Chapter 10. Transition
Intermediate level courses in say secondary school should build a mastery
of logic, of basic trigonometry, and of the symbolic and algebraic way of
writing and thinking. The exposition of trigonometry is simplified by a
previous or concurrent description of complex numbers. Here set theoretic
concepts are welcome but the mastery of the symbolic-algebraic writing
and thought is more important. It is needed even for the exposition of
set theoretic concepts. Intermediate level instruction builds on the
previous mastery of counting, arithmetic and simple formulas (proven or
not). The first objective need not be the deductive or Euclidean
representation and derivation of mathematics but simply a familiarity
with logic and the algebraic way of writing and thinking. The latter may
stem from many illustrative strands of deduction and algebraic thought.
The aim here is to extend the common knowledge of mathematics. Ease of
exposition will be the guide. Topics not mentioned below could also be
added to the core proposed below. The companion works Three
Skills for Algebra and Why Slopes and More Math includes much,
if not all of the following material. The following words may be regarded
as their defense.
Explaining Logic
For logic and pattern based thought, there are algebra-free lessons on
implication rules, deception, chains of reason, longer chains of reason β
mathematical induction, and islands and division of knowledge. One shows
students how to apply rules and patterns one at a time or one after
another. Another shows students the need to read statements and
definition in a precise fashion β every word counts. And one, the essay
Islands and Divisions of Knowledge offers a model for rule and
pattern based thought that can be easily understood before the study of
Euclidean Geometry, the original model. These lessons in full should be
understandable to the typical fifteen year old but only to a precocious
ten year old.
The discussion of logic, the use of the terms and, or, not etc.,
can be further illustrated with Venn Diagrams and with the symbolic or
notational description of sets. The discussion of truth tables for
implications and the logical interpretations of the operation NOT and the
connectives AND and OR provide a further symbolic or algebraic
perspective of logic.
The symbolic perspectives should be presented in full after and not
before the algebraic way of writing and thinking is introduced. See the
next section. To keep these comments on how to explain logic in one
place, truth tables or a variation of them are discussed in the next
paragraphs.
Truth or Obeyance Tables. Entries in the truth table for a
material implication if A then B has left many instructors, yours
truly included, at a lost for words. With this, students have been told
to accept the entries as is, without question. But the three notions of
an implication rule being obeyed, disobeyed or not disobeyed provide
justification for the entries. In particular, an implication rule A
implies B or if A then B is said to be false in situations
where it is disobeyed and it is said to hold (or be true) in those
situations where it is obeyed or at least not disobeyed. Finally, the
implication rule is said to be always true in the circumstances of
interest, provided it is never disobeyed in those circumstances[1].
There is a distinction to be made between describing instances where a
rule or conditional statement IF A THEN B is obeyed, not disobeyed
or disobeyed and identifying the respective circumstances in which the
rule or statement is never disobeyed. See the logic chapters common to
the companion books Three Skills for Algebra and Pattern Based
Reason. They describes how the three notions of a rule being obeyed,
not disobeyed or disobeyed can be used to describe and explain or justify
the entries in truth tables for material implications.
[1] An implication rule may be stated with
the understanding that it only applies in a given set of circumstances.
Those circumstances need to be identified in implication rules which might
otherwise be quoted or applied out of context.
Explaining Algebra
Following a mastery of counting, arithmetic and the use of simple
formulas, the algebraic way of writing and reasoning can be introduced by
talking about and then illustrating the following three skills for
algebra:
1. We can describe numbers and quantities.
2. We can describe calculations that are done or might be done. This
description can be done with words or with a symbolic (algebraic)
shorthand notation, that is formulas.
3. We can change how a number or quantity is computed. Here
symbolically described rules of arithmetic (properties of real numbers)
say how.
