In America they say "PEMDAS" and the British often use "BODMAS" and some decry any mnemonics at all, but everyone teaches essentially the same rules of evaluation.

Consider the poor Algebra I or II student, who has seen and heard the following in his first few weeks of school:

"Multiplication and Division are handled as equals, from left to right."

"When we say 6

*a*we mean 6 times a, where a just represents any number that we might choose to be use in place of it"

And then later we write on the board 6a

^{2}/ 3a = ?

and so the student says, "I get it, that means 6 times a

^{2}then I divide by 3 and multiply by a. It must be 3a

^{3}."

Then, wanting to follow instructions, he types the expression into his Ti-84 + silver edition calculator after storing a value for x, and sure enough, he gets the value of 3x

^{3}.

But of course, you explain, that's wrong. when we divide like this we mean that the 3a is intended to be a monomial term, taken as a single unit.

Your student is compliant, so he/she just nods and murmurs "Ok" and as you walk back to the front of the room, turns to their neighbor and says "You understand that?" and gets a shrug and a side to side head-shake... all just part of the mystery of mathematics.

If you realize what is confusing them, you may become very conscientious about writing problems with the fractions written out with horizontal fraction bars $\frac{6x^{2}}{2x}$ and talk about "implied parentheses". Unfortunatly many of your students have only heard the term "implied" used in prejudicial situations;"He implied I had stolen it!" or similar, and really have no idea what you mean. So maybe you get REALLY conscientious and write every fractional expression using actual parentheses $\frac{(6x^{2})}{(2x)}$.... and then that night on the homework, the book does NOT use them.

The repercussions of this order of operations confusion leads to students unsure of whether to write the linear term in an equation as $\frac{5x}{2}$, or 5$\frac{x}{2}$ or $\frac{5}{2}$ x.. but sometimes that even leads to $\frac{5}{2x}$ .