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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www.whyslopes.com >> Volume 2 Three Skills For Algebra >> Chapter 7 Prep for Calculus Arithmetic Exercises Next: [Solutions For Arithmetic Exercises.] Previous: [Chapter 6 Change of Language.] [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] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42]
Chapter 7, Arithmetic Review Problems with hints of Algebra
Volume 2, Three Skills for Algebra
Here are some arithmetic review problems with hints of algebra. I would
give these problems at the start of a calculus or precalculus course to
check skills and correct common mistakes. Further, some arithmetic
patterns below provide experience with geometric sums and quick,
algebraic summation, formulas.
There is more to mathematics than just doing arithmetic carefully, but
in arithmetic you must master addition of fractions with least common
denominators and the cancellation of common divisors in fractions by
themselves or in products to do well in algebra and in calculus.
Calculus requires algebra and exact arithmetic with great strength and
precision. Answers or solutions provide the correct answers and
format for those answers.
2. Arithmetic and Algebra Review Problems
This large set of arithmetic and algebra review problems may help
you to check or diagnose your arithmetic skills and some algebra skills
as well  if you have studied algebra before. Answers will be found at
the end of this book. Doing these problems and seeing their answers
checks for gaps in your understanding, and may even fill some. Watch for
arithmetic or algebraic patterns in the last of these "review" problems.
Some calculations are slightly repetitive to help you spot such patterns.
2.1 Basic Stuff
Perform the indicated calculations by hand. Then check your calculations
with the aid of a calculator.
 Find the sum of the three numbers 456 and 76 and 312.
 Find the product of 176 and 86.
 Subtract 2396 from 4892 and check your answer.
 Compute 1416 divided by 813 to 3 decimal places.
 Compute 2396 4892.
2.2 More Basic Stuff
Compute if possible the value of the following
Remember in calculations that operations inside parentheses $( )$ or
brackets [ ] are to be done first. Use your calculator as little as
possible.

$A = (4 \div 5)\div 3$

$B = 4 \div \frac53$

$C = 4 \times (5 \times 3)$

$ D = (4 \times 5) \times 3$

$E = (4  5)  3$

$F = 4  (5  3)$

$G = 4  5 3$

$H = \sqrt{3^2}$

$I = \sqrt{(3)^2}$

$J = \sqrt{ 4^2}$

$K = \sqrt{ (4^2+3^2)}$

$L = \sqrt{ (4^2 + (3)^2)}$

$ M = (\frac54) \div [ (\frac87)\div (\frac95) ]$

$N = [(\frac54) \div (\frac87)] \div (\frac95)$

$O = \frac{5}{4} \times [ \frac78 \times \frac95 ]$

$P = [\frac{5}{4} \times \frac78] \times \frac95$

$Q = \frac{5}{4} \div \frac78 \div \frac95 $

$R = \sqrt{16} + \sqrt{9}  \sqrt{25}$

$S = (3.1416)^0$

$T = 3.1416  \frac{22}7$

$U = \pi  3.1416$

$V = \sqrt{4^25^2}$
2.3 Calculator Button Exercises
Put your calculator in degree mode. Now find or compute the following
quantities.

$A=\sin (90 ^\circ)$

$B=\sin (180 ^\circ)$

$C=\sin (0 ^\circ)$

$D=\sin (270 ^\circ)$

$E=\sin (90 ^\circ)$

$F=\sin (720 ^\circ)$

$G=\cos (90 ^\circ)$

$H=\cos (180 ^\circ)$

$I=\cos (360 ^\circ)$

$J=\cos (0 ^\circ)$

$K=\cos (90 ^\circ)$

$L=\cos (720 ^\circ)$
Put your calculator in radian mode. Now find or compute:

$a=\sin (\frac12\pi \mbox{ radians})$

$b=\sin ( \pi \mbox{ radians}) $

$c=\sin (0 \mbox{ radians})$

$d=\sin (\frac32\pi \mbox{ radians})$

$e=\sin (\frac12\pi \mbox{ radians})$

$f=\sin (4\pi \mbox{ radians})$

$g=\cos (\frac12\pi \mbox{ radians})$

$f=\cos (\pi \mbox{ radians})$

$h=\cos (2\pi \mbox{ radians})$

$i=\cos (1.5 \pi \mbox{ radians})$

$j=\cos (\frac12\pi \mbox{ radians})$

$k=\cos (4\pi \mbox{ radians})$
Observe that the numerical values computed by the sine,
cosine, tangent and all other trigrelated function
buttons, all depends on the units used for angle measurement.
2.4 More Calculator Button Work
Compute or find the following quantities:

$ A= \exp( 2 \ln(5))$

$B=e^{2 \ln(5)}$

$ C= 10^{ 2 \log(5) }$

$ D= 10^{\log(25)}$

$ E= \ln(\exp(6.2))$

$F= \ln( e^{6.2})$

$G,$ the sixth root of $(16)^{12}$

$ H= \left[(16)^{12}\right]^{\frac16}$

$ I= 1+3+3^2+3^3+3^4+3^5+3^7$

$ J= \frac{1+3^7}{[1+3]}$

$ K= \frac{[13^7]}{[13]}$

$ M= 1+(1.06)+(1.06)^2+(1.06)^3$

$ N= \frac{[1+(1.06)^4]}{[1+1.06]}$

$ P= \frac{[(1.06)^41]}{[1.061]}$

$ Q= \frac{[1+(1.06)^4]}{[0.06]}$

$R= [1+(1.02)^{1}+(1.02)^{2}+(1.02)^{3}+ (1.02)^{4}]\times
(1.02)^{(4)}$

$ S= \frac{[(1.02)^{(5)}1]}{[1.021]} \times (1.02)^{(4)}$

$ T= 1+(1.02)^{(1)}+(1.02)^{(2)} +(1.02)^{(3)}+(1.02)^{(4)}$

$ U= (1.02)^{(4)}+(1.02)^{(3)}+(1.02)^{(2)}+(1.02)^{(1)}+1$

$V= \frac{ (\frac1{1.02})^{(5)}1}{(\frac1{1.02})^{(1)}1}$
2.5 More Arithmetic Examples
Answer the following without the use of a calculator  One of the
following is not defined. See Answers.
 Simplify, if defined, \[A= [(\frac4{5}) \div (\frac{24}{35})]
\div (\frac2{7})\]
 Simplify, if defined, \[B=(\frac4{5}) \div [(\frac{24}{35})
\div (\frac2{7})]\]
 Simplify, if defined, \[ C=(\frac4{5}) \div (\frac{24}{35})
\div (\frac2{7}) \]
2.6 A Summation Shortcut
Before the shortcut is given, we will tackle two suggestive tasks. The
sum of the cubes of the integers 1 to 4 is S =
1+2^{3}+3^{3}+4^{3}. Your first task is to
compute S. Your second task is to compute
\[a=[(\frac12)4(4+1)]^2\]
Now compare the values of big S and little a.
Now here is the shortcut: if n is a positive integers then the sum
of the cubes of the integers 1 to n is \[ S(n)=[\frac12n(n+1)]^2
\]
Why it holds is an intellectual debt: It can be justified or proven with
the help of mathematical induction and the ideas in the chapter Some
Finite Mathematics. With this formula, and also without if you like,
find
 the sum of the cubes of the integers 1 to 5
 the sum of the cubes of the integers 1 to 15
 the sum of the cubes of the integers 1 to 30
In the last problem, what requires the least amount of arithmetic, use
of the formula $S(n)=[\frac12n(n+1)]^2$
or directly adding the 30 cubes $ 1^3, 2^3, 3^3, \ldots
(29)^3, (30)^3?$
2.7 Algebraic Exercises
Some may be harder than the previous ones. If the algebraic exercises are
not understandable now, try them later.

Simplify $\frac{1+x+x^2+x^3}{x1}$ if possible.

Factor $x^2+5x+6,$ Hint: try to use the algebraic pattern $(x+a)(x+b)
= x^2+(a+b)x+ab$ with $ab=6,$

Solve $0=(x1)(2x+4)(3x),$ Hint: There are three numbers in the
answer.

Factor $x^3x,$

Simplify $4(x+1)(x+3x^2)[(x+1)x+(x+1)3x^2)],$

Simplify $13^25^2(13+5)(135),$

Simplify $7 \sqrt{(3^2+4^2)},$

Simplify $[\left(\frac37\right)^{13} \times \left( \frac{(4x^2)}{(3^2
\times 7^3)}\right)^5],$

Simplify $[(9x^2+3)(4+4x+4x^2)][(x1)(2x+2)2x^2+2)],$

Compute $f(4)$ if $f(x)=\sqrt{25x^2}.$

Find: $ (x,y)$ if $x+y =\pi$ and $yx=1.$

Express $$[(2 \times 3^2\times y^3z^{(3)}t^3)^{(2)}] \times
[3^3x^4y^{(5)}]^2$$ with positive powers only.

Find: $ x$ if $(x10)(x3) = 0$ and $ x > 4,$ Also, how many
acceptable solutions would there be if the requirement $x>4$ was
replaced by $x >12,$ or $x > 2$ or $x > 99$? Consider the
three alternate cases separately.

Find $ x$ if $4 =\frac1{(x+1)}$

Find $ z$ if $z=2x+3,$ $t=3^2,$ $wt=10$ and $x= 4t+1$ and $y=y^2,$
Observe some information is not needed.
www.whyslopes.com >> Volume 2 Three Skills For Algebra >> Chapter 7 Prep for Calculus Arithmetic Exercises Next: [Solutions For Arithmetic Exercises.] Previous: [Chapter 6 Change of Language.] [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] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42]
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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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