Talking about these three skills compensates say for the nonverbal nature
of the symbolic or algebraic description of formulas β better seen and
read silently than spoken aloud. The initial discussion and illustration
of the three skills should show and emphasize that there are two notions
of variables, one with and one without symbols, that there is more to
mathematics than just doing arithmetic; and that the symbolic or
algebraic description of calculations is more compact than a word
description. The algebraic description however requires an understanding
or definition of the symbols involved β their roles. And as motivation
for algebra, instructors may observe to students that the main service of
this algebraic notation is to describe calculations or help change the
way that they are done in all numerical disciplines. Discussing and
illustrating the three skills also provides a mechanism for alleviating
math phobia[2].
How to Present the Skills
First Skill: Talking about Numbers and Quantities. We can talk
about numbers and quantities and, in doing so employ the everyday
meanings of the adjectives constant, variable, known, unknown, given,
forgotten, .. in the description of numbers and quantities. Some words or
examples may be needed to show the difference between numbers and
quantities. The latter involves units of measurement. Use of units and
calculations involving quantities needs to be sanctioned, since we are
not necessarily concerned with treating math as solely dealing with
numbers. Quantities appear in every day life and many computational
disciplines, economic, technological or scientific.
Second Skill: Describing Calculations. We can describe
calculations that may be done or postponed or never done. This
description can be done with words alone or with formulas, that is
algebraic shorthand notation. A few words and examples are appropriate to
show the equivalence in some simple cases and to show the advantages of
the shorthand description in other cases. The aim here is to emphasis
that advantage, while mentioning the cost: algebraic formulas are better
seen and read silently. The latter has been a nonverbal obstacle to the
discussion of algebra and algebraically (or symbolically) derived and
described results in mathematics.
When students are not yet familiar with complicated formulas, that is
calculations hard to describe with words alone, the lessor aim is just to
give the following understanding: formulas for numbers or quantities are
simply symbolic or algebraic ways of describing calculations that might
be done. Then when complicated formulas appear, the tongue in cheek
exercise of describing them with words only can be given. Language
teachers in the later years of secondary school can give students the
essay option of clearly describing the evaluation of the calculations
encoded and represented by the quadratic formula or the compound interest
formula.
Third Skill: Changing Calculations. We can change the way a number
or quantity is computed. Here students may be familiar with cases where
two different calculations give the same result. In such situations, one
computation, preferably the simplest, can replace the other to lessen the
amount of computation. The arithmetic review problems [3] contain a few examples or hints of such computational
shortcuts. A further motivation is the message: formulas for numbers and
quantities may be obtained from other formulas by replacing one
computation by another that gives the same result, or by interchanging a
calculation with a symbol that represents its result. The interchange
here is possible in two ways. Students can be given examples to support
this message.
Describing and Changing Calculations
Three topics may further introduce and illustrate the algebraic way of
writing and thinking. Given the difficulty that many have had in
mastering IT, saying too much or redundancy in the explanation is better
than saying too little. The algebraic way of writing and thinking is not
obvious. It involves some rationalizations which the naturally adept may
acquire via osmosis, but which others, artificially adept, can understand
as well. The topics are as follows.
1. Area and Volume Calculations β the substitution concept. Here
two formulas for the volume of a box are shown to be interchangeable.
Here the area formula A = WL is solved for L and
W. See below.
2. Comparison of arithmetic and algebraic methods for using the
compound interest formula A=P(1+i)n. The methods
are employed to obtain the values of the quantities A, P and i, or formulas
for them. The resulting examples also provide motivation for roots and
powers. [4] Here arithmetic and algebraic solutions
may be given and compared at length until students impatiently call for
only the algebraic solution.
3. The solution of linear equations with one, two and more unknowns
using numerical or symbolic coefficients. Here there is a message
that sometimes working with the numerical coefficients is better β less
involved. Here numbers go from being unknown to known. The expression
find the unknowns could be better rephrased make the unknowns, known.
These three topics provide opportunities to show and illustrate various
substitution and replacement ideas or laws. Their presentation and
discussion at length following the three skills should I conjecture give
a mastery of the algebraic way of writing and reasoning sufficient to
understand the symbolic and algebraic statement of the properties of real
numbers.
