I LOVE teaching Order of Operations. It is one of those topics in math that students are initially totally overwhelmed by, but once they get the hang of it, they love it too! It is so fun having students tell me how easy it is and to watch their eyes light up as they “get it.” Sadly, not every student “gets it” right away. After teaching this topic for the past ten years, some mistakes and misconceptions have become so common, I just had to write about them, and what I do in my classroom to overcome them.
Common mistakes when students solve Order of Operations problems:
Working ONLY left to right – Students are DYING to work from left to right, no matter what. If you think about it, this makes sense. This is how we all read and these kiddos are only 10 or 11. They have been forcing their brains to work left to right for the past 4-5 years and now we are asking them to look at math problems in a whole new way! In order to break this habit, it is important to explicitly teach students the rules “PEMDAS” or “GEMDAS” and to explicitly explain what they mean and demonstrate using multiple problems as examples. When I first started teaching, I made the mistake of thinking that just teaching the rules would be enough. It turns out that there will always be several students that don’t understand that the list is actually giving an ORDER of operations. I know, it seems a bit ridiculous to us math teachers, but kids just don’t always “get it” on the first try with this one. One way I help kids understand is by actually having them write First, Second, Third, Fourth next to the acronym as we learn it, like in this notebook page:
I also make sure that my students build up to larger problems slowly. I like using this domino game in my classroom either as partner work or in centers so that students get the practice they need with smaller expressions before moving on to longer, more complicated ones.
Always Multiplying or Adding First – The next hurdle is once we get kids broken of the “don’t work left to right” rule is that we now WANT them to work left to right when doing multiplication/division or addition/subtraction. This can be infinitely frustrating for the students. It is important to not just teach students to memorize Multiplication, Division, Addition, Subtraction, but that we teach them to memorize Multiplication Division left to right, Addition Subtraction left to right. In addition, this is a part of what we write in our notes and what goes up on my anchor chart.
Double-using Numbers – When students are solving a portion of the problem, at times they may bring down a number they just used into the next part of the problem. Here is an example:
In the example, the student used the three twice. Then, he didn’t know what to do next. At this point, students may guess or even combine the numbers into one long number. To combat this problem, I have students cross off any numbers and operations they solve, write the solution to that part of the problem, and re-copy any portions of the problem they have not yet used like this:
Not following the order of operations within parentheses – Since we teach students to solve the parts of the problem within parentheses first, when there are multiple operations within those parentheses, we will often see students reverting back to doing everything left to right. It is important to teach them that the multiplication/division and addition/subtraction rules still apply inside parentheses. This is yet another rule that needs to be taught explicitly. Do not assume that your students will just automatically apply the rule within the parentheses.
Multiplying the Base by the Exponent – It is in student’s first nature that when solving exponents, they will multiply the base number by the exponent (instead of by itself). If you are using exponents in your order of operations problems, be sure that the students have a full grasp of exponents in isolation before they move onto this topic. Once it is learned fully, this one is not too much of a problem, but I don’t recommend just teaching students how to deal with exponents once they see them in an order of operations problem.
Not Re-Writing all Parts of the Problem Correctly – Students have the most difficulty with this as they work through problems with the nested grouping symbols. They will solve one portion of the problem and then as they copy the problem to the next layer, they will incorrectly transcribe the problem from the line before. I most often see them omitting the grouping symbols which, of course, completely changes the meaning of the problem. In my experience, this happens most often when students are rushing. I try to make sure that students work through their order of operations as slowly as necessary, and I even let some students check their work after each question for accuracy. I also like to give my students a variety of questions with nesting symbols in them so they get plenty of practice. This set of task cards is perfect for making sure the students get the practice they need.
Do you see other common mistakes in your class, or have other things you do to overcome mistakes? Comment below to share!