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 Megan Termini. Her topic, from Pre-Algebra: order of operations.

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

The order of operations appears in pop culture in many different ways. An example is the song “Cupid Shuffle” by Cupid. There are certain steps that you do in a specific order. If you do not follow the order, then it is no longer the cupid shuffle. An activity would be incorporating the order of operations into the “Cupid Shuffle”. For example, the chorus is,

“Parentheses, Parentheses, Parentheses, Parentheses,

Exponents, Exponents, Exponents, Exponents,

Now Mult. or Div., Now Mult. or Div.

Now Add or Subtract, Now Add or Subtract.”

There are certain dance moves to go along with each step in the song. Here is a video of some students doing the song and dance (Reference A). This is a very effective way of teaching the students the order of operations(PEMDAS) because many students love music and dancing, and they are more likely to remember the song and dance moves than just memorizing the order itself.

How could you as a teacher create an activity or project that involves your topic?

There are tons of activities that you could do that involve the order of operations. As the teacher, you would want to create an activity that is fun and engaging for the students. Something that involves everyone in the class and not just a few students. One activity that would-be fun is Order of Operations War. Many students love playing the card game war. Now it is the same game just involving the order of operations. Each student will get a deck of cards and evenly deal them. Then they will get note cards with each of the operations on it. They will each flip 3 cards, arrange them with the operations and try to get as close to the target number as they can. The person who gets the closest is the winner of the round. This game would be a great way of getting all the students involved and a good way of learning the order of operations. (Reference B)

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

Learning the order of operations is very important for the students to learn, especially for their future courses in mathematics or science. The order of operations is used is almost every mathematics course from then on and most of the science courses. That is why is it very important to understand how it works. You know that you will use them in math and science course, but also you will use the idea of order of operations in computer sciences courses. When programming, the code has to be in a specific order to work. Just like a math problem, if you don’t apply the operations in the correct order, then you won’t get the correct answer.

References:

A. (2014, March 11). Retrieved September 01, 2017, from https://www.youtube.com/watch?v=EfgtWthLvk4

B. Order of Operations War With Just A Deck of Cards. (n.d.). Retrieved September 01, 2017, from http://us9.campaign-archive2.com/?u=3c5f5b9960a466398eccb35f8&id=cf58289e69&e=c87fd3cb28

Thoughts on Silly Viral Math Puzzles

I’ve seen silly math puzzles like this one spawn incredible flame wars on social media, and for months I’ve wanted to write an article about how much I’ve grown to loath these viral math posts.

Of course, after months of dilly-dallying, someone else beat me to it: http://horizonsaftermath.blogspot.com/2017/08/sick-of-viral-math.html. I encourage you to read the whole thing, but here’s the post’s outline of the myths perpetuated by these puzzles:

Math is just a bag of tricks.
Math is memorizing a set of rules.
Math problems have only one right answer.
Being smart means solving problems quickly.
Math is not for you.

Engaging students: Order of operations

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 Lisa Sun. Her topic, from Pre-Algebra: order of operations.

How could you as a teacher create an activity or project that involves your topic?

Given that my students have knowledge on the topic of Order of Operations, I will provide them a project where they must apply their knowledge and present it in front of their peers. Students will each receive a number from me and they must create a mathematical problem, an equation, using all of PEMDAS (Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction).

Students will then present individually in front of their peers at the board. The presenter’s role is to be the teacher. To have the ability to clearly vocalize his/her thought process to achieve their given number with the use of PEMDAS. As each student presents, the audience will be following along on a sheet of paper where they must also solve the equation that the presenter created. This paper must be turned in along with their own project to document that they were paying attention. The audience’s role is to be the grader. To make sure that the presenter’s use of PEMDAS was correct to achieve the number that was given to them. If the presenter’s use of PEMDAS is incorrect, I will select an audience member to explain. The presenter will then have to come present their project again to me before or after school so that I can make sure there is no misconception regarding the Order of Operations.

To help motivate the students’ to be precise with their project, I would state that if all students were able to display their use of PEMDAS correctly, everyone would receive 5 extra points on the upcoming test. I believe that this project would be great for students to strengthen their knowledge on Order of Operations. As they are taking up on their roles as the grader, they are physically and mentally reinforcing their knowledge by solving problems after problems. As the teacher, they are verbally reinforcing their knowledge.

How has this topic appeared in pop culture?