Area and Volume Calculations
These first illustrations from the book Three Skills for Algebra
are based on the calculation of the area A of a rectangle of width W and
length L. These examples introduce the replacement and substitution
operations in settings that should be accessible and familiar to students
who have mastered in primary school, the use of simple formulas.
In shorthand notation, A = W L. This gives a means for computing A from
the other two quantities. Next I note that W = [(WL)/(L)] due to the
multiplication reversal by division rule. And then the equality WL = A
suggests W = [(A)/(L)]. This provides a first example of algebraic
reasoning. It based on the multiplication reversal by division rule and
the replacement of WL by its equal A. The letter A can be thought of as
shorthand for the result of W times L. Solving for L provides an another
example and hints at the interchangeability of the roles of length L and
width W. For mathematical novices, that is, students, I would not invoke
this interchangeability or symmetry. It would be an afterthought - a
comparison of two similar computations and a suggestion that we could
have used a symmetry argument.
The second illustration is the calculation of the volume V of a box of
height H whose base has area A, width W and length L. The volume of the
box is V = H ×W×L, where the order of the product and the grouping in it
does not matter. Students can be asked to verify the latter for an
example or two. The grouping V = H (WL) implies V = HA by the replacement
(again) of WL by its equal A. So there are two very different ways to
compute the volume: one using H and A and the other using H, L and W. The
reverse replacement, that of A by WL in the formula V = HA yields V = H
(WL). This is another example of algebraic reasoning: it reinforces the
replacement idea.
Before or during both of the above examples, students can be told the
following verbal rule: multiplying a first number by a second nonzero
number and then dividing the product by the second, yields the first
example. A few examples can confirm it. This multiplication reversal by
division rule can be given in verbal form. Remember the students are
still perhaps prealgebraic.
Before or during both of the above examples, students can be told the
following verbal rule: multiplying a first number by a second nonzero
number and then dividing the product by the second, yields the first
example. A few examples can confirm it. This multiplication reversal by
division rule can be given in verbal form. Remember the students are
still perhaps prealgebraic.
Interest Formula Examples
A third illustration employs the compound interest (or investment)
formula. (The simple interest formula could be used instead). Here the
derivation or justification of the formula in the work may be done before
and/or after this illustration β repetition here is not harmful, and
there may be alternative viewpoints.
The compound interest formula A = P
(1+i)n can be employed to compute the final
amount A from a knowledge of the principal P, the interest
rate i, and the number of periods n invested. But the formula can
also be used to solve for
|
P
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=
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P (1+i)n
(1+i)n
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=
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A
(1+i)n
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| using the multiplication reversal by division rule. Do this once
with numbers given for A, i and n, and once with the
letters instead to show how the arithmetic and algebraic patterns agree,
but that one is more general. In the arithmetic solution, I postpone all
evaluations of arithmetic operations and all simplification of fractions so
that the arithmetic-algebraic pattern is more obvious.
After a discussion of powers and roots, the compound interest formula
A = P (1 +i)n can be used to
obtain the value of i in the event that numbers are given for the
other quantities, or a formula for the interest i in the general
case. Here again, one may solve an arithmetic problem in a manner that
resembles the more general solution. The latter is given second.
These manipulations of the compound interest formula further lead
students to an algebraic way of thinking. They show that the algebraic
way has the potential to give a formula or pattern to solve many similar
problems at once. To this end, I may insist on following all the steps in
an arithmetic problem, exactly as in the algebraic solution, until
students impatiently suggest that we use the algebraic formula (and
thereby omit the chain of reasoning that lead to it). This marks a
turning point in their comprehension.
The foregoing represents an inductive and a psychological approach to the
explanation and comprehension of the algebraic way of writing and
thinking. What is missing now are examples to reinforce it, and rules
formally stated to say when two different calculations give the same
result.
