The Instigator
Marshall-Abarca
Con (against)
Losing
0 Points
The Contender
Bannanawamajama
Pro (for)
Winning
8 Points

Should Algebra be necessary for a high school diploma?

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Post Voting Period
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after 2 votes the winner is...
Bannanawamajama
Voting Style: Open Point System: 7 Point
Started: 7/18/2013 Category: Education
Updated: 3 years ago Status: Post Voting Period
Viewed: 2,529 times Debate No: 35755
Debate Rounds (5)
Comments (12)
Votes (2)

 

Marshall-Abarca

Con

People often times confuse different types of math with algebra, for example I've seen people say that doing taxes is a form of algebra. This is in fact false, taxes relies on basic numbers and doesn't really use integers nor variables which Algebra does use. It's also worth it to look at the fact that doing your taxes involves % now it may have been just me but I don't recall learning anything about % in either my Algebra classes 1 or 2. Just go look up "basic algebraic equations" and you'll see they have nothing to do with your daily life's use of numbers and math. It's for this reason that we shouldn't have Algebra as a requirement because of the fact that unless you are going to specialize in it and become a teacher or something else, it has no practical use in life for the most part. We should be focusing funds for classes on more vocational classes like maybe a "daily life" class where people learn to do taxes, or tips for paying off mortgages, or this could be placed into "economics."

Now let me say right away that I'm sure there may be some form of
BASIC Algebra that can be used in life like say "x+4=10 find 'x'" that can be seen as both basic Algebra and possible normal math as it could be written as "10-4=?" but overall most algebraic equations are not likely to be used in day to day life.[1]

The definition of algebra is as follows via Merriam-Webster
Algebra - A generalization of arithmetic in which letters representing numbers are combined according to the rules of arithmetic.[2]
In practically ever website I've looked at problems relating to Algebra are not something people would use in their daily life, just look at these lists of general areas algebra is seen. Some include things like "exponents" "polynomials" "radicals" etc. It's not hard to see based off of these examples that Algebra isn't something that the average person would use in their daily life, therefore it shouldn't be something that's required for a highschool diploma or to be promoted from grade to grade.[3] [4] [5] [6]

Just go through some of the sources and look at some of the problems that Algebra has in store, and ask can these problems be placed into daily life?

[1]http://www.washingtonpost.com...
[2]http://www.merriam-webster.com...
[3]http://www.coolmath.com...
[4]http://www.math.wsu.edu...
[5]http://www.freemathhelp.com...
[6]http://www.math.com...
Bannanawamajama

Pro

I. INTRODUCTION
I hearby accept your debate. As we discussed, I will start my arguments in the first round. I apoligize that some of this may be redundant to you, but since we are continuing a conversation we already started elsewhere, I want my main points on this page for reference, so I may repeat things you have heard before. I look forward to a civil debate.

II.MAIN PREMISE
Now to start, I would like to address your claimm that taxes rely on "basic numbers" rather than variables or integers.

First, I'm not sure what you mean by basic numbers, however, Integers are simply numbers that do not contain a decimal component. They are in fact one of the most basic sets of numbers you can consider.Second, I would like to provide you with a link to the US National Finance Center, where you can look up the tax calculation for any state [1]. For brevity, I won't repeat what is typed verbatim, but the method of calculating taxes is as follows:

Taxes due = F(annualincome) - (biweeklyincome - SavingsPlancontribution - FederalHealthPlan -(2250* 26*biweeklyincome))

Where F(x) = the taxes you have due based on a piecewise taxbracket function. This is the official given method of calculating your due tax and potential exemptions given by the bureaucracy. As you can see, it is a mathematical function that includes many values which are not given. These values, which I state in terms of words to make easy to understand, correspond to your individual conditions. I intend to argue that this in fact does make it algebraic in nature. When I do so, it should prove the necessity of algebra, as you yourself admit in your opening the importance of such tasks as tax payment as relevant to real life.

III. DEFINITION OF ALGEBRA
I will agree to your use of the webster dictionary as a credible source. However, an alternate definition is given on the same page for Algebra, defining it as "Generalized version of arithmetic that uses variables to stand for unspecified numbers. Its purpose is to solve algebraic equations or systems of equations. " [2]. To explain this supposed inconsistency, we must define a Variable. Looking at the term Variable from a variety of source, in the fields of Physics, Computer Science, and Math, along with the Webster definition, we get the following trend:

"a symbol (usually a letter) standing in for an unknown numerical value in an equation."[3]
"deliberately changed by the experimenter"[4]
"a way of naming information for later usage"[5]
"A symbol for a number we don't know yet"[6]

From these we see a common definition emerge: A variable is a symbolic representation for a value we do not know yet. It is either something we are looking to find out, or something that we choose before computing a specific solution. Source [4] defines this distinction as "Dependent" and "Independent" variables. Important things to note are that variables CHANGE, as implied by its root of vari-. Also, source 3 states that while a variable MAY be a letter, it is not always. Indeed, in Computer Science, variables are just as often words or conjunct letters to clarify their purpose. "Age" for example is a common variable used to store the age of a person in a database.

We can now reconcile our definitions of algebra. I assert that your definition of "arithmetic in which letters representing numbers are combined" is refering to variables. This is evident by the word "representing". Including letters is not sufficient for algebra, those letters must be indicative of a distinct value, the value of which will modulate the answer to the question. By applying this, we can see that the two definitions we provide line up and can be used interchangably. The lack of use of the word "variable" in your definition was likely just to avoid confusing people unfamiliar with the term.

By combining these definitions with the explanation of variables I have given, we could define Algebra as an application of mathematics where certain values of an equation or equations can be changed or chosen by the operator without invalidating the equation as a whole. That is to say, it is math where you can control a certain value and the problem will still have a correct solution. This value is a variable. Where there is more than 1 variable, like y=mx + b, variables would be determined to be "dependent" or "independent" as mentioned before, where you choose "independents" to solve for a "dependent". For future reference, I would here note that a value which is not a variable or operation is known as a "constant"[7].

Looking back at the taxation equation I provided earlier, you see that this fits this derived explanation of algebra. There are independent variables which the user inputs, such as gross income or deductions, which provide a value for a dependent variable, the taxation. The use of an algebraic system is that while "10-4 =" is applicable to only one situation, "10-x =y" can be used with any x to find a y. This is important because hopefully you will continue to earn more money over time, and as such your taxes would increase. Rather than putting out a 100000-row table which gives you your taxes for every possible amount of money you can earn, an algebraic equation gives a method that will apply universally.

