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 >> Arithmetic and Number Theory Skills >> 7 Arithmetic and Fractions with Units
Notes
In my school days, mathematics courses were too pure to
mention or show how to work with units in arithmetic and algebraic
calculations. So calculations with units in chemistry and physic course
appeared without sanction from my mathematics courses. But mathematics
courses should support the use of units, and calculations with units
may appear in the application of trigonometry and the development of
calculus. This subsection of the fraction folder provides a
remedy.
Arithmetic and algebra with units represents alternate title for this
subject of fractions with units.

An operational command of calculations with units could be
sufficient for further use in the representation of rates and
proportionality constants and for further use in calculation in
chemistry, physics and money matters. In this arithmetic with units
of measurement, products and quotients of monomials may be formed
and simplified with monomials that contain units to unlike powers.
In contrast, sums and difference of monomials to have a physical
context may only be formed and simplified only when the monomials
are real multiples of each other. Inclusion of this topic will help
later in examples of exponent addition and subtraction with
monomials in one to several variables x, y, z, ... and in their
products or quotients.

Unit of measurement are part of applied mathematics and external to pure
mathematics. Yet calculations involving units of weight, mass, length,
time, money (hand it over) and so on appear in the measurements and
calculations of daily life alone or as part of calculations with rates
and proportionality constants.

Origin and
Addition of Some Units: Measurements system appear in daily life,
science and technology. They involve calculation with units. This
lesson introduce standalone units, and indicates how to add subtract
multiplies of them in a way that resembles the forward and backward use
of a distributive law for counting and measuring.

Units and Equal Signs:
Mathematical practice requires the equality sign to have the "forward
and backward" reflective property that a = b when and only when b = a.
Here may read the equality sign to mean the same as or is equivalent
to. Yet in daily life, the statement 3 apples + 4 oranges is, gives or
equals 7 fruits, departs from this practice. We are not going change
the habits of daily life, but should be aware of the difference here
between the technical and common use of the word equals or the symbol =
for it.

Products of Units
and Numbers: The first part of this lesson shows how to add
subtract numerical multiplies of them in a way that resembles the
forward and backward use of a distributive law for counting and
measuring. Carries and borrows in addition and subtraction provide a
unitfree form of conversion. In counting and arithmetic, conversions
are further present in expressing counts or numbers in groups of ones,
tens, hundreds and so on, and in adding, subtracting and multiplying
such counts. The second part of this lesson talks about
changing and converting the units used to measure or keep track
of quantities. Your fortune may be measured in pennies. or in dollars?
The third part show how to form products of quantities. The
operations here are similar to work with monomials where the product of
4xy with 5 xy^{3}z is 20 x^{2}y^{4}z. Instead
of using letters x, y and z, we use measurement units and counting
units as well, and could be more meaningful for students.

Fractions With Units.
This lesson introduces fractions with units and extends the concept of
equivalent fractions to fractions with units. These fractions with
units are analogous to ratios of monomials and their
simplification:
12 xy^{3}
8 x^{2}y^{2}z

=

3 y
2 xz


Simplification
of Fractions with Units. Simplification of fractions with units is
analogous to to simplification of ratios of monomials (fractions with
monomials) in one to several variables.
24 yw^{8}
9 w^{2}y^{4}z^{22}

=

8
3

w^{6}
y^{3}z^{22}


Reciprocals,
Division and Compound Fractions for Fractions with Units. This
two part lesson shows how to (i) write the Reciprocals of Fractions
with Units, and how to (ii) divide Fractions with Units, and how to
(iii) evaluate the corresponding compound fractions where numerators
and denominators are fractions with units. Equivalent fractions
reappear here as part of the simplification of results. The operations
here analogous to calculating reciprocals of ratios of monomials in
variables x, y and z, etc; to dividing such ratios and to evaluating
compound fractions with those ratios as numerators and denominators.
three part lesson shows how to (i) write the Reciprocals of Fractions
with Units, and how to (ii) divide Fractions with Units, and how to
(iii) evaluate the corresponding compound fractions where numerators
and denominators are fractions with units. Equivalent fractions
reappear here as part of the simplification of results. The operations
here anonymous to calculating reciprocals of ratios of monomials in
variables x, y and z, etc; to dividing such ratios and to evaluating
compound fractions with those ratios as numerators and
denominators.
 50167_Converting_or_Changing_Units.html">Conversion of Units in
Fractions With Units: A physical quantity may be described in different
units. Here we show how speed (it is written as a fraction with units)
written in distance per hour may be expressed in distance per minute. The
trick of multiplying by a fraction with value 1 (a fraction with units
which is equivalent to one) appears here.
To Learn More about how about how fractions with units are treated and
occur in daily life and physics, see the Units in
Calculations pages in site Volume 3: [7 Velocity]
[7
Varying Velocity Example] [7. Velocity
Calculation] [7 Changing
Units] [7 Same Velocity
Motions] [10 Slopes without
Units.] [10 Units &
Slopes] [10 Units in Cost
vs. Quantity] [10 How Units
Appear] [10 Unit
Elimination] [10 Partial
Elimination][10 Interest &
Units] [12 More on
Units]
In general quantities in daily life involve basic
units of measure, that is of time, length, mass (or weight) and of
money too  an artificial concept. The units and their numerical
multiples may be multiplied together form monomials  expressions
equal to a real number times a product of units to nonnegative
powers. Writing these monomials as numerators and denominators in
fraction like expression (they look like fractions) gives fractions
with units. This section on arithmetic and algebra with units shows
how form and simplify expression for monomials form by products of
units and fractions formed by taking monomials for numerators and
denominators.
Advanced Students: The presentation here is
informal, but it could be codified in a formal way. See the first
or second chapter of Henri Cartan's work below for a model. . That
codification coupled with invariance ideas leads in the physical
sciences to similarity analysis and similarity requirements for
solutions of equations.

End Notes for Teachers
 In applied mathematics calculations with proportionality, multiples
of simple and compound units will serve as proportionality constants and
rates with physical or social meaning. An operational command of
calculations with units could be sufficient for further use in the
representation of proportionality constants and for further use in
calculation in chemistry, physics and money matters where units
appear.
 Operations with monomials involving units and their quotients are
similar to operations on monomials in variables x, y, z etc and their
quotients. The latter too may represent formal operations on expressions
that have no meaning for students other being marks on paper. In
contrast, the previous note implies a role for polynomial like operations
on units in calculations.
 In the modern axiomization of mathematics as seen in school and
colleges, the codification is limited to pure numbers. Units are not
discussed. But as a service to the physical and social sciences, money
matters included, the algebraic role of units in their computations can
be codified or modeled. That can be done informally. Henri Cartan's work
Elementary theory of analytic functions of one or several complex
variables, translation of THEORIE ELEMENTAIRE DES FONCTIONS ANALYTIC
D;UNE OU PLUSIEURS VARIABLES COMPLEX, could be extended to provide a
formal codification of operation with units.
www.whyslopes.com >> Arithmetic and Number Theory Skills >> 7 Arithmetic and Fractions with Units
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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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