Linear Equation Examples
Solving linear equations provides a further confidence building component
for the algebraic way of writing and reasoning. Here to a first number
adding and then subtracting a second yields the first. Physically this
echoes the notion that adding and then subtracting the same number of
marbles to a bag of marbles leaves the count of marbles in the bag equal
to its original value. The algebraic pattern is (a+b)-b=a This be can
confirmed with a few examples (with b positive if students are not yet
familiar with the subtraction of negative numbers).
Next containers for numbers can be labelled with letters, and the letters
used as shorthand symbols or abbreviations for the contents. This allows
us to speak of numbers or quantities without them having much physical or
economic significance.
In explaining the solution of linear equations, we may start with several
numerical examples of one equation with one unknown, for instance,
5x+7=28, and then solve them in following the algebraic pattern used to
solve ax+b=c. Following the algebraic pattern means that arithmetic
operations involving the coefficients are recorded but not done nor
simplified, even if that tries the patience of student. The numerical
examples again lead to and corroborate the algebraic solution. The latter
can be derived following the same pattern or reasoning process employed
in the postponed arithmetic examples. The solution formula x =
(c-b)/a is seen to solve
many similar problems at once. This offers more incentive for the
algebraic way of writing and thinking.
Following one equation with one unknown, we may offer numerical of
triangular systems with two to several unknowns. These triangular systems
can be solved in a forward or backward substitution manner. (The term
triangular stems from the location of the nonzero coefficients in matrix
representations of such systems). Their solution builds confidence and
puts students at ease with working with several unknowns. Students can be
told or shown that the algebraic pattern becomes complicated, and that
arithmetic approach requires less work than obtaining a formula β a
limitation on the algebraic way of writing and reasoning has appeared.
With the triangular systems, we may include systems of equations
equivalent to a triangular system after a change in order of the
equations. This presents a slight variation on the theme of backward or
forward substitution for solving lower and upper triangular systems.
Next, the reduction of general linear systems to triangular systems to
solve them, can be shown for two and more unknown numbers.
At some point in this solution process, the idea of checking results can
be emphasized. The solution of linear systems follows long chains of
reason prone to error.
The foregoing steps increase the confidence of students and makes them at
ease with looking for numbers that are initially unknown. Solving a
consistent system of equations can be characterized as changing the
psychological state or knowledge of some numbers or quantities from
unknown to known.
Calling the unknowns in a linear equation variables is a somewhat
objectional, yet presently standard abuse of language. The frequently
employed letters x and y are used to represent the numbers, known or not,
in a linear system. They may be called variable if it is accepted that
for a given linear system, they represent fixed numbers in the solution
of the system, but they and the solution may change or vary from system
to system. Use of the term variable should be justified, otherwise the
use is an abuse of language, a too common one.
Arithmetic Rules for Algebra
After some or all of the previous topics, students should be thinking
algebraically. These topics, together with a nascent algebraic thinking,
provide a context for the comprehension of the algebraically described
properties of real number arithmetic. Most can be introduced or described
as assumptions or rules which say when two different computations give
the same result. Calling an assumption an axiom or law may disguise its
humble origin: an assumption is an assumption is an assumption.
In the companion book Three Skills for Algebra, the chapter Arithmetic
Rules for Algebra, henceforth called the chapter, illustrates the
computational significance of each rule, that is it provides an
interpretation for each one. For instance, the commutative law for
multiplication represents the idea that the order of the factors does not
affect the results. It can be stated for a pair of real factors or
several. The rule for several can be derived, if one wants to complicate
matters, from the rule for a pair. The chapter omits the rule for
several (that could be rectified) but it does illustrate the rule for a
pair with decimal numbers and it emphasizes that the factors could be
given by the results of one or two formulas.
The chapter emphasizes that the properties described algebraically
and symbolically, imply methods or rules for changing the way
calculations are done as well methods for simplifying arithmetic. And in
both, substitution may be employed.
The chapter also emphasize the while the state laws involve only
addition and multiplication, laws of arithmetic for subtraction and
division follow as subtraction can be regarded as the addition of a
negative (additive inverse) and as division can be recast in terms of
multiplication by the reciprocal (multiplicative inverse).