IV. FUNCTIONALITY OF HIGHER MATH
Since your contention is with higher forms of algebra, it would be appropriate of me to provide you an example that shows that these more complicated procedures have real-life applications just like leser problems. Your given source 4 names exponentiation as a concept belonging under algebra, so I will provide evidence that exponentiation is useful.

Let's assume you 5 years from now. You've worked for a decent amount of time, and you have $3000 dollars that you don't need for anything now and don't expect to need in the near future. You want to invest this money. One option of course, is a bank. Banks usually provide a rather low interest rate, so lets go with 0.5% to make calculating easier. Alternatively, a buisnessman is offering you the chance to invest in the startup for an entrepeneureal project. He gives you an iron clad contract that says if you give him your $3000, in 10 years he will give you it back plus a 5% bonus. Now obviously 5% is much better than 0.5%, but which investment should you choose?

Clearly the way you decide is to figure out how much each investment will give you and pick whichever is higher. Someone looking at this problem superficially might assume at first glance that the returns are equal, since 10 years times 0.5% equals 5%. This however, is not accurate. When you calculate out the resultant gains, it is seen that a higher return comes from storing money in the bank. How is this result reached? By using exponents. Using the rules detailed on your own given site[7], the return from a bank is D*R^n with D, R and n denoting your deposit, rate, and number of incrementations. Now you may argue that a "common sense" method of calculating this value could be reached, by simply multiplying your base 3000 by 1.005 10 times in succession. But you would find that this, in fact, is the very definition of exponentiation. It is a source that you yourself provided which confirms this, as well as classifying its operation as an aspect of algebra. It is necessary, by this definition, for you to use exponentiation to project your future financial situation if you ever need to deal with interest rates. This includes mortgage, bank loans, car loans, penalty fees, and a number of other issues that all adults in modern society face.

1.https://www.nfc.usda.gov...;
2.http://www.merriam-webster.com...;
3.http://www.merriam-webster.com...;
4.http://www.nuffieldfoundation.org...;
5.http://cscircles.cemc.uwaterloo.ca...;
6.http://www.mathsisfun.com...
8.http://www.math.wsu.edu...;
Debate Round No. 1
Marshall-Abarca

Con

Thank you to my opponent for accepting my challenge. Now to the debate

Re: Main Premise

Integer - "A number with no fractional part. Includes the counting numbers {1, 2, 3, ...}, zero {0}, and the negative of the counting numbers {-1, -2, -3, ...}."[1]

What I meant by "basic numbers" were basically positive integers, I used the term "basic numbers" because those are the first numbers we learn to use. However negative numbers aren't something that we use usually, while I agree they should be taught I believe we should really emphasize problems relating to positive numbers more.

The method of calculating taxes however won't support your argument for Algebra, and here's why. You say "it is a mathematical function that includes many values which are not given." True in that formula no specific numbers are given, however look at the math problem "a+b=c" that is basic math right there but because it's just a outline of how math (in this case addition) works doesn't mean it's algebra. If we knew the specific economic conditions of someone all of those points could easily be filled in with numbers, which then make the method of tax calculation a normal math problem.

RE: Definition of Algebra

Your method of defining algebra by using the definition of a variable is flawed and here's why, a variable in biology and other lab sciences is different from definition used in Algebra. The mathematics definition of variable, which is the only definition relevant to this topic.

Variable -a symbol for a number we don't know yet. It is usually a letter like x or y.[2]

going off of the mathematics definition of what a variable is, you see that it's something that we have to calculate basically hence the basic goal of most algebra problems being "what is x." To further support the fact that different fields have different definitions of variable here's one from "thefreedictionary.com"

1. Something that varies or is prone to variation.

2. Astronomy A variable star.

3. Mathematics

a. A quantity capable of assuming any of a set of values.

b. A symbol representing such a quantity. For example, in the expression a2 + b2 = c2, a, b, and c are variables.[3]

This all being said, the majority of you argument defining algebra collapses, you also go off and make your own definition of Algebra based again off of irrelevant information.

You also say "That is to say, it is math where you can control a certain value and the problem will still have a correct solution."
However this is flawed in the sense that in algebra we have predetermined variables, for example: x + 4 = 10, now we obviously know that x=6 however we had no control over that whatsoever, the same goes for more complex equations, if I may make an analogy it's like a game of sodoku, where you don't know which numbers go where but that lack of knowledge doesn't change the fact that a "9" will go in the most upper right hand corner.

Another quick argument is you say that the word "age" can be a variable when calculating perhaps statistics on a computer, and that is 100% true, however you fail to connect how this has anything to do with Algebra while statistics is a form of math it isn't the same as Algebra, for example how often is the word "age" used as a variable in Algebra?

To end this part, you look back on the tax equation you provided however this is in no way Algebra because of the fact that all of these numbers are pre-determined by said individual’s current situation.

here's the method of taxes.
Taxes due = F(annual income) - (biweekly income - Savings plan contribution - Federal health plan -(2250* 26*biweekly income))

However this isn't algebra because of the fact that all the numbers needed for a sum are available. In Algebra the sum is usually the available number along with one of the parts of said sum, let's try plugging the numbers in.
Taxes due = (120,000) - (5000 - 1200 - 800 -(2250* 26*5000) *rough estimates I know, they may not make sense but this is just off the top of my head*

These numbers again are already available for the calculator when he/she needs them, there's no unknowns other than the sum, however this again is just basic math.

Re: Functionality of Higher Math

Ok, you use the example of bank loans and exponents to show how Algebra is useful in daily life; however you made some mistakes here. For example your layout of how the calculation would happen if flawed.

Let's start with a very basic problem of what an Exponent is
example: 2^3= 2*2*2=8[4]

Correct me if I'm wrong but this seems different than what you said.
"Now you may argue that a "common sense" method of calculating this value could be reached, by simply multiplying your base 3000 by 1.005 10 times in succession. But you would find that this, in fact, is the very definition of exponentiation."

If we go off of your example that 3000(d) * .005(R)^10(N) then the first thing we'd do is .005*.005*.005*.005*.005*.005*.005*.005*.005*.005 = , this is because of PEMDAS (Parenthesis, Exponent, Multiplying, Dividing, Adding, Subtracting)

What you said "mulitplying you base 3000 by 1.005 10 times" ignores PEMDAS, not to mention I don't know how you got 1.005 since

.5% = .005^10 = 0.000000000000000000000009766 &
5% = .05^10 = 0.00000000000009766

Not only does your example not make sense, but even if the numbers you listed worked it still wouldn't make sense since you didn't use exponents correctly. This is not a legitimate example of exponents, and therefore not a legitimate example of Algebra being used in daily life.