Figure 1: Pokémon Center Lego

Pokémon Go is the craze among society today and I believe it would be fitting to engage the class with both Pokémon and Legos. I would present this to the class, preferably one that is physically available to the class and ask the following questions:

When building this Lego figure, do you think procedures need to be followed sequentially?
What happens if they are not? (Display to the class what the Lego figure would look like).

I would discuss why doing things in order is important tying it with Orders of Operations. Display a problem with Orders of Operations but solve it by not following PEMDAS (Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction) and state that the solution comes out to be incorrect. Similar to how the Lego that was built in the wrong order didn’t match up with the picture on the box.

I believe YouTube can be a great learning tool in the classroom when it comes to engaging students. People of all demographics post helpful tools on this site that are so easily relatable to students today. Below is a video of a PEMDAS rap song.

I will be playing this PEMDAS rap song as students are walking into class to quickly engage the students. Once class has officially started, I would play this video again as students are reading the lyrics and following along the two examples the video provided. This video is to aid the students to remember the Orders of Operations by the use of PEMDAS (Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction). To engage the students even more, I would have the students sing along the chorus. “Parentheses first, exponents next, multiplication and division in the same step. Addition and subtraction, if you got the nerve, from left to right, first come first serve”. Hopefully, this song will be catchy enough for the students to have it be stuck in their head for a while.

References:

http://www.purplemath.com/modules/orderops.htm

http://www.pbslearningmedia.org/resource/mgbh.math.oa.ooo/order-of-operations/

Engaging students: Order of operations

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

This student submission again comes from my former student Dorathy Scrudder. Her topic, from Pre-Algebra: order of operations.

B2. How does this topic extend what your students should have learned in previous courses?

Students should know how to add, subtract, multiply, divide, and use exponents by the time we cover this topic. To begin the class, I will have students split into two groups. Both groups will be given a series of one-step equations that correspond to a multi-step equation; however, one group will be given the steps out of order. We will then discuss why the two groups were working on the same multi-step equation but have different answers. The students should find it interesting and ask a few follow up questions such as, a) how do we know which answer is correct – it is correct by use of the order of operations which was decided on by mathematicians in the 1600s; b) how do we know what order to do the operations in – we use the acronym PEMDAS which stands for Parenthesis, Exponents, Multiplication/Division, Addition/Subtraction; c) how are we supposed to remember the acronym – we can either pronounce it pem-das or use the saying Please Excuse My Dear Aunt Sally.

C2. How has this topic appeared in high culture (art, classical music, theatre, etc.)?

The order of operations is also utilized in many areas outside of math. Take a play for example. To begin, an actor or actress must first audition for the desired role. Once they have been hired, the actor/actress must learn their lines and then rehearse with the other actors and actresses before the opening night of the play. The actor/actress cannot perform the opening night if they have not learned their lines yet. The set designers must also follow the order of operations. They must first design what they want the set to look like and then decide what materials they need and how much to buy. Once they have the materials, they cannot start painting intricate details until they have constructed the set. Following the order of operations is an important concept and hopefully these examples will help the students understand why we need to follow the steps in the given order.

D2. How was this topic adopted by the mathematical community?

Multiplication before addition has been a common practice since before algebra was written, however, it was always an assumption and mathematicians never felt that it had to be proved. The earliest printing that we have where multiplication comes before addition is from the early 1600s. Dr. Peterson, from Ask Dr. Math, has stated that he believes the term “order of operations” has only just come into common use within the past century by textbook authors. Sarah Sass, from University of Colorado in Denver, has found that students have trouble when it comes to the multiplication and division step and again at the addition and subtraction step of the order of operations. She suggests that instead of using “please excuse my dear aunt sally,” in which students often assume all multiplication comes before division and all addition comes before subtraction, Sass suggests that we teach “Pandas Eat: Mustard on Dumplings, and Apples with Spice.” This allows the students to understand that the mustard and dumplings, or the multiplication and division, go together at the same time, while the apples and spice, or addition and subtraction, are completed at the same time, all from the left to the right.

References:

http://www.math.ucdenver.edu/~jloats/Student%20pdfs/4_Order%20of%20OperationsSass.pdf

http://mathforum.org/library/drmath/view/52582.html

Engaging students: Order of operations

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 Theresa (Tress) Kringen. Her topic, from Pre-Algebra: order of operations.

