Counting For Parents
A. Similarity Between Learning 123 and ABC
Children ages 3+ to 6 years of age may be given and learn to say
and write letters of alphabet
abcdefghigjklmnopqrstuvwxyz
ABCEDEFGHIJKLMNOPQRSTUVWXYZ
one letter at a time, one after another, in the given sequence,
Likewise, they be given and learn to say and write numbers or
counts
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, ...
one at a time, one after another, in sequence. Mastery of that sequence
begins before any discussion or perception of decimal notation and
decimal place.
B. Simple and Compound Numbers - A nuance
Numbers given single digit numbers 1 to 9 are simple - formed by a
single symbol. In contrast, whole numbers more than 9
10,11, ..., 45, ... 381, ...., 4530
are compound or composite numbers in that they are given by multiple
symbols concatenated (that is written side by side) with place
value. We write numbers in this decimal "compound" form in order to
count or track how many are present in terms of ones, tens, hundreds,
thousands and larger groups. So we do not have one unit for counting, we
have several as we count 0 to 9 groups of ones, tens, hundreds, thousands
etc that may be formed in tracking how many. Each group has a power of
ten for its number. The foregoing counting is subject to the
convention that more than ten of any group is grouped to form a larger
group. Counting here is mixed because 2 and more digit decimal
numbers count in terms of units - groups of different sizes that in
principle may be formed from each 2 or more digit number.
C. A First Counting Practice/Principle
In counting, the possibility of error appears. Different people
or counters may obtain different counts of how many are present. Even
a single or counter may obtain different results when counting how
many. When such disagreement occurs, a recount is needed. We may
teach children to count twice in order to catch disagreement and to see
whether or not an other count is needed. In this practice, we are
introducing a principle in the form that counts should be independent of
(a) the counter and (b) when done. Beyond that, counting should be
independent of which order the counts are done. For example the sixth
item in one count could be the thirtieth item in another count.
First Counting Principle (Assumption): Counts are or should be
independent of how counted
Nuance: This counting principle or assumption has consequences
for addition and multiplication of whole numbers, since addition and
multiplication may be viewed as counting methods or shortcuts. In
particular when the number of objects may be found by addition or
multiplication, the number in question should be independent of
the order in which the the addition or grouping is done.
Commutative and associative properties of addition and multiplication
operations with whole numbers may be drawn from this counting
principle. Beyond that, this counting principle may be employed to
simplify or rearrange complex expressions whose value can be seen as a
count.
D. A Second Counting Practice/Principle
Counting Names, Labels and Marks in place of Objects
(taking advantage of one to one matching or correspondence)
A class of students may be counted directly, one head a time, one head
after another. Alternatively, each student in the class has or should
have a unique name. To count the student, we have to form a list of
their names and count the names. If there students with the same
name, we may write their names twice in the list. Here there is a
pairing or one to one correspondence between the students being counted
and their names.
There is a shorter method to count the students. Instead of making
a list of names, we may write a mark (a tally mark) instead of or besides
each name, and then count the tally marks. Thus the number present
may be described by tally marks, one per student. With one tally
mark per students, the count of students would equal the count of tally
marks.
Recording Counts with Tally Marks: In days before people knew
about how to count with decimals in groups of ones, tens, hundreds and so
on, the number of persons, items or animals present might be
tracked by tally marks on paper or sticks, one per person or item or
item. The tally marks by themselves were enough to do that tracking, with
more marks being added if more people or objects were added, and tally
marks being erased (subtracted) if people or objects left. With the
current knowledge of decimals, the foregoing could be done by by
counting, adding and subtracting directly with or without tally marks be
added or erased.
E. Place Value in Groups of Three - North American Development
The exercise of reading decimal aloud with several places before and
after the decimal point can provide comic relief in a mathematics class
while developing and reinforcing place value comprehension.
A compound number like 345325 may be expressed at length in
words as three hundred thousand, four ten thousands, five
thousands, three hundreds, two tens and five ones. It may be expanded
numerically as
3 × 105 + 4 × 104 + 5 × 103 + 3
× 102 + 2 × 101 + 1 × 100
where 100 = one, 101 = ten, 102 = one
hundred, 103 = one thousand, 104 = ten thousand and
105 = one hundred thousand. Now 325 and 345 may be read aloud
as 3-2-5 and 3-4-5 respectively. With that convention, we express
345325 as 345 thousands and 325 ones. More generally, a large number like
38, 782, 456, 876, 765, 304, 289, 533, 450, 514, 613
may read aloud backward as
613 ones, 514 thousands, 450 millions, 533 billions, 289 trillions, 289
quadrillions, 304 quintillions, 765 sextillions, 876 septillions,
456 octillions, 782 nonillions and 38 decillions
and so save the decimal places to last.
Reference: Names
of Big Numbers
Likewise,
103.038, 782, 456, 876, 765, 304, 289, 533, 450, 514, 613
may be read forward direction as
103 ones, 38 thousandths, 782 millionths, 456 billionths, 876
trillionths, 765 quadrillionths, 304 quintillionths, 289 sextillionths,
450 octillionths, 514 nonillionths and 613 decillionths.
It is simple an exercise to express the numbers and fractions decillions
=1030 to decillionths =
10-30 in power of ten.
In college and senior high school courses in science
Avogadro's number =
6.0221415 × 1023
may be introduced as the number of Carbon-12 atoms in 12 grams of the
substance.
Now 1021 = one septillion. So Avogradros
numbers is 602.21415
sextrillions or
602 sextrillions plus 214 quintrillions plus 15 quadrillion
to the nearest 10 quadrillion. The foregoing discussion may help.
F. Place Value in Groups of Three - Metric Development
This treatment is for countries where a billion is a
million million. My personal preference is to develop
comprehension of decimal value in groups of three, and not in groups of
six because multiples (or powers + & -) of a 1000 may be easier for
students than multiples of a million.
Now 325 and 345 may be read aloud as 3-2-5 and 3-4-5 respectively.
With that convention, we express 345325 as 345 thousands and 325 ones.
More generally, a large number like
876, 765, 304, 289, 533, 450, 514, 613
may read aloud backward as
613 ones, 514 kilo-units, 450 megaunits, 533 gigaunits 289 teraunits,
289 peta-units 304 exa-units, 765 zetta-units 876
yota-units
and so save the decimal places to last.
Reference: SI
Prefixes
Likewise,
103.038, 782, 456, 876, 765, 304, 289, 533, 450
may be read forward direction as
103 ones, 38 milli-units, 782 mirco-units, 456 nano-units, 876
pico-units, 765 femto-units, 304 atto-units, 289 zepto-units and 450
yocto-units ,
It is simple an exercise to express the numbers and fractions decillions
= 1030 to decillionths =
10-30 in power of ten.
In college and senior high school courses in science
Avogadro's number =
6.0221415 × 1023
may be introduced as the number of Carbon-12 atoms in 12 grams of the
substance.
Now 1021 = one septillion. So Avogradros
numbers is 602.21415
zetta units or
602 zetta-units plus 214 exo-units plus 150 peta-units
to the nearest 10 peta-units.
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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.
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