Just for more support here's some Algebra problems, please tell me how these can be used in daily life.[5] [6] [7]

[1] http://www.mathsisfun.com...
[2] http://www.mathsisfun.com...
[3] http://www.thefreedictionary.com...
[4] http://www.coolmath.com...
[5]http://www.coolmath.com...
[6] http://www.coolmath.com...
[7] http://www.math.wsu.edu...

Bannanawamajama

Pro

NUMBER THEORY
I concede that yes, natural numbers are simpler than integers. However, I can't imagine that anyone is truly confused and frustrated by the concept. Adding a negative is like subtraction, they both even use a similar symbol (-) to make it clear.

However, integers are still widely used. For example, temperature uses negative numbers, as do bank accounts. It is not simply enough to say that your account has $0 if you have overdrawn from it, you actually owe money to the bank and only negative numbers express that. In addition, when you get a bank statement, some banks, at least mine, use positive numbers to denote deposits and negative to denote withdrawals when you are looking at your account summary[1] (In the image you can see the negative numbers). It saves space this way, because while storing a negative or positive integer takes the same space in a database, storing both a number and a value that describes it as a withdrawal or deposit takes potentially twice as much space per transaction.

In any case, I do not see this as a matter of Algebra, it is under a broader study math that deals with classification. You can tell this because you can have likely done problems like "-4 +8/2 =", before you ever took an algebra class, so learning integers is not specifically an algebraic concept.

VARIABLES (CONT.)
I agree that the words are used in different fields, but all incarnations of "Variable" show consistency in purpose. The word derives from the Latin root Vari-[2] which means "change". In each of the definitions you state in your argument, and in those I gave previously, there is always an element of change in each given definition. The construction of the word itself supports my interpretation, for to not change would make the label incorrect.

As for the "X+4" argument, consider: If X was a stable value, then you would never have to solve for x again, since you know it equals 6 based on that equation. Obviously this is not the case. X is defined by on the parameters imposed on it, in this case the x+4=10. For taxes, yours is a valid application of the equation, but what happens when someone with the same pay as you but different deductions uses the same equation? He pays different Taxes. This is because the variable of deductions changes, and yet he can use the same equation to find his solution.

To clarify, my "age" comment was only to establish that a variable is not always a single letter, it can be denoted in any way, such as a word. I wanted to see if you agreed with that is all.

READRESSING OF INTEREST CALCULATION
Alright, I think we have misinterpreted each other on the calculation thing. Allow me to restate what I meant.

Yes, 2^3 is 2*2*2. This is successive multiplication, where you are multiplying 3 2's together, one after the other. That's what I meant by that. Now where I think you are getting confused is using 0.005 for the interest value. While this is the value given by the bank for interest, you notice that if you multiply any number by a number less than 1, you will get a smaller number. Gaining interest on say a bank account actually increases your money though. For this reason, you use 1+ your interest when calculating, which is how I got 1.005. The reasoning behind this is that every time you accumulate interest, you end up with 100.5% of the money you had before, which gives an increase. So if you are accumulating interest over 10 years, you will have 10 times that your money increases by 0.5% of what it was before. Here is a full explanation of the process I am using[3].

I would ask how you would calculate this out in a way that does not use exponentiation or successive multiplication. Keep in mind that for a $3000 deposit and a .5% interest, your answer after 10 years should be approximately $3153.42. This is consistent with exponentiation, because for my proposed (3000*1.005^10), if you take 1.005^10, you get 1.0511, and if you multiply that by 3000, you get the correct answer. This even follows PEMDAS, as Exponentiation has precedence over Multiplication. However, without exponents, I don't see any simpler or more efficient way to get the solution.

ECONOMICS AND ACADEMICS
You've mentioned your intent to replace Algebra with Economics as a class to teach mathematics that you consider relevant. There exists already an economics course in many high schools in the US, in fact there are two AP Courses, Micro and Macroeconomics[4]. Since AP is the closest to a national standard by which certain high school courses base their curriculum, I will use it as an example of the class subject matter. By looking at the course description, we see that both Micro and Macroeconomics are based on the study of market and market trends, including supply and demand and elasticity. Nothing is mentioned, however, about how to manage personal finances or calculate mortgages. The study of economics, in an official academic setting, is based on business considerations, and as such has nothing to do with the life skills you are advocating learning.

This brings up the question of where exactly you define "relevant" education. Surely after 2nd grade once you can spell English classes are meaningless, and History is never used on a daily basis. Neither is basic Chemistry or Physical Education. Since your resolution is concerned with relevance, I would ask you to define which, if any, courses in current academics you believe to be worthy of necessitating before I continue.

EXAMPLE PROBLEMS
5. Using the distributive property shows up when you're dealing with both variables and constants. Say you need to build 5 fence segments, and a fence needs 2 nails per post plus an extra 4 for the ends. So you write this as 5*(2x+4) nails. X is the number of posts per fence segment. The distributive property lets you deal with elements you can't combine yet collectively.

6. Multiplying radicals is directly necessary to resolve Geometrical problems, specifically when they include circles or trigonometry. The most common angles we use for most things are 0, 30, 45, 60, 90, and so on all the way around to 360, which is a full circle. We use trigonometric functions Sin and Cos as a way of figuring out the distance something covers when pointed at an angle. As you can see here[5], the values for 30, 45, and 60, all use radicals. When in a 3D scenario, you have to multiply 2 angle projections together to deal with the 3 axes, which leads to multiplying radicals. This is why you need to know how to deal with radicals. While this may sound esoteric and complex, it actually shows up in most careers that involve space, so if you want to be a mechanic, a pilot, a farmer, an engineer, an architect, or a number of other things you may need this. Since it is so ubiquitous, it is considered somewhat of a general purpose skill.

7. I'm not sure what to say about this. The page the link takes me to is a page of formulas, which are inherently useful in real life. I don't feel that I need to provide an example when they are themselves examples.

Rather than continuing to address individual cases, I offer these examples of sources that discuss how applied algebra is used to solve relevant problems. These are proposed examples by University math departments, so I believe they can be trusted to discern the differences of math, so these examples can be assumed to be under the family of algebra[6][7][8].

