The Order of Operations & a freebie!

When I was a middle schooler (a few years ago now!) I learned the order of operations with the acronym PEMDAS. We remembered it with the pneumonic phrase "please excuse my dear aunt Sally." I definitely remember the phrase, and I always remembered what they stood for... but I didn't really conceptually understand what I was expected to do. I followed the order strictly as it was written- parentheses, exponents, multiplication, division, addition, and subtraction. I remember struggling through algebra and wondering why sometimes division came before multiplication. My mind as further blown when a student moved in from another state and knew the order of operations as BEDMAS. How could that be the same?? The problem is pervasive- check out some of the popular meme pages for math problem postings. Grown adults solve problems incorrectly regularly, and argue about their strategy. Even those that solve it correctly generally back their answer up by sayings "I followed PEMDAS." That's troubling- because that is an acronym, not a mathematical procedure. 

I have been introducing the order of operations in a new way to my kiddos to try to prevent these types of misconceptions.

The two triangles above are what I give my kiddos now. They paste both into their math journals to refer back to. Why two different organizers? Well, I think it illustrates the fact that division and multiplication (or addition and subtraction) are to be completed in the order they appear, from left to right. If students constantly hear the operations in a specific order, many of them will memorize that and inadvertently solve problems in that order. The use of two graphic organizers with the operations reversed in each section helps reinforce that equal weight is given to "MD" and "AS" in each step. I also usually have my kiddos draw an arrow from left to right in the three lower sections of the triangle just to make it a bit more clear.

In fifth grade, we do not have to learn about square roots, but I give my kiddos the graphic organizer with square root on it anyway. I read an interesting article a few years back about "rules that expire" which inspired me to create it this way. The basic idea of the article was that throughout a child's life, they learn things in math that continually "expire" as they go on. For instance, when you multiply by 10, just add a zero! Well, that "expires" in fifth grade when they begin multiplying decimals. 42.5 x 10 is not 42.50! Conceptual understanding clears this up, of course, but trying to avoid the rules that "expire" is very helpful, too. In this case, the order of operations "expires" when square roots come into play if the students have not heard about them before. 

I love that the triangle shape takes the form of the problems as they are being solved, too! Each step along the way makes it smaller and smaller, until the work is eventually in the shape of a triangle. 

If you are interested in using this graphic organizer in your classroom, check the freebie here. I have posted several versions of the organizer, including as pictured above, with and without square roots, and a blank triangle to be filled in.  Enjoy!