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 >> More Algebra >> 3 Quadratics Geometrically >> 1 quadratics graphing exercises Next: [2 quadratics graphing in general.] Previous: [ Quadratics in 10 steps.] [1] [2][3] [4] [5] [6] [7] [8] [9] [10] [11] [12] Graphing QuadraticsStep I. Graphing Quadratics from Standard formThe numerical experience obtained in doing or completing the optional but recommended exercises below will help with the practice and theory in further steps. Inclass or homework exercises like these would likely be given if you were in a course given by me.Each quadratic expression y = ax^{2}+bx+c can be rewritten in the standard form y = a[(xh)^{2}+k] The latter is good form for graphing. How to rewrite is the subject of further lessons in steps II and III. The points on the graph of y = a[(xh)^{2}+k] can be obtained from by the following operation on the graph of y=x^{2 }or its points. The following numerical exercise should help your understand the shifts and vertical scaling.
At the end of the exercise, you may see how the first two may be combined into a single shift. Numerical Exercise

x  y=x^{2} 
2.0  4.00 
1.5  2.25 
1.0  1.00 
 0.5  0.25 
0.0  0.00 
+0.5  0.25 
+1.0  1.00 
+1.5  2.25 
+2.0  4.00 
Observe the yvalues are symmetric about x = 0. The minimum value in the table is at x = 0.
This first table of values leads to plot points
(2,4), (1.5, 2.25), (1, 1), (0.5, 0.25), (0,0), (0.5,0.25), (1,1), (1.5, 2.25) (2,4)
Exercise plot the points on graph paper where the horizontal xscale includes the interval [3,3] in divisions of 0.25, and the vertical yscale includes the interval [1, 5] also in divisions or jumps of 0.25. Use solid dots for these points Next join them by a smooth curve. to approximate the graph of y=x^{2}
Second. Apply the three steps above.
Recall h = 0.25, k = 1 and a = 0.5 = ½.in
y = a[(xh)^{2}+k] = ½[(x0.25)^{2}+1]
Step 1: Plot x in the hshifted interval
[2+h, 2+h] = [1.75, 2.25]
x  x 0.25  y=(x½)^{2} 
1,75  2.0  4.00 
1.25  1.5  2.25 
0.75  1.0  1.00 
0.25   0.5  0.25 
+0.25  0.0  0.00 
+0.75  +0.5  0.25 
+1.25  +1.0  1.00 
+1.75  +1.5  2.25 
+2.25  +2.0  4.00 
Observe the yvalues are symmetric about x = 0.25 = h.. The minimum value in the table is at x = h.
This second table of values leads to plot points
(1.75,4), (1.25, 2.25), (0.75, 1), (0.25, 0.25), (0.25,0), (0.75,0.25), (1.25,1), (1.75, 2.25) (2.25,4)
Notice that these points come from the first table points shifted to the right by 0.25 units.
Exercise plot the points on the same graph papers before. Use hollow circles for these points with a dot in the center.. Next join these dotted circles them by a smooth curve to approximate the graph of y=(x½)^{2}
Step 2: Recall h = 0.25, k = 1 and a = 0.5 = ½.
The previous step was independent of the value of k and a. This step depends on k = 1. We add k = 1 to the ordinate, that is second coordinate or y coordinate, of each point.
x  x 0.25  y= (x½)^{2} +1 
1,75  2.0  5.00 
1.25  1.5  3.25 
0.75  1.0  2.00 
0.25   0.5  1.25 
+0.25  0.0  1.00 
+0.75  +0.5  1.25 
+1.25  +1.0  2.00 
+1.75  +1.5  3.25 
+2.25  +2.0  5.00 
This third table of values leads to plot points
(1.75,5), (1.25, 3.25), (0.75, 2), (0.25, 1.25), (0.25,1.0), (0.75,1.25), (1.25,2), (1.75, 3.25) (2.25,5)
Notice that (i) these points come from the second table points shifted up by 1 units, and (ii) these same points also come from the first table points shifted up by 1 unit and to the right by 0.25 units, at the same, or with one shift followed by another.
Steps 1 and 2 could be combined together as a shift or translation not necessarily parallel to the coordinate axes.
Step 3: Recall h = 0.25, k = 1 and a a = 0.5 = ½.
The previous step(s) were independent of the value a. This step depends on the
value a == 0.5 = ½. In it, we multiply each yvalue of the previous step by the
value of a to obtain the graph of
y= ½[(x½)^{2} +1] = a[(xh)^{2}+k]
x  x 0.25  y= ½[(x½)^{2} +1] 
1,75  2.0  2.50 
1.25  1.5  1.625 
0.75  1.0  1.00 
0.25   0.5  0.625 
+0.25  0.0  0.50 
+0.75  +0.5  0.625 
+1.25  +1.0  1.00 
+1.75  +1.5  1.625 
+2.25  +2.0  2.50 
Observe the yvalues are still symmetric about x = h = 0.25 with a minimum at x = h = 0.25.
This fourth table of values leads to plot points
(1.75,2.5), (1.25, 1.625), (0.75, 1), (0.25, 0.625), (0.25,0.50), (0.75,0.625), (1.25,1), (1.75, 1.625) (2.25,2.5)
The yvalue, ordinate of these point come (i) from the third table yvalues multiplied by a = 0.5; or (ii) the first table points shifted or translated by (h,k) = (0.25,1) followed by a multiplication of the yvalue points by a = 0.5 = a vertical scale factor.
www.whyslopes.com >> More Algebra >> 3 Quadratics Geometrically >> 1 quadratics graphing exercises Next: [2 quadratics graphing in general.] Previous: [ Quadratics in 10 steps.] [1] [2][3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
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Pattern Based Reason
Online 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 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:
2001  Math Forum News Letter 14,
2002  NSDL Scout Report for Mathematics, Engineering, Technology  Volume 1, Number 8
2005  The NSDL Scout Report for Mathematics Engineering and Technology  Volume 4, Number 4
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 trigformulas for dot and
crossproducts.
LinesSlopes [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.