Algebra Islands - First Set
Past or present algebra learning difficulties can be mostly eased or
avoided by introducing smaller and extra steps for skill and concept
development, and by expanding the role of words in providing an informal
understanding and mastery of the shorthand roles of letters and symbols
in mathematics. Difficulties may be avoided by showing the following
algebra islands
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How to evaluate formulas and arithmetic expresions in an observable
and verifiable show work manner, step by step;
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How to solve linear equations from one equation in one unknown to
triangular systems, essentially one systems of equations, and systems
in 2 unknowns - more if wanted;
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How to recognize equivalent computations before and then in axiomatic
description of the properties of numbers.
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How to using formulas and proportionality relations forwards and
backwards, numerically and then algebraically - a.k.a, literally.
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How to talk about and describe numbers, amounts, quantities and
calculations with words before and besides symbols and marks on
paper, formal concepts included;
These islands are described one at a time, one after another below. The
logical structure of algebra emerges with the recognition that some
computation rules are equivalent to each other, or imply each other; and
with the comprehension of axioms and how these implication rules or
patterns are used in identifying equilavent numerical expressions and
equivalent computation rules. The logical structure further emerges with
the recognition that proportionality relations are used forwards and
backwards, often in the same problem.
Learning how to talk about numbers and calculations extends the oral
dimension of mathematics. The latter may and should include the numerical
or arithmetic practices of finding counts, sums and products by forming
and adding or multiplying subcounts, subtotals - subsums, and subproducts
directly or iteratively. The latter should also include the naming of
formulas, and the exact or suggestive description of calculations by
phrases and slogans. Learning to talk about numbers, measures calculation
and equations may extend common and technical skills and know-how for
people meeting with or sans the ability to show each other on paper, what
each is thinking. The introduction of more words in understanding and
explaining mathematics and logic may help the blind as well.
The mastery of linear equations ax+c = cx+d in one unknown is usually
introduced in an algebraic manner. However as an early part of geometry,
letters may be used to identify points, line segments and their lengths
in maps, plans, diagrams and geometric formulas. The names square and
cube for x2 and x3 echo the initial role of letters
in denoting lengths. Following the historical development of algebra, the
site introduction to solving linear equations begins with letters
denoting the length of a visible line segment or stick before shifting
away from that to allow a letter to denote a unknown number, one that may
be without physical significance.
- Stick Diagrams
The site introduction of fractional operations on stick diagrams takes
advantage of earlier acceptance of letters in denoting lengths of visible
line segments, known or not. The stick diagram approach introduces a
three column method for solving linear equations with physical or
fractional operations on line segments - the sticks. That provides a
visible and hands-on or manipulative mechanism to slowly introduces
algebraic reasoning without use of stick diagrams. In practice, I have
seen this approach works well for most if not all. The further advantage
of the stick method is that it visibe encompasses addition, subtraction,
division and replication or multiplication operations on line segments,
all in a way that improves fraction skills and sense to the point that
linear equation ax+c = cx+d with fractional and not integral coefficients
a to d are easily solved. There other hands on or manipulative approaches
to solving linear equations currently in use - the stick diagram approach
here appears to have the least overhead.
The stick diagram approach besides opening an easier route into solving
linear equations also builds and re-inforces fraction skills and sense
through addition, subtraction/cutting, division and duplication or
multiplication operations on line segments.
- Triangular, Essential and then General Systems in 2 Unknowns
After students have learnt to solve linear equations ax+c = cx+d in one
unknown, skill is easily provided in solving triangular or essential
triangular linear systems directly, and in solving systems of linear
equations in essentially one unknown via substitutions that result in one
equation in one unknown. Many word problems in secondary mathematics that
can be cast as one equation in one unknown may be cast with less trouble
and thought as systems of equations in essentially one unknown. The
latter casting will make word problems easier to understand and solve.
Following mastery of systems of equation in one unknow via substitution
and thuse elimination of all unknowns except one, students may be shown
how to rewrite systems of equations in two unknowns as a system in
essentially one unknown.