1. http://infocenter.bankofamerica.com...
2. http://wordinfo.info...
3. http://math.about.com...
4. http://apcentral.collegeboard.com...
5. http://easycalculation.com...
6. http://www.wtamu.edu...
7. http://www2.math.umd.edu...
8. http://www.colorado.edu...
Debate Round No. 2
Marshall-Abarca

Con

Re: Number Theory
Yes, that makes sense, I will agree that even negative numbers have a place in daily life. I also agree that those aren't specifically linked to Algebra.

Re: Variables pt2
I'd first again like to address that no other meaning of variable other than the meaning used in Algebra matters. Since they have different meanings and we're only talking about Algebraic variables, those are the ones that matter, now about the example. While it is true that "X" changes to "6" in the example I made again please refer to my links defining variable
3. Mathematics
a. A quantity capable of assuming any of a set of values.
b. A symbol representing such a quantity. For example, in the expression a2 + b2 = c2, a, b, and c are variables.[1]
"X" is again a symbol representing a quantity "6" that quantity never changes, sure when you solve and find x=6 the variable changes to a constant, however the value of the variable itself doesn't change, this goes against the non-mathematical versions of the word from the same source.


1.
a. Likely to change or vary; subject to variation; changeable.

b. Inconstant; fickle.


2.
Biology Tending to deviate, as from a normal or recognized type; aberrant.

3. Mathematics Having no fixed quantitative value.



1. Something that varies or is prone to variation.

2. Astronomy A variable star.[1]

Again while "x" will change to "6" that's the extent of it's change, however it is consistently going to be "6" in this problem.

Now about the tax deductions, while it is true someone in my situation with just different tax deductions will have different taxes, the deductions themselves aren't changing or really all that unknown per say again one just has to plug in the set amount of said deduction(s) and subtract that from the total, this is in no way resembling Algebra.

Re: Readressing of Interest Calc & Example Problems
I will concede this, that if one wants to see how much money they'll obtain when/if their deposit hits maturity, then yes that formula is efficient, however it is not necessary per say. What I mean is that it's not the only method of figuring things out. For example while your exponent method is probably the more efficient method for figuring how much you'll have at the end of the deposits maturity, there's still the fact that one can multiply .005*3000 add that product to 3000 and repeat that process 10 times and still come to the same conclusion. Then there's the fact that the method I just said is also pretty useful for say figuring out how much you'll have in a few years if say one wants to know just in case for emergency.

5. This fence example doesn't really hold up in real life though when you think about it, there are too many unknown factors. For example with nails, we don't know how many nails we'll successfully nail in as there's always room for error in things like craftsmanship. Your algebra formula doesn't take this into account, not to mention when it comes to building things you only need a few things. First a plan for what you want to build, you'll need things like measurements which require one to read a ruler or some instrument of measurement, this is something I'm for being taught in class. Then the next thing you'll need is the materials to build, which means that based off of the measurements you took of the area you want the fence to cover, you measure the width of a set of fence posts, and you decide how much to buy accordingly, then you buy the nails and you just buy extras because you don't know if when you go to hammer the nail if it will bend. These are real life factors your use of the distributive formula doesn't take into account, which further supports my claim that overall Algebra doesn't work in real life, but let's look at the others as well.

6. Your use of the radicals is flawed in some areas.
"We use trigonometric functions Sin and Cos as a way of figuring out the distance something covers when pointed at an angle."
" Multiplying radicals is directly necessary to resolve Geometrical problems, specifically when they include circles or trigonometry."
-I've worked in drafting classes before, which are basically pre-architect like classes, and really all you need to know is how to read a scale (a ruler with different forms of measuring, like the imperial, and metric system) not ONCE did I ever use a radical to figure out any measurements even when I was using circular or angled drawings, we used a protractor and/or scale for those things. We never used a calculator which is a necessary tool to use "sin," "cos" and "tan". I can seriously say this from experience, and from watching a lot of home improvement shows that they never use anything like this, maybe in something that specializes in "sin" "cos" and "tan" but again in general building of houses and such, it's not really needed. Again, your relation to daily life here is more towards fields that specialize in high end mathematics, and when you do draw some relation to other fields it's not really accurate especially this one, again this is coming from personal experience, from my dad who worked in a architect like field, and from watching home improvement shows. Not the most credible I can understand, but I think you'll be hard pressed to find a lot of drawings and designs that involve radicals.

7. This page takes one specifically to a formula for calculating distance and what not, however even this formula isn't all that useful in real life. Mainly because this would only take into account a scenario where's there's no traffic, and come on, how many times have you driven someplace and there's no traffic, I wish that was the case but it's not. A lot of the formulas that Algebra uses, are good for on paper like examples, however add real life factors like traffic and a lot of the formulas fall apart.

Re:Economics & Academics
Allow me to elaborate on how I think the school system at least in terms of mathematics related courses should work. First I believe that from around k-6(maybe even 7) no computers allowed just full on numbers in the head, where kids work with basic math (add, subtract, divide, multiply), how to read measurements as that's something that's very necessary if one wants to be self-dependent, this could save money. I also believe that kids should have to learn critical thinking problems like off the top of my head here, "Julie has 22 apples she gives Ben 8 and eats 4, how many does she have? 10." You know things like that, and for the life of me I can't remember the fourth lesson I believe is needed, I believe it was more on how to manage money and such in the real life, basically a much more in depth lesson on economics. In relation to economics, yes it's already in schools but it's only for seniors and for a semester, at least that's what it was for me, I think that it should be stressed for sophomores through seniors along with a government class (but that's unrelated).

Now to address some of your concerns.
"Nothing is mentioned, however, about how to manage personal finances or calculate mortgages."
-I don't know what kind of economics you were taught, however I was taught some of this stuff it just wasn't emphasize in detail because we only had a semester I again think we need more of that.
"The study of economics, in an official academic setting, is based on business considerations, and as such has nothing to do with the life skills you are advocating learning."
-Actually economics has to do a lot with daily life from banking and mortgages it also has to do with how one can use deductions for taxes and bonds to make some guaranteed money. It all depends but I see this as more useful than Algebra.

I apologize I can't address links 6-8 from this round because I'm running out of space, I'll try to get to them in the next round though, if you can please re-post them so at to remind me. Also sorry for the lack of links this time this one was more of a response type. Thank you.

[1] http://www.thefreedictionary.com...
Bannanawamajama

Pro

I'm going to try to start shortening this a bit since I notice we're both hitting the character limit

VARIABLES RE-RE-RE-REDUX
As you seem very intent on sticking to literal dictionary definitions here, I need to ask you to detail the differences. In the definitions you post, each contains, respectively, "likely to change", "Inconstant", "Having no fixed...value", "prone to variation", which all have identical meaning. Changeable. Refering to Math Variable 3.b: if you were to designate the variables a, b, and c as 3, 4, and 5, you could change a to 1, and b to 1 freely, so long as you also change c to sqrt(2).

