The first part of this lesson shows how to add subtract
numerical multiplies of them in a way that resembles the forward and
backward use of a distributive law for counting and
measuring.& Carries and borrows in addition and subtraction
provide a unit-free form of conversion. In counting and arithmetic,
conversions are further present in expressing counts or numbers in
groups of ones, tens, hundreds and so on, and in adding, subtracting
and multiplying such counts. The second part of this
lesson talks about changing and converting the units used to measure or
keep track of quantities. Your fortune may be measured in
pennies. or in dollars? The third part show how to form products
of quantities. The operations here are similar to work with monomials
where the product of 4xy with 5 xy3z is 20
x2y4z. Instead of using letters x, y and z, we
use measurement units and counting units as well, and could be more
meaningful for students.
A. P roduct of Numbers with Units
A quantity in the first instance is given by a number of units, a
number times a unit, that is a product of a number with a unit. In
measurement, such products appear to count many units are present. The
count may be fractional. The count is called a coefficient.
Three times the amount 7 dishes is 7 dishes + 7 dishes + 7
dishes. So we write
3 x (7 dishes) = (3 ×7) dishes = 21 dishes.
Here multiplication by 3 is just repeated addition.
In general, if a and b are whole number, fraction or decimals, we assume
a (b units) = (ab) units
and use this equality to compute the left hand side a (b units) --
read as a times b units.
Moreover, if a is nonzero, we assume
For instance a quarter of 12 cups, all identical, is given by 3 cups
where 3 is a quarter of 12. In symbols
(1/4) 12 cups = (12/4) cups = 3 cups.
Like wise 12 cups divided by 4 is again three cups:
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12 cups
4
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=
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12
4
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cups = 3 cups
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Division by four (4) is the same as multiplying by a quarter (¼).
B. Changing and Converting Units
Recall 1 metre = 100 centimetres and 1 decimetre = 10
centimetres. Here the unlike units metre, centimetre and decimetre
are of the same type [L] for length. So
9 metres + 8 decimetres + 3 centimetres
= 9 ×100 centimetres + 8 ×10 centimetres + 13
centimetres
= (900 + 80 + 3) centimetres
= 983 centimetres.
Alternatively 983 centimetres = 98.3 decimetres = 9.83 metres.
In calculations involving lengths, we can replace
- metres by 100 centimetres or 10 decimetres;
- decimeters by 0.1 metres or 10 centimetres; and
- centimetres by 0.01 metres or 0.1 decimetres.
Here I have use decimal notation for the fractions (1/10) and (1/100) for
convenience while typing this page. That is, we could use fraction
notation and mixed number notation as well in the foregoing.
Similarly, we can use the equations
1 minute = 60 seconds,
1 hour = 60 minutes,
1 day = 24 hours
to go back and forward between unlike units of measurement of time
[T].
Further examples could follow using units of mass and force or
weight.
C. Products of Quantities
The operations here are similar to work with monomials
where the product of 4xy with 5 xy3z is 20
x2y4z. Instead of using letters x, y and z, we
use measurement units and counting units as well. The monomials
below involve units of measure or counting in place of "variables" x,
y, z and so on.
The following examples illustrate the multiplicative computation
conventions for units.
(10 cm)(5 sec) = 10x5 (cm)sec = 50 cm sec
(3.4 cm)(2.0 kg) = 6.8 cm kg
(4 hours)(3 hours) = 12 hour2
So we take
(a unit1)(b unit2) = (ab)
(unit1)(unit2)
We take or declare a product of units
(unit1)(unit2) to be new unit, a compound
unit.
All the foregoing considerations with simple units of measurement work
with compound units. Division of units yields more compound
units and more compound fractions. Examples follow to show
how. The appearance of units alone, in products and in quotients
(fractions) in the description of rates and further proportionality
constants - exact or approximate - will provide a context for these
operations. Have patience.
Here we multiply the units and their numerical coefficients separately.
Unit multiplication behaves like monomials. We take
hour2hour5 = hour2+7
= hour7
Saying how to write the product of units or quantities of like and
unlike kind defines the operation. The operation here is a notational
convenience. In general we take
(a unitm)(b unitn) = (ab) unitm+n
So the multiplication rule for quantities is to add exponents and
add coefficient of identical units.
Operations with units and their numerical coefficient represent
exercises on paper in the first instance, but again they
are useful later in the discussion of rates, proportionality
constants and computations with physical quantities.
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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
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May 2012, Composition Starting:
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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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McCainian: drill, drill, drill then Toronto
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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
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Is your child able to add, subtract and multiply amounts
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Arithmetic
and Number Theory Skills
Algebra
Starter Lessons
Geometry
- maps plans trigonometry vectors
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Algebra
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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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