2. Proportional Reasoning - Algebraic View
Definition (1). A single quantity Y is proportional to a
second quantity X when and only when there is a non-zero
constant K such that Y = K X.
Here the direct use of Y = KX is to calculate the value
of Y from those of K and X. But the typically two step problem gives
the values (X1,
Y1) first, from which the value of the
proportionality K can be computed via a backward use of the formula.
And after K is known, the formula Y = K X can be used directly or
indirectly to compute Y or X respectively. The foregoing represents a
two step recipe for finding and then using the proportionality constant
K. The discussion of rates of changes can be included in this subject
along with development of algebraic computation skills with
units. See the site section Fractions, Ratios, Rates,
Proportions & Units.
Students may have met proportional reasoning unknowingly in the
following nine examples or situations. The proportionality can be
suggested by numerical examples or questions, and the graphing of one
quantity by another. Pick and choose the examples you like for
presentation in class, and then give the rest or further ones in
exercises.
- Average speed S for a journey is given by distance D traveled divided
by time T taken for the journey. Whence the distance traveled is
the product of speed and time.. That is D = ST. Here S is the
proportionality constant.
In the forward use of the formula D = ST, the values of
S and T are given and the value of D is computed. In the backward use,
the value of D and one of S and T are given. A typical two step
problem may say an object travels at a constant average speed over a
time interval of length T2 and ask how far the object has
traveled if the time T1 to travel an given distance
D1 is known. The first step of the solution computes
the proportionality constant K =S from the given values of (D, T) =
(D1,T1). The second step uses the formula D
= ST directly using T2 and the computed value of
S.
- The length S of arc of a circle of radius R subtended by a central
angle is proportional to the number of degrees N in the subtended
angle. The foregoing relation S = KN can be suggested via drawing
small angles and then considering multiples of them. The proportionality
constant K can be found from the fact that semi-perimeter (number of
degrees N = 180) is pR
where R is the radius of the circle. So
pR = K 180
Whence
and hence
is proportional to the product RN and hence jointly proportional to
both quantities N and R. Mastery of the latter formula means being
able to describe the suggestive geometric proportionality involved in
its derivation, and being able to use the formula
directly and indirectly, that is backwards and forwards. See Volume 2, Chapter
20. and express the calculation in chapter 20 in terms of degrees
only (not radians)
Definition A single quantity Z is jointly proportional
to two quantities X and Y when and only when there is a
non-zero constant K such that Z = K XY.
- The area A of a sector of a circle of radius R is proportional
to the number of degrees N in central angle.
The foregoing relation A = KN can be suggested via drawing
small angles and then considering multiples of them. The
proportionality constant K can be found from the fact that area
of a full circle, the case where the number of degrees N =
360 is pR2 . So
pR2 = K 360
Whence
and hence
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S
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= (
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pR2
360
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) N
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= (
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p
360
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)N R2
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is proportional to the product N R2 and hence jointly
proportional to the number of degrees N in the central angle and the
square R2 of the radius R. Mastery of the latter
formula means being able to describe the suggestive geometric
proportionality involved in its derivation, and being able to use the
formulas
directly and indirectly, that is backwards and forwards.
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Division of Fractions Example: The question of how many times T
a line segment of length X unit lengths can be divided in line segments
of fixed length D unit lengths can be viewed from a proportionality
perspective. Geometric drawings suggest that T = KX.
To find K observe T = 1 when X = D. So the proportionality
equation K in T = K X satisfies 1 = K D. Hence K =
1/D. So T = (1/D)X.
In the case D = A/B, the relation 1 = K(A/B) implies K =
B/A and hence
T = (B/A)X = X (B/A).
The foregoing argument supports the rule that division by a fraction D
= (A/B) has the same effect as multiplying by its reciprocal B/A.
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From Direct to Inverse Proportionality: The work W done in
many situations is jointly proportional to the number of workers N and
the interval of time T worked. That can be suggested by a few
well-posed questions. So
W = KNT.
That being said, this joint proportionality relationship can be used
backwards to find the value of K from values of N, T and W. Then with
the latter value of K it can be used directly or indirectly to find any
one of the quantitiies W, N and T when the other two are given or
implied by the circumstances at hand.
Now the algebraic view of the backward use of equation W = KNT.
implies the time T required to accomplish work W with N workers is
So the quantity T is proportional to W and inversely proportional to
N, jointly
Now the algebraic view of the backward use of equation W =
KNT. implies the time N of workers required to accomplish work W
in a time interval of length T s
So the number N required is proportional to W and inversely
proportional to time interval T worked.
The foregoing shows students who have mastered the algebraic
viewpoint of solving equations from earlier topics how inverse
proportionality relations may follow from direct proportionality
relations.
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When are two simple or compound fractions equal? The proportionality
connection: The question of when a fraction C/D, compound or
not, has the same value as another fraction A/B, that is the
question of when
has a simple answer. Put
Then
and so the numerator
C = KD
is proportional to the denominator D and the proportionality constant
K = A/B.
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Proportionality and change of units. Show students that
the number of centimeters in a length is proportional to the number of
meters, and vice versa with a proportionality constant k. Show students
that the number of square centimeters in a length is proportional to
the number of square meters, and vice versa with a proportionality
constant K2. The foregoing could lead to the discussion of
the relationship between lengths and areas in scale drawing, that is
plans and maps, and the actual lengths or areas. A further
generalization in exercises, if not in class, see next item,
might connect material use, volumes, areas and lengths in scale models,
larger or smaller, to unit or full scale models.
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Proportionality and Map or Model Features. In maps and plans,
and 3D models, the scale of 1 to K implies lengths and distance
in the plan, map or model is one K-th (1/K) of the actual lengths or
distance, or that the latter are K times the former. In consequence,
the area of actual real regions or surface is =
K2 the area of the corresponding map, plan or 3D model
region In consequence, the volume of actual real solids are
K2 the volumes of the corresponding map, plan or 3D model
region or representation.
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Binary and Multiple Ratios and Rates. A discussion of
binary ratios a:b and multiple ratios a:b:c appears in this
section's section
Ratios & Fractions Lessons The notation a:b and
a:b:c is archaic but still in common use. While I am quite content to
use ratio as an alternative term for fraction - all fractions are
ratios, but some ratios (those of parts to parts) are not
fractions. Something more needs to be said here. I would emphasize the
difference between the ratio of part to whole (identifiable with a
fraction) and the ratio of complementary or overlapping parts of a
whole (not identifiable with a simple fraction). To make the
distinction between ratios and fractions even clearer, I would
discuss, time permitting, multiple ratios and multiple proportions.
However, the discussion of ratios is, as indicated, an archaic topic in
mathematics courses, one that remains due to later requirements and
common conventions in society. To add to the confusion, or lack of
distinction between fractions and ratios, the ratios of a pair of
numbers, whole or not, may be called a fraction, a habit I still
keep. The site author needs further schooling in this matter.
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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
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:
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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
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.
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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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