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ADAM CLEMENTS SME 430 Discussion Post #8 3/23/2010

Complete the readings for this week and then post your reflections to the prompts below:

Writing Algebra with Symbols

What is your own definition/description of algebra?o A part of mathematics that uses letters or symbols to represent quantities.

Is there a formal definition of algebra? (If so, what?)o A question about numerical operations and relations in which an unknown

quantity must be deduced from known ones.

o www.m-w.comdefines algebra as the following: 1: a generalization of arithmetic in which letters representing numbers are

combined according to the rules of arithmetic

2: any of varioussystemsor branches of mathematics or logic concernedwith the properties and relationships of abstract entities (as complex

numbers, matrices, sets, vectors, groups, rings, or fields) manipulated in

symbolic form under operations often analogous to those of arithmetic compareboolean algebra

What are the characteristics of algebra that distinguish it from other branches ofmathematics?

o Its a simple universal language (if all symbols are understood by the reader)where as something like logic requires more knowledge of content to understand

what the question is even asking.

o Symbolic symbols are used and rhetorical style is not used as it still is in otherforms of math.

o In an algebra problem, regardless of how its written, is a question aboutnumerical operations and relations in which an unknown quantity must bededuced from known ones.

How is a symbolic style different from a rhetorical style in algebra? Give an example ofeach, and state one advantage of each approach.

o Symbolic style The style of algebra that we are more commonly familiar with today x3 5x2 + 7x = x +6 Advantage: takes up less space, more precise, less confusing, follows

universal language of math, can represent more than one unknown o Rhetorical style

Uses words to write the equation The cube and seven things less five squares is equal to the root of six more

than the thing.

Advantage: You need to understand what the symbols mean and any onewho knows the language should be able to read it and understand what

you are trying to say.

http://www.m-w.com/http://www.m-w.com/http://www.merriam-webster.com/dictionary/algebrahttp://www.merriam-webster.com/dictionary/algebrahttp://www.merriam-webster.com/dictionary/algebrahttp://www.merriam-webster.com/dictionary/boolean+algebrahttp://www.merriam-webster.com/dictionary/boolean+algebrahttp://www.merriam-webster.com/dictionary/boolean+algebrahttp://www.merriam-webster.com/dictionary/boolean+algebrahttp://www.merriam-webster.com/dictionary/algebrahttp://www.m-w.com/

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Solving First Degree Equations

Solve the following problem using the false position method: A quantity and its fifthbecomes 24

o x + 1/5x = 24o

say its 5, so 5+1= 6o so we got 6 and we 24 so we need to multiply by 4o so (5)(4) = 20 and (20) + (1/5)(20) = 24 so x = 20

Solve this same problem using the traditional algorithm: A quantity and its fifthbecomes 24

o x + 1/5x = 24o 6/5x = 24o x = (24)(5/6) = 20

How is the mathematics used in these two solutions the same? Different?o Same:

You get the same solution

You are using standard algorithms of multiplication and additiono Different:

The first method allowed for simpler computations and using smallernumbers, but uses trial and error. The second uses a more direct path, but

you have to have more complicated and larger number and complication.

Will the false position method work for all first degree algebraic problems? How doyou know?

o Yes, but it only works if the equation is Ax+B. If instead the equation were Ax+C = B, the it is no longer true that multiplying x by a factor causes B to change

by the same factor, and this simple version of the method breaks down. We might

try subtracting C from both sides, but that isnt always as easy as it sounds,

because the expression on the left side might initially be very complicated, andfinding the correct constant to subtract would require us to simplify it to the form

Ax +C.

Create your own double false position problem and solve it . Be sure to include youroriginal problem that you started with as well as a description of all the steps to took to

solve it.

o A bowl of 200 grapes is divided among four friends, A, B, C, and D, so that B has3 more grapes than A, and C has 3 times as many as B, and D has the same as A.

o 200 = A + B + C + Do 200 = x + (x+3) + 3(x+3) + xo So I guess x = 20o So A has 20, B has 23, C has 69, and D has 20 which added up is 132 so we are

off by 32.

o So we guess lower, x = 15o So A has 15, B has 18, C has 54 and D has 15 which added up is 102 so we are

still wrong and off by 2

o 20 32X

15 2

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o 20 x 2 = 40o 15 x 32 = 480o 480 40 = 440 / (132-102) = 14.66666o So person A gets 14 and 2/3 grapes