Algebra Starter Lessons
Lessons 1 and 2 above may be easy reads. Folders 2, 3, 4 and 7 include
the key steps.
Notes for Students and Teachers
The following sixs folder and their 60 or so lessons provide an
efficient and effective path to introduce and develop algebra
skill in an observable and verifiable manner.
Steps 1, 2 and 5 could be sufficient in the first instance. Step 6 comes
after 5. Steps 3 and 4 could be placed after 5. The discussion of Algebra-Rules
in chapter 18 of Volume 2, Three Skills for Algebra, complements steps 3 and 4 and could
be recommended reading, early for gifted students and later for students closer to calculus.
Algebra Skill Development
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Formulas Forward Use In geometry, formulas appear for
perimeters, areas and volumes. Skill here defined by
requiring students to do and record steps, one at a time, one
after another, so the writer and others may see and check as
done or later. The resulting written work shows whether or
not the ability to do and write steps clearly - some would
call that communicate - is present or in need of repair. In
evaluating formulas, solving equations and eventually giving
proofs, the objective is to do and record figuring and
reasoning steps in ways that the doer and others can see and
confirm or correct, as done or later. The recommended format
in practice not only demonstrate skill in mathematics, but
also allows the domino effects of care and mistakes to be
seen.
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Stick Diagrams and Solving Linear Eqns. begins with a
stick diagram approach to solve linear equations. Learners
ill at ease with saying let x be a number after exposure to
geomtric formulas where letters denote lengths are more like
to be put at ease with saying let x be the length of a stick
or line segment, even when the length is unknown. The three
column stick diagram approach here provides a bridge for
student to cross, so that they become at ease with the notion
of letting a letter denote a number or quantity in a
context-free environment. The stick diagrams here not only
provide that bridge. They also demonstrate and exercise
fraction skills and sense. Here again the three colum format
for doing and recording the steps implies the solver and
others may see and check steps as done or later. Remember the
stick diagram is a bridge. The desired destination is the
ability to solve linears equation without drawing stick
diagrams. Note too, a format is included for checking
solutions - when check are emphasized, students may learn to
self-correct their own work before submissiong. In this, when
the check indicates an error, students need to be told that
the error itself may be between the start of their solution
and the end of their check.
2005 -
The NSDL Scout Report for Mathematics Engineering and
Technology -- Volume 4, Number 4:
... section Solving Linear
Equations ... offers lesson ideas for teaching linear
equations in high school or college. The approach uses stick
diagrams to solve linear equations because they "provide a
concrete or visual context for many of the rules or patterns
for solving equations, a context that may develop equation
solving skills and confidence." The idea is to build up
student confidence in problem solving before presenting any
formal algebraic statement of the rule and patterns for
solving equations. ...
Once students have master the stick free, algebraic approach
to solving linear equations, easily solved simultaneous
systems of equations may be introduced. Examples here systems
which are triangular, and systems in essentially one unknown.
Many junior high school words problems which can be
translated into one equation in one unknown may be more
easily translated into a system in essentially one unknown, a
system easily solved. The latter two step route will be
easier for some to follow. Show both routes to student, and
give them a choice. Here again how to check solution needs to
be emphasize to make students more independent. In the case
of word problems, it is important to verify the original
problem is solved and not just the equation[s] used to solve
it - errors in the translation of words into equations are
possible.
The discussion here of easy systems provides another bridge,
here to the solution of simultaneous linear equations in two
unknowns. That latter step in equation solving, easy given
the earlier ones, lies in the mastery of one to three
Gaussian elemination methods for solving two equations in two
unknowns. The key lies in showing how to convert these
systems into a system of equations in essentially one
unknown, a system that is also triangular.