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

Order of operations is commonly used in most mathematics problem that involve more than one operation or when parenthesis are involved. It would be easy to show the students what the answer to a given problem, say 5+20/5, would be when using the proper order of operations, then solve the problem by solving left to right as you would read a book. It is clear, to a math major, that the answer is 9. For someone who does not know the order of operations, they most likely would come up with the answer of 5. The difference in the correct answer and the incorrect answer is only 4, but the problem is only working with numbers less than or equal to twenty. It would then be beneficial to point out that when dealing with more complex problems, that this answer may become even larger. If the class was working on given problems, I would give them a few word problems to solve. Once they solved them on their own, I would show them that the difference between the correct way to answer the given problem and the incorrect way to answer the problem to help them connect the concept to why it is important to compute answers in the way.

How does this topic extend what your students should have learned in previous courses?

This topic extends what students should have previously learned by allowing them to use their skills of multiplication, division, exponents, addition, and subtraction to solve more complex problems. When learning how to solve problems more complicated than what they have been given in the past, they use this topic to guide them through to the next step. They must already be familiar with all of the operations by themselves prior to using the order of operations to solve a problem. Once they are accustomed to using the order of operations, the will be given more challenging problems and their math skills will build upon itself. It is clear that if a student is unable to solve a simple problem, such as an exponent problem or a more complicated division problem, they will not be able to use the order of operations for problems that contain what they have not learned.

How did people’s conception of this topic change over time?

It is believed that the idea of using multiplication before addition became a concept adopted around the 1600s and was not disagreed about. The other operations took their place in the order over time, beginning in the 1600s. It seems that although it was not documented well, most mathematicians agreed upon the same order. It wasn’t until books stated being published that it was important to document the order of operations. The notation may have been different depending on who was writing on the subject, but the concept was the same. It seems that although it was not documented well, most mathematicians agreed upon the same order. Once books were being published, the order, PEMDAS (Parenthesis, Exponents, Multiplication, Division, Addition, and Subtraction), was put into print. Now, teachers use the phrase Please Excuse My Dear Aunt Sally as a way for students to remember the acronym and are able to put it to use.

http://jeff560.tripod.com/operation.html

http://mathforum.org/library/drmath/view/52582.html

Engaging students: Order of operations

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:

Resources: http://www.glencoe.com/sec/math/algebra/algebra2/algebra2_05/extra_examples/chapter5/lesson5_1.pdf

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.

Resources: http://archives.math.utk.edu/visual.calculus/2/chain_rule.4/index.html

Student misconceptions about PEMDAS

Simplify $6/2*(1+2)$ .

A Common Incorrect Answer. According to PEMDAS, we should handle the parentheses first. So $6/2*(1+2) = 6/2*3$ . Next, there are no exponents, so we should proceed to multiplication. So $6/2*3 = 6/(2*3) = 6/6$ . Finally, we move to division, and we obtain the answer $6/6 =1$ .

The above answer is incorrect and (even worse) arises from a natural but unfortunate misconception of the way that children are commonly taught order of operations. If you don’t see the misconception, please give it some thought before continuing.

The mnemonic PEMDAS, commonly taught in the United States, stands for

Parentheses

Exponents

Multiplication

Division

Addition

Subtraction

I personally never learned this memorization trick when I was in school. What I do remember, from learning BASIC computer programming around 1980, was the mnemonic My Dear Aunt Sally. I’m told that in the United Kingdom (and perhaps elsewhere in the English-speaking world) schoolchildren are taught BIMDAS, where B stands for Brackets and I stands for Indices.

Unfortunately, all of these memorization devices suffer from a common flaw: they do not indicate that multiplication and divison have equal precedence, and that addition and subtraction have equal precedence. In other words, the order of operations really are

Parentheses

Exponents

Multiplication and Divison (left to right)

Addition and Subtraction (left to right)

Therefore, the correct answer to the above problem is

$6/2*(1+2) = 6/2*3 = (6/2)*3 = 3*3 = 9$ .

In brief, though not intended by teachers, PEMDAS and BIMDAS perhaps promote the misconception that multiplication takes precedence over division and addition takes precedence over subtraction. To avoid this misconception, one of my colleagues suggests that PEMDAS be taught more visually as

P
E
MD
AS

so that students will have a better chance of remembering that MD and AS should have equal precedence.