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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 >> Volume 3 Why Slopes - A Calculus Intro Etc >> Fall 1983 Calculus Appetizer Next: [Chapter 1.Introduction.] Previous: [ Foreword.]   [1] [2] [3][4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29]

A Calculus Preview

Volume 3, Why Slopes and More Math.

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If you have ever been or gone skiing, if you have ever walked over hills, then you know about slopes and you have also met or felt basic ideas in calculus before the use of symbols. Calculus in the first instance is the subject of slope computation and interpretation, and the reversal of slope computation with its applications. Slopes (rises/runs) appear whenever one quantity is mapped against another. Height versus horizontal movement is just one example.

Recall the slope of a straight line or line segment is given by the rise over run of a right triangle with hypotenuse on the segment, and sides horizontal or vertical.


Slope Interpretation

 For travel along a line segment, the slope m is positive for uphill motion. It is negative for downhill motion. Finally, it is zero for horizontal motion.
 

[Image: Slope Interpretation (Drawn March 26, 1997)]

 [play realplayer video]  80 seconds: Slope Sign Interpretation for Linear Functions

Meet the Skier

While skiing or walking you can observe and feel when you are walking uphill from the slope of your ski or heel. Likewise, you can feel when you are walking downhill. Alex the skier shown in the diagrams has a similar skill. It is his picture - the stick diagrams -- that you see above and below.

[Image: Meet the Skier (Drawn March 26, 1997)]

 The slope of Alex's ski is positive when he is heading uphill, negative when is he heading downhill, and zero when he on a horizontal portion. We assume he travels from left to right -- traveling the other way would reverse the sense of uphill and downhill.

Formulas for Slope

 Ski hills y =f (x) usually do not consist of a single straight line segment with a single slope. In consequence, the slope m of his ski varies with his position. 

[Image: Slope Dependence (Drawn March 26, 1997)]

 Height y at x is given by a formula or function f(x) involving x. So we write y = f(x). Likewise, when the skier Alex is above x at height y on the hill, the slope of his ski may be given by a formula or function m = g(x). It depends on x.

Note we also write g(x) = f'(x) -- read f prime of x -- to say or suggest that the formula for slope m can obtained or derived from the formula for f(x). Rules for slope computation (differentiation) say when. Calculus courses may call formulas for slopes obtainables or derivatives -- one of these names is correct. The other is not.

For the following diagram, answer the following questions. Assume forward motion in the direction of increasing x.

  1. Where (above what intervals) is the slope m m = g(x) =f'(x),
     (a) positive?
     (b) negative?
     (c) zero?
  2. Where is the slope increasing? In other words, where is the slope becoming more positive or less negative?
  3. Where is the slope decreasing? That is, where is the slope becoming more negative or less positive.
    Where does the hill become steeper? (That is, identify the intervals or hill portions where with x increasing, motion forward, the slope become more positive or more negative?)

[Image: On the Slopes (Drawn March 26, 1997), Repeated]

 In this ski trip, when x = b, is the skier at a hilltop or at the bottom of valley (or depression)? Is the value y = f(b), the least or greatest value of f(x) for x between a and c?

[play realplayer video]  2¼ minutes:  Slope Interpretation for a 2D ski hill y = f(x).

Slope (or Derivative) Tests  for High and Low Points

In first calculus courses, you may be given a formula for y =f(x). From this formula, you may then obtain a formula for the slopem = g(x) = f'(x) at each point x on the curve. By factoring the expression for m, if that be possible, you may see where the slope m is positive, negative or zero. This allows you to say where are the maximums (greatest value) and minimums (least values) of the original function. Slope sign analysis can be done whenever one quantity y is graphed against another x.

In graphs of height versus distance, the slope has no units, but in graphs of distance versus time the slope has a unit of the form distance over time. Slopes with units appear when the abscissa and ordinate are multiples of different units of measurement.

In Summation (to say that again)

In skiing or walking you can tell where the path is going up, down or is on the level. The slope is positive on uphill portions, negative on downhill portions and zero on flat portions. Knowing the sign of the slope gives information about the hill. The slope changes from positive to negative in crossing a hilltop. It changes from negative to positive in crossing through a low point (a valley). Just knowing the sign of the slope is enough to identify the uphill, downhill and flat portions of the path, and then location of high points and low points.

