns that arose in the previous lessons or examples to calculate remainders, modulo 10, for many sums and products.

5. Remainder Arithmetic Modulo 5 calculates the remainders or values, modulo 5, of whole numbers 1 to 10. There-in lies an introduction to Remainder Arithmetic, modulo 5.

6. Remainder Arithmetic Modulo 5, Properties describes the properties in an algebraic way, and then gives an example to illustrate the product property.

7. Remainder Arithmetic Modulo 5 Examples I calculates the remainders, modulo 5 for 10, 230, 345, 348 and 347. The remainder for 347 is left as 7, modulo 5. The latter equals 2, modulo 5.

8. Remainder Arithmetic Modulo 5 Examples II calculates the remainders, modulo 10 for 14583 and from the latter, the remainder modulo 5 for a 14583. The exercise is repeated for 84767 = 7 modulo 10 or 2 modulo 5. The conclusion is that each whole number its last digit modulo 5.

9. Remainder Arithmetic - Divisibility by 5. The foregoing discussion of remainder arithmetic, modulo 5, sets the stage for recognizing divisibility by 5. There in lies a justification of the divisibility rule - a whole number is divisible by 5 when the last digit of its decimal representation is a 0 or a 5. Note the number nineteen has a decimal numeral representation 19 with a last digit 9, but the Roman numeral or reprsentation is XIX, a representation to which the divisibility rules does not apply.

10. Remainder Arithmetic - Long Division by 5 Quickly. For each whole number, division by 5 can be done by long - or short - division methods. Here is alternative to both based on the decimal representation of the whole number.

11. Remainder Arithmetic - Long Division by 5 Quickly more. The alternative in the previous lesson is shown again for another example.

12. Remainder Arithmetic Modulo 10 Example gives an arithmetic expression 56 + 13×28 and calculates the remainder modulo 10 for it.

13. Remainder Arithmetic Modulo 5 Example takes t expression 56 + 13×28 and calculates the remainder modulo 5 and modulo 9 for it. This example should be read after the next lesson on Remainder arithmetic, modulo 9.

14. Remainder Arithmetic Modulo 9 Example . This lesson show 10, 100, 1000 and 10000 all have remainder 1, modulo 9. Then it calculates 8362 modulo 9.

15. Remainder Arithmetic Modulo 9 Example /b>. Calculate 537, Modulo 9

16. Remainder Arithmetic Modulo 9 Example 2. Illustration of the sum of digits rule for calculating remainders, modulo 9.

17. Remainder Arithmetic Rule of 9 for checking sums, Example I illustrated for the sum of 537, 821 and 674.

18. Remainder Arithmetic Rule of 9 for checking sums, Example II illustrated for the sum of 824, 560, 301 and 24.

19. Remainder Arithmetic Rule of 9 for checking sums, Example III illustrated for the sum of 421, 321, 567 and 222.

20. Remainder Arithmetic Rule of 9 for checking sums IV illustrate for sum of 44375 and 82621.

21. Remainder Arithmetic Modulo 3 shows that 10, 100, 1000 and 10000 are all equal to 1, modulo 3. Then calculates 8432 modulo 3 from its decimal expansion in terms of 10, 100, and 1000 - a sum of digit rule results.

22. Remainder Arithmetic Modulo 3 more - the question of when is a number a multiple of 3, and if not what is the remainder. Remainder arithmetic calculation, modulo 3, is applied to 4856.

23. Remainder Arithmetic Modulo 2 calculates remainders on division by 2 for the whole numbers 1 to 10. Then calculates the remainder modulo 2, for whole numbers 3453, 5674, and 47, and arrives at a last digit rule.

24. Divisibility Tests for 2 3 5 9 10 - Each number has a decimal form. Based on that form, we have rules for recognizing when a whole number is divisible by when of the divisors 2, 3, 5, 9 and 10.

25. Divisibility Tests for 2 3 5 9 10 Examples - which are the numbers 45, 55, 31, 4 and 369 are multiples of 2, 3, 5, 9 10. Divisibility rules and remainder arithmetic are applied.

26. Divisibility by 2 3 5 Examples. Test numbers 560, 382, 184, 962, 866 and 118 for divisibility by 2, 3 or 5.

27. Divisibility by 2, 3, 6, 5, 9 , 10 and 100 Questions. Calculate the remainders of the numbers of the numbers 242, 1500, 173, 180, 37 and 600 on division by 2, 3, 6, 5, 9 , 10 and 100

www.whyslopes.com >> Arithmetic and Number Theory Skills >> 4 Remainder Arithmetic and Divisibility

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.