Units in Computation
The exposition of mathematics in secondary school should acknowledge,
support and sanction this computational role of mathematics in other
disciplines. The chapter also emphasizes that the properties of
real numbers also apply to quantities β that is, real numbers times units
of measurement for weight, mass, speed, distance, time, temperature or
monetary amounts, etc. Currently mathematics courses, except examples in
trigonometry, only discuss real numbers and forgo or avoid any discussion
of the units or quantities that appear in science, technology or
commerce.
It is possible to formulate all computations without units, but without
any extra work, units can be carried through computations algebraically
or symbolically in the same manner as indeterminates. For the sake of
algebra across the curriculum, the sanction in mathematics courses of
units in computations is recommended. Otherwise, there is a void.
Mathematics curricula need to sanction and teach the algebraic abilities
required by other subjects.
Implication Rules
When mathematics is only described and not derived, implication rules and
logic are missed or not noticed. It is possible to identify them or at
least the presence of some reasoning process. Every use of the terms or
phrases such as therefore, thus, hence, and from this
signals the drawing of a conclusion. Further, any multistage rule-based
process which yields a result or conclusion gives an example of a chain
of reason. Beyond this, the appearance of implication rules and their
contrapositives in the statement of axioms or assumptions can be brought
to the attention of students. The virtual absence of synthetic geometry
in schools and college makes this extra attention to implication rules
(conditional statements) and their contrapositive necessary. Reminders
follow.
1. The zero product law for the product of nonnegative numbers
is the contrapositive of the following rule.
If a and b are both positive numbers then their product ab is
nonzero.
The latter is implied by the rules of multiplication in
decimal arithmetic. When students learn about arithmetic with negative
numbers, the generalization of the previous rule, namely,
If a and b are both nonzero real numbers then their product ab is
nonzero.
follows. For this rule to be never disobeyed, when a product
ab of real numbers a and b is zero, then the statement
a and b are both nonzero real numbers
must be false. This gives a link with the elementary knowledge
of arithmetic gained say in primary school. Students can be alerted
(forewarned) at the time the zero product rule is formulated, that this
rule can be used to solve equations. Examples in mathematics of the latter
are provided by the solution of polynomial equations by factorization. The
latter includes the derivation of the quadratic formula.
The quadratic formula can be derived by completing the square, or by
showing the expansion of
|
a
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/
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|
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\
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x-
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2a
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\
|
|
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/
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/
|
|
|
\
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x+
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2a
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\
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/
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equals ax2+bx+c after a
simplification. (Note the expansion reverses the factorization of the
quadratic, and can even be used as motivation for the factorization or the
completing the square process.) Situations where the discriminant
b2-4ac < 0 provide
motivation for the discussion of complex numbers, and in absence of any
knowledge of complex numbers show how algebraic considerations can go
beyond what is understood β the first employers of algebraic analysis did
not have a large repertoire of numerical quantities.
2. A discussion of unlimited accuracy in computations leads to
concepts of continuity and limits. Limits on accuracy lead to the
concepts of discontinuity in functions or their machine based
computations. See the chapter Limits, Error Control and Continuity
in the companion book Why Slopes and More Math. Gifted students in
high school should able to follow and discuss the error control
perspective of continuity, and its contrapositive formulation. Otherwise
this topic provides college level material.
[2] Primary school math instruction
corresponds to teaching children how to wade in a paddling pool not deep
enough for swimming. Post primary math instruction corresponds to
teaching students to swim, that is to employ deductive reason and
algebraic ways of writing and thinking. Both appear and are required
after arithmetic in mathematics.
Swim instruction today favours non-swimmers entering the shallow end of a
pool to bounce up and down while exploring and practicing swimming
motions. Many of these non-swimmers then gain a dynamic sense of balance
or buoyancy and so begin to swim. Following this, their technique or
strokes can be refined.
