Internalizing the Order of Operations

  • Internalizing the Order of Operations

    By Tina Cardone, Posted February 16, 2015 – 

    As we review in my classes, I notice some quirks as students evaluate expressions. One student distributes rather than completing the operation enclosed in the parentheses first. It may not be the simplest way to complete the exercise, but it is mathematically valid. Another student, stuck in equation-solving mode, wants to use opposite operations despite the lack of an equal sign. Other students correctly compute the operations but in the wrong order. I see students calculate the expression from left to right regardless of the operations, as well as students who tackle the exponent first, but are determined to add before they subtract, regardless of the order the operations appear in. These students have yet to internalize the order of operations.

    My students arrive in class with PEMDAS memorized well. The problem is, they didn’t learn it. They do not understand why there is an order, or that multiplication and division are so interrelated we would get some absurd inconsistencies if multiplication always came first.

    Students interpret the acronym PEMDAS in the order the letters are presented, leading them to multiply before dividing and add before subtracting. For example, students often incorrectly evaluate 6 ÷ 2 × 5 as follows:

    mtms_blog-2015-02-16-ART-1_resize

    Isn’t that equally correct? What happened?

    Let’s figure it out. Division is the inverse of multiplication; you could say, “divide by two,” and I could say, “multiply by one-half”, and we would both get the same result. If we do that in this example, we produce an interesting expression:

    mtms_blog-2015-02-16-ART-1-resize

    Why is this expression interesting? Because multiplication is the only operation, and it is both commutative and associative, meaning we can multiply any two numbers and then multiply that product by the third number.

    mtms_blog-2015-02-16-ART-3_resize

     

    What if we encouraged students to be creative in their problem-solving strategies rather than demanding they evaluate all expressions in order? (Would you rather take half of six or half of five?)

    Students should know that mathematicians need a standard order of operations for consistency, and collectively, through experience, decided on exponentiation, multiplication, and addition. The most powerful operation should be completed first; exponentiation increases or decreases at a greater rate than multiplication, which increases or decreases at a greater rate than addition. Sometimes we want a different order, so we use grouping symbols to signify “do this first.”

    Instead of practicing meaningless examples, have students practice in situations where they will see how the standard order makes sense. For example:

     

    “Start with 5 pennies and then add 3 cups of pennies.”

    Have students figure out how many pennies they have in all, given a variety of quantities in the cup. If you ask them to describe the process they are repeating, they will realize the expression 5 + 3c represents this situation.

    Some students might want to say that 5 + 3c is equivalent to 8c, but what would that mean? If you return to the context of our expression, it means having 8 cups! What happened? The five and the three mean very different things in this expression. The five represents five pennies. The three represents three cups of pennies. We must multiply first to find the number of pennies in three sets, then we add to find the total number of pennies.

    When students try to generalize the order of operations, I suggest using GEMA.

    G is for grouping, which is better than parentheses because it includes all types of groupings, such as brackets, absolute value, expressions under radicals, the numerator of a fraction, and so on. Grouping also implies more than one item, which eliminates the confusion that students experience when they try to “Do the parentheses first” in 4(3).

    E is for exponents. This includes radicals, as they can be rewritten as exponents.

    M is for multiplication. Division is included in this stage, as well. Since only one letter appears for both operations, emphasize the important inverse relationship between multiplication and division. For example, discuss the equivalence of dividing by a fraction and multiplying by its reciprocal.

    A is for addition. Subtraction is included in this stage, too. Again, since only one letter appears for both operations, emphasize the important inverse relationship between addition and subtraction. A useful definition for subtract is “add the opposite.” Just as with multiplication, rewriting the expression using only addition allows students to use the commutative and associative properties to add terms efficiently.

    We need to excuse our dear Aunt Sally and replace her with a method or mnemonic that won’t confuse students. Math shouldn't be magic; math should make sense.

    MTMS_blog-2015-02-02_AU_PIC_s

    Tina Cardone, @crstn85, is a high school teacher at Salem High School in Salem, Massachusetts. She is the author of Nix the Tricks and blogs about her teaching at DrawingOnMath.blogspot.com.

     

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    Joie Brannan - 2/26/2015 2:07:11 PM
    I like it. It's streamlined, it's a better reflection of what is really happening mathematically, and since Gema can be a woman's name, you still have the opportunity to use some kind of story or visualization that would help students remember it.