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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 >> Geometry - maps plans trigonometry vectors >> 5 What is Similarity

This folder What is Similarity provides a general and unified treatment of the likeness or similarity of squares, circles, triangles and arbitary regions in the plane. The lessons here can be covered after section 1 on maps, plans and measurement and after the introduction of Cartesian coordinates. The objective here is to explain and reconcile different characterizations of similarity.

The key question is how to recognized similarity. In modern life, objects are similar by design if they stem from the same plans but are built to different scales. That reflects a coordinate view point of similarity objects. Two objects are similar if we can attach coordinate system to each so that the set of coordinates for the points for one object essentially provides the plan for the other as is or after the application of a scale factor - a dilatation. This set of coordinate development easily explains how and why circles, squares and rectangles - those with a common aspect ratio - may be similar. The coordinate perspective of similarity in the case of similar triangles and more generally in the case of similar polygons implies corresponding angles are equal and corresponding sides are proportional. A partial proof of the converse is included in the section what is similarity - a full proof is left to later as site development to do.

Preamble and Context

Similarity is an artifical concept. It main comes from the construction of rectangular, circular and triangular or polygonal shapes for walls, doors windows and layouts or floor designs of buildings large and small; and from the construction of furniture tops, sides and legs. Where construction is is based on plans and drawings, actual lengths and surface areas, and volumes are by design proportional via a scale factor, its square and its cube to corresponding parts on the plans and drawings. In this corresponding angles are equal. Further circles and squares in the drawings correspond to circles and square areas or objects in the construction The proportionality still holds when the same plans or drawings are implemented at different scales, albeit too great variation in scale may lead to structural instability. In modern times with the advent of digital plans and drawings, planned objects are described in terms of coordinates - sets of coordinates or the data needed to determine those sets. In this, the digitilized plans and drawings may be displayed at different scales. In nature, similarity appears in the form and inner components of larger living beings, but as beings grow, the measures and dimensions of the "similar shapes" are not proportionality. No doubt there will be a few exceptions.

Primary School Geometry - Like Shapes

Primary level instructions may ask students to identify like shapes, years before the secondary level mention and introduction of similarity. Indeed, young children may recognize like shapes from their senses of vision and touch, and from the function of objects in the plane or space. The ability to recognize like shapes allows children and people in general to recognize others and navigate their way their local environments, all without or all before any formal acquaintance with the concepts of similarity.

This late primary or secondary level geometry may introduce maps and diagrams drawn to scale. At home or in a library, students can be shown maps and plans of their community and its surroundings. Maps may be employed in the description of routes followed by family members, cars, buses, trains, and planes, etc. School going children and teenagers will likely see maps of school building and terrains. On these planar maps - when drawn to the same scale horizontally and vertically, actual rectangular, circular and triangular regions appear as rectangles, circles and triangular regions respectively. The study of maps may be employed to point and recognize similar shapes before any formal discussion of similarity. That being said, this level may show and imply that angles on these maps and drawings equal the corresponding angles in the drawn objects. When a unit length on the map coresspond to a unit length in actuality, this level may further slowly show that the number of map unit lengths needed to cover a drawn length and map unit squares needed to cover a drawn rectangular region equals the number of real unit lengths and real unit squares needed to cover the actual lengths and region. The use of the scale factor or it reciprocal as is or squared then gives a proportionality constant between drawn and actual lengths and areas. Most likely, the length case and the area cases should be treated differently.

Late primary and early secondary quantitative skill development should emphasize measuring skills with rulers, tape measures and protractors for measuring lengths and angles in the environment and on maps and plans drawn to scale. Tutors and teachers may observe that the map measurements are often easier to make - require less movement. This level may show students how to recognize different kinds of triangles and quadrilaterals and connect the latter to parallel lines or line segments. All the foregoing may be done on paper with maps, plans or drawings. Coordinates signed and unsigned should be introduced along line segments and for maps and plans. The game of Battleship may be adapted to test mastery of coordinates. Students may be further given a sequence of coordinates for points in the plane to join to test and reward coordinate mastery. The joined points or dots may form a picture of objects or animals in the local environment.

Familarity with maps and diagrams drawn to scale should make the assumptions or axioms of coordinate free Euclidean more self-evident. An operational mastery of maps and plans drawn to scale sets the stage for site simplified treatment of Euclidean Geometry, one reserved for the keener students in senior highschool, one sufficient to introduce students to the use of deductive reason in mathematics.

Finally, secondary level geometry instruction may tell and show students that all circes are similar and all squares are similar; it may provide students criteria for the similarity of triangles, criteria based on equality of corresponding angles, or proportionality of corresponding sides; and it may criteria for the similarity of rectangles based on equal aspect ratios or equivalently based on the on the proportionality of corresponding sides. The foregoing rules and criteria are consistent with each other. To the foregoing, I would have secondary school geometry add the coordinate perspective: Two objects in the plane or space are similar if coordinates systems can be choosen for both such that the set of coordinates for one are proportional [scaled versions] of the other. The latter criteria echo the modern day use of plans and drawings as indicated above to digitize actual or concieved objects in two and three dimensions, that is, in planes and in space. The coordinate perspective is timely if or when secondary level geometry talks about the proportionality of corresponding lengths, area and/or volumes in the discussion of similar objects in a plane or in space.

Proportionality of Map and Actual Measurements Explained

A map unit length corresponds to a given length, a unit length, in the real world. Here the number N of map unit lengths needed to cover a path on the map is the same as the number N of given lengths needed to cover the corresponding path in the real world. The ratio of the number N of given lengths to the number N of map unit lengths equals the the ratio of the given length to the map unit length, that is the map scale factor. Likewise, the number M of map square units needed to cover a region in the the map equals the number M of of squares with sides provided by the given lengths needed to cover the corresponding region in the real world. The ratio of the number M of given lengths squared to the number M of map unit squares that equals the the ratio of the given length squared to the map unit square, that is the map scale factor, squared.


www.whyslopes.com >> Geometry - maps plans trigonometry vectors >> 5 What is Similarity

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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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