Learning how to swim by wading and bouncing about in the shallow end is
not always successful but it may be more successful and encouraging than
the old fashioned approach of getting students to jump in the deep end.
This old-fashioned approach leads to the identification of people with a
natural talent for swimming. Yet it immediately discourages others
and leads them to the notion that they have no natural talent for
swimming. So it should be avoided. Yet besides people with a natural
talent, thus recognized, there are others with a cultivatable talent,
those who can learn to swim by wading in the shallow end to gradually
practice swimming motions in the hope of obtaining a (dynamic) sense of
buoyancy.
There may be a similar situation in mathematics. The presentation and
illustration of the three skills for algebra in particular give another
approach for cultivating the mastery of algebraic writing and reasoning
skills while avoiding the perils and phobias of sudden immersion.
[3] in the companion work Three Skills for Algebra
[4] The use of the compound interest
formula assumes that students are familiar with it β simpler examples to
show the importance of solving for certain quantities could be based on
the simple interest formula I=Pit or with the formula A=P(1+it).
[5] The discussion of how this state
changes when the system is inconsistent is left to the reader :)
[6] In retrospect, the chapter should
called Symbolically or
Algebraically Described Rules of Arithmetic.
[7] Note again, intermediate level
instruction in the 1960s or 70s modern math curriculum did not
acknowledge the decimals representation of real numbers and provides no
alternative. Here real numbers of great interest and service were not
immediately defined in that their decimal representation is not
sanctioned (nor fully exploited). Following these modern math curricula,
the precise set-theoretic definition of real numbers is left to a college
courses seen by very, very few students. All others are left in
suspense.
Sets
The mathematics curriculum has been influenced by the set theoretic and
logical, that is the thought-based, codification and framework for
mathematics. This framework is indispensable or at least appears
sufficient for the rigorous discourse of most present-day professional
mathematicians β whether or not this perspective is in favour a hundred
years hence is another question. But for novices and for computations in
many disciplines, this perspective is not required. In colleges where
courses for mathematics students may emphasize or be presented in the set
theory framework, mathematics courses for students in other disciplines
will forgo the set theoretic wrappings and emphasize computation or the
mastery of algebraic and geometric concepts. Intermediate level
instruction in secondary school need not be more faithful to the set
perspective than most college mathematics service courses β those given
to students not specializing in the discipline.
Intermediate instruction should emphasize sets and the allied concepts of
membership, unions, intersections, ordered pairs and complements, only
where their use shortens exposition, or provides a second higher
math perspective. We recall a few examples where some benefit may be
present.
Venn diagrams have a slight advantage in illustrating logic and linking
logical terms (and, or, not) with set theoretic terms (intersection,
union, complement). Venn diagrams, sets and subsets provide a framework
to simplify the computation of probabilities or the counting and
representation of ways in which objects can be combined or arranged.
Functions can be defined by means of a computation rule (for instance a
formula) or by means of a set of ordered pairs which satisfies the
vertical rule property. The two ways of defining functions are equivalent
or interchangeable. For a set of points in the plane, the vertical rule
property, if satisfied, can be employed to define a computational rule
for a function, and a computational rule for a function can be employed
to define a set with the vertical line property. Acknowledging this
equivalence entails a comparison of a set theoretic definition, that of a
function, with a non-set theoretic perspective of mathematics. The
set-theoretic perspective is but a subset of the discipline.
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|
Teachers & Tutors: Site pages offer better or best practices for providing skills -
simpler than expected & comprehensive but for exercises. For your charges, your duty is to study them alone or in
groups and develop skill building exercises & activities to share. Start now. The effort here is the best I can do.
Others are welcome to refine or exceed it. Please do.
Secondary
Mathematics for Ages 11+, A Practical Approach for home-tutoring or -schooling, or for schools & colleges
with local curriculum control. Study how to include site content - its skill development how-TOs and innovations
into present or future lesson plans - some reading required.