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Computation Rules. A formula or other methods
which says how to obtain or compute one number from another
or several others may be called a computation rule. In
geometry, examples are provided by formulas for perimeters,
areas and volumes. In number theory, examples may be provide
by formula 2n and 2n+1 for even and odd numbers, or by
formulas 3n+ 2 for whole numbers equal to two, modulo
three. We informally introduce function notation for
these computation rules, and to indicate when one number or
quantity depends on another. In the next item, a computation
rule viewpoint of axioms is given to ease or avoid common
difficulties in their comprehension. And in the later study
of functions, equivalent computation rules - those that give
the same result - may represent the same function, but have a
different form. Food for later thought: A function may
be given or calculated from its graph. It may be also given
by a formula when the latter is available.
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Axioms - Computation View: Many axioms or assumed
properties of whole, rational and real numbers indicate when
two different computation rules give the same result. The
distributive law a(b+c)=ab+ac is geometrically implied for
unsigned numbers to illustrate and introduce the concept.
This approach shifts the context of discussion from
properties of numbers to properties of computation rules. In
learning to master the axioms, students may find the idea
that two computation rules f(a,b,c)= a(b+c) and g(a,b,c) =
ab+ac easier to understand and numerically verify - perhaps
with the aid of calculators - than the drier statement:
a(b+c)=ab+ac whenever a, b and c are real numbers. The
suggested shift in context may make axioms easier to
understand and explain. The logical shortcomings, if any, of
this shift may be left to reflection in the later university
level education of would-be mathemticians.
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Formulas Backwards. The lessons here slowly, carefully
and informally in great detail introduce the numerical and
algebraic-literal backward use of areas formulas and the
compound interest formula. The treatment here is informal in
the sense that axioms are not invoked to justify every step.
None the less, the aim is to introduce patterns numerically
and then rephrase the patterns algebraically as a bridge to
the general algebraic-literal solution of equations without
the bridge to become self-evident. The site discussion of
solving equations also repeats very similar numerical
solution in order to provide a bridge to the
algebraic-literal solution. In the UK, the backward use of an
equation A=f(W,L) in which the subject A to obtain a formula
for say W would be called changing the subject.
Talking about the backward use of formulas and rules is not
limited to formulas and equations. In logic, a backward use
of an implication rule A if B would be the assumption Not B
if Not A. In calculus, the backward use of differentiation
rules and methods leads to formulas and methods for
integration. In the study of proportionality relations, alone
or interrelated, backward use of the associated will be
needed find the associated proportionality constant. That may
be a first step in the furrther use. In all mathematical
skills and subjects, Illustration of the direct or forward
use of rule, pattern or formula needs to be accompanied by
the verbal declaration: Sooner or later, we will be using
this rule, pattern or formula backwards. Every teacher and
tutor may emphasize that to introduce more words into
understanding and explaining rule and pattern use, and to
alert students that doing things backwards is a deliberate
part of mathematics and logic mastery.
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Proportionality Back/Forwards: In numerical examples of
proportionality, use of a proportionality relation-equation
may begin backwards with the determination of the value of a
proportionality constant. After that the proportionality
constant may be used in a further direct or indirect use of
the proportionality relation-equation. In case of similarity
by design in 3D, the lengths, areas and volumes of similar
objects are proportional to a scale factor, its square or its
cube. Three simultaneous proportionality relation result.
Each may used to obtain the scale factor - a backward use.
After that, each may be used directly or indirectly in
finding lengths, areas and volumes of corresponding parts. In
two dimensions, the situation is simpler with two
simultaneous proportionality relations instead of three. Here
distance=speed times time and many formulas in physics may be
cast as proportionality relations. Given one proportionality
equation, related ones can be algebraically-literally derived
by changing its subject. So the forward and backward use of
equations is everywhere.
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Notes for Instructors
For skill development, the topics marked (**) are recommended in full.
Read all the marked items before use in lesson planning and delivery.
Reasons for each step in them are given Before writing began I saw and
sensed gaps in course design and delivery. Here are my remedies. Before
starting algebra skill development, or beside it, you want want to check
and solify student arithmetic skills. These starter lessons reflect and
support inductive principles for skill development online in site Volume
1B, Mathematics
Curriculum Notes.
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Working With Sets: Sets skills and concepts need to be
covered. But when is not clear. Here is a very rough draft of lessons
to appear here. Ignore them or suggest how to improve them.
