This is part of a continuing series of key math topics in various grades. It will eventually be a book (Traditional Math: An Effective Technique that Teachers Feel Guilty Using), to be published by John Catt Educational. (Readers are encouraged to provide examples of mistakes that students will make for the particular topic being discussed. They will be incorporated into the ever-evolving text, so you can be a part of this next book!)
An expression such as 6 + 4 x 3may be interpreted as 10 x 3, or as 6 + 12. This is where “order of operations” comes in-- an agreed upon hierarchy that eliminates ambiguity.
I recall learning about order of operations in my algebra class. What stuck with me after learning the rules, was Mr. Dombey’s final advice: “When in doubt, use parentheses.” This simple advice is how most ambiguities in math are resolved. The need to interpret an expression that does not have grouping symbols such as parentheses or a fraction bar doesn’t come up all that often.
And—full disclosure here—order of operations is not my favorite topic. In real life, people don’t write multiplication or division problems in ways that special rules are needed. Moreover, algebraic symbols take order of operations into account by virtue of rules for algebraic symbols. For example, if you had to find the value of ab + 6, where a = 2, and b = 3, you would know to multiply the a and b values together, since that’s what ab means, and then add 6 to the product to obtain 12. You would not multiply a by the value of b + 6, since that’s not what the algebraic expression represents.
All that said, students are expected to know the rules, and they do show up on standardized tests. In the end, however, Mr. Dombey’s advice has prevailed.
Warm-Ups. Warm-Ups for this lesson should include some translation from words to algebra:
1. Translate into symbols. Five more than three times a number (Answer: 5 + 3n)
2. Translate into symbols. Six less than two times a number (Answer: 2x-6 )
3. Translate into symbols: Four more than two times three (Answer: 4 + 2 x 3)
Punctuation Marks in Algebra. Problem 3 of the warm-ups serves as an entre into the day’s lesson. I will ask the class what the numerical value is of 4 + 2 x 3. Chances are that they have had order of operations in sixth grade and most will come up with the right answer. In that case, I will ask a student to tell what rule he or she followed in coming up with the correct answer.
Most of them have learned the mnemonic PEMDAS, which stands for Parentheses, Exponents, Multiplication, Division, Addition and Subtraction.
We have not yet covered exponents, so that will not be in today’s lesson, but the others will. I also clarify that the first category, parentheses, can also be called “groupings” because they include such things as a fraction bar.
I like to use the comparison of order of operations in math to punctuation marks, which is how it is described in the algebra textbook by Dolciani (1962). I write the following sentence on the board:
paul said the teacher is very intelligent
This can be punctuated in various ways. One way is: “Paul,” said the teacher, “is very intelligent.”
Or: Paul said, “The teacher is very intelligent.”
The difference in punctuations lend different meanings to the same sentence, just as we had different meanings for the numerical sentence. In math, the main punctuation tool we use is the parentheses, which is called a “symbol of inclusion”. It is called this because it encloses, or separates, an expression for a particular number.
In the case of 4 + 2 x 3, we have some choices for how we want to punctuate the expression. Since the English sentence in the warm-up said “Four more than two times three”, it is evident that we are adding four to the product of two and three. I will ask the class where we would place the parentheses to make it clear that’s what we want to do. The answer is 4 + (2 x 3).
I will then ask, “What if wanted to punctuate it so that we are multiplying the sum of 4 and 2 by 3? The answer would be (4 + 2) x 3.
At this point I digress a bit to introduce a convention in algebraic expressions, which is that we indicate multiplication as (4+2)3, or 3(4+2), leaving out the times symbol.
I will then ask for them to state the English equivalent of what various expressions mean. I give an example followed with the answer so they can then do it on their own afterward. For example, the expression 5 + (3 x 2) can be translated into “Five more than the product of three and two), or “The product of three and two increased by five”.
Other examples may include:
(8 + 3) + 5. (Possible answer: The sum of eight and three, added to five)
(5+3)2 (Possible answers: “The sum of five and three multiplied by two”, or “the product of two and the sum of eight and three”.
Order of Operations. Just as my teacher, Mr. Dombey, made clear to me the importance of parentheses, I make it clear to students that parentheses are the main method for clarifying the meaning of a numerical expression. But in the case of punctuation marks being omitted, the order of operations hierarchy is the agreed upon rule for interpreting such expressions.
The rules for order of operations are:
1. Simplify the expression within each grouping (i.e., parentheses, or fraction bars);
2. Perform multiplications and divisions in order from left to right;
3 Finally, do the addition and subtractions in order from left to right.
To provide the students facility in using parentheses as well as interpreting expressions according to the rules of order of operations I’ll write some expressions on the board and ask students to place parentheses so that the expressions now follow the rules for order of operations.
For example, the expression 5 + 3 x 2 means 5 + (3 x 2) under the order of operation rules.
I work two more with the class and then have them write the answers in their notebooks or on mini-whiteboards:
3 x 4 + 7 (Answer: (3 x 4) + 7
30 ÷ 10 x 3 (Answer (30÷10) x 3
5 - 2+4 x 3 (Answer: (5-2) + (4x3)
Additional Problems. Finally, I will ask students to place parentheses in such a way as to obtain a particular value for the expression. For example, if we want the expression 4 x 6 + 8 to equal 56, the parentheses would be placed as follows: 4 x (6+8). If we wanted it to equal 28, it would be (4 x 6) + 8.
If we want 30 ÷ 6 + 4 to equal 3, the answer would be 30 ÷ (6 + 4).
Others:
4 + 6 x 2 – 8; Value = 12: Answer (4+6)2 – 8; Value = -60: Answer (4+6) x (2-8)
Value: 8; Answer 4 + (6 x 2) – 8
The homework assignment should have a mix of problems that ask to apply the order of operations, as well as those which ask for parentheses to be placed to obtain a specified value.
Reference:
Dolciani, Mary P., Berman, S. L., Freilich, J. (1962). “Modern Algebra: Structure and Method”. Houghton Mifflin. Boston