Road
Safety Messages and Questions: When and why should you face
traffic when walking along a road or cycle path? Is it a good
idea to hang limbs outside of cars etc? What gives more
protection in a crash: a car, motorbike or bicycle?
See too, the BBC-Belgium story Texting and
Driving - texting & the impossible test - the article links to a gruesome utube video on the subject
The Logic of Injustice:
How Texas sent
an innocent man to his death - The wrong Carlos. Some judgments are irreversible. Procescution: Where and when prosectors play to win rather than for
justice, guilt beyond a reasonable doubt goes unrespected due to prosecutors who putting winning
first, those innocence before the law may be convicted. Some procescutors offices in continuing to accuse after a pardon
due to reasonable doubt or innocent being shown, may sucessfully oppose compensaton for false convictions
by asserting a pardon individual is still under suspicion. Then the pardoned individual or the latter's estate
is not compensation for years or decade
of improper or false imprisonment, or for execution. Site chapters on Logic
and some in Pattern
Based Reason may slowly lead to greater precision in reading, applying and
writing laws.
May 2012, Composition Starting:
Pre-School and Primary Mathematics - Quantitative Skills, An
Intellectual View, Feedback Welcome:
The 8 Most Popular Site Inlinks
Parent Center: Help your child or teen
learn:
Parent-friendly
Work Booklets for ages 3+ to 13 Use these or others to check
or build skills. Other booklets are available but these booklets
allow parents unsure of themselves in mathematics to help their
children. The selection acquired in Canada is published in the
USA. So it has a US orientation. In retrospect, the selection
shows parents what to check with the booklets or by other ways,
the choice is theirs. But in retrospect, the selection does not
cover integral and fractions liquid weights and measures - ask
the publishers to correct that! For ages 9 to 12 say, parents may
compensate by showing boys and girls how to use weights or mass,
and further measures in food preparation. Beyond that children
may be shown how to measure and calculate angles, lengths and
areas [proportional amounts too] directly or by using maps and
plans drawns to scale. Learning how to gather and measure all the
ingredients, pots and pans for a dish or a meal, along with
cleaning up sets the stage for like activities or experiments in
science courses, and in developing organizational skills,
gives boys and girls a head start. Good luck. At the other
extreme, more comprehensive than light, if your motto is
McCainian: drill, drill, drill then Toronto
mathematician and actor John Mighton's jump math organization has jump math
workbooks for at least grades 3 to 8 for at-home and in-school
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Fractions for
Grades 3-4 & Grades 5-6 [Read] Free Resources grades 1 to 8
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Mathematics
Skills For Ages 3 to 14 - technical!
Skills with take
home value - A few ideas
Basic skills include
time-date-calendar Matters; money matters; map, plan and
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Is your child able to add, subtract and multiply amounts
of money, work with fractions, work with clocks and calendars,
work with maps and plans, and measure length, weight-mass and
volume? Schools may promote your son or daughter without
providing basic skills in reading, writing and
arithmetic.
Arithmetic
and Number Theory Skills
Algebra
Starter Lessons
Geometry
- maps plans trigonometry vectors
More
Algebra
70
Calculus Starter Lessons
Calculus Lessons Elsewhere:
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How to Ace Calculus: Street Wise Guide - Mostly
Text.
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Flash
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They cover basic topics in ways likely to complement your
notes, your textbooks and site material. When Goldilocks
trespassed in the house of the three bears, she found three bowls
of porridge, two not to her liking, and one just right. Different
bears have different tastes. As invited guest here and elsewhere,
if one or more explanations is not to liking, try another. It may
be better or just right.
Unsolicited Advice
Learning to do and high marks if it comes to easy is often
deceptive - light rather than deep. For that reason, students
with learning difficulties determined not to let it get in their
way may go deeper and farther than those with none. High marks,
if the come easy, may be deceptive - provide a too light and not
a deep mastery. That could have been your problem in secondary
school, one that leads to comprehension shock or difficulties in
calculus and more generally in the first year of college. Bon
Appetite.
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