Set language appear in the description of real numbers, rationals
numbers, integer, natural numbers and whole numbers. Set skills and
concepts appear in the expression of probability concepts and for
probability theory, in the counting of possibilities. Set skills and
concepts also appear in the discussion of functions after these algebra
starter lessons.
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Formula Forward Use - Evaluation (**). Lesson here provide
format for doing and recording the formula evaluation steps in an
observable and confirmable or correctable manner. There-in a remedy
for students not knowing how to show work.
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Solving Linear Equations(**). This folder in four subfolders
provides a [step by step] path for developing algebraic reasoning.
The first step in showing how to solve some linear equation
geometrically with fractional operations on pairs of line segment -
sticks diagrams has two purpose. From formula evaluation, students
are use to letter denoting lengths that may be known or given. In the
first steps, letters denote the unknown length of sticks, visible
stick, a length that is to be found. That may make the role of
letters to denote unknowns easieer for students to accept and employ.
For students whose fraction skills and sense may be weak, the
multiplication and division of line segments by whole numbers in the
stick diagram approach may provide or c consolidate skills and sense.
The first step is optional. But all four steps show how good format
in doing and showing work, provides steps for the solver or others to
check, confirm or correct. The format further makes the domino effect
of student care and errors easier to see.
Requiring students to check their work and showing how implies
students will know whether or not their solutions are in need of
correction before solutions are submitted. When a solution check
fails, we may tell students that the mistakes are located between the
start of the solution and the end of the check. The mistake may be in
the check and not in the solution - so proceed with caution. Students
need to be warned against erasing work in tests and finals. Tell them
to cross out work (one or two lines is sufficient) and to leave it in
place, so that instructors may see it and even give credit for it.
Skill when it appears, cross-out or not, should be recognized.
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Computation Rules and Function Notation(**). The function
notation y = f(a,b,c, ...) is introduced to indicate or emphasize the
dependence of a quantity or number y on other numbers or quantities.
Computation rules here are given by formulas involving one or more
quantities. The intent here is to shift algebraic thought of students
from operations on letters, letters whose role in denoting numbers is
often too complex for students to grasp, to operation on computation
rules. The objective is to make algebra easier to understand and
explain.
In the physical sciences, writing y = f(r,s) implies y depends on the
quantities r and s, but without an explicit computation rule for
f(r,s) - the function f(r,s) may be given by many different
computation rules. At each point of it domain, each computation rules
that work are expected to agree, else further discussion of what is
the value of y is required. See the dicussion of equivalent
computation rules - the site alternative to the discussion of
identities - identities of the computation kind that is, there be no
other kind in secondary level mathematics. In modern
mathematics curricula, functions may be introduced using different
computation rules over all or part of a domain. Then the question
arises whether or not the function is well-defined, that is,
consistently defined.
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Real Numbers. These lessons present and review whole numbers,
natural numbers and rational numbers before introducing real numbers.
Following that arithmetic operations on real numbers are introduced.
The last lesson covers comparision.
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More Less Greater Than Inequalities and Comparison. Before the
introduction of signs or signed numbers, numbers and quantities were
compared by size or magnitude, and not by position. This folder
explains both kinds of comparison for real numbers, and explains the
difference.
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Axioms Logic and Equivalent Equations. The real number axioms
for associative, commutative and distributive properties of real
numbers are intepreted as rules for saying when different computation
rules give the same result.
Equivalent Computation Rules, Arithmetic Identities and
Algebra. For a, b and c positive, the distributive law a(b+c) =
ab +ac says two different computation rules f(a,b,c) = a(b+c) and
g(a,b,c) = ab+ac always gives the same result.
This distributive law may be implied by saying the area of a
rectangle with sides of length a and b+c is the sum of areas of two
subrectangles, one with sides of length a and b, and the other with
sides of length a and b.
The casting of real number axioms for associative, commutative
and distributive properties of real numbers as rules for saying when
different computation rules give the same result may make some axioms
easier to understand and explain. Computer or calculator programming
which employs formulas may re-inenforce this viewpoint.