This brings me to a point which I have been avoiding up till now because I didn't want to confuse the issue further, however now I have no choice. Your examples like "4 + 10 = X" aren't really algebra. The fact X here is necessarily a fixed value means that it is not in fact a variable, since it is not "capable of assuming any of a set of values". Such problems as these fall under the category of "Pre-Algebra"[1.]. The link specifically uses a 1 variable fixed equation on the page. They exist as a method of introducing the concept of letters being representative of values to students before learning actual Algebra. The equation given, though, is pure calculation and arithmetic, no algebraic manipulations are done.

I would like to submit now an image[2]. This is a standard x-squared function of no significance. It's a visual depiction of an algebraic function. Notice that all the points in the domain are concurrently defined and expressed. When you evaluate a square function at say x=2, you move along the x-axis to 2, then check where the function aligns with the y axis to get your answer. However, in choosing x=2, the remaining x values still exist. This is what evaluating an algebraic function is. It's looking at a particular point in a function to find the output, but it doesn't affect the function itself. Now look at the tax problem. When you input your income and deductions(independents), it gives a specific output(dependent). However, this doesn't in any way change the tax equation as a whole, it rather just isolates a specific point on the function for you to look at. The function remains fluid and changable so that another person can still use it and get a different value for his needs. In this way an algebraic function is constantly "changeable" in that it can be evaluated at more than one set of independents.

EJEMPLOS
Your process, while successful, is far less efficient. What happens when you invest in a 50-year bond(Yes thats a thing)?Will you repeat your addition 50 times? I can find the answer in 1 line. Math is all about efficiency. To use an analogy, Exponentiation:Multiplication::Multiplication:Addition. To say Exponents are useless is to say multiplication is as well, and I maintain that the efficiency of multiplication makes it very useful indeed.

If nails are bothering you, replace them with carriage bolts, those are reusable. The purpose of that expression was to relate number of posts you need to number of bolts you need. If you know the length of fence, like you suggest, you use that to find how many fence segments you need, and then this expression is still applicable.

You may have taken a drafting class, but as an engineer I can tell you that a minor portion of design. Much of the work involved was probably spared on you in an elementary drafting class, but determining the constraints of how you are supposed to construct something is where this math comes into play. Likely you were given the necessary constraints to work with, but calculating those is the majority of a designer, architect, or engineer's work. I have had to design equiptment and have worked along side civil engineers, we all use trigonometry.

Using traffic: Have you ever noticed that stoplights spend more time green for one direction than another? Thats becasue there isn't a person watching every light. Instead, in addition to sensors that see when a car is present, a technician programs the light with various equations that represent the traffic flow rate vs. time for each direction of traffic. The computer then balances this equation using algebra to find the optimal balance of time spent in each position to optomize traffic for a certain part of the day[3].

PURPOSE OF HIGH SCHOOL MK.II
Now I understand you probably ran out of space on the last round, so I'll ask again: What current required High School classes, if any, do you feel are necessary? The topics you listed are all things which are covered by the time you finish middle school. Likely you would reject most all current classes, because Formal Writing, Chemistry, and learning French aren't skills you will use in daily life. Which brings up this point: High School in general isn't necessary for "Real Life". There are plenty of people who drop out and still live their lives. To prove this, consider the GED. This test is used as a High school replacement for those who never finished. As their history details, the point of the GED, and by syllogism the point of High School, is to demonstrate proficiency in higher fundamentals to show capability for more mentally involved careers[4].

Your argument is centered around the idea that Algebra isn't used in everyday life, but that is not valid because Elementary School is meant to teach you skills for everyday life, like reading. Thats why its "Elementary". High school teaches you advanced material that is applicable if not necessary for average use, but crucial for white collar work. Manual labor or service industry doesn't require a high school education because it is just that, labor. A high school diploma is only needed for people looking for intellectually involved work or admission into college. As I mentioned how algebra can be involved in white collar work in the above section, I won't elaborate further.

To your critique of my explanation of Economics, yes I agree that banking is involved with it, however, scholastic study of economics, as a class, does not. Im basing this not on personal opinion, but the nationally accepted Collegeboard standards from my source last round. If you wish to incorporate the things you listed into a class, it could just as easily go into a Math class as into an Economics class.

SOURCES
1. http://www.mathplanet.com...
2. http://www.sparknotes.com...
3. http://en.wikipedia.org...;(I know it's wikipedia, but I'm not gonna make you read a dissertation on Flow Theory, so you can make do with this)
4. http://www.gedtestingservice.com...

Also, you mentioned that my first round had bad links, I would like to try to re-post them in case anyone else ends up reading this debate and wants to check them.

1. https://www.nfc.usda.gov...
2. http://www.merriam-webster.com...
3. http://www.merriam-webster.com...
4. http://www.nuffieldfoundation.org...
5. http://cscircles.cemc.uwaterloo.ca...
6. http://www.mathsisfun.com...
7. http://dictionary.reference.com...
8. http://www.math.wsu.edu...
Debate Round No. 3
Marshall-Abarca

Con

Re: Variables & Examples

Firstly let me say that you again have used the definition of a variable from different and irrelevant topics, like here
[http://www.nuffieldfoundation.org...][http://cscircles.cemc.uwaterloo.ca...].You link Algebra to many different aspects of real life. You defined it with your definition of variables, something that we "do not know yet. It is either something we are looking to find out, or something that we choose before computing a specific solution." R1 quote


My question is then can you give me a specific algebraic problem that relates to real life while maintaining your definition of algebra and keeping the variables unknown? because my problem "4+x=10" (that's what it really was by the way) as you said "X here is necessarily a fixed value means that it is not in fact a variable, since it is not 'capable of assuming any of a set of values'."
The challenge here is to see if you can establish a problem that 100% is relevant to real life, and your tax one doesn't work because going by your logic those variables must never be defined, which if you were to apply an individuals real life numbers to those variables would be defined. This is a blatant contradiction to your logic.
I will concede that the exponent problem does show relevance and efficiency in the field of banking, and somewhat of bonds, but that's about it and it's only one aspect of Algebra though again by your definition of Algebra this makes no sense because not only would the exponent be determined but the other variables themselves are determined also.

