March 11, 2015

Order of Operations in Math: not open to interpretation

Some things are debatable.  Open to interpretation.

Like evolution.  Religion.  White chocolate or milk chocolate.  Chicken or beef.  

Other things are static.  They just are.  There's no debate.  No different interpretations.

Like photosynthesis.  DNA.  Gravity.  And Order of Operations in mathematical equations.

You can debate gravity all you want.  You can claim you have a different interpretation of how gravity works until you're blue in the face.  But, at the end of the day, you're still going to meet an impactful death if you jump off the Empire State Building.

Likewise, you can debate Order of Operations all you want.  Also known by its acronym, PEMDAS (which stands for Parentheses, Exponents, Multiplication/Division, Addition/Subtratction), you can claim that everyone interprets PEMDAS differently until you're blue in the face.  But, at the end of the day, if you try to solve an expression in any way you feel like solving it, without following the rules of PEMDAS, you're not going to arrive at the correct solution.

Despite what Common Core methods are taught nowadays, it's not okay to get the wrong answer as long as you can show how you arrived there.  There is only one correct answer to a math problem.  We need to know and understand how to arrive there, but we also need to know and understand how to arrive there correctly.

In division, for example, more than one method exists for solving the problem.  Long division.  Short division.  Other methods I'm not even aware of.  But as long as all those methods arrive at the same answer, each method is correct.  6 divided by 2 equals 3.  Always.  150 divided by 5 equals 30.  Always.

In mathematical expressions, where more than one operation exists, how you arrive at the solution is of utmost importance.  There is only one method by which you can arrive at the correct solution.

For example, (5+5)-5x5+5/5x(5-5).

In the above expression, one can not simply work from left to right without taking into account the different operations that appear in the expression.  Let's simplify the expression working from left to right:


25 is, by the way, not the correct solution.  In order to understand why it is not correct, we need to understand PEMDAS.

"The basic rule that multiplication has precedence over addition [Order of Operations] appears to have arisen without much disagreement as algebraic notation was being developed in the 1600s"  ~Greg Vanderbeek, University of Nebraska, 2007.

PEMDAS (Parentheses, Exponents, Multiplication/Division, Addition/Subtraction) tells us that, when solving a mathematical equation or simplifying an expression, there is a particular order in which we must approach the operations.  If parentheses are present, we must work with the operation inside the parentheses first.

Next, we tackle any exponents.  (Exponents are a shorthand way to show how many times a number is multiplied times itself.)  In our example above, there are no exponents, so we move on to the next step.

Next, we address the multiplication AND division, in the order they appear from left to right.  Multiplication does not take precedence over division or vice versa.  They are equal as far as which we must do first.  The only deciding factor is which appears first when reading from left to right.

Finally, we work with the addition AND subtraction.  As with multiplication and division, one does not take precedence over the other.  We work from left to right.

Following the Order of Operations, then, here is the correct way to simplify our above example:

(10)-5x5+5/5x(0)  Parentheses
10-25+1x0  Multiplication/Division
10-25+0  Multiplication
-15+0  Subtraction
-15  Addition

The solution to our expression is -15.

What if there are no parentheses or exponents in the expression?  Can we then just work from left to right?

For example:


It seems logical, doesn't it?  However, whether or not parentheses or exponents are present in an expression does not change the fact that PEMDAS must be followed.  In this case, we would simply begin with step 3:  Multiplication/Division, and follow it up with Addition/Subtraction.  Like this:

5+5-25  Multiplication
10-25  Addition
-15  Subtraction

The solution to our expression is, again, -15.

If a group of students are asked to measure the circumference of an inflated balloon, chances are good that they each will come back with slightly different measurements, because one may stretch the measuring tape tighter than the other, or the balloon may lose air between measurements.  The circumference of a balloon, though still a mathematical solution, is debatable.  It will be interpreted in different ways.

The solution to a mathematical expression, however, is static.  Only one method exists for arriving at the correct answer.  We need to know and understand this method in order to properly teach our children.  Unless, like Common Core methods taught in public schools, you feel it doesn't matter whether or not they arrive at the correct answer, as long as they can explain how they got there.

Personally, I think that does our children a disservice.  I'd even go so far as to say it's a failure.

I certainly don't want an architect building a house for me who was taught that way.  Can you imagine?  If he measured and did his calculations incorrectly, no matter how well he could explain how he got those calculations, the construction of the house would not be sound.  At the very least, it would be crooked.  At worst, it would cave in on itself.  Either way, finding the correct solution to a math problem really does matter.  I prefer my houses still standing.

What do you think?  Is gravity, or is math, open to interpretation?

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