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Welcome. The explanation of solving linear equations in chapter B to F provide
one pillar for algebra mastery. Chapters 8 to 12 and 14, 16 & 17 in
Volume 2, Three Skills for Algebra, provide a second pillar.
The following pages offer a new path for learning and teaching how to
solve linear equations in high school or college.
[ Proper Use of Equal Sign ] [ A. Letters and Lengths ] [ B.. Solving Linear Eq'ns - WS ] [ C. Solving Linear Eq'ns - No Sticks ] [ D. Almost One ] [ E: 2D Systems - Sub Method. ] [ E: Continued ] [ E: Still More ] [ F. Larger Systems ]
The path reinforces arithmetic and algebra skills and concepts. The path
may begin with stick diagrams, but it ends without them.
| Teachers:
The use of stick diagrams is a temporary measure. Remedial
mathematics education may employ stick diagrams but for students
entering calculus, the stick diagram approach to solving linear
equations should be a distant memory. |
Notes
Item A. Proper Use of Equal Sign
is a must in solving linear equations, and in testing or verifying found or
given solutions.
Item B, Stick Diagrams, leads to
sequence of pages explaining how to use stick diagrams to solve some linear
equations.
Stick diagrams by themselves 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 before any formal algebraic statement of the rule
and patterns for solving equations.
Using geometry to develop algebra and fraction skills and
sense. For students starting algebra, it far easier to allow a
letter to stand for length of a given or drawn length that has yet to measured
or than is to allow a letter to denote a number yet to be given. That makes
stick diagrams more amenable to the development of algebraic skills and sense
through a three column presentation methods which equates stick diagrams and
operations on them with linear equations and operations on them.
Cryptic Summary: Stick
Diagrams provide a new invention for the visual solution of a single
equations in one unknown, say x, starting with a pair of equi-length
sticks to represent the left and right side of the equation and then
proceeding through a sequence of operations which yield further pairs of equi-length
stick corresponding to simpler and simpler equations until the last pair gives
the value of the unknown x. Stick diagrams provide a visual guide and
introductory crutch for solving special linear equations which students will
eventually abandon in order to solve the general linear equation in one
unknown while learning the necessary skills on route. The sequence of
operation on stick diagrams illustrates and strengthens arithmetic with whole
numbers and fractions in a visual manner. Examples here use letters
other than x to as the unknown length in an equation. The sticks in the
diagrams might be replaced by vectors when negative constant terms are
permitted - an extension that might be useful, yet may not be needed due to
intended transient nature of the stick diagrams.
Learn More
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The optional start, lesson A.
Letters and Lengths, material drawn from a chapter 9 in site volume
Three Skills for Algebra, shows or recalls how letters may denote
lengths in the computation of areas of rectangles, triangles and circles. the
notion of describing or denoting a length by a letter is key to mastery of
stick diagrams, a notion that may be obvious to some and not need further
explanation.
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Site pages on Fractions,
Ratios, Rates, Proportions & Units are for
students who want know-why besides their know-how. They are also for
teachers and tutors. Explore the pages one at a time and one after
another. Skip those not to your liking. Pages 13 to
15 cover ratios, simple or multiple. Pages 16 to 18 introduce units in
calculations and provide a setting for the discussion and definition of
rates and proportionality constants.
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Exercises on Mostly
Fractions will test fraction know-how.
Nibbling
at all in parallel is an option until their digestion is complete.
Teachers - the stick diagram method makes solving linear
equation more accessible, but not universally accessible. : Some students may follow the stick
diagram reasoning without immediately seeing how to write the corresponding equations.. For such students,
we might accept mastery of the stick
diagrams while pushing the algebraic equation viewpoint and give exercises
in translating the stick diagrams into equations to remedy this difficulty or as
a preventive measure at the very start of the coverage of stick
diagrams. Simplification; An example of the form x +
a = c where a < c are whole numbers would provide a simpler introduction to the use of stick diagrams. Some
students need it.
A story to repeat
In learning to read, write and spell all letters in the
alphabet must be learnt. We would not tell a student who is learning to read
and write that he or she only had to learn 20 letters in the alphabet, and
that the rest of the alphabet could be skipped. That lead would lead to
substandard reading and writing. Likewise in mathematics, we
cannot let students believe that parts of arithmetic and parts of algebra are
optional, and can be skipped. College, consumer and workplace
mathematics require arithmetic with decimal and fractions at full strength for
employment in calculations or simply as a sign of the ability to follow
methods and conventions precisely and fully, with care and patience. In
arithmetic and algebra, there is chance to develop the habit of following and
applying rules and methods carefully (or paying attention to details) in
a way that leads to repeatable, reproducible, verifiable and reliable results.
The latter has value in and beyond mathematics. That provides a first
motivation for mathematics.
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Solving Linear
Equations
|(Feb 14, 2005)
with & then without stick diagrams plus
testing solutions -do not hand-in untested solutions;
solving word problems; and solving systems: - essentially one
unknown, essentially triangular & general
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Skill in arithmetic with fractions is a must for
algebra.
Folder Chapters -
lesson groups
Proper Use of Equal Sign
A. Letters and Lengths
B.. Solving Linear Eq'ns - WS
C. Solving Linear Eq'ns - No Sticks
D. Almost One
E: 2D Systems - Sub Method.
E: Continued
E: Still More
F. Larger Systems
Area Intro
(i) x + 20 = 29 WS
(ii) 2x + 5 = 20 WS
(iii) 3x + 10 = 32 WS
(iv) 5a + 16 = 3a+ 24 WS
(v) (½)x + 8 = 24½ WS
(vi) (¾)a + 16 = (¼)a+ 24 WS
(vii) (¾)q + 17 = 32 WS
(viii) 13 =[2/3]x +7 twice WS
(ix) Animated Examples WS
(a) Integral Coefficients (A)
(b) Integral Coefficients (B)
(c) Fractional Coefficients
(d) With parameters
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