Logic Review and Decimals etc (87-98)
-
Logic Development in the context of improving reading and writing
skills (grade 9 and above). (I) Master the difference
between one-way implications A if B and two implications A if and only
if B. (II) Learn about the contrapositive form of one way
implications: that IF A then B requires IF NOT B then NOT A. The
assumption that NOT (NOT A) implies A then implies the equivalence of a
one way implication and its contra-positive. (III) Learn about
syllogisms and short chains of reason: the use of one ways implication
If A then B and IF B then C to imply IF A then C. (IV) Learn about
longer chains of reason and mathematical induction in reason and in the
recursive definition of sequences or functions; (V) Talk about
consistent story telling and extension: How an possible extension if is
inconsistent or implies an inconsistency with an earlier part of the
story then the extension cannot be included. (VI) learn about islands
and bodies of rule and pattern based knowledge, and the thought that
different bodies may may different entry points, equivalent or not.
(VII) Talk about logic ideals: the derivation from a minimal set of
assumptions or axioms, and the hope of avoiding inconsistencies or
contradictions.
Aims: Master the use of direct chains of reasons with
the implication rules IF A THEN B. Understand the equivalence (same
meaning) of the latter with the pattern B if A and the difference
between the one-way implication rule B if A and the two way
implication rule B if and only if A. Recognize that in circumstance
where IF A then B never fails, then If NOT B then Not A must
hold.
-
First Logic Application (grade 9 and above). The area
interpretation of products of a pair of unsigned numbers implies the
product of pair signed numbers is nonzero if the factors are nonzero.
The contrapositive of that is the implication rule: If the product of a
pair of signed numbers is zero, then at least one must be zero.
-
An Algebraic-Geometric Proof of the Pythagorean Theorem (grade 9
and above). Use the geometric view of the distributive law and the
Chinese Square Dissection Proof to imply the previously given and used
Pythagorean Theorem. Students should learn or review the Pythagorean
3-4-5 and 5-12-13 triples, and about the real number triples associated
with the isosceles right triangle and the 30-60-90 right triangle - a
triangle derived from the bisection of an equilateral
triangle. Introduce in grades 8 or 9 say.
-
Algebra and Geometry, Forwards and Backwards (grade 9 and
above). Understand how to apply the Pythagorean theorem
(contrapositive form, 2nd Logic Application) backward to recognize when
a triangle is not a right triangle, and how to determine the value of a
missing side - leg or hypotenuse.
-
Property of Decimals & Coordinates - Recognized and Sanctioned
(grade 9 and above). Assume whole numbers, natural numbers, rations
and real numbers may be (i) identified with sets and (ii) identified
with points on a real line. In or with the foregoing, note or observe
that decimal fractions have finite decimal expansions, observe by long
division that other fractions have periodic decimal expansions. Learn
(be told that) irrationals have infinite, non-periodic decimal
expansions.
Completeness: In terms of coordinates, view infinite
decimal expansions as a sequence of approximations to the location (its
limit) of a point on a real line or coordinate axis. In this, assume
the Decimal Axiom: Each real number may be given or represented
by a sign prefixed to a finite or infinite decimal expansion, a decimal
representation of its unsigned part.
Known Ambiguity: A point located by terminating decimal
may also be viewed as the limit of an infinite decimal expansion, one
that terminates with the digit 9 repeating.
-
Error Control and Convergence/Limits (grade 9 and above).
For measures and unsigned Numbers: Learn about percentage error,
relative error and significant digits in decimals, in measures and in
the following computations: sums, difference, multiplication and
division. Be aware of error control in approximate calculations with
measures. Be aware of error control and convergence (continuity) in
the definition or discussion of arithmetic with infinite decimal
expansions. This topic is a prequel to the decimal view of error
control, limits, convergence and continuity in calculus. Geometric
context may be provided by the calculation of perimeters, areas and
volumes, and use of the triangle inequality.
-
Expression of Repeating Decimals as Fractions (grade 9 and
above). Method (1) Do arithmetic with infinite decimal expansions
to find the limit. For first example example, let L= 0.723723723 be a
repeating decimal fraction with limiting value L It has period 3.
Assume 103L= 727.723723723 ... in a repeating manner.