- Checking Solutions
By showing students how to check solutions to linear equations and to
word problems, students may see how to find and correct their own
mistakes before the submission of written work for correction. During
solutions and solution verification errors are possible. Students may be
told that when a check fails, the error in their work may be seen between
the start of their solution and the end of their check. A check failed
means more checking of both solution and checked is needed. It does not
always imply the solution itself is in error. That's life.
Further Reading: Chapter 15,
Solving Linear Equations in site Volume 2, Three Skills for Algebra,
introduces the algebraic or literal solution of one equation ax+b = c by
writing and rewriting numerical examples with solutions to imply an
algebraic pattern. The popularity of the latter chapter with search
engines undermines my notion that the site section on solving linear
equations should be read first.
Algebra Island: Order of Operations to Equivalent Computation Rules
In learning to evaluate arithmetic expressions, formulas included,
students should learn that the order of operation matters, that care has
been taken in evaluation, and that different expressions or formulas
often give different results. Site algebra starter lessons begin with
formula evaluation examples to introduce a repeatable and reproducible
format for doing and recording calculation steps in ways that can be seen
and check as done or later. Being careful in that, and avoiding the
domino effects of errors becomes an end and value for evaluation
exercises.
Each algebraic formula represents a different computation rule. However,
the arithmetic properties of numbers imply that different calculations
will always give the same result. For example, the distributive law
$a(b+c) = ab+ac$ say that different computation rules are equivalent and
even interchangeable because they lead to the same result. Where students
of higher mathematics may see that $f(a,b,c) = a(b+c)$ and $g(a,b,c) =
ab+ac$ define the same function because of their knowledge of the
distributive law, students at a lower level will not. Instead they may
and should see two different computation rules. The calculations are
clearly different even we may know the give same results. That knowledge
may be confirmed or implied by giving students a collection of different
computation rules to evaluate by hand or with the aid of a computer
program.
With the aid of geometry and with the computer programs that compute
different formulas, students may be able to understand and interpret
most axioms for real numbers as statement of when different computation
rules give the same result. Computer programming or on paper
calculations may introduce the notation that substitution of equivalent
computation rules into equivalent computation rules leads to equivalent
computation rules. Whence axioms or assumptions about equality and
replacement may follow explicitly. Or, instruction may quietly employ
the accompanying practices. The former route is more complete but it is
more detailed and hence overwhelming for some. The latter is not a
complete, but less theory or etail may be best for some students. There
is no pleasing all.
Most algebraic stated field axioms for real numbers may understood as
statements about the equality or equivalence of different computation
rules. That gives a functional or computational perspective clearer an
easier for many to grasp, and hence more inclusive. That shift in
perspective provides an operational command of these axioms good enough
to serve the needs of all students outside of pure mathematics. .
The shift sets the stage for the silent or explicit discussion of
substitution principles in terms of equivalent computation rules, or
functions.
Money Matters
In the context of money matters, formulas for compound interest or
growth and formulas for geometric sum may be introduced numerically and
used forwards and backwards, algebraically and numerically. Before the
study of mathematical induction, the formulas may be introduced and
confirmed in numerical exercises or by
numerical examples.
Arithmetic exercises in site Volume 2, Three Skills for Algebra,
Chapter 7, section 2.4, numerically hint at formulas for geometric
sums.
The forwards and backward use of formulas numerically and algebraically,
may be presented as a unifying theme for the role of algebra in
mathematics and science. Talking about forward and backward use of
formulas and proportionality relations, numerically and algebraically, or
literally, also represents a larger unifying theme from addition and time
tables in arithmetic to differentiation and anti-differentiation in
calculus.
Repetition: We may emphasize how every rule and pattern in logic, in
mathematics and it application in daily life or science will be
eventually used forwards or backwards. For example, in the study of basic
skills, the backward use of addition and times tables may introduce or
strengthen subtraction and division abilities. In logic, the indirect or
backward use of an implication rule A IF B says IF NOT A then NOT B.
The forward and backward use of formulas, algebraically or literally, is
called changing the subject of an equation in the United Kingdom.
So the concepts is not new. However, site chapters and steps explaining
it try to make the hard easier to learn and teach. If that is the case,
related formulas given earlier may be shown to imply each-other by
changing the subject of an equation or formula. So fewer need to be
mastered.
