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. Early High School Arithmetic
Deciml Place Value  funny ways to read multidigit decimals forwards and
backwards in groups of 3 or 6. 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? Early High School GeometryMaps + 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 humanmade 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 >> 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 PreviewVolume 3, Why Slopes and More Math.
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.
[play realplayer video] 80 seconds: Slope Sign Interpretation for Linear Functions. Meet the SkierWhile 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. 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 SlopeSki 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.
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.
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?
Slope (or Derivative) Tests for High and Low PointsIn 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).
Slope or Derivative Tests for High Points and Low PointsAdvanced 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 warningCalculus 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 ReviewIn 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,
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] 
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 headtotail
addition of arrows in the plane. Click and drag complex numbers A and B to change their locations.
Pattern Based ReasonOnline Volume 1A, Pattern Based Reason, describes origins, benefits and limits of rule and patternbased 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 storytelling view of learning: [ A ] [ B ] [ C ] [ D ] to suggest how we share theory and practice in many fields of knowledge. Site Reviews1996  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; patternbased 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
crossproducts, 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 howtos 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. 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. 