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 6. Slopes and Vertical Shifts Next: [Chapter 7 Slopes and Velocity.] Previous: [Chapter 5. Slope Sign Tests.] [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 6. Slopes and Vertical ShiftsVolume 3, Why Slopes and More Math. The vertical shift or motion of a curve y = f(x) adds a constant d (the displacement) to each point. It yields a new curve y = f(x)+d. The next diagram indicates an example. A simple vertical shift or motion may change the height of a skier, but not the slope of his or her skis.
Constant Difference Theorem
What can be said for sure about two functions when they have the same
slope (or derivative) everywhere? One response is given by the following
assertion.
Note the assertion says that there is a constant d. It does not say how to find it. Here is an analogy: Saying there is a needle in a haystack, does not say how to find it. Note also the word every. If there is a point x_{1} in the interval (a,b) where the slope is not defined then no conclusions can be drawn from the Constant Difference theorem. Remark. If f_{1}(x) and f_{2}(x) satisfy the conditions (hypotheses) in the Constant Difference theorem with f_{2}(c) = A and f_{1}(c) = B at some point c in the interval (a,b) then at x = c,
Remark. Mathematical assertions and theorems which say that a number (or limit) is not defined or does not exist actually mean that a finite number (or finite limit) does not exist. The vertical motion theorem given above applies only when the common slope is finite in the interval (a,b) of interest.
Different vertical displacements over different portions of the trail are possible. The next diagram gives examples of this. Upshifts have been made in the topmost curve of the previous diagram. Between a and b, between b and c and between c and d, the two trails y = f_{1}(x) and y = f_{2}(x) shown have the same slope but not the same heights. At the ski jump and cliffs in the upper trail y = f_{1}(x), the slope is not defined. Where Slopes are Not DefinedOn a ski trail y = h(x), there may be a few places where the slope is undefined or a single slope to the graph or trail does not exist. In the following diagram, the slope is undefined at the ski jump above the point x = a. At sharp peaks and kinks, a short ski may pivot or rotate while keeping in contacting with one point, the kink or peak on the trail. So a single slope to the trail cannot be defined there. Where the trail has some vertical jumps, the graph ceases to be the graph of a function. The slope is said to be undefined or not to exist here, even though we might say it is infinite, +¥ or .
Consider the next diagram. At the points a,b,c,d and e, the slope function or derivative m = h'( x ) cannot be given a single value. In general, a single value cannot be assigned to slopes at sharp changes in the direction of a curve y = h(x).
But a single slope may sometimes be defined before and after such kinks or sharp turns in the graph of a function.
The ski at the sharp peak is shown pivoting, that is rotating, as the ski passes over.^{1} In pivoting at the peak, a ski can have many orientations or slopes without intersecting the curve y = h(x) on either side. Problem for Advanced Students. A graph with vertical segments is not the graph of a function, but it may be the image of a parameterized curve $(x(t),y(t))$ where $t$ belongs to some interval. Show that if $(x_j(t),y_j(t))$ for j = 1 and 2 are two continuous curves parameterized by $t \in [a,b]$, then \( (x_1'(t),y_1'(t))=(x_2'(t),y_2'(t)) $ for all $t\in (a,b)$ implies there are constants $d_1$ and $d_2$ such that $(x_j(t),y_j(t))=(x_j(t),y_j(t))+(d_1,d_2)$. This problem is both a consequence and an extension, a generalization, of the constant difference theorem just stated. It applies to some graphs with vertical segments. www.whyslopes.com >> Volume 3 Why Slopes  A Calculus Intro Etc >> Chapter 6. Slopes and Vertical Shifts Next: [Chapter 7 Slopes and Velocity.] Previous: [Chapter 5. Slope Sign Tests.] [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. 