A significant problem for many students is that they type math expressions into their calculators unaware than the calculator will interpret it differently that the students intend.
-22 = ? For instance, take the squaring of a negative number. Suppose x = -2. What is x2? That would be -2 times -2, which equals 4. Everybody knows that.
So does that mean that “negative two squared equals four”? Not so fast.
Many students will calculate this on their calculator by typing the negative sign, then the two, then the squaring function. And the calculator says that the answer is -4. How is this possible?!
The answer lies in the difference between the order of operations, which the calculator recognizes, and the natural human tendency to hear “negative two squared” and interpret it as a negative number that is getting squared. After all, “squared” comes at the end; shouldn’t it wait it’s turn? In reality, however, the order of operations says that the exponent after the two has a higher priority than the negative sign in front of the two, and thus the two gets squared first apart from the negative, and then the negative applies to the 22, creating a -4.
If students want to square a negative number on their calculator, they must overrule the priority of the exponent by using a grouping symbol or parenthesis, like this: (-2)2
In summary: -22 = -4 (-2)2 = 4
8 = ? 2x Dividing by a product causes problems as well. Most people would pronounce the above expression as “eight over two x,” which naturally sounds like the “eight” is upstairs (in the numerator) and the “two x” is downstairs (in the denominator). The problem is that students will type these directly into the calculator as “8/2x” and expect the calculator to understand that the 2x is downstairs. But what happens?
While some calculators will think that way (i.e. Casio fx-300MS), many others will not (i.e. TI-83 and many others). These latter calculators will “understand” this expression as “8/2•x” because there is (as everyone agrees) an implied multiplication sign between the two and the x; and if that is the case, the order of operations says that the division takes priority over the multiplication because it comes before it because it is left of it. Thus the expression gets understood by the calculator as "(8/2)•x," because the x is getting multiplied by the 8/2 ultimately.
A simply remedy for this is, again, to use parentheses:
"8/(2x)" will convince the calculator that you want the 2x calculated and placed in the denominator.
There are many other instances where students and their calculators fail to communicate. And most, if not all, of them can be traced to the order of operations. Students must learn how their calculators work, whether by experience or by reading the instructions.