9 Proportionality Backwards and Forwards
1 What is Proportionality
2 Algebraic View
3 Proportionality Examples
4 Rates Ratios and Proporitionality
5 Proportionality in Equivalent Fractions
Proportionality Relations
forwards and backwards
The following pages summarize the main ideas. That sets the stage
for a more detailed or elementary approach - another site to do.
Proportionality Concepts
and Practices- Three plus Kinds of Proportionality Relations,
Forwards and Backwards: The lesson says what is (defines)
Direct, Joint, Inverse Proportionality and describes how to
shift or generate proportionality relations from each
others. In a proportionality relation (or
equations), algebraically interchanging the dependent quantity
with an independent one via a backward use of the relation leads to
further proportionality relations of the same or different type.
The use of proportionality relations begins with the backward use
problem of finding the value of a proportionality constant. Once
its value is known, the proportionality relation can use in the
forward direction to find values of the dependent variable, or in the
backward direction to find values of a so called independent
variable.
Proportional
Reasoning, algebraic perspective
Twenty or so
Examples of Proportionality and Multiple Ratios or Proportions:
Many examples of proportionality relations appear in high school
mathematics and physics. Here is a list of some (most if
not all) that may be met. Remember each proportionality
relation will be used forward and backwards in multiple
ways.
Two
and Multiple-Term Ratios, a proportionality constant viewpoint.
Fraction and ratios are overlapping concept and have overlapping roles
in arithmetic, but they are not identical even though fractions a/b
where a and b are whole numbers may be called ratios. In mathematics
ordered pairs of whole numbers a and b may appear in coordinate form
(a,b) or [a,b]; in ratio form a:b and in fraction form.
Proportionality
Constants for Equivalent Fractions: The numerator is
proportional to denominators in any fractions equivalent to a given one
- a simple matter.
An Algebraic Pre-Requisite
The Forward and
Backward Use of Formulas and Equations introduce
a universal & unifying theme in the mathematics and science.
This theme first appeared in Volume 2, Three Skills
for Algebra, chapters 10 and 14.That being said, the algebra
starter lessons, provide a
newer treatment of this theme.
The two equivalent phrases Forward and Backward Use (or
Direct and indirect use) voice, identifies and emphasized what
has hitherto been a silent theme in the teen and adult mathematics
education. The phrases spoken repeatedly in the classroom will alert
students to this common thread and the need to understand and master
it.
Chapter 10 considers the backward use of the rectangular area
formula A = WL where W denotes the width and L denotes the length of a
rectangle
Direct or forward use of the rectangle area formula A = WL calls
for the value of A to be calculated from given value of W and L.
A first backward use of this formulas will find the value of the width
W from the values of area A and length L. Finding the length L from the
values of A and W would be another backward or indirect use of this
formula. Chapter 10 does that exercise algebraically in the hope that
readers will follow.
Chapter 14 in Volume 2 employs the more complicated Compound
Interest formula in the form A = P(1+i)n directly
and indirectly (forwards and backwards), and gives both arithmetic
(numerical) and algebraic (literal) solutions to solve backward use
problems. Every formula met in high school and
college mathematics and science is likely to be used backwards and
forwards. The arithmetic approach to this may be easiest or most natural
for students in the first instance, but the algebraic approach and it
ability to solve many problems at once points to a power of algebra.
Mastery of the algebraic approach with that power is the objective.
The algebraic approach is essential, not all powerful.
For Right triangles, the Pythagorean identity c2
= a2+b2 between leg
lengths a and b, and hypotenuse length c is never
used directly. The near forward use would obtain c from the
principal square root of a2+b2 before or
after substitution of values for a and b. The arithmetic solution would
involve substitution first, while algebraic solution would
involve substitution after. A backward use find a, given b and c
values, would obtain a from the principal square root of
c2- b2 before or after substitution of values for
a and b in the identity.
Between the forward and backward use of formulas for area of
rectangles and compound growth A = P(1+i)n, formulas for area
of triangles, squares, r circles, trapezoids, parallelograms and
polygons; for volumes of spheres, cylinders, cones, pyramids, and boxes
(parallelepipeds); and for perimeters of triangles, rectangles, circles
and so on, provide opportunities to illustrate and reinforce the backward
use of equations using arithmetic and algebraic solution methods.
Algebraic expressions for systems of linear equations in 3 or
more unknowns, can be derived, but the derivation and their expression
is so complicated numerical methods for solutions are preferred (except
in special cases).
|
|
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.
|
|