7. Numerals, Fractionals and Quantals
Recall whole numbers may be written in different forms. Each
of those forms is called a numeral. For example
345 = CCCXXXXV in Roman Numerals
= 3 hundred + 4 tens +
5 ones in expanded form
90 = 9 tens = 4 score and 1 ten.
Here we giving number or describing as in terms of groups of hundred, ten
and ones, with the cosmetic convention that there should be 0 to nine of
each multiple of ten.
Decimal form for whole numbers or counts describe the latter in
terms of ones, tens, hundreds and larger multiples or powers of ten. As
a unit for counting the latter are all different. In the early
days of decimal notation, the numbers 1 to 9 were considered simple
numbers because they had single digit decimal forms while numbers 10
onward were considered compound or composite numbers. The
number 345 is composed of 3 hundreds, 4
tens and 5 ones.
Whole Numbers and Fractions too may be written in different forms by
raising and lowering terms. Each form may be called a numeral or a
fractional. For an first example, the whole number
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4 =
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8
2
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=
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12
3
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=
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16
4
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=
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100
25
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For a second example
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3
5
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=
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6
10
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=
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30
50
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=
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9
15
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Measures too may be expressed in different forms.
4 metres = 40 decimeters = 400 cm = 4000 millimetres
4500 seconds = 75 × 60 seconds = 75 minutes = 1 hour
and 15 minutes or 1¼ hours
$3.75 = 3¾ dollars or 375 pennies.
Each form may might called a quantal. We may prefer or like some forms
over another due to various conventions. For example, expressing
fractions in lowest is a convention.
Equivalence: When we do arithmetic with counts, fractions and
quantals, we assume or hope that results (except for form) are
independent of the form in which we write or record or describe whole
numbers, fractions and measures. Different forms of the same number,
fraction or measure are taken to be or assumed to be equivalent not
only for their description, but also in arithmetic. The question
of why or the question of providing a formal justification for this
practice is left for higher studies or research.
Observe, the time 1 hour and 15 minutes is expressed in mixed units
of time measure, that is hours and minutes. In general many mixed
units of measure can be used in calculations. It is a matter of
convention, habit and convenience, which units of size are employed
in measuring, in counting or in describing
fractions.
Fractions with unlike denominators represent multiples of different
unit numerator fractions: one half, one third, one quarter, one fifth,
one sixth and so on. To add, subtract, compare and even divide
fractions, we may express each over a like or common denominator in
order to do the arithmetic. So the form in which an expression is
given or expressed may help calculations.
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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
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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
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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;
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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.
-
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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