A Fourth Skill For Algebra
Volume 2, Three Skills for Algebra
Direct and Indirect Use of Formulas, or Forwards and Backward Use of
Equations
Every formula met in mathematics, accounting, science, technology etc may
be used directly and indirectly, that is forwards and backwards.
The simple message that the forward and backward use of formulas
(direct and indirect use) is part of high school mathematics and
beyond names a required skill and allows us to recognize,
identify and thus emphasize the most frequent pattern in high school
mathematics and beyond.
This message needs to be given explicitly and early in secondary
mathematics. Otherwise the underlying skill become part of the hidden, or
silent and unspoken, agenda in mathematics courses.
Teachers: Consider combining the www.purplemath.com a two page lesson on
solving
literal equtions with the message above and the examples and
exercises indicated below. The page banner above was Forward and
Backward use of equations but it now reflects the purplemath
lesson, Solving Literal Equations.
First Site Example
Direct and Indirect Use of the Rectangle Area Computation Formula
Chapter 10 in discussing Direct use of A =WL assumes W and L are given.
Indirect use assumes A and one of W and L is given, and leads to the
calculation or formulas W = A/L or L = A/W. The explanation of
those formulas is a step towards algebraic reasoning - the direct and
indirect or forward and backward use of formulas.
More Examples: Formulas for perimeters and areas
of squares, circles, triangles, rectangles etc can be used
forwards and backwards. Finding the value of a proportionality constant
k say in an equation y = k x represents an indirect or backwards
use of an equation, a pre-requisite to further forward and backward use
of the equation y = kx. The calculation of parameters a and b in
y = ax + b (or y = mx +b) represents another backward use of a formula
or equation. Quebec students in secondary III have met the
forward and backward use of the Pythogorean equation
c2=a2+b2 where c is the length of the
hypotenuse and the two numbers a and b are the lengths of the other two
sides (legs) of a right triangle.
To Do: : Post some details and exercises
here to further illustrate and emphasize the forward and backward use
of common formulas.
Going Further (More on Substitution)
Chapter 10 before the forward and backward use of a formula goes further
in showing how to describe a the calculation of a box V = H(WL) and show
how to employ substitution (a new concept for students) to go between
this formula and V = HA where A = WL. Details are
given in the chapter. The details may be easier to grasp if
numerical examples are added to this exposition.
Seeing how a box volume formula V = hA and V
= h (WL) can be transformed into each other illustrates and may
introduce the notion of equivalent expressions. The law applied here is
A = WL is a geometric law rather than an algebraic law (like the
distributive law). None, the idea that an expression represents a
number or quantity and that there may be more than one ways to compute
the number or quantity is key to the notion of equivalence.
Students thus see how substitution in formulas leads to new
formulas, how arithmetic patterns may be used to use formulas
directly and indirectly, and how algebraic solutions may be more
general or powerful than arithmetic solutions.
Algebraic Exercises:
- Find a formula for the area of square in terms of its perimeter
(easy)
- Find a formula for the area of circle in terms of its perimeter
(easy)
- Find a formula for the perimeter of square in terms of its
areas (harder)
- Find a formula for the perimeter of circle in terms of its
areas (harder)
See www.purplemath.com two page lesson
on solving
literal equtions for hints or to learn more.
The exercises could be easier after reading the first sections of Chapter
15 and Chapter 14 here in Three Skills for Algebra. The chapter 15
material may be easier.
The first sections in Chapter 15, >Solving Linear
Equations derives an algebraic formula for the solution of equations of
the form ax + b = c, and so emphasize the use of algebraic shorthand
reasoning to imply solutions for many problems of a given form at once.
All the foregoing emphasizes the power of algebra, or the shorthand way
of writing and reasoning with letters in place of numbers. That being
said, numerical experience is still required with formulas and their
graphs, otherwise the connection between numbers and algebra may too
weak.
A Deeper Site Example
for now or later or never.
Chapter 14 introduces the direct and indirect use of
the compound interest formula A = P(1+i)n.
Chapter 14 presents algebraic and arithmetic solutions that may be used
to check the calculator skills of students while developing the
algebraic way of writing and reasoning. In the compound
interest formula A = P(1+i)n three of the four amounts
A, P and i and n are assumed known, and the problem is calculate or
find a formula for the missing fourth. The use of this formula is
indirect when the left hand side quantity A is given or known, and the
task is to find the value of the principal P, the interest rate i or
the number of compounding periods n. Add to chapter 14
coverage, numerical confirmation that the algebraic solution
works. The algebraic solutions for the indirect use of formulas involve
substitution and assumes the pattern (AB)/B = A.
Coverage of Chapter 14 is recommended as part of the next topic:
exponents and radicals.
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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:
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.
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