Adding Words to Algebra
While some calculations may be described with words - think of the
perimeter calculation example, many calculations can be represented or
given by an algebraic expression or formula that is best seen and grasped
in a silences in a glance. Formulas like the compound interest or growth
formula and like the quadratic formulas fall in this category of being
seen and grasped in silence. The use and development of formulas for
lengths, areas, volumes, speed, distance, time, simple interest, compound
growth and even or odd numbers, introduces students to the shorthand role
of letters and symbols in mathematics.
A. Naming and Identifying Formulas
Outside of algebra, a picture is worth a thousand words. Inside, formulas
to awkward to read aloud term by term may also be worth a thousand words.
However, both pictures and formulas may be named or identified with
words. Outside of algebra, the phrase Mona Lisa identifies a painting of
Leonardo da Vinci. Inside of algebra, formulas may be identified by name
- the compound interest formula
- the quadratic formula
Inside algebra, descriptive or identifying phrases also identify
formulas.
- Triangle area calculation formula
- Square area calculation formula
- Circle Perimeter Formula
- Circle area formula
- Sphere or ball surface area formula.
- Sphere volume calculation
- Cube surface area expression
- Box volume formula
In these phrases, the words formulas, expression and calculation may be
used interchangeable.
In dealing with fractions, we may speak of general and/or efficient
methods or formulas for adding, subtracting, multiplying and dividing. At
the higher level in the mathematical subject of calculus, the phrases
product rules, quotient rule, chain rule, integration by parts
identify operations with words.
Words to name or identify formulas and operations is a simple way to
expand the role of words in mathematics.
B. Talking about Numbers, Amounts and Quantities
Following the appearance of algebra and logic in and apart from
mathematics education, the oral element introduce above may expand. The
first skill for algebra which I introduced in fall 1983 emphasized that
we can talk about numbers, amounts and quantities without doing any
arithmetic and even apart from or parallel to the use of letters and
symbols to denote them. So we may talk about and describe numbers,
amounts and quantities as being known or not, measure-able or not,
private or not, confidential or not, forgotten or not, constant or not,
and varying or not. Some of these descriptive terms are not usually part
of mathematics, but their presence suggests how we may understand and
explain the concepts of a numbers, amount or quantity being known or not,
constant or not, or variable or not. I have written an essay What is
a Variable that informally introduces and expands the concept before
algebra begins. While pure mathematics and logic introduce technical
definitions of what is a variable, those definitions are too complex for
people just learning algebra. In the first instance, when a letter that
denotes a number, amount or quantity that is unknown, constant or
variable then the letter too will be called an unknown, constant or
variable, respectively. And in the use of formulas, letters often denote
numbers or measure whose value is to be given or found. What I am
advocating here is a simpler use of language, a use closer to everyday
use. While numbers, amounts and quantities may be described or talked
about apart from algebra or the use of letters to denote them or their
values, doing so while denoting them by letters or symbols expands the
role of words in algebra and so in mathematics.
C. Using Formulas etc Forwards and Backwards.
A theme that transcend algebra. A Fourth Skill For Algebra.
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
Volume 2, 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.
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)
The exercises will be easier after reading the first sections of Chapter
15 and Chapter 14 in Volume 2, Three Skills for Algebra.
Remark. In arithmetic to calculus and beyond in mathematics and
the physical sciences, and in the use of implication rules in or from
logic, all rules and formulas will be used forwards and backwards, some
time bothways in the same problem, especially those involving
proportionality constants. In the latter, finding one represents
a backward use of a formula, after which the formula
may used forward or backwards.
use.
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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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