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48÷2(9+3) = ?

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Postby greeney2 » Fri Feb 08, 2013 1:26 pm

My son is law is right, you are removing the division sign improperly on your last step.


48÷2(9+3)
48÷2(12) (Parentheses) Divide either the product(24 into 48) or you can divide them both one at a time(2 into 48=24, followed by 12 into 24)
24(12) (Division) The division sign is not removed at this point, the parentheses are--The answer is 2


You are wrong, my son in law is right.
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Postby at1with0 » Fri Feb 08, 2013 1:47 pm

Division and multiplication are calculated with the same priority from left to right. That is because division is multiplication by the reciprocal.


48÷2(9+3)
48÷2(12) parentheses first

Now there is a division and a multiplication. By the definition of division, dividing by 2 is the same as multiplying by the reciprocal of 2 which is 1/2 or .5.

Then we have
48x.5(12)

These two multiplications can be carried out in either order to get

24(12)

which is 288.

How does your son in law define division by 2?
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Postby greeney2 » Fri Feb 08, 2013 2:35 pm

Sounds like one of your trick questions At1 :lol: You guys know more than me, this is my Son in Laws answer by email to me.

You don’t have to work it out from left to right, and it is dependent to what rules you are willing to accept and what the context is in math(algebra, arithmetic, etc.). If you accept left to right after the work in parenthesis, then he is correct, if you don’t then my answer is. We do not teach what was stated by your opponent, we say “multiplication or division, left to right”, there are other properties that are taught that conflict with his premise, he must know these or has an awareness of them. What is associated with the parenthesis is usually worked on next, rather than going from left to right.

Theres no correct determination in this exercise given multiple competing understandings, as its sole purpose is bait –for him to get attention and hold you and others hostage to changing and conflicting contexts.

I don’t have a masters in math, I have one in educational management, and a course shy of a second one in political science/public administration. My bachelor of science degree is in criminology. I have enough math courses for a minor in math and to qualify for a teaching credential in math(which is enough units to qualify in most schools for a major in math).

We have done this same problem, as a problem of the week in 6-8th math for the past 16 years, and before I became a teacher as well. It’s a classic, but has more than one interpretation.
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Postby at1with0 » Fri Feb 08, 2013 4:22 pm

The PEMDAS convention, coupled with the definition of division and subtraction, gives one unique evaluation for every well-formed expression. Using them, the result is 288.
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Postby greeney2 » Fri Feb 08, 2013 7:41 pm

Pretty obvious the question, was going to present more than one answer or why post it at all, other than to do exactly what my Son in Law could see? Fooling someone like me is as easy as the shell and pea game. Like I said a kind of trick problem. As my Son in Law stated as such, depending on what premise you are sticking to, and he also stated you would know that, so lets be honest, you know what he is saying. You also know what he is saying, in how students are taught the order in which they would solve this problem in school, the parentheses first, followed by what pertains to that next.

As he said with your thinking you are correct, but with his thinking 2 is correct, and given the way it would be taught, seems like 2 is not only right, but most proper.
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Postby at1with0 » Sat Feb 09, 2013 11:00 am

Yes, I have stated that the "answer" depends on what convention you use. However, 2 is not the result of the standard PEMDAS convention. The PEMDAS convention will uniquely evaluate every well-formed expression. Your son in law is interpreting 48÷2(9+3) as 48÷(2(9+3)) which is a distinctly different expression. You should ask him whether or not division by two is, in his book, the same as multiplying by 1/2 (by definition).

It's a little scary if people are teaching students that PEMDAS will not give a unique answer for every well-formed expression. A well-formed expression is by definition unequivocal. Given any well-formed expression, PEMDAS as well as any other convention, leads to one result.

If PEMDAS did NOT lead to a unique answer, then computers would not be able to evaluate expressions.
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Postby greeney2 » Sat Feb 09, 2013 12:50 pm

I posted exactly what he wrote me back, I am far from a Math expert, you both are light years ahead of me. I'm sure you are right, I do not even know what the convention is you are talking about, I'm sure he knows, as he intimated to. As he stated if you work the problem as you say, 288 is right, if you work it the other way, the answer is 2. To my thinking I thought the answer of 2 was correct, and that 288 resulted from making an error. Even my son in law states that is not so, if there are 2 approaches. When I said 2 may be the proper answer, that was me talking, not my son in law, so I do not know if they are not teaching as you ask, he probably does. As you know as a teacher, he has to abide to the school districts agreed way to teach each subject.

What I do know is that if that was you bank calculating you bank fees of $288 from that equation, you would argue to the death, it should be $2. :lol:
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Postby Sojourner » Thu Sep 11, 2014 4:16 am

Exactly the kind of problem arising from a mathematical expression open to interpretation. The result only leads to confusion and relativity. The purpose of an expression should be to convey information accurately and definitively rather than being reliant upon the invocation of a particular subset of rules. Thus the majority of our mathematical advancements are further scrambled with every additional method of interpretation we erect. The answer to such a problem is to change the structure of our mathematics into a method which there is only one possible interpretation. This can be achieved through utilizing the same operation throughout the expression without variance. As one can easily see when you mix and match operations you conclude with differing results according to the order in which you approach them. It is vital that we not allow confusion to redirect us from the solution an expression is meant to convey.
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