Fence ex: If your solution is "replace them with carriage bolts, those are reusable." That's not really an Algebraic response, so I can't see this as a legitimate answer to this topic. As for the purpose being "that expression was to relate number of posts you need to number of bolts you need." Again that expression didn't take into account messing up and also it wasn't that logical to begin with here you say..


"Say you need to build 5 fence segments, and a fence needs 2 nails per post plus an extra 4 for the ends."

Not only does this again go against your logic of variables and Algebra as a whole, but it also fails in real life, again all one needs is the measurement of the total area they want the fence to cover, the width of the pieces and then maybe divide the total area by the width of the pieces. Now as a counterargument you can say "This is only a hypothetical", however this goes against the whole purpose of the "pro" position which in this case is to prove Algebra overall is useful in real life.

Design ex: That's pretty convenient of you to just assume we had "constraints" placed on us in tech drafting classes. Even though we were doing things like designing (bottle of course)rockets, bridges, cars and catapults. None of which used Algebra, again though being an engineer and a student you must have your books or some form of example readily available, so please tell me how you used Algebra not Trigonometry, but Algebra specifically and please try not to break away from the logic you've established in the early rounds. Also about Trig, your an engineer learning trig's going to be a given wouldn't it?

Traffic ex: Could you possibly give me that specific formula because this link [http://auto.howstuffworks.com...] would say otherwise. Because Think about it when you pull up to a light and there's no one coming from any other side the light changes pretty much instantly, because it sensed the pressure of the car, and as my link states that some lights even run on timers, where's the Algebra being used there? I mean all one has to do is set the timer for like 3 Min's and that should be more than enough assuming we're talking about one light. Though you could probably have the same thing done for an entire city have one side going every 3 Min's or so.

Re: Purpose of High School Mk. II

I refuse to go into detail about how I feel the school system should be ran in the main debate, because of the fact that we're talking specifically about Algebra. I'd rather we focused only on Algebra and not get sidetracked by another much more lengthy debate, if you like I can start a whole new debate or I can post in the comments section how I think things should be run in American schools.

Side Note: I'm using a new method of posting my links now, I'll see whether or not I like it in this round but we'll see, just so you know.

Bannanawamajama

Pro

ONLY ONE "VARIABLE"
I admit to your statement. I did call upon definitions of "variable" which came from different topics. However, I take issue with them being irrelevant. Indeed, my point of using different topics as reference was to point this out: Variable is not a Math exclusive term like "differentiate", which has a unique meaning in the context of mathematics. Rather, it is a word being properly applied to refer to what something in math is. The same is true with "division". In cell replication, division is the step after DNA has been replicated[1]. It is a process where 1 big thing splits into more than one smaller thing. Just like in math.

Why do I care that a word is consistent in its usage? Because this shows that there is a definite meaning that is not subjective to circumstance. "Variable" means what I claim it means because that is what it always means. I challenged you to explain the distinction between definitions from different scientific branches, but you have not done so, likely because there is no such distinction to be found.

AN "ALGEBRAIC" EXPRESSION, AND CHANGE
Now let me explain why taxes is different to 4 + x = 10. Taxes can change by circumstance. The rates change, the people change, the actions you take change. It is only for the moment of calculation that you solidify your values, and for a moment, the variables become constant. It is inaccurate to say variables must be defined, rather they are defined only when specified.

y=x^2 is defined at x=2, but not GENERICALLY given a distinct value. Evaluating an equation makes its values constant there, because what you are doing is taking a snapshot of a mathematical function at a specific point, and looking at it. Just like a photograph, it never changes, but that which it captured always does.

Conversely, x in the other equation must always be 6. It CAN NOT change. Ever. By the parameters you've set with that problem, it is inherently a constant, because it does not conform to the most literal possible explanation of a variable by its root terms vari- and -able, "able to change".


EXAMPLES
I agree exponents don't really count, but it was in your list of problems from round 1. We'll forget about it though.

FENCE: I meant use carriage bolts from the start, instead of nails. If you mess up with bolts, you don't need to buy more, as you pointed out as an original flaw. For a fence, a certain length of a fence is not just 1 plank, its maybe 5, which all need to be connected by bolts to vertical posts, hence you need 5 bolts for each 2 feet, if your fence has a post every 2 feet. However the end posts are only connected on one side. So instead, you just need bolts for a latch or something, maybe 2. Those 2 are constant because you only ever have 2 end posts, but the sets of 5 you need multiple of based on how many vertical poles, and as such the length, your fence has to accomodate. So its something like 5*X + 2(Thats not the expression I used before but its the same idea, I just forgot my old numbers)

TRAFFIC: Yes, sensors are involved, but those alone cant be used. If they were, a light would change exactly when a car had passed, even if another car was 3 seconds away from the light and should have made it. This would undermine the purpose of traffic lights. Your proposition for this is constant 3 minute timers. But this too is bad. What happens at an intersection of a busy road and a nonbusy one? Its illogical, because obviously one way has more people and should get more time. But what if one road is busier in the morning, but the other road is busier in the afternoon? Accomodating these issues is what algorithms are for. They are equations that have time as an independent and traffic as a dependent, showing which light should be green for longer at a given point in the day. Here is an example, but as I warned before, its kind of complex[2].


HIGH SCHOOL
In asking that, was simply offering you a simple way to refute the argument I made, and am making again now. NOTHING you learn in high school is intended to help you with mundane household tasks. High school is education which prepares you to enter the Professional world. To answer the question of "Should Algebra be necessary for a High School diploma?", you need not ask how you will use it in personal use, but how you will use it in a job. Not flipping burgers at a fast food place, you don't need a diploma for that. You need a diploma to get an office job that will make you money to live comfortably. Corporate jobs, buisness, construction. If I show you need algebra to do these things, I've proven my position. I believe I've done so above, but I promised you one good real example, so here it is:

DESIGN
You mentioned in your drafting class that you designed bridges. I don't really know what exactly you did in that class, but building bridges seems fairly straightforward. Now a "constraint" is a functional limit to your product. Everythign you design has constraints. Now there are several constraints to a bridge, but one is obvious: It should hold a certain weight before breaking. How you managed to figure this without math I'll never know, but for well made structures, theres no guesswork. Either you do the algebra, or people die.