 Here the sign of the slope indicates where the path is going up (ascending) or going down (descending). Positive slope corresponds to going up while negative corresponds to going down. Moreover, and this is whether matters become complicated, the slope in changing may increase or decrease. Here a positive slope may increase by becoming more positive and a negative slope may increase by becoming less negative. Likewise, a positive slope may decrease by becoming less positive and a negative slope may decrease by becoming more negative. And in all these cases, the steepness or slope of the curve changes.

 The steepness is given by the absolute value or magnitude of the slope. Problem: What can you say about the slope behavior when the steepness of the path is increasing? (The answer will depend on whether the path you are following is ascending or descending).

[Image: On the Slopes (Drawn March 26, 1997), Repeated]


Slope or Derivative Tests for High Points and Low Points

Advanced Topic -- take a break before proceeding.

(???)In travelling over an interval a < x < b interval the over downhill travelling is (s)he that and negative slope b, a when uphill going positive observes ski her or his of feel from skier c,> In this ski trip, when x = b, is the skier at a hilltop or at the bottom of valley (or depression)? Is the value y = f(b), the least or greatest value of f(x) for x between a and c? How would your conclusions change if the words positive and negative were interchanged? In first calculus courses, you may be given a formula for y =f(x). From this formula, you will obtain a formula for the slope m = g(x) = f'(x). Then by factoring the expression for m, if that be possible, you may see where the slope m is positive, negative or zero. This allows you to say where are the maximums (greatest value) and minimums (least values) of the original function. This analysis can be done whenever one quantity is graphed against another.

Algebra and Logic in Calculus - A warning

Calculus employs at full strength the algebraic way of writing and reasoning. Students who have done well in previous math courses without fully understanding the algebraic way of writing and reasoning will find calculus stressful. Memorization of formulas or rules for differentiation by itself is not enough. Understanding is required.

The computations in calculus employ very finely and carefully, constants, variables and algebraic shorthand notation (formulas) to discuss and describe calculation that might be done. A few are even performed. In Volume 2, chapter 8 Three skills for Algebra and the  logic chapters before it (or chapters 4, 6, 7, 8 and 12 in Volume 1A) should be read and mastered, preferably before you take calculus. Mastering the logic appetizers should help read the definition in calculus precisely, and follow the chains of reason provided by your teacher or textbook. (Reading the text is advised -- it gives a second opinion.)

A Review

In skiing or walking you can tell where the path is going up, down or is on the level. The slope is positive on uphill portions, negative on downhill portions and zero on flat portions. Knowing the sign of the slope gives information about the hill. The slope changes from positive to negative in crossing a hilltop. It changes from negative to positive in crossing through a low point (a valley). Just knowing the sign of the slope is enough to identify the uphill, downhill and flat portions of the path, and then location of high points and low points.

Now in walking along a path, you can also tell when or where the steepness or slope of the path changes. For instance,

  • Along one portion of a path that the slope could be positive and becoming more positive. On this portion, you are walking up hill and the slope is increasing.
  • On another uphill portion of the path, the slope could be decreasing --- becoming less positive and less steep.
  • On yet another portion of the path, the slope could be negative. So as your walking along, the height of the path is decreasing. Now as you walk along this downhill portion of the path, the slope may become more negative or less negative. Where the slope is becoming more negative, your downhill path of descent is become steeper; and where the slope of the path is becoming less negative, your downhill path is becoming less steep.

Here the sign of the slope indicates where the path is going up (ascending) or going down (descending). Positive slope corresponds to going up while negative corresponds to going down. Moreover, and this is whether matters become complicated, the slope in changing may increase or decrease. Here a positive slope may increase by becoming more positive and a negative slope may increase by becoming less negative. Likewise, a positive slope may decrease by becoming less positive and a negative slope may decrease by becoming more negative. And in all these cases, the steepness or slope of the curve changes.

The steepness is given by the absolute value or magnitude of the slope.

 Problem: What can you say about the slope behavior when the steepness of the path is increasing? (The answer will depend on whether the path you are following is ascending or descending).


www.whyslopes.com >> Volume 3 Why Slopes - A Calculus Intro Etc >> Fall 1983 Calculus Appetizer Next: [Chapter 1.Introduction.] Previous: [ Foreword.]   [1] [2] [3][4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29]

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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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