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Numerical Solution of Algebraic Equations

By Frances Reyes,2014-06-17 01:48
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Numerical Solution of Algebraic Equations ...

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    Visual Algebra for the Early Grades

    George Mondras

    There are two ways to solve an algebraic equation: 1) the axiomatic method that will be

    taught in the later grades and 2) the visual method that can be taught in the early grades.

    Visual Algebra is a method that was prevalent at the dawn of the mathematical sciences. At

    the University of Alexandria, 310 B.C.E., geometric methods were used in the solution of ?algebraic equations. For the equation: 2 x = 8, each term in the equation was depicted as an

    area and a geometric diagram was constructed. In the first textbook on mathematics, The

    Elements, by Euclid, the fourth axiom, “Things which coincide with one another are equal to one another,” is compelling evidence that algebra was geometric in that era. A student would ??conclude that equality requires that the two rectangular areas are equal; therefore length x must

    coincide with length 4.

    ?The geometric solution: 2 x = 8

     ?? 2 x ?? 2 8 2

     x4

    ?????? Symmetry in algebra is achieved when an equation is reformulated such that the form of

    the equation is the same on both sides of the equality. Visual algebra does not solve for the

    unknown value of the variable, but applies the axioms to reformulate and render an equation

    symmetrical. The solution is then found implicitly by a one-to-one matching of terms.

The reflective symmetry of the human form is a natural concept that is intuitively understood, is

    ingrained in everyone and was once utilized in the early geometric development of algebra. The

    visual method is geometric algebra, with symmetry substituted for geometry. Reformulating an

    equation to obtain symmetry utilizes the student’s most primitive ingrained concepts.

The solution using symmetry:

    ?2 x = 8

    ??2 x = 2 4 x = 4, from the one-to-one matching of terms

     ??The concept of one-to-one matching is most likely deduced from the result of a simple

    experiment that everyone does very early in life: match thumb to thumb, index finger to index ????finger, and so on to obtain a perfect left to right hand match. A classroom demonstration that

    one-to-one matching is within the purview of the students; ask anyone to respond if they think

    the students in the room is greater than the number of chairs. No one will respond. The students

    need only observe one empty seat or that everyone has a seat and they can say with certainty that

    the number of chairs is greater or equal to the number of students. There is a more advanced way

    to determine the facts, count all the chairs and count all the students then compare the two

    numbers. Counting is advanced and requires prior knowledge whereas one-to-one matching is

    primitive.

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    Definition of Algebra:

    Algebra is the language of higher mathematics.

    1) The numbers are the nouns of the language.

    ?, ?, ?, ?, ?, , and so on, tell us what action to take, so they play the 2) The operators,

    roll of verbs in the language.

    3) Assumptions that are postulated for algebra are the grammar and syntax of the language;

    five assumptions were postulated to develop the early geometric algebra: 1) Things that are ??equal to the same thing or to equal things are equal to each other. 2) If equals are added to equals, then the

    results are equal. 3) If equals are subtracted from equals, then the remainders are equal. 4) Things that

    coincide with one another are equal to one another. 5) The whole is greater than the part.

    Algebra is the first level of abstraction in mathematics and requires the use of literals to

    represent numbers. In an equation, the literals initially have unknown or variable values that can

    be determined by algebraic means. The literals are generally the letters of an alphabet: A, a, B, b,

    C, c, X, x, and so on; using capitals letters first and lower case later in parallel with student early

    learning. Avoid the use of the literals I, i, O, o, and lower case ell l as they are too easily

    confused with zero and one. Do not overuse the literal x and, if the numbers have a tangible meaning, try to choose a literal that acts as a memory aid: N for number, M for Mary’s age, V for

    velocity, T for temperature, etc.

    One possible way to introduce visual algebra is to initially have the students conduct the

    finger-matching experiment (see page 1). The pedagogy can then proceed through equations that

    are already written in translational symmetry form. The K-1 students would only be asked to

    identify and state the unknown value of the variable by the one-to-one matching of terms. As

    skills develop, the equations can require arithmetic operations to achieve translational symmetry.

Define Visual Algebra:

     Apply the axioms to one side of an equation to reformulate and achieve symmetry. Solve for

    the variable implicitly using a one-to-one matching of terms. Check any modified expression.

    Visual Mantra: Rewrite the equation until both sides look the same.

     Apply the axioms to one side of the equation to achieve final symmetry a?x?b?c

    c?bc?b Translational symmetry; x = by one-to-one matching of terms a?x?b?a?()+baa??c?b Reflective symmetry is a possible reformulation but is not likely a?x?b?b?()?aa

     ????Examples: Acronym: TS = Translational symmetry

    3?x?5?3?2?5a) ? Translational symmetry; x = 2 by the one-to-one matching of terms ??

    3?A?5?3?(1?2)?5b) ? TS requires 1 operation; note that A is now 3 3?B?5?3?(2?2)?2?3c) ? TS requires 2 operations; note that B is now 4 ??

    3?D?5?3?(3?4?2)?7?2d) ? TS requires 3 operations; note that D is now 6

    ??3?d?5?3?(6?2?4?2)?6?1e) ? TS requires 4 operations; note that d is now 5

    ??3?X?5?3?(3?6?2?10)?4?1f) ? TS requires 4 operations; note that X is now 1

    ??3?a?5?3?a?5g) ? Equations, written in translational symmetry form, can be reformulated

    ?? in many ways to enable the students to acquire both arithmetic and algebraic skills.

    ??

    ??

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    Visual Solution of Equations:

    Acronyms: LHS/RHS = Left/Right Hand Side.

    English sentence: Three times the quantity x, in addition to five is eleven.

     x + 5 = 11 Algebraic sentence: 3 ?

    3?x?