Engaging students: Order of operations

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Alyssa Dalling. Her topic, from Pre-Algebra: order of operations.

C. How has this topic appeared in pop culture (movies, TV, current music, video games, etc.)?

Hannah Montana is a Disney series that aired from 2006-2011. On this episode titled “Sleepwalk This Way”, Miley’s dad writes her a new song which she reads and doesn’t like. She decides to keep her dislike of the new song to herself causing her to start sleepwalking. In order to not tell her dad what she thinks of the song while sleepwalking, Miley stops sleeping which causes her many problems. One such problem occurs when Miley gets dressed in the wrong order causing her to get an unwanted result.

I would start out the class by showing the first 46 seconds of this Hannah Montana scene. (Editor’s note: Trust me, this is hilarious.) This scene is perfect for the engage because it is a way to relate the order of operations to getting dressed. After watching the scene, the teacher would explain that just like getting dressed in the proper order is important, the order of operations when doing math is as well. The students would learn PEMDAS (parenthesis, exponents, multiplication, division, addition, and subtraction) and try different problems to get them better acquainted with the concept.

B. How can this topic be used in your students’ future courses in mathematics or science?

The order of operations will be used in almost every math class following Pre-Algebra. One example is in Algebra II when students start working with problems involving simplifying numbers and multiple variables. One example is

$\left( \displaystyle \frac{18a^{4x} b^2}{-6 a^x b^5} \right)^3$

Start out the class by asking students how the order of operations says to answer this question.  Most students will follow method two below. Upon completion of this lesson, students will learn multiple methods of problem solving which expand their previous knowledge of order of operations.

The first method students can use is to raise the numerator and denominator to the third power before simplifying. By raising each variable to the third power, no rules in the order of operations will be broken showing the student there is more than one way to use the order of operations. (Reference Method One below).

The method most students will originally think of is simplifying the fraction before raising it to the third power. The student would follow their previous knowledge of PEMDAS in order to simplify the equation to the reduced form. (Reference Method Two below). In either case, the students will see that the solution can be found by using a variety of different means that all fall under the order of operations.

Method One:

Method Two:

B. How can this topic be used in your students’ future courses in mathematics or science?

An understanding of the order of operations is relied upon in Calculus as well. One application is when learning the chain rule. The following YouTube video does a fun job at explaining the chain rule by using a catchy song. The students are able to learn the rule and see examples that they can use to help them with this concept. Start it at 1:32 and end it at 2:10 (shown below).

The chain rule is used to find the derivative of the composition of two functions. So if $f$ and $g$ are functions, then the derivative of $f(g(x))$ can be found using the chain rule. Using the example $F(x) = (x^3+5x)^2$ , the chain rule states that the derivative will be $F'(x) = f'(z) g'(x)$. Following this definition, the student finds the derivative to be $2(x^3+5x)(3x^2+5)$ . This is where the order of operations comes in. The student must use their previously acquired skills from Pre-Algebra as well as Algebra II to simplify the expression. From their previously acquired knowledge, the student would know they would have to multiply the $2$ by each expression in $f'(z)$. Also, if a question asked the student to find the derivative when $x=3$, the student would have to use their knowledge of the order of operations to find the solution after applying the chain rule.