 
Mildred was at home correcting a test that she'd had in math class that day. The teacher had marked a question incorrect and Mildred couldn't figure out what she'd done wrong."I followed the order of operations" she said. "I did what's in the brackets in the first step, and then did the brackets first in the second step, multiplying 2 by 4". Fortunately I was visiting and was able to help. Her father could have helped, but he was at work. Her mother had been pretty good at math too, but she was busy in the basement cleaning up the remains of the chair that the dog had eaten. "Where did I go wrong?" Mildred asked me. "Brackets always come first, right?" I told her: "The problem is in this line":  "In the order of operations, doing 'brackets first' means doing what's inside the brackets first". I pointed at the 4. "That's just a number now. It doesn't really have to be in brackets any more, although lots of people will still use them. "You know that in the first line there is an invisible multiplicaton sign between the 2 and the bracket?" I drew an arrow to show her. She agreed. 
"Once what's in the bracket is reduced to a single number, it can be multiplied. The line really looks like this."  "The brackets don't have to be there". "So these are the same thing?" she asked me.
"Yes. Which means you do division and multiplication in order, left to right". "Eight divided by two is four, times four is sixteen", she said, completing the problem correctly. I summarized it for her. "'Doing brackets first' doesn't apply when the brackets just contain a number. The brackets don't have to be there". She wrote out a correct solution. 8 ÷ 2(2 + 2) = 8 ÷ 2(4) = 8 ÷ 2 x 4 = 4 x 4 = 16 |