I have been noticing lately that my students are making mistakes involving the use of parentheses. Sometimes parentheses are overused and other times they are missing, and errors are also made while using calculator technology. Using symbols and notation correctly is part of SMP #6, “Attend to Precision”, and is also a component of mathematical communication, since so much of math is written in symbols. I want my students to be efficient and accurate in their work, and I hope their notation supports their conceptual understanding. So I’ve been contemplating the purposes of parentheses…
Purpose #1: To Provide Clarity with Negative Integers. Negative integers can be set off with a pair of parentheses for addition and subtraction, as in these examples, but the expression’s value is unchanged if the parentheses are not used:
- (–4) + 6 = 2
- 6 – (–4) = 10
With an exponent on a negative integer, however, the parentheses are essential. We are working on sequences and series in Algebra II. When a geometric sequence has a negative common ratio, the explicit formula has a negative number raised to an exponent:
- The sequence 2, –6, 18, –54, … has explicit formula An = 2·(–3)n-1
To convince my Algebra II students that the parentheses are required, consider –32 vs. (–3)2 on the TI-84+ calculator:
The calculator executes the order of operations: exponents are evaluated before multiplication. Since the negative sign actually represents –1 times 32, the 32 is evaluated first. Although I prefer students to focus on conceptual understanding and not merely procedural rules, I say to “always use parentheses for a negative base”.
Purpose #2: To Specify the Base for Exponentiation. Another class is studying exponents and logs, and students notice that using parentheses has mathematical meaning for the result.
- (2x)3 vs. 2x3
- Each component of the fraction within the parentheses gets raised to the power; these are all different (and the TI-Nspire CAS handles them nicely):
Attending to precision is essential for students, and by doing three similar but different problems as a set, they get practice analyzing how the notation changes the results.
Purpose #3: To Properly Represent Fractions. Fractions generally don’t need parentheses when written by hand, and I’m direct with students about my strong preference for a horizontal fraction bar rather than a diagonal bar when writing fractions on paper or on the board.
Complications can occur when students try to enter the fraction into a calculator without using a fraction template. Pressing the DIVIDE button to create the “slash”, as in 3/4, has the advantage of connecting a fraction with the operation of division but the drawback of the diagonal bar. For anything more complex than a simple fraction, parentheses are needed to “collect” the numerator and denominator so that the fraction is computed correctly. For example:
- Find the mean of these three test scores: 85, 96, 77.
- Graph a rational function
Thankfully, fraction templates are readily available, so errors using parentheses are avoided. On any TI-84+, set the mode to “MathPrint” and press ALPHA and Y= to access the template. On a TI-Nspire, press CTRL and DIVIDE or select the fraction from the template palate.
This was especially useful for finding the sum of the following geometric series; notice the error on the first try due to missing parentheses, and then the corrected version:
And the calculator comes to the rescue! I encourage students to enter complicated expressions all at once. Making separate entries for each part is taking a risk:
Purpose #4: To Indicate Multiplication. Probably the area in which I am observing the most “overuse” of parentheses is for multiplication. At some time before students reach me in high school, they have been taught that in addition to using the × symbol to multiply, they can also use • , a raised dot. A third alternative is to use parentheses to indicate multiplication, especially for negative integers or to distribute multiplication over addition:
- (–4)( –6) = 24
- 2x(x + 5) = 2x2 + 10x
I’ve seen some students “over-distribute” if they rely on parentheses instead of the raised dot for multiplication:
11. (–4)(x)(3x2) should be –12x3 ; however what if a student “distributes” the –4?
[One more pet peeve of mine: when students utilize the × symbol for multiplying even when using the variable x. I strongly suggest that once they are in Algebra I, students should “graduate” to the raised dot • to symbolize multiplication.]
When using the Chain Rule in Calculus, students sometimes make the error of “invisible parentheses” and then lose them entirely in their subsequent algebraic simplification.
- Find the derivative of (3x2 – 4x + 5)–2
Notice the missing parentheses for (6x – 4) and how the error carries through.
Purpose #5: Operator Notations. My final category of parentheses usage is as part of the notation of certain function operators. Students are familiar with using parentheses in function notation f(x), where the independent variable x is the input for the function expression. Other functions such as logs and trig functions can use parentheses to set off their “arguments”, and the calculator supports this use by providing the left parenthesis. Entering the right parenthesis is optional on the TI-84+, but a good practice for students:
If students get in the habit of using the parentheses, it enables them to correctly apply the “expand to separate logs” and “condense to a single log” rule. Here the parentheses are not “needed” to indicate the argument of the log, but helpful for this student.
- Solve each equation:
And in these last two examples, the parentheses helps the student get the correct result:
- Expand to separate logs:
- Condense to a single log:
One final note: I want my students to harness the power of parentheses to support their conceptual understanding and mathematical accuracy. Being precise about notation is not about “doing it my way” but instead about doing it in a way that helps them grasp the purpose of the symbols they use to clearly communicate their mathematical thinking.
NOTES & RESOURCES:
For more about the “loss of invisible parentheses”, ambiguous fractions and other common math errors, see this site.
For one teacher’s approach to using parentheses to evaluate function values, read this blog post: An Algebraic Oath.
And here is one teacher’s elegant and simple definition of parentheses: Parenthetically Speaking.