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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 >> 6 Fractions and Ratios

## Notes

Lessons below may revise or consolidate fraction skills and concepts with operations on lengths. In the process, algebraic descriptions of operations are indicated not as requirement for fraction mastery but as enrichment option. Methods for comparison, addition, subtraction, multiplication and division are all developed or justfied below by raising terms to transform a general case into simpler cases. Algebraic view of those operations, click here.

#### Step-by-Step Skill Development

While all methods are likely to be well-known in the mathematical sense, the methods themselves may be new to many students and teachers. These raising term methods may overwhelm some students but help others. Instructors and tutors should emphasize fraction sense and skill mastery by rote when and where explanations of why methods work do not help. Partial or full comprehension may be left to older or gifted students, and to students who need to understand the development or origins of methods before using them.

1. What is a Fraction: Meaning of a Fraction - A whole number counts how many ones, A fraction counts how many parts of equal value. Algebraic Description included.

2. Fraction Multiplication I. What is a Unit Fraction of a Unit Fraction? What is a half of a third? What is a unit fraction of a unit fraction? A unit fraction has one in the numerator. Algebraic Description included.

3. Fraction Multiplication II. Unit Fraction of a Simple Fraction: what is a half of two thirds? What is a quarters of seven tenths?

4. Fraction Multiplication III. : What is a Fraction of a Fraction: what is seven quarters of three tenths?

5. Equivalent Fractions: What is the difference between two quarters and one half? What is the difference between six eights and three quarters? What is the difference between an apple and four quarters of the apple? The thought that there is no difference, that different fractions may describe the same amount, quantity, leads to the idea of equivalent fractions. Examples of fraction simplifying and equivalence included.

6. Multiplication Algebraic Development. The first, easy case treats the case where the denominator of one fraction is a divisor of the numerator of the second fraction The general case follows by raising terms in the second factor to apply the easy case.

7. Mixed Numbers and Equivalent Fractions: We may describe a distance as 3 half meters or we may describe the same length as 1½ meters. Physically, there is no difference in the distance, only its descriptions. The descriptions 3/2 meters and 1½ meters both have the same value physically. So we declare the numerals 3/2 and 1½ to have the same value, or to be equivalent. Likewise 3 fifties (half-hundreds) is the same a 1½ hundreds. There is another instance where the fraction 3/2 and the mixed numeral 1½ may be identified as adjectives for the same count or measure.

8. Comparison of Fraction and comparison of Mixed Numbers: Algebraic Description included of the first included.

9. Fraction Addition I: Easy case of like denominators - the easy case Algebraic Description included.

10. Fraction Addition II: General Case of unlike denominators. the general case follows from raising terms (as little as possible) to use the easy case.

11. Examples to show "raising terms" similarities between comparison, addition and subtraction of fractions.

12. Fraction Addition III: Methods for adding and subtracting Efficiently - Questions and Problems: (a) How is the list method used to obtain a least common denominator = the least common multiple of a pair of denominators? (b) How can the prime number decomposition (also known as factorization) be used to calculate the LCM and GCD of a pair of whole numbers? (c) How can Euclid's algorithm be used forwards and backwards to calculate the GCD (the normal result) and then the LCM of a pair of whole numbers? (d) Employ the M-method to find the sum of eighteen 21st and nine 14-ths? Say which of the latter is more than the other, and find how much more.

13. Fraction Multiplication IV: Efficient Ways to Multiply Fractions: Learn how to calculate a few products of fractions with and with the cancellation methods described below for "efficient" multiplication, or more precisely efficient or easier simplification after (cross) cancellation of common factors.

14. Fraction Division, Compound Fractions and Reciprocals: A Physical Introduction to Fraction Division: Explanations, twice-over, are given in the next lesson.

15. Fraction Division Methods Explained. See the previous page for an introduction of the fraction division methods or formulas below.
- Two Step Development Option - Explanation in two smaller steps, first with the easy like denominators case and second with unlike denominators (Raising terms in dividend and divisor fractions turns the second case into the easy case).

16. How to do Arithmetic with Rational Numbers, that is signed fractions, signed whole numbers and signed mixed numbers. Here a model for introducing arithmetic with real numbers.

#### A. Ratios And Fractions Similarities and Differences

Fractions may be identified with two term ratios, and vice-versa as well, sometimes. Two fraction are equal or equivalent when and only when the corresponding two term ratios are equal or equivalent. But fractions can be added, subtracted, multiplied and divided while the same operations are not defined for two- and multiple term ratios. While we may call a fraction, a ratio or a rational number, ratios are different. Triple term ratios exist, but triple term fractions do no exist. Three and more -term ratios cannot be identified with fractions.

The following lesson cover the properties of two term and multiple term ratios.

1. Fractions As (two term) Ratios and Fractions Versus Ratios: Fractions are often called ratios, and vice-versa. But the vice-versa only holds for two term ratios. This lesson identifies fractions with two-term ratios and contrasts the properties of fractions and two-term ratios. (Ratios cannot be added, subtracted or compared, but like fractions, the terms in ratios can be raised or lowered).

2. Implied or Derived Ratios - New Fractions and Ratios from Old: If two fractions are equivalent then their reciprocals are also equivalent. Likewise if a pair of two-term ratios are equivalent, interchanging the first and second terms of each ratio in the pair leads to a pair of equivalent ratios. Beyond that, more equivalent ratios can also be generated from a pair of ratios. Food for thought: How may equivalent fractions or ratios may be formed from the relations ad = bc?

3. Multiple Ratios: Multiple Term Ratios - Three Term Ratios to be precise. We read the triple ratio a : b :c as a to b to c. We further write

a : b: c :: A: B: C

to say two triple ratios a : b: c and A: B: C are equal or equivalent when and only when

 a b = A B and b c = B C

there are other ways to say when two triple ratios are equal or equivalent.

Note: Triple ratios or triple proportionalities occur between the sides of similar triangles. More generally, multiple ratios or proportionalities occur between the sides of similar triangles.

The discussion of ratios or multiple ratios is best understood besides a discussion of proportionality.

Inner Versus Outer Terms - small point: In the discussion of equality of ratios a : b = A: B written in that order, the inner terms are small b and big A while the outer terms are small a and big B. In contrast, if we rewrite the equality as A: B = a : b, we find the inner and outer terms are interchanged. However, the equality requires the product of the inner and outer terms be equal, that is aB = Ab. That equality is not affected by rewriting a : b = A: B as A: B = a : b, and the resulting swap of inner and outer terms

www.whyslopes.com >> Arithmetic and Number Theory Skills >> 6 Fractions and Ratios

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.