In sum, this folder talks about addition and multiplcation axioms -
the context of computation rules appears here; gives a zero-product
axiom in direct and contrapositive form; explains how to rewrite
division and subtraction, so that addition and mutliplication axioms
may be applied; talks about equality in algebra and equivalent
systems of equations.
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Unifying Theme for Algebra (**) -Using formulas and Rules
Forwards and Backwards. The statement that a person is doing
something backwards has previously be employed a critical
criticism or correction. Yet in mathematics, logic and science, rules
and formulas have to be mastered forwards and backwards. The concept
that rules and formulas will have to be used Forwards and
Backwards with both directions occuring alone or together in
routine problem solving may serve as a unifying theme in the learning
and teaching of science, of logic - see implication rules and of
mathematics at the college and upper secondary level, if not before.
For example, subtraction and division may be done or introduce via a
backward use of addition and time tables. In upper level,
pre-university, secondary mathematics, the backward use of functions
leads to inverse functions. And in calculus, the backward use of
differentiation methods provides integration methods while the
backward use of an implication rule provides criteria for divergence.
Using rules, formulas and tables backwards may be seen from counting
to calculus. in the
Using formula backwards may be a fourth skill for algebra. The
lessons here show how to so numerically and then algebraically or
literally. The latter process represents the power of algebra to
solve many like problems at once. The numerical solution in the first
instance are developed, and follow by an algebraic solution. The
pattern of the first solution provides a pattern for the algebraic
solution. The aim here is to help students make the transition from
arithmetic solutions to algebraic [also called literal] solutions.
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Adding Words to Mathematics. Advanced and frontier level
studies and research in mathematics express or codify pure if not
applied mathematics in terms of symbols, letters and diagrams. This
codification has been called tongue-in-cheek marks on paper. Many a
true word has been spoken in jest. Advanced level avoids the use of
physical ideas and avoids commentary that goes beyond those pure
marks on paper. There-in lies a silence that amplifies the natural
tendency for silence in mathematics. In arithmetic and in algebra,
many expressions and formulas are too complicated and long to be read
aloud, term by term, in a manner that the human mind can grasp or
see. Yet a person with sufficient skill or training in algebra way of
writing and reasoning, may view an arithmetic or algebraic expression
and understand it in a glance. Just as a picture may be worth a
thousand words, an expression or formula better seen and read or
understood in silence, may also be worth worth a thousand words -
being too long or difficult to grasp when read term by term, the path
that computer programs follow. Yet the two words Mona Lisa in
naming and identifying a picture of Leonardo De Vinci may represent
the picture and invoke its image in the mind of one who has seen.
Thus two to several words in naming or identifying a picture or an
expression may also be worth a thousand words.
A balance is required. Rules and formulas once named in mathematics,
logic or science or in any field, may discussed or talk about in
conversation and in chains of reason or communication. Are you
familar with a rectangle or circle area formula? Are you familar with
the compound growth and decay formulas? Do you know the quadratic
formulas? If yes, the names or identifying phrases here may bring
those formula or images of them to mind. Otherwise, the names and
phrases identify objects to study. Mathematics skill development may
become stronger if names and descriptive phases are repeatedly and
redundantly employed to identify rules and formulas. That may permit
longer conversations with and without writing instruments: If you
remember the quadratic formula with letters a, b and c being the
coefficients of x-squared, x and the constant term respectively, then
replacing a, b and c respectively by .... leads to the answer.
Besides identifying expressions and formulas with names and phrases,
we may talk about or comment on numbers and quantities; describe
calculations with words and symbols, or both, which ever is the least
difficult; and through the ideas of substitution and equivalent
computation rules, move from one expression to another for numbers
and quantities. The site essay on what is a variable talks
about numbers and quantities in a pedagogical sound, informal manner,
which may serve as a prequel to the formal specialize, abstractions
of what is a variable in logic and mathematics - abstractions that
serve technical ends, not skill development ends. Words can also be
included to introduce and explain how letters and compound symbols
may represent numbers and quantities, and how they should do so
unambiguously.
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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:
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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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