8. Mixed Measures and Mixed Numbers
Examples of proper fractions are: one third, two-quarters, three
sevenths, and nine tenths. Proper fractions with numerators smaller than
denominators - tops smaller than bottoms - appear in counting the
subdivision of an object into equal parts. But measurement of length,
time and mass [or weight], may be given by a whole number of units plus a
proper fraction of a single unit. Examples are given by one and a half
meters, six and 2 tenths minutes; and four and three quarter kilograms.
Now a single meter equals two half meters. So one a half meters is the
same as three half meters or 3/2 meters. Likewise, six and 2 tenths
minutes is the same as sixty-two tenths of a minute since 10-tenths make
a whole. Finally, four and three quarter kilograms is the same as 19
quarter kilograms since 4 quarters make a whole and so 16 quarters make 4
whole units. In general, proper fractions appear from counting
subdivision while improper fractions appear from expressing a
measurement, say whole of number of unit plus a proper fraction, in terms
as a count of subunits.
Mixed Measures: In describing lengths of time, we may
talk about days, hours, minutes and even seconds, and by convention all
mixed units in this description. Thus we speak of 2 hours and
15 minutes without convention requiring a conversion into a large number
of minutes, or into a small mixed number 2¼ of hours. The
length seven quarters of a meter describes the same
length as 1.75 meters and 1 meter 75 cm. In describing long thin
rectangular, we say the length is 5 m and the width is 80 cm. For some
that description may be more pleasing than saying 500 cm by 80 cm, or 5 m
by 0.8 m. How we describe measures does not affect them but
the numbers in the description depend on the choice of units. The
area of the foregoing rectangular is given by the product of it
dimensions in square meters, in square meters or even as a number
of meter-cm. The latter would be the area of a rectangle with
length one meter and width one cm. The area of the rectangular is
given by three different expressions
A = 5 m × 80cm
A = 5 m × 0.80 m
A = 500 cm × 80 cm
The foregoing leads to three answers: 400 m × cm, 4
m2 and 40000 cm2 for the area. Each has the same
value since 1 m2 = 10000 cm2 and 1 m ×
cm = 100 cm2
There is no harm in using mixed units of measures in evaluating formulas
as long as the unit carried through the steps. Conversion of the
the different units for a measure may be done in any step or in the
original data. What is important is that a measure be describe as a
number of units and not by by a number alone. In general, arithmetic with
measures may be done with mixed unit of measures alone or multiplied and
divided by others. While the form of a result may vary, its values will
not. The following are example of addition using and keeping mixed units
of time measure:
5 hours, 30 minutes + 4 hours, 20 minutes = 9 hours, 50
minutes
2 hours, 50 minutes + 3 hours, 50 minutes = 6 hours, 40
minutes
In the latter, a conversion of 100 minutes into 1 hour, 40 was
don.
An operational mastery of fractions with units will help
Note: While pure mathematics may avoid the carrying of units in
and through calculations by selecting a a consistent system of units for
calculations, the intellectual overhead in selecting that consisting
system and converting all units of measure to it may be avoided by using
and carrying units of measure through calculations. That is the
practice in senior high school and college courses in chemistry and
physics. Moreover, fraction skill with units present, and manipulations
with products and quotients of units of measure in general, useful
for the description of speed, rates and proportionality constants.
There is more immediate motivation and context to this
manipulations with units of measure as is or multiplied by numbers, than
there is to the combination of monomials in letters w, x, y and
z in products and quotients.
Mixed Numbers: In the early development of decimal notation,
the digits 1 to 9 represent simple numbers while two or three digit
numbers like 42 and 368 represented mixed or compound
numbers. The latter represent the sum of 4 tens and 2 ones, and the
sum of 3 hundreds, 6 tens and 8 ones respectively. So we count in
mixed groups: hundreds, tens and ones. Now the fraction 5
quarter- meters represent a whole number of the unit one
quarter meter. We may write 5 quarter meters as one meter and one
quarter meters. That mixes the unit of length measure one meter with the
unit one quarter meter. Now the mixed number 4½ stands for 4 wholes
and one half a whole. The mixed units of counting here are ones and
one half. Now in counting or measuring we may find ourselves with 4
wholes, ¾ of a whole, and ½ a whole. The total count or
measure will be 5¼ wholes. The underlying notion here is that we may
count and measure with whole and with multiples of unit-numerator
fractions, more easily written here in word form as one half, one third,
one fourth, one fifth and so on. A mixed number or measure is equal
to a whole number of ones plus a proper fraction: multiples of
fractional units. In general, we add, subtract, multiply and even
divide mixed numbers and measures of ones and units where the units of
counting and measure may be different. The practice of raising
terms for the sake of addition, subtraction or comparison is resembles
the conversion of mixed units of measure into multiples of a common
unit. For example (in a long format chosen to illustrate
ideas)
2
3
|
+
|
3
4
|
=
|
2
|
×
|
1
3
|
+
|
3
|
×
|
1
4
|
|
|
|
|
|
=
|
2
|
×
|
4
12
|
+
|
3
|
×
|
3
12
|
|
Convert one third units and one quarter wholes
into one twelfth units
|
|
|
|
=
|
|
|
17
12
|
|
|
|
|
|
Count number of twelfths : 8 = 9 = 17
|
|
|
|
=
|
|
1
|
5
12
|
|
|
|
|
|
Convert 12ths into wholes
(that resembles the conversion of more than 10 tenths
in wholes in addition with decimals.
|
Remark: Notions of mixed numbers and measures underlying
many arithmetic operations with counts, decimals, fractions and
measures. Clarification of those notions may help us to decide
whether or not, or to what extent we should discuss them in developing
mathematics skills.
|
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.
|
|