A very commonly used method of bridge building is "truss structures"[3]. These are the bridges made of little triangles of steel bars. You've probably seen industrial bridges made like this, or those highway road signs that point out upcoming towns. They appeal to constructors because the strain on a bar only goes in 1 of 2 ways. To find the strain, we apply Newton's second law, which says an object under net zero force won't accelerate. For a static structure, like a bridge, this implies that the forces on any part are balanced. However, since we live in 3-D space, we must take that into account, so we write 3 equations, one for each of the spacial dimensions:

Sigma(Fx) = 0
Sigma(Fy) = 0
Sigma(Fz) = 0


The "Forces" I mentioned earlier come from either other parts of the bridge, and their weight, or outside forces, like a guy standing on top of your bridge. But not all trusses in a bridge are perfectly aligned to an axis, in fact for structural integrity its kind of important you dont do that. So when you put your weight on a truss pointing diagonally down, you maybe have 1/6 of that in the "x", 1/3 in the "y", and 1/2 in the "z" direction. This happens again with each truss you involve. So you end up with these 3 equations all relying on unknown values of the other 2. So how do you figure out the stress on one direction? By using the algebra "elimination method"[4] of solving a system of equations, which is DEFINITELY algebra and not trig or anything else. You can't guess or use common sense or anything else, elimination is the only way you'll get that.

But the problem isn't done. We've just reduced a 5 variable equation to 2. Now there are two variables, the weight on the bridge, and the strain on a part. There is no "definite answer" here, there is no "right" amount of people that are going to stand on your bridge, and theres no "right" amount of strain a part can handle, because different materials handle different amounts and kinds of strains. Instead you say "I can make this bridge hold 10 people if its made of expensive steel, or 5 if its made of cheaper wood." or "If I want this bridge to hold 10 people, this part will need to stand 2000 N of force." Thats varying the variables. That is NECESSARY if you want to build a shed or a porch, or work in construction, or get a job drawing technical specs. Relevant to real world use. And you couldn't figure this out without algebraic equations or methods.

1. http://www.phschool.com...
2. http://par.cse.nsysu.edu.tw...
3. http://vcity.ou.edu...
4. http://www.algebra.com...
Debate Round No. 4
Marshall-Abarca

Con

Re:Variable and Algebraic expressions
For starters let me again say that you defined Algebra as "an application of mathematics where certain values of an equation or equations can be changed or chosen by the operator without invalidating the equation as a whole." R1

The above definition is proven wrong by this problem[1] notice you say "operator" but if you look at the problem "4x-7=5" if you do the math x=3 however if you change the operator which you say is valid, then you automatically must come to a different conclusion of what "x" is. Meaning that your definition of Algebra was inherently flawed from the beginning, you also used the term "variable" to support this claim. You said and quote "A variable is a symbolic representation for a value we do not know yet. It is either something we are looking to find out, or something that we choose before computing a specific solution."

You also say in relation to my example of 4+x=10 that "X here is necessarily a fixed value means that it is not in fact a variable, since it is not 'capable of assuming any of a set of values'." However as [1] shows in its example that x regardless of it being “fixed” is still a variable because it represents the unknown. This goes back to the free dictionary definition

3. Mathematics Having no fixed quantitative value.

a. A quantity capable of assuming any of a set of values.

b. A symbol representing such a quantity. For example, in the expression a2 + b2 = c2, a, b, and c are variables.

“Fixed” in this case when added to [1] means that “a” won’t =1 and “b” won’t = 2, it doesn’t mean that they won’t represent a number in a given situation, like in my example of “4+x=10”

You’ve admitted that in different scenarios words can hold different meanings and I’ve made this point since round 2 that since we’re only talking about the mathematics version of “variable” we need to use the mathematics definition. However you kept insisting on using these irrelevant forms and even said that I didn’t “explain the distinction between definitions from different scientific branches” However I did do this, proof

“going off of the mathematics definition of what a variable is”R2 & “I'd first again like to address that no other meaning of variable other than the meaning used in Algebra matters” R3

So yes I did say constantly that those other v. didn’t matter but you ignored this and have now contradicted yourself.

As for the tax problem, yes taxes change per the individual, that doesn’t make them unknown to the individual themselves, you can take any individual person’s situation and plug it into that formula, and you can still do the basic methods of math and find how much money you owe the IRS or whomever. As for my example I’ve already explained with sources that regardless “x” is still a variable.

[1] http://www.mathsisfun.com...
[2] http://www.icoachmath.com...
[3] http://www.thefreedictionary.com...

Re:Examples

Fence: “For a fence, a certain length of a fence is not just 1 plank, its maybe 5” – 2 things

  1. 1. This goes against what you just said that these numbers are variables, because they will not change, in the specific situation that’s being given
  2. 2. When I said take the width of one plank I mean that say one plank’s width is 2ft and you want to put them right next to each other so no space in between. That means if say your backyard is 20ft long you’ll need about 10 planks on both the left and the right side of your back to satisfy your wants. This is basic math and measuring that doesn’t need Algebra at all.

Traffic: Here you mistake Algebra for statistics from your own [2] after hitting CTRL+F and typing "formula" I found that this is about “probability” that means this is statistical not Algebraic, it’s a different field of math because of the fact that Algebra doesn’t deal with statistics and probability.

[4] http://par.cse.nsysu.edu.tw...

RE:High School

I will address this briefly; first I never said talked about “mundane household tasks” or “flipping burgers at a fast food place” the whole position of my argument was that Algebra in the vast majority of fields is not even needed a bit. Again, I won’t go into details because of the fact that it’s irrelevant to whether Algebra specifically is needed or not, if you want to have a debate on that then I suggest after this one start another debate about the whole school system.

Re:Design

I will concede that you were able to show a good example of Algebra being used while I am still skeptic as to whether or not some details are being forgotten by one/both of us, I can see how in the field of design algebra can hold some relevance. However the whole purpose of the “pro” position in this debate was to disprove my statement from R1 “overall most algebraic equations are not likely to be used in day to day life.” While again you made a good point in that you showed a good example this doesn’t prove that Algebra is usable in most forms of everyday life.

Conclusion
Kudos to my opponent for a civil debate, I think that both sides did well here. However overall the whole purpose of Con was as stated in Round 1 that "overall most algebraic equations are not likely to be used in day to day life." While my opponent has proven that Algebra holds some usage in banking with perhaps bonds and such, along with design, he has not shown enough evidence that in depth learning of Algebra classes is needed to be able to have a successful life, while I have a different opinion of Algebra as whole my stance on this issue still stands, that Algebra is not necessary for highschoolers to learn in order to lead a successful life.

Bannanawamajama

Pro

As we are at the end of the debate, and my opponent can no longer respond, I will not address his last post too thoroughly, but rather give a general summation of my position.

