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 >> Chapter 2. Slopes and Ski Trails Next: [Chapter 3. Slope Sign Analysis.] Previous: [Chapter 1.Introduction.] [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] Chapter 2 . Slopes and Ski TrailsVolume 3, Why Slopes and More Math.
Slopes of Line SegmentsA tour of calculus begins. Recall how the slope to a straight line is computed. The slope m of a straight line segment between two points ( x_{1}, y_{1}) and ( x_{2}, y_{2}) may be calculated as follows.
The pointslope formula for a line
The latter says that the change $\Delta y$ in $y$ is proportional to the change in $\Delta x$ in $x.$ For a straight line segment, the slope m is a constant of proportionality between \[\Delta y = y  y_1\] and \[\Delta x = x  x_1\]
A CrossCountry Skier and Her Trail[Play Video] 2¼ minutes: Slope Interpretation for a 2D ski hill y = f(x). (Appeared earlier in Why Slopes Appetizer)Meet the crosscountry skier, Barbara:
She has only one ski. Alternatively, you can imagine she always travels with both skis parallel. Travel with one ski was the way in which both alpine and crosscountry skiing began. Also meet the Jack Rabbit ski trail y = h( x) (see below) which she skis, always in the direction > from left to right.^{5} That is, she travels in the direction of increasing x.
Imagine or suppose that the hill is smooth enough, so that, at most points, a ski can lie flat against the hill surface. The slope beneath a foot or ski gives what should be the slope of a tangent line to the hill. The slope of a ski can in principle be measured any time by freezing a skier in place, or equivalently taking a photograph (snapshot) and then measuring the slope from the photograph.^{6}
The vertical line segment in the above graph represents a jump or cliff. The above diagram strictly speaking consists of the graph of a function y = h( x) plus a vertical portion to represent a ski jump.
The shorthand for the word positive is +ve. The
shorthand for negative is similarly ve. A Skier in MotionImmediately below are a few snapshots of Barbara on another portion of the skill trails and hills y = h( x). In each snapshot, the slope of the ski is assumed to be equal to the slope of the hill at the ski midpoint.
In the diagram observe:
1. Local HighPoints or Maximums
High points of bumps and hills are called maximums. As the skier Barbara moves up a hill (or a local bump), and then down the other side, the slope of her ski changes from positive on the uphill side to negative on the downhill side. At the very top of hill, her ski is horizontal and its slope is zero.^{7}
There can be several hills, with different and varying steepness on each side. From the slope of her ski, Barbara knows even without looking when she crosses over the top of a hill or a bump: the slope of her ski changes from positive to negative. The next diagram shows the sign of the slope (of the one ski) before, at and after a high point.
Knowledge of the sign of the slope does not provide all information about the ski trail y = h( x). In particular, unless she measures and records the height h( a) of each hilltop, she can not say which is the highest. Every hilltop gives a local maximum for her height. It has a height greater than or equal to ( ³ ) all nearby heights. A point with a height greater than or equal to ( ³ ) all other heights, is called an absolute maximum for the portion of the trail or curve y = h( x) being examined.^{}
Definition. (Highest of the High Points). A point with height greater than all other heights in a given portion of the trail, more precisely not less than all other heights in the portion, is called an absolute maximum for the portion or interval in question. In the event of a tie, each of those points having or sharing the greatest height is called an absolute maximum. 2. Local LowPoints or Minimums
Low points of depressions are called minimums. As with high points (hill tops or maximums), several depressions or valley bottoms may be met and crossed in following a ski trail. From the slope of her ski, Barbara again knows even with her eyes closed when she crosses the valley bottom: the slope of her ski changes from negative to positive. Note again that unless she measures and records the height h( a) of the low points in each depression or hollow, she can not say which is the lowest. The bottom of each depression or hollow gives a local minimum for her height. It has a height less than or equal to all nearby heights.
Definition. A point with a height less than or equal to ( < ) all other heights in a portion of the trail is called an absolute minimum for the portion of the trail or curve y = h( x) in question. In the event of a tie, each of those points having or sharing the least height is called an absolute minimum. xDependence of Slope
As Barbara, the oneski skier, moves along a trail y = h( x), both the height of the ski midpoint y and the slope m of the ski depend on its x coordinate. The slope of her ski at x = a, x = b, x = c and x = 2 in the following diagram are all different. The slope of her ski midpoint depends on her location.
Notations for Slopes and DerivativesCalculus has several expressions for the slope m of the ski or hill function y = h( x). Some follow. \[m = g(x)=h'(x)=\frac{dy}{dx} = lim_{\Delta x \to 0} \frac{\Delta x}{\Delta y} \] Still more may be seen. Different notations exist because calculus was discovered and employed by different people in the past four or five centuries.www.whyslopes.com >> Volume 3 Why Slopes  A Calculus Intro Etc >> Chapter 2. Slopes and Ski Trails Next: [Chapter 3. Slope Sign Analysis.] Previous: [Chapter 1.Introduction.] [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. 