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Home < Archives < Progressive Observable Motivated Mathematics Education << 2 arithmetic with signed numbers
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<<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN">
Arithmetic with Signed Numbers (18-25)
Signed Numbers: Use signed
numbers as coordinates along a line, in coordinates for points in the
plane, as indication of depth above or below a reference point for
water level or temperature, as a convenience for recording assets and
debts. Add and subtract signed numbers. Add signed numbers using
subtotals. Compare signed numbers using the concept of more than and
less than by a unsigned amount. Multiply and divide signed numbers by
unsigned and then signed numbers. Multiply signed numbers using
sub-products. Explain that subtraction needs to be converted into
adding the additive inverse (negative) in order to add using subtotals
with an arbitrary grouping of addends. Emphasize again that division
needs to be converted into multiplying by the multiplicative inverse
(reciprocal) in order to mutliply using subproducts with an arbitrary
grouping of factors.
- Master the use of signed numbers as coordinates. Examples may be
provided by depth above and below sea level or a fixed water level, by
temperature scales, by the amount of money in a bank account where
deposits are counted as positive and withdrawals give by negative
numbers.
- Learn the use of signed numbers as displacements. Learn the addition
of pairs of these displacements with like and unlike sights. Learn the
multiplication of these displacements by unsigned whole numbers (a form
of repeated addition) and unsigned fractions (a whole multiple of a unit
numerator fraction). Learn how to divide signed numbers by unsigned
numbers and by another signed number in the like sign case. Introduce
multiplication by a + sign as the identity operation, and by a - sign as
a sign reversal operation - interpret the latter as changing the
direction of a displacement. Then multiplication by an signed number
becomes multiplication by its sign combine with multiplication by its
unsigned part.
-
Addition of Signed Numbers: The sum of signed numbers with the
same sign has the common sign prefix to the sum of their unsigned
parts. The sum f(a,b) = a + b of two signed numbers a and b with
opposite sign is given by the sign of the longest prefixed to the
difference, that is the biggest unsigned part - smallest unsigned part.
The sum is be zero of in the cases where the addends a and b are
additive inverses (have opposite signs with unsigned parts equal)
Clearly f(a,b) = f(b,a). Subtraction of a signed number b equals the
addition of its negative inverse: a - b = a + (-b). The
triangle inequality for signed numbers may be understood as follows:
The unsigned part of sum of two signed numbers is less than the sum of
their unsigned parts and greater than the difference of the largest
minus smallest unsigned parts.
Algebraic Formulation: If absolute value |x| is the unsigned
part of a number x then | |a|-|b| | < |a+b| < |a| + |b| for signed numbers
a and b.
-
Product of Signed Numbers: The product of signed numbers is the
product of their signs prefixed to the product of their unsigned
parts. The multiplicative inverse of a signed number b equals its sign
prefix to 1 divided by its unsigned part. In symbols, sign(b) prefixed
to 1 ÷ b With that, a ÷ b = a × the multiplicative inverse of
b.
-
Master and simplify fractions with signed numbers in numerators and
denominators, and units of measure too.
-
Sum Rule (Generalized Commutative- Associative Property of
Sums): Sums of signed 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.
Social Implication: Adding assets and debts (liabilities) will not
change the net amount one has or owes.
-
Product Rule (Generalized Commutative- Associative Property of
Products): Products of signed 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. Prime Factorization
employs this property.
- Rule of Thumb for Sensitivity Analysis and Greater Accuracy in
Computations:
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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
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May 2012, Composition Starting:
Pre-School and Primary Mathematics - Quantitative Skills, An
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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
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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
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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