Order of Operations vs. Associative Property
Date: 10/10/2002 at 12:39:38 From: Jeff Horn Subject: Order of Operations vs. Associative Property I volunteer to teach a weekly extra-credit math course in my daughter's 5th grade class. Today I taught a lesson on the correct "Order of Operations" in mathematical expressions (i.e.: parenthesis, exponents, multiplication/division from left to right, and lastly addition/subtraction from left to right). That very day, her regular math teacher taught a lesson on the associative property, in which operations in expressions only consisting of addition or multiplication can basically be performed in any order, with the same result. In fact, her homework assignment included some mental math problems in which she was to rearrange the order of operations in multiplication problems in order to more easily solve the equation in her head. For example, 4 x (6 x 25) might be rearranged to (4 x 25) x 6, which is perfectly okay under the associative property. My question is: does this make the associative property an exception to the order of operations, a corollary, or something totally unrelated? Thanks for whatever help you can give me and thanks for the invaluable help your site gives me every week as I prepare my lesson plan (and try to keep one step ahead of the students!).
Date: 10/10/2002 at 15:28:23 From: Doctor Peterson Subject: Re: Order of Operations vs. Associative Property Hi, Jeff. The ideas of Order of Operations, and properties of those operations such as commutativity and associativity, are distinct concepts. The former deals with the mere syntax of how we write algebraic expressions; the latter deals with the semantics - what the operations mean. It's as if I taught you that you should say "I ate a pizza" rather than "Me eat pizza" (teaching grammar), and the next day told you that you could change that sentence to "The pizza was consumed by me" (restating the same fact using different words and word order). These are not contradictory, but two different levels of language understanding. So when we write 4 * (6 * 25) that means that we multiply 6 times 25, then multiply 4 by that value. It's just a set of instructions for calculating. But when we say that 4 * (6 * 25) = (4 * 6) * 25 that says that if we make two different calculations, the results will be the same. This is an equation, asserting that this fact is true. And if we say that FOR ALL a, b, and c, a * (b * c) = (a * b) * c this is a theorem, presenting a fact that we can then make use of whenever we want. It says that because of the ways numbers work, when we see this particular arrangement of operations, we can safely rearrange the order in which we do things, and will still get the same answer. It doesn't mean that these expressions don't mean different things (in terms of the operations they say to do), only that their values will always be the same when we carry out those operations. So, combining the two concepts, if we see an expression like 4*6*25 without parentheses, although we know that the rules of grammar say we should multiply from left to right, first doing 4*6 and then 24*25, the rules of algebra say we can change the order to make the work easier. We can use the commutative property (swapping the order of the 4 and 6) and say that the expression has the same value as 6*4*25 and then using the associative property we can do this by first multiplying 4*25 to get 100, then multiplying by 6 to get 600. The Order of Operations rules told us what this expression _meant_; the properties of multiplication told us tricks we could use to do the calculation (or, later, to rearrange an equation in algebra). I assume you have found discussions in our archives of order of operations, such as Explaining Order of Operations http://mathforum.org/library/drmath/view/57199.html and also discussions of properties, like Properties Glossary http://mathforum.org/dr.math/faq/faq.property.glossary.html Thanks for letting me know you find our site useful! I am always especially happy to be able to assist teachers (of any sort), which multiplies the usefulness of our efforts. If you have any further questions, feel free to write back. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
Date: 10/10/2002 at 15:29:14 From: Doctor Roy Subject: Re: Order of Operations vs. Associative Property Thanks for writing to Dr. Math. The left to right rule really only applies to division and subtraction. This is because they don't satisfy the associative property of equality. To be really precise, it is because we are really "loose" about the definition of subtraction and division. If we define subtraction properly, it is adding by an inverse. For example, (3 - 5) + 7, is really 3 + (-5) + 7. By writing it this way, we can rewrite to get 3 + ((-5)+ 7), since addition is associative. We get the same result as before. The same applies to division. By dividing, we are really multiplying by a multiplicative inverse. For example 5/3 is really 5 * (1/3). Written this way, we can use the associative property. The "left to right" rule in the order of operations really exists because elementary school/middle school students are generally too young to understand the abstractions that would make the explanation make sense. They are also generally too young to understand the concept of an additive inverse and a multiplicative inverse. So the "left to right" rule allows us to keep subtraction and division as they are without changing the basic rules of math that students are taught (i.e. that subtraction and division are really addition and multiplication with different names). Even now, whenever I see something like (3-5)-2, I have no problems associating 3 + [(-5) - 2] to get the same answer, or even switching the order to get (-5) - 2 + 3. It's all really the same, even for multiplication/division. So, really, by introducing of the associative property (and all the other field axioms), we are introducing concepts like additive identity (i.e. negative numbers) and multiplicative identity (multiplying by 1/3 instead of dividing by 3). This is the point when the somewhat artificial "left to right" rule can be abandoned for a more formal explanation. I suppose I have given a rather long-winded response, but the question deserved a very full answer. Simply put, you can think of the associative property of multiplication and addition to be an exception to the order of operations, if it makes arithmetic easier. But in a more real sense, the order of operations going from left to right was already more an artificial structure that isn't properly needed except when we want to use the very familiar constructs of division and subtraction. I hope this helps. - Doctor Roy, The Math Forum http://mathforum.org/dr.math/
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