Origin of Units.
Whole numbers and fractions (pure numbers without units) may appear in
counting as number or multipliers to describe many items are present.
For example, 256 represent the idea that a set of
objects may be grouped into 2 sets of one hundred, 5 sets of 10 and 6
objects left over - a set of six perhaps. Two people may reach the same
decimal description or count 256 via groupings the object differently.
The object that appears in one person's first group of one hundred may
appear in another person's second group or in one of the sets of 10 or
6 the other forms in the count, sets formed explicitly or not. But we
assume (a counting principle or practice) that any two people counting
a set of 256 objects will reach the same decimal description 256, but
not necessarily with the same grouping of the objects into sets of 100,
10 and the 6 leftover.
Simple units of measurement may appear to identify what we a counting.
5 apples, 10 oranges, 10.5 centimeters, 5.34 kilograms, 10 degrees
Celsius (temperature measure), 90 degrees (angle measure)
A number times (or written besides) a unit of measurement is called a
quantity.
In daily life, science and technology, there are systems of measurements
for length, mass, time, money, and so on
Measurement systems in the physical involve units of length, mass and
time:
- cgs system: centimeters, grams and seconds
- mks system: meters, kilograms and seconds
- imperial system(?): feet, slugs and seconds.
Unit of measurement are part of applied mathematics and external to pure
mathematics. Yet calculations involving units of weight, mass, length,
time, money (hand it over) and so on appear in the measurements and
calculations of daily life alone or as part of rates and further
proportionality constants.
Pure mathematics deals only with pure numbers in what is
called dimensionless or context-free manner that leads to a separation
of mathematics from motivations and considerations that may lead to
false conclusions. That is not say, the logic in pure mathematics is
perfect. Problems still remain. You can investigate them if you become
a mathematician.
Addition and Subtraction of Quantities
(Symbolic or Algebraic shorthand
description/form)
When we have 5 apples and 6 bananas and 1 orange in a bag, the expression
5 apples + 6 bananas + 1 orange
represent this collection of objects or fruit. The units here apples,
bananas and oranges. We write units in singular or plural form in
accordance with language rules. But in writing expressions, we do not
care or distinguish units written in singular or plural form. So in our
calculation with units, we write 5 penny means the same as 5 pennies. The
expression looks like a sum. Now from bag of 5 apples and 6 bananas and 1
orange in a bag, we may remove 2 apples, 3 bananas and 1 orange. The
result would be 3 apples, 3 bananas and zero oranges. We may write the
foregoing in shorthand form (algebraic or symbolic form) as
(5 apples + 6 bananas + 1 orange) - (2 apples + 3 bananas + 1 orange)
= (5-2) apples + (6-3) bananas + (1-1) oranges
= 3 apples + 3 bananas + 0 oranges
= 3 apples + 3 bananas
In this subtraction we are combining like terms: those involving apples,
bananas and oranges, respectively.
On the other hand if I have 10 dimes and 4 pennies and you have 6 dimes
and 20 pennies, together we have
(10 dimes + 4 pennies) + (6 dimes + 20 pennies) = 16 dimes + 24
pennies.
This addition combines like terms - terms with the same units. A dime is
coin worth ten pennies. Changing dimes in pennies or vice-versa is
optional here.
In general, for a units of a quantity plus another b units of the same
quantity together give (a+b) units of the same quantity. Symbolically, we
write
a units + b units = (a+b) units
provided of course the unit of measurement in all terms are identical. In
the same circumstances,
a units - b units = (a-b) units
Thee grouping represents the distributive law for working with numbers
and quantities.
Remark: In primary school mathematics, student
may have learnt or accepted that 2 + 3 = 5 from the question of how to
describe the result of combing 2 units and 3 units gives 5 units,
simply by counting how may units there are in total. Student met
examples like the following drawn instead of written
-
2 rabbits plus 3 rabbits give 5 rabbits (by
counting)
-
2 dots plus 3 dots give 5 dots (by counting)
-
2 pies plus 3 pies give 5pies (by counting)
Many examples like this for the pair of digits
(multipliers) 2 and 3 may lead students to accept or proclain that 2
units + 3 units = 5 units for like units, or 2 ones plus 3 ones = 5
ones (here one serve as a pronoun for a unit - so we could also write 2
its + 3 its =5 its) and finally, arrive at 2 + 3 = 5. Similar
considerations lead use as young students to fill in the addition table
for all pairs of digits 0 to 9. That process along with decimal value
notation leads to addition, comparison, subtraction and comparision of
quantities and pure numbers. See the development of arithmetic skills
and concept in site pages.
Examples with like units
- 5 kilogram + 4.5 kilograms = 9.5 kilograms
- 4 hours + 8 hours = 12 hours
- 20 seconds - 16 seconds = 4 seconds
- 8 centimeters + 6 centimeters + 2 centimeters = (8+6 +2) centimeters
= 16 centimeters
- 10 meters - 18 meters = - 8 meters
The last makes sense if 1 meter represented one step to the right and -1
meter represented one step to the left.
Examples with unlike units (Read + as and)
- (8 apples + 4 pennies) + ( 3 pennies + 2 apples) = (8+2) apples +
(4+2) pennies = 10 apples + 7 pennies
- (4 oranges + 3 bananas + 2 lemons) + (2 oranges + 3 lemons) = 6
oranges + 3 bananas + 5 lemons
These calculations symbolically represent the addition of stocks of
different kinds of fruit in one calculation involving unlike units
instead of separate calculations involving like units. That being said,
the counting of apples, oranges and so on would as a matter of practice
be done in separate calculations (separate lines) where only one kind of
unit appears.
|
|
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.
|
|