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. 
1 versus 2way implication rules  A different starting point  Writing or introducting
the 1way implication rule IF B THEN A as A IF B may emphasize
the difference between it or the latter, and the 2way implication A IF and ONLY IF B.

Deductive Chains of Reason  See which implications can and cannot be used together
to arrive at more implications or conclusions,

Mathematical Induction  a light romantic view that becomes serious. 
Responsibility Arguments  his, hers or no one's 
Islands and Divisions of Knowledge  a model for many arts and
disciplines including mathematics course design: Different entry
points may make learning and teaching easier. Are you ready for them?
Deciml Place Value  funny ways to read multidigit decimals forwards and
backwards in groups of 3 or 6. 
Decimals for Tutors  lean how to explain or justify operations.
Long division of polynomials is easier for student who master long
division with decimals. 
Primes Factors  Efficient fraction skills and later studies of
polynomials depend on this. 
Fractions + Ratios  See how raising terms to obtain equivalent fractions leads to methods for
addition, comparison, subtraction, multiplication and division of
fractions. 
Arithmetic with units  Skills of value in daily life and in the
further study of rates, proportionality constants and computations in
science & technology.
What is
a Variable?  this entertaining oral & geometric view
may be before and besides more formal definitions  is the view mathematically
correct? 
Formula Evaluation  Seeing and showing how to do and
record steps or intermediate results of multistep methods allows the
steps or results to be seen and checked as done or later; and will
improve both marks and skill. The format here
allows the domino effects of care and the domino effects of mistakes
to be seen. It also emphasizes a proper use of the equal sign. 
Solve
Linear Eqns with & then without fractional operations on line segments  meet an visual introduction and learn how to
present do and record steps in a way that demonstrate skill; learn
how to check answers, set the stage for solving word problems by
by learning how to solve systems of equations in essentially one
unknown, set the stage for solving triangular and general systems of
equations algebraically. 
Function notation for Computation Rules  another way of looking
at formulas. Does a computation rule, and any rule equivalent to it, define a function? 
Axioms [some] as equivalent Computation Rule view  another way for understanding
and explaining axioms. 
Using
Formulas Backwards  Most rules, formulas and relations may be used forwards and backwards.
Talking about it should lead everyone
to expect a backward use alone or plural, after mastery of forward use. Proportionality
relations may be use backward first to find a proportionality constant before being
used forwards and backwards to solve a problem.
Early High School Geometry
Maps + 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
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whyslopes.com >> Arithmetic and Number Theory Skills >> The 20 Times Table Next: [Expression Evaluation how to show work.] Previous: [The 12 Times Table Visually.] [1] [2] [3] [4] [5][6 swf] [7 swf] [8] [9] [10]
20 by 20 Multiplication Table
During the school year, practice filling in the 10 or 12 times
table at least three times correctly without a calculator. Observe how the
numbers increase by a constant amount in each row and in each column.
* 
1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
1 
1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
2 
2 
4 
6 
8 
10 
12 
14 
16 
18 
20 
22 
24 
26 
28 
30 
32 
34 
36 
38 
40 
3 
3 
6 
9 
12 
15 
18 
21 
24 
27 
30 
33 
36 
39 
42 
45 
48 
51 
54 
57 
60 
4 
4 
8 
12 
16 
20 
24 
28 
32 
36 
40 
44 
48 
52 
56 
60 
64 
68 
72 
76 
80 
5 
5 
10 
15 
20 
25 
30 
35 
40 
45 
50 
55 
60 
65 
70 
75 
80 
85 
90 
95 
100 
6 
6 
12 
18 
24 
30 
36 
42 
48 
54 
60 
66 
72 
78 
84 
90 
96 
102 
108 
114 
120 
7 
7 
14 
21 
28 
35 
42 
49 
56 
63 
70 
77 
84 
91 
98 
105 
112 
119 
126 
133 
140 
8 
8 
16 
24 
32 
40 
48 
56 
64 
72 
80 
88 
96 
104 
112 
120 
128 
136 
144 
152 
160 
9 
9 
18 
27 
36 
45 
54 
63 
72 
81 
90 
99 
108 
117 
126 
135 
144 
153 
162 
171 
180 
10 
10 
20 
30 
40 
50 
60 
70 
80 
90 
100 
110 
120 
130 
140 
150 
160 
170 
180 
190 
200 
11 
11 
22 
33 
44 
55 
66 
77 
88 
99 
110 
121 
132 
143 
154 
165 
176 
187 
198 
209 
220 
12 
12 
24 
36 
48 
60 
72 
84 
96 
108 
120 
132 
144 
156 
168 
180 
192 
204 
216 
228 
240 
13 
13 
26 
39 
52 
65 
78 
91 
104 
117 
130 
143 
156 
169 
182 
195 
208 
221 
234 
247 
260 
14 
14 
28 
42 
56 
70 
84 
98 
112 
126 
140 
154 
168 
182 
196 
210 
224 
238 
252 
266 
280 
15 
15 
30 
45 
60 
75 
90 
105 
120 
135 
150 
165 
180 
195 
210 
225 
240 
255 
270 
285 
300 
16 
16 
32 
48 
64 
80 
96 
112 
128 
144 
160 
176 
192 
208 
224 
240 
256 
272 
288 
304 
320 
17 
17 
34 
51 
68 
85 
102 
119 
136 
153 
170 
187 
204 
221 
238 
255 
272 
289 
306 
323 
340 
18 
18 
36 
54 
72 
90 
108 
126 
144 
162 
180 
198 
216 
234 
252 
270 
288 
306 
324 
342 
360 
19 
19 
38 
57 
76 
95 
114 
133 
152 
171 
190 
209 
228 
247 
266 
285 
304 
323 
342 
361 
380 
20 
20 
40 
60 
80 
100 
120 
140 
160 
180 
200 
220 
240 
260 
280 
300 
320 
340 
360 
380 
400 
Two message to help you build skills and confidence
Do you know about the domino effect of mistakes and errors in
arithmetic? If not, ask parents, teachers or fellow students to explain
what that might be. If yes, if you know about it, then you know about the
need to be careful in each arithmetic step you take. The domino effect
of errors appears both in and outside of arithmetic. In it, an error
in one step likely makes all following steps and any result wrong. Taking
care to avoid the domino effects is a must for building skills and
confidence in all skill based subjects at home, at work and in school.
Good luck.
Understanding exactly or precisely what is said or written will avoid
confusion when you are following instructions at home, at school and at
work. Learning to write and speak exactly or precisely will avoid confusion
of others when you explain or give instructions as well. When
you are old enough (or now), read the site math free chapter on two
logic puzzles. The chapter may help you read, write
and speak with precision. That in turn may help you avoid the domino
effects in many subjects.
Good Luck.
Some good practices for skill development in arithmetic appear in site
arithmetic and number theory steps. Where these good practices are not
best, say so.

