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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.

30 pages en Francais || Parents - Help Your Child or Teen Learn
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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www.whyslopes.com >> Algebra Starter Lessons >> 3 Adding Words To Arithmetic Next: [4 A Brief Story of numbers and algebra.] Previous: [2 What is a Variable.]   [1] [2] [3] [4][5] [6] [7]

Words before Symbols: In the first instance, the use of letters in formulas to denote lengths or amounts stems from their shorthand role in providing a more compact description of a calculation. But that shorthand role of letters and symbols has limitations. For example, calculation of the perimeters of a triangle, quadrilateral and polygons in general may be simply given by the instruction: add the lengths of the sides. Before the introduction of algebra, that instruction can be understood and followed. In contrast, the algebraic description of the calculation of these perimeters introduces many letters and symbols, alone or with subscripts, and in doing so raises the level of complexity. That introduction of letters and symbols is has a role in the introduction of algebra but when the aim to show how to compute perimeters, algebraic expressions for perimeters are not needed.

Before algebra begins, words may be also used to say when different counting or arithmetic methods lead to the same result. Here again the words may be simpler to understand and follow than the corresponding and far more complicated algebraic descriptions of the same mathematical rules or patterns. In particular, the following sequence of phrases describe common practices in primary and secondary mathematics more easily explained and understood with words. In mathematics lessons given by teachers not fully versed in algebra, the use of words in place of symbols makes instruction simpler with little or no loss of content and rigour.

Explanations of why these inter-related practices work - all or some can assumed as axioms and those not assumed may explained in terms of the others.

Oral Rules for Arithmetic

Each rule or pattern in below has value in its own right. But the earlier ones lead to the last two. Moreover, these rules or patterns may developed and understood verbally before any algebraic description of properties of numbers, whole to real. The rules wordily given and explained reflect and extend the common know-how in ways that may have take-home value.

  1. Counts are independent of the order in which elements of a (finite) set are counted. In practice, people may count twice in the hope of detecting mistakes. Recounting is required when earlier counts disagree.

  2. Counts may be obtained by forming and adding non-overlapping subcounts in any order. In practice, addition of whole numbers may be introduced as a counting shortcut. People may add twice in the hope of detecting mistakes. Disagreement between additions will require a recount - here another addition. Counting by addition is the practiced in elections where polling stations report their totals for inclusion in larger totals.

  3. Measures of lengths, areas and volumes may be obtained by forming and adding non-overlapping sub-measurements - measurement (counting how many units and then how many fractions of units) is a form of counting. Counting by addition justifies measurement by addition.

  4. In accounting, money may be counted may be obtained by forming and adding non-overlapping subtotals. In metric or decimal based, money is counted in terms of whole units (dollars, pounds, francs, Yen and so on) and in terms of one hundredths of those units (cents and pennies).

  5. In arithmetic, sums of whole numbers, fractions and decimals may also be obtained by forming and adding non-overlapping subtotals. If one looks carefully enough, this practice or principles is another consequence of the ability to count wholes by addition.

  6. In accounting, sums of revenue (positive amounts) and costs (negative amounts) may be obtained by forming and adding non-overlapping subtotals. The subtotals themselves may be zero or signed. Here adding twice is a process to check for mistakes. Students may be told that the uniqueness of the total means in daily live that the sum of assets and debts can be made more nor less by subtotaling in different ways. That knowledge has take home value. This property may be presented as special case of the next, or it may be employed to introduced the next.

  7. In arithmetic with signed numbers (integers and then rationals, sums integers and rationals may be obtained by forming and adding non-overlapping subtotals. Note when students use both signed coordinates and arrows to represent movements on a map or plan, the observation that the head to tail addition of arrows in sequence may be done by subtotalling and that addition is commutative informally implies most of this property and the previous one with signed numbers.

  8. In Arithmetic, Products of numbers may also be obtained forming and multiplying non-overlapping subproducts. This rule find application when students are shown how to group like primes in the the prime factorization of whole numbers. It has also has application in the discussion of decimals - the optional explanation of how or why decimal methods for multiplication work. Extension: This rule also applies to division as division by a number is replaced by or identified with multiplication by its reciprocal in the case of fractions and a multiplicative inverse.

  9. In arithmetic, products of nonzero numbers are nonzero, but if one or more factors is zero, the associated product is zero. This rule may answer questions about whether or not a product is zero. This rules may be used to speed the calculation of a product in which one of the factors, one given by the value of an expression, happens to have the value zero. The rule may be implied by observations about how and why the product of two nonzero counts or two nonzero lengths cannot be zero.

The first two of the last three rules or patterns imply that sums and products of terms and factors may calculated and grouped (carefully) in different ways even before algebra begins.

The distributive property of arithmetic with real numbers is introduced elsewhere with the aid of geometry and coupled with a column methods for calculating products of sums. See using geometry in algebra. Mastery of the algebraic form of properties of real numbers etc may be left to courses in pure mathematics. The modern mathematics course designs seen in my student days emphasized the algebraic form of arithmetic properties, but presented as axioms for real numbers etc. The use of algebra in that manner raise the level of complexity beyond the level of many students and teachers. Furthermore, in the senior high school development of mathematics, the necessary extensions to aid if not justify polynomial addition, subtraction, multiplication were indicated orally and not written algebraically. Thus the use of oral rules to justify if not explain has been part of secondary mathematics previously, that being for ease of exposition.

With the expansion of the role of words before and in algebra to expand and enrich the common know-how or knowledge in mathematics, Upper high school mathematics and calculus instruction may have balance the verbal and algebraic description of arithmetic properties of numbers, whole to real or complex. Course design and delivery will have to adjust.


www.whyslopes.com >> Algebra Starter Lessons >> 3 Adding Words To Arithmetic Next: [4 A Brief Story of numbers and algebra.] Previous: [2 What is a Variable.]   [1] [2] [3] [4][5] [6] [7]

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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.

2005 - The NSDL Scout Report for Mathematics Engineering and Technology -- Volume 4, Number 4

... 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.


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