Then
103L = 727 + L. So (103 -1)L = 727 and so
Method (2) observe
|
L =
|
727
103
|
(1+10-3 +(10-3)2 +
(10-3)3 +
(10-3)4 +
(10-3)5+ ...
|
|
|
is given by a geometric sum with limit
|
|
=
|
727
103
|
1
1-10-3
|
|
=
|
727
999
|
|
The foregoing requires a discussion of geometric sums and their
limits.
Remark: In general a decimal expansion such as
M = 345.5658787878787 ...
which eventually repeats may be written as the sum of trunk and a
repeating part. In the latter example:
|
M
|
=
|
34.565 + 0.0008787878787 ...
|
So the repeating part may be find by method (I) or (II)
-
Existence of Irrational Numbers (grade 9 and
above). The principal square root of 2 can be approximated
with the aid of a calculator. As more decimal places are included, the
square of the approximation appears to approach 2. That numerically
suggests the principal square root of 2 exists and has a decimal
expansion. Application of the Pythagorean to a isoceles right triangle
with two sides of unit length implies the length of hypotenuse is
sqrt(2) times the unit length. Thus sqrt(2) exist geometrically.
Logic Application: the observation that the possibility the latter is
rational is inconsistent with previous knowledge of prime numbers and
fractions. Thus the decimal representation of sqrt (2) cannot finite
nor repeating.
-
Rules for Exact Arithmetic (all grades). Do arithmetic with
whole numbers and fractions exactly in a way that avoid decimal
approximations. If well known irrationals like p are present, carry it symbolically through exact
calculations with whole numbers and fractions - do not replace with
decimal approximations. Like if square roots and cubes of whole numbers
and fractions are present, simplify them - there are conventions for
them, and like p, carry the square and cubes
roots of primes through calculations algebraically and simplify.
|
The number p is not exactly 3.14
nor the fraction
|
22
7
|
|
even though some primary and secondary text say use these values
for p instead of more carefully and
more precisely say use these values as approximations to
p . There are other and better
approximation to p . For example,
calculators may display to several decimal places.
|
-
Rule of Thumb for Minimizing Errors in Calculations (all
grades). In initial and further calculations where exact
arithmetic is not possible, for approximations to numbers and
measures, Where one the results of one step are used in the next,
carry the greatest number of decimals that the step may provide - do
not introduce new approximations.
-
Rule of Thumb for Estimating Accuracy of Calculations (all
grades). Do the calculation with great care. That in place of
values with the greatest accuracy, use numbers with less accuracy and
see how the result changes. Then may show the sensitivity of the
calculation to errors or carrying fewer decimals in the steps of the
calculation. That sensitivity in general must be found by trial and
error. It may sometimes imply which digits in a result are
significant.
-
Errors in Measurement (Grades 8 and up): When measurement of
physical quantities are done approximately, measurement should be done
to the greatest accuracy possible, so that the last decimal recorded
has an uncertainty (maximum error) of less of half a unit. In that
case, the last decimal and all before it are said to be significant.
Besides significant digits, there are other ways to indicate the
maximum possible error in an approximation - one may say that the true
value is within an interval containing the approximation, one may give
the maximum possible (absolute) error. One may also describe the
maximum possible relative error or maximum possible percentage error.
Worse, instead of knowing the maximum possible error, it may have to
estimated. College c ourses on numerical analysis (with their rules
for computation and rules or rules of thumb for error control) may
explain more.
|
|
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
use - training sessions for teachers available. Jump math has
been expanding to cover older students. Jump Math Samples: plus
Fractions for
Grades 3-4 & Grades 5-6 [Read] Free Resources grades 1 to 8
[unread - likely to be good]. and
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
scale diagram matters;counting, measuring and figuring;
decision making with logic and likelyhood; being careful and
being aware of the domino effect of mistakes; reading and
writing with precision.
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:
-
How to Ace Calculus: Street Wise Guide - Mostly
Text.
-
Flash
Video for Calculus Phobics
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.
|
|
Return to Page Top
Home < Archives < Progressive Observable Motivated Mathematics Education << 7 logic review and decimals an odd combination
[1] [2] [3] [4] [5] [6] [7] [8][9] [10] [11] [12]
All trademarks and copyrights in this are owned by their
respective owners.
Copyright to comments & contributions are owned by the Poster.
The Rest
© 1995-2011, by site author, Alan Selby, Ph. D., Montreal,
All Rights Reserved ---
Skype
or Email to contact.
|