Definition of Primes, Simplified : A simpler definition of
prime numbers which takes advantage of the 12 times tables to
identify small primes upto 13. In particular, a whole number is said
to be prime if it is not the product of two smaller whole numbers.
With the word smaller in the definition, the whole numbers 11 and 13
are prime because it is not given by a product inside the 10 and 11
times table. [This was the small example given above]

Quick Prime Factorization of Small Whole Numbers: Emphasis of
a square or square root rule to provide QUICK prime factorization
skills for whole numbers less than 169 = 13^{2}. In
particular, a whole number less 169 is prime if and only is it is not
a multiple of the primes 2, 3, 5, 7 and 11 less than 13. Simple
divisibility rules and calculators (an overkill) here may be used to
recognize multiples of 2, 3, 5 and 11. Quick prime factorization of
whole numbers is a key to exact and efficient fraction practices
employed in mathematics from algebra to calculus. There is no escape.

Fraction Operations Explained: A thoughtbased development of
addition, comparison, subtraction, multiplication and division
operations starting with simpler cases where operations are easily
explained, and continuing on to general cases where all operations
are justified by raising terms. In higher mathematics, if not
elementary mathematics, comprehension of why methods work is highly
valued, it is part of the spirt of mathematics mastery. Understanding
how and why operations are justified should move you away from
learning by rote. Reference:
fraction operations by raising terms

Arithmetic and Fractions With Units: Figuring with denominate
numbers, that is multiples of units of measure for physical
quantities and units of value for monetary quantities. This practical
value for calculations involving speed, rates in general and
associated proportionality constants in daily life and also in
practical and applied arts and sciences. (In algebra taught by rote,
you may see similar figuring with multiples and powers of variables
in products and quotients. The path here has more meaning and is very
practical)

Oral Dimension of Arithmetic: Verbal description and extension
of common practices for finding counts, totals and products by
forming and adding or multiplying subcounts, subtotals and
subproducts. Here calculation practices are introduced and described
orally instead of symbollically, the latter being harder for many to
grasp. For many, how to calculate averages and how to calculate
perimeters of polygon are best described with words, the use of
letter or symbols being to complicated to understand in the first
instance. Mastery of common practices for counting, totaling and
multiplying do not have to wait for their algebraic description.
Instead, the verbal forms can be given. [These arithmetic notes
expand on part of the big example given above.]

Place Value Revisited: An exposition of place value in
decimals with places before an after the decimal point in groups of
three may amuse and inform. In it, students in North America may
learn how to read aloud and write on paper the decimal
6,571,045,375,905,333,034,412.450,033,870
as 6 sextrillions, 571 quintrillons, 45 quadrillionths, 375
trillionths, 905 billionths, 333 millions, 34 thousands, 421 ones,
450 thousandths, 33 millionths and 870 billionths. In contrast,
students elsewhere may use the following "SI" (system international)
method how to read aloud and write on paper the decimal form of 6
zettaunits, 571 exaunits, 45 petaunits, 375 teraunits, 905
gigaunits, 333 megaunit, 34 kilounits 421 ones, 450 milliunits, 33
microunits and 870 nanounits.

Addition, comparison, subtraction, multiplication and division of
decimals: The site development may covered more lightly than
presented. The development of place value methods for all but long
division is thoughtbased. Why methods work are both indicated. Long
division method is given without justification, but with a method to
check results. In all methods, students will meet the domino effects
of care and mistakes. Avoiding the latter provides an end, value and
tool for skill mastery, an echo of the old fashion idea that figuring
well is a sign of practical intelligence.

Signed Numbers: The site description of arithmetic with
integers and arithmetic with signed numbers is not bad. The site
objective so far has not been to cover everything in mathematics, but
to develop and express ideas on how mathematics should be learnt or
taught. That being done, a clearer account of arithmetic with signed
numbers is due.

More Steps To Elaborate  not in site material: Talk about
scientific notation, arithmetic with, and arithmetic with mixed
decimals  that is, decimals with multiple places before and after
the decimal point. Relate foregoing to fraction skills and practices.
Explain the comparison, addition and subtraction of scientific in
terms of of finding a common factor or denominator.
whyslopes.com >> Arithmetic and Number Theory Skills >> The 20 Times Table Next: [Expression Evaluation how to show work.] Previous: [The 12 Times Table Visually.] [1] [2] [3] [4] [5][6 swf] [7 swf] [8] [9] [10]
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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 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:
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. ...

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.
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.
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