Related Skill: Volume 2, Chapter 15, Solving Linear Equations,
implies the algebraic or literal solution of the linear equation ax+b =
c for x by solving the equation for three or four different sets of
coefficient a, b and c, always in the same format. Later on in
calculus, we will follow the same pattern for the algebraic evaluation
of limits that depend on parameters. The evaluation of parameter
dependent limits sets the stage for deriving formulas for derivatives
$f'(x)$ from formulas for $f(x).$ So instructional methods of
introducing and extending the algebra way of writing and reasoning may
be used and re-used at different levels.
Proportionality Relations
Proportionality equations from direct to inverse square and joint all
represent formulas that appear forwards and backwards in a single
problem, or in the mathematical description of a topic outside of
mathematics, say in chemistry or physics. Once a proportionality relation
is written as a formula or equation, finding the value of the
proportionality constant there-in represents a backward use. Once the
proportionality constants is known, the proportionality relation may be
employed forwards and backwards to find missing values. That is a
numerical application. Or, with the proportionality constant known or
assumed, the subject of the proportionality equation may be changed
algebraically or literally to imply a related proportionality relation.
Scale Factors in Geometry : In the algebraic study of scale
factors K, K2 and K3 for length, hieght, volume
and proportional quantities, simultaneous proportionality relations
hold in which the scale factors K, K2 and K3
serve as proportionality constants. Finding the value of the scale
factor K directly or indirectly, then set the stage for many forward
and backward use problems.
Emergence of Logical Structure in Algebra
Comprehension of how formulas or equations may be emphasized as a source
of strength in skill development in a multilevel manner. That is,
students may thrive with the latter combined and detailed axiomatic
account of it. Others will be overwhelmed. From my perspective there is
no rush: the students who survives calculus well will not be overwhelmed
by the details, if or when they are given. However, at this level, the
aim is not to derive all formulas and statements from a minimal set of
axioms, but merely to provide a consistent and sufficient set of
assumptions and practices to imply and support common needs.
In showing how some formulas may be derived from others, and in talking
about the forward, backward, isolated and chained use of implication
rules, connections between formulas start to emerge and so one aspect of
the logical structure of secondary mathematics emerges.
Algebra Island - Enlarging the Oral Dimension
Duplicate Paragraph: Learning how to talk about numbers and
calculations extends the oral dimension of mathematics. The latter may
and should include the numerical or arithmetic practices of finding
counts, sums and products by forming and adding or multiplying
subcounts, subtotals - subsums, and subproducts directly or
iteratively. The latter should also include the naming of formulas, and
the exact or suggestive description of calculations by phrases and
slogans. Learning to talk about numbers, measures calculation and
equations may extend common and technical skills and know-how for
people meeting with or sans the ability to show each other on paper,
what each is thinking. The introduction of more words in understanding
and explaining mathematics and logic may help the blind as well.
In the past, the existence of arithmetic and algebraic expression better
seen and read in silence than read aloud, term by term, has been an
obstacle to oral communication in mathematics. In mathematics mastery,
there is no hidden agenda, but there has been silences, a lack of oral
tradition for understanding, explaining and rationalizing skills and
concepts. Given that obstacle, the role of letters and symbols in
mathematics has not been verbally introduced or described. Mathematics
has consisted of marks on paper and geometric activities, written or seen
while verbal development of skills and concepts has been missing. Site
Volume 2, Three Skills for Algebra, provides a correction in the
following chapters:
In this phase or the next two, talking about three skills for algebra and
illustrating with examples in Chapters 8 to 10; and then talking about
the shorthand roles of letters and symbols in Chapters 11 and 12,
altogether extend the verbal rationalization and mastery of algebra. That
may enrich skills and comprehension - fill some gaps in people's command
of algebra. The essay What
is a variable provides a simple, explanation of the underlying
concept - one verbally more basic than the use of letters to denote
numbers and quantities that may be constant or variable.
The correction is continued in one online postscript
A Fourth Skill For Algebra and five more chapters:
Chapter content provide a mix of lessons and background information for
use in lesson planning or for assignment as extra reading for students.
Chapter 18 is long. It contains some sections I would not cover in class,
but might be good for extra reading by students who want to learn more.
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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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