Arithmetic With Unsigned Numbers (Items
1-17)
Master Operations with whole numbers, fractions, decimals,
percentages and radicals. In mastery of fractions and radicals (square
roots and cube roots) employ prime factorizations (see below) to aid
operations. For the sake of algebra, an efficient mastery of exact
arithmetic with whole numbers, fractions and radicals without a
calculator is expected.
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Exact and Efficient Arithmetic with Decimals (grades 4 to 8):
This will employ place value and place-value based methods for
counting, addition, comparison, subtraction, multiplication and
division. Avoidance of the domino effect of errors and approximations
should be emphasized here. Learn how to add using subtotals. Learn how
to multiply using subproducts.
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Reading Large and Small Numbers Aloud: Develop number sense and
humour with the ability to read out aloud, decimals with many, many
decimal places before and after the decimal point.
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Prime Factorization of Whole Numbers (grades 7 & 10): with
use in operations with fractions, in simplifying radicals - converting
to standard form, counting and generating integral factors of whole
numbers. Explain that products may be calculated using subproducts to
explain how to group prime factors. Use Prime Factorization to find
gcd, lcd for the efficient addition and reduction of fractions. Use
Factorizations to generate all whole number factors of a whole number.
For quick prime factorization, use the property that a whole number is
composite (non-prime) if and only if has a prime divisor less than the
square root of the whole number.
Prime Factorization Methods etc. For an exact and efficient
command of algebra students need an efficient and exact operational
command of arithmetic with decimals, fractions and primes sans
and with calculators. A command of place value methods for
multiplication and division of decimals eases the way for a similar
command of column methods for multiplication and division of
polynomials. The ability to find the prime decomposition of whole
numbers less than 200 quick say is a must for exact arithmetic with
fractions and the expression of surds in standard form. Part or all of this material would be covered in junior high
school in other mathematics programs.
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Arithmetic without and with Calculators (grades 4 to 8): Be
able to do exact arithmetic with decimals and fractions without a
calculator. That being proven, use Calculators to do approximate and
exact calculations. For example for one number divided by a smaller
number, take the integral part of the approximate quotient provide by a
calculator to find the number of whole times one number goes into a
larger one. Take the fractional part times the divisor to estimate the
remainder. Have the calculator display a sufficient number of digits to
avoid wrong results.
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Exact and Efficient Arithmetic with Fractions (grades 4 to
8): This will employ least common multiples, greatest
common divisors, least common denominators and various methods with and
without prime factorization, and may Euclid Algorithm to find them. Add
using subtotals. Multiply using subproducts. Identify Ratio of Part to
Whole with a fraction. But explain that some ratios (part to part, and
multiple term ratios) cannot be identified with fractions.
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Multiplicative Inverse (grades 8 or 9): The multiplicative
inverse of an unsigned number equals 1 divided by its unsigned part.
With that, a ÷ b = a × ( 1 ÷ b ) in the like
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Social Arithmetic (grades 7 and 8): Employ arithmetic with
decimals, fractions and percentages in common problems from daily life
at home and work, or in buying and selling goods and services. Use
subtotals to calculate sums of assets and debts.
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Working With Unit of Measures (grades 7 and 8): Master
arithmetic or fractions operations when units are present in numerators
and denominators. In particular, work with numerical multiples of
monomials in units of measure (length, time, mass, etc) alone or in the
numerators and denominators of fractions. Use units, products of units
(monomials in them) alone and in fraction with units to represent
physical quantities and in general to learn about arithmetic with
units: multiplication and division and then where possible, addition,
subtraction and comparison. Learn that use of units, monomials in them
and fractions with units is a notational convenience. Express per unit
costs, speed, rates of change in terms of fractions with units.
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Estimation (grades 7 onward): Estimate and approximate
arithmetic results by rounding numbers to the nearest unit, ten,
hundred, etc. Steps should be done and recorded on paper using say the
approx equal sign. Exception: Advance mathematics implies that
rounding in approximate calculations should be delayed and that all
intermediate calculations should be done with the greatest number of
digits that can be carried for results with the greatest
accuracy.)
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Use and understand mean, median, mode (grades 7 and 8): For
small set of numbers, this may be an exercise in exact arithmetic with
whole numbers and fractions. For larger sets, this may be an exercise
in calculator used. Talk about average wage calculations. Require the
input for these calculations to be recorded and along with a term or a
few words to say what is be computed, so that what is being computed is
fully described.
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Counting Practice - Recognized and Sanctioned (grades 7 and
8): Assume the number of elements in a set may be
calculated as the sum of subcounts with the understanding that the set
may be partitioned into subsets in arbitrary manner to obtain those
subcounts. After mastery of set language, a precise symbolic
version of this law may be understood. (young children students may
learn to count and check counts by recounting in different orders, with
or without the aid of grouping. That common practice needs to be
recognized and sanctioned.
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Addition Rule (Generalized Commutative- Associative Property of
Sums) Recognized and Sanctioned (grades 7 and 8): Assume sums of
unsigned numbers may be calculated as the sum of subtotals with the
understanding that addends can be partitioned into subsets in arbitrary
manner and order for the calculation of subtotals. After mastery
of set language, a precise symbolic version of this law may be
understood.
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Product Rule (Generalized Commutative- Associative Property of
Products) Recognized and Sanctioned (grades 7 and 8): Assume
Products of unsigned numbers may be calculated as the products of
subproducts with the understanding that factors can be partitioned into
subsets in arbitrary manner and order for the calculation of
subproducts. After mastery of set language, a precise symbolic
version of this law may be understood.
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Direct Sum (Additive) Area Calculation (grades 7 and 8):
First, learn how to calculate areas of regions decomposable (some
observation required) into regions whose areas are given by formulas
may be calculated by subdivision and adding subareas. Subareas may be
given by fractions of circles: half, third, quarter. Then calculate
total areas via the sum of subareas. Students have to be told to watch
for situations which the foregoing aids area calculation.
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Indirect Sum Area Calculation (grades 7 and 8): Imagine the area
of a region may be calculated by a direct sum of subregion areas
(subareas). Then a union of some subregions may be calculated by
subtracting from the area of the region, the direct sum of all
subregion areas for subregions not in the union. Students have to be
told to watch for situations which the foregoing aids area
calculation.
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Direct Sum (Additive) Volume Calculation (grades 8 and
9): First, learn how to calculate volumes of solids
decomposable (some observation required) into regions whose volumes are
given by formulas may be calculated by subdivision and adding
subvolumes. Subvolumes may be given by fractions of shapes with known
volumes. Then calculate total volume via the sum of subvolumes.
Students have to be told to watch for situations which the foregoing
aids volume calculation.
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Indirect Sum Volume Calculation (grades 8 and 9): Imagine the
volume of a solid may be calculated by a direct sum of subregion
vvolumes (subvolumes). Then a union of some subregions may be
calculated by subtracting from the volume of the region, the direct sum
of all subregion volumes for subregions not in the union. Students
have to be told to watch for situations which the foregoing aids area
calculation.
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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
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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
[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.
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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.
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