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 7 Slopes and Velocity Next: [Chapter 8. Slope Interpretation.] Previous: [Chapter 6. Slopes and Vertical Shifts.] [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 7. Slopes and VelocityDistance Versus TimeSigned Distance along a RoadSuppose in traveling along a road, position at time t is given say by d = f(t). The coordinate d gives a signed distance to the origin or point of reference. Assume positions on one side of this origin have a positive dcoordinate and the positions on the other side of this origin have a negative d coordinate. The absolute value or magnitude of d, that is d, gives the unsigned distance to the origin or point of reference. The coordinate d will just be called the distance or signed distance hereafter.
Constant Speed and Velocity ExampleHenry Snail walks along a path away from its origin for four hours at a rate of 6 kilometers per hour. He started at 1 p.m., eight kilometers from the origin. Problem: Graph his distance d to the origin versus the time t since he began. Solution. His speed s = 6 kilometers per hour = 6 ·[km/hour]· Thus he travels
Harry travels a distance Dd = s Dt in time Dt. This information allows us to form the following table,
Here the slope m has the role of speed or velocity. That
is,
everywhere on the graph. Observe that
First Varying Velocity ExampleProblem: Graph the distance d to the origin of a path versus time t for the following journey of Harry Snail.
Solution. The trip has five segments. Comments on each segment or portion follow. 1. Before his trip begins, he could be stationary, that is, not moving. This possibility, a suspicion which cannot be confirmed, is represented by the horizontal dashed line. The slope of this speculative dashed line is
2. The first described portion of the trip starts at the point A = (2 hrs,50km). He reaches the point B = ([3½] hr, 200km) after traveling at 100 kilometer per hour for one and a half hours. The slope of this portion of the trip m = 100[ km/hr] = 100 km per hour. 3. The second described portion of the trip lasts for one half hour. By remaining stopped (stationary) for [1/2] hour, his (t,d) coordinate changes from B = (3[1/2]hr,200km) to C = (4hr,200km). The slope or speed m here is again zero. 4. By traveling at 75 kilometers per hour back towards the origin for two hours, his position coordinates (t,d) change from C = (4hr,200km) to D = (6hr,50km). The slope
5. Finally, he does not move for 2 hours. This gives the last portion of the graph with d = 50km and slope m = 0. Changing Units (Digression)The measure or unit of slope, speed and velocity in the last example is given by the ratio [ km/min] or km per minute, or km per one sixtieth of an hour. But other units (or ratio of units) are possible. The question becomes: what units do you like for the expression of these quantities. It is possible to change units of time and distance.
Multiplying by the number 1 does not affect the value of an expression,
but the number 1 can be written in many ways. Some, not all, are helpful.
These different ways can sometimes help in changing the units used for
the expression of a quantity. A few slope or velocity based examples
follow. Note that [60min/1hr] = 1 implies the following
Remark. An alternate method is to substitute 1 min = [1/60] hr into an expression. For example
Motion with the Same VelocityTwo small problems follow, to further examine the situation where two graphs have the same slope everywhere. The slope in distance versus time graph is given by speed or velocity. Information for the first problem. Paul starts at 3 kilometers north of his home. From there, for two hours, he walks northward at 5 kilometers per hour. Then he stops for a one hour rest. Next for one hour, he travels southward at 6 kilometers per hour. First Problem: How far is Paul from his starting point? Information for the second problem. John starts at a distance d_{0} north of Paul's home. He matches Paul's speed and movements. From his starting point, like Paul, he walks for two hours northward at 5 kilometers per hour. Then he stops for a one hour rest. Next for one hour, he travels southward at 6 kilometers per hour. Second Problem: How far is John from his starting point? Solution to both problems. A graph of Paul's motion is easily drawn with the help of the following table:
In this graph
Paul thus travels 7 km3 km = 4 km from his starting point. A graph of both Paul's and John's respective motions can be obtained from the next table.
The distance of John from his starting point is also 4 km. The scale has been omitted from the vertical axis in the following graph since d_{0} is unknown. This graph shows the case d_{0} > 3 km. How would the graph change if d_{0} was £ 3 km? Questions
www.whyslopes.com >> Volume 3 Why Slopes  A Calculus Intro Etc >> Chapter 7 Slopes and Velocity Next: [Chapter 8. Slope Interpretation.] Previous: [Chapter 6. Slopes and Vertical Shifts.] [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. 