ALGEBRA
As I have presented it, "algebra" is the branch of mathematics which deals with "variables", at its simplest level. My opponent, as far as I can tell, agrees with this much. We agree to some point on what constitutes a variable. I define it as "a value which can change". I believe that this stands up to scrutiny, due to it agreeing with presented definitions of the term, and following logically based on both on the etymology of the word and its common use in a variety of scenarios and fields. My opponent has challenged this in two ways, once by offering "x + 4 = 10" as an example, which I dismiss on its inherent immutability and thus exclusion as a variable, and later by asserting that an undefinable variable is useless. I respond to this by adding a corollary that variables can be defined, but only temporarily. This is achieved by having 2 or more variables present in a given equation, which I claim is necessary to be algebraic.

REAL LIFE EXAMPLES
In addition to the use of algebra alone, the subject is a necessary prerequisite to learning other useful mathematics, such as calculus and statistics. Statistics is considered by many to be the most "real life" use of math, as it bridges the absolutism of theoretical mathematics to the uncertainty of real life. My opponent mentioned my last example given was of statistics, not strictly algebra. However, linear regression[1], which is the basis of "functional statistics", is itself a process of algebra.

My opponent has noted, with validity, that there are alternative methods of solving many of these problems. I offer this as a response: not only are some problems impossible without algebra, but those which are not are often solved more efficiently through algebra. This is the reason all math is learned, to improve efficiency. An analogy for this would be the area of a circle, pi*r^2. It is possible to cut a circle into square pieces and measure its area manually, but this is far more tedious and time consuming than using a generic standardized process. This is what algebra offers to the layperson, a way of quickly optimizing scenarios with minimal effort. While not necessary per say, it is certainly useful, and worth learning.

HIGH SCHOOL
I admit that my opponent and I seem to have confused eachother on the exact point of this debate. His main challenge seems to be my examples do not have application in "real life". However, I point out now that the challenge of this debate was "necessary for a high school diploma". As explained earlier, High School is not meant for learning skills purely for daily use. It is intended to show an aptitude necessary for certain jobs or admission to higher education. While I also believe algebra can be used in real life, this is not necessary for me to prove. If I instead make the case that many jobs which require a high school education also involve algebra, that would satisfy the terms of this debate, as most people who take algebra in high school would be using it on a day to day basis, in their job. It is unreasonable to try to apply this to all people, for there is an exception to everything. My position doesn't stand up when considering a man who gives up his life to God and becomes an ascetic monk, but in that case him going to high school didn't matter to begin with.

High school involves a certain amount of specialization to skills that not EVERY person needs, but rather those that are needed by society as a whole. I have tried to provide examples of application in diverse fields such as construction, city planning, finance, and physics, but this is not all. There simply isn't enough time for me to detail examples for every possible career using math, but there are plenty of them out there. My burden is not to show that all people will certainly need algebra, but merely that a broad spectrum of mid- to upper-class careers will to prove that algebra warrants being taught to the general populace. I believe that I have done this throughout the debate.

To offer a more generic case, any buisness manager uses algebra to deal with inventory. Supplies coming in compared to how much can be used or how much can be sold, and how much can be stored is a decision which must be handled precisley, or else wasteful expenditures will result. Algebra's ability to exactly relate certain quantities or values to others on which they depend is the general application which makes it useful in economics, healthcare, or governmental policymaking.

CLOSING
To end, I would like to make my case as thus: Algebra is math that relates varying values. Varying values can be used in complicated problems to make life easier, and to achieve difficult tasks. Becoming qualified to do more difficult tasks is the purpose of going to High School, and. As such, algebra would be useful to a wide variety of people, and should be necessary for a high school diploma.

I would like to thank my opponent for a well argued debate. Though I still hold firm to my position, he has defended his own admirably, and offered far more convincing an argument than I expected to come in this debate. Whoever wins, I respect his debating skill and his restraint in using logic to challenge me rather than falling into emotionally driven fallacy. Thank you.

1. http://www.the-idea-shop.com...;
Debate Round No. 5
12 comments have been posted on this debate. Showing 1 through 10 records.
Posted by Bannanawamajama 3 years ago
Bannanawamajama
Oh cool. Ive only been in 1 before too so I guess I can't either. I just wasn't sure if people nomally do or not
Posted by Marshall-Abarca 3 years ago
Marshall-Abarca
I never vote on my debates, and even if I could I wouldn't. I can't because I haven't completed enough debates, you have to have completed three debates.
Posted by Bannanawamajama 3 years ago
Bannanawamajama
Also, I haven't debated much here so I'm not sure if theres a precedent to this, but shall we agree that neither of us should vote on this since that would obviously be biased?
Posted by Bannanawamajama 3 years ago
Bannanawamajama
All right, thats fine with me
Posted by Marshall-Abarca 3 years ago
Marshall-Abarca
The conclusion will be next, I'd like to just end it with us wrapping up our main points.
Posted by Marshall-Abarca 3 years ago
Marshall-Abarca
Oh ok my apologizes I see them now, I didn't see 5-8 before. :P
Posted by Bannanawamajama 3 years ago
Bannanawamajama
Footmarks? I don't understand. I numbered the links to correspond with the brackets in the text that refer to them, isnt that what you're supposed to do?
Posted by Marshall-Abarca 3 years ago
Marshall-Abarca
Also this time you forgot to leave footmarks as to which links gave what information. LOL
Posted by Bannanawamajama 3 years ago
Bannanawamajama
I apologize, I double checked my links on this last argument, they all worked this time for me, hopefully you will be able to see them as well. I'll look for my first round's sources again and try to repost them in my third.
Posted by Marshall-Abarca 3 years ago
Marshall-Abarca
Also your links at the bottom didn't work, I don't think it was your fault I had issues with my links for my 2nd argument as well. Just a heads up though if you see a ";" at the end of your link that means it most likely won't work and you'll need to paste it again.
2 votes have been placed for this debate. Showing 1 through 2 records.
Vote Placed by Nataliella 3 years ago
Nataliella
Marshall-AbarcaBannanawamajamaTied
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Total points awarded:05 
Reasons for voting decision: Both used reliable sources (Government sources! Good job to both of you!), but Pro used more. Spelling and grammar were equal. Some of Con's arguments were very fallacious, giving arguments to Pro. Conduct was equal.
Vote Placed by Piccini 3 years ago
Piccini
Marshall-AbarcaBannanawamajamaTied
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Total points awarded:03 
Reasons for voting decision: Good debate, but Pro has proven that algebra is useful, while Con tried to change the meaning of Algebra to make it useless.