Writing Expressions
Task
Write an expression for the sequence of operations.
 Add 3 to $x$, subtract the result from 1, then double what you have.
 Add 3 to $x$, double what you have, then subtract 1 from the result.
IM Commentary
The instructions for the two expressions sound very similar, however, the order in which the different operations are performed and the exact wording make a big difference in the final expression. Students have to pay close attention to the wording: “subtract the result from 1” and “subtract 1 from the result” are very different.
Solution

This problem can be done stepbystep. We first add 3 to $x$:
$$ x+3. $$Then we subtract the result that we just got from 1:
$$ 1(x+3). $$We then double, meaning we multiply this entire expression by 2:
$$ 2(1(x+3)). $$If we choose to simplify this expression, we use the distributive, commutative and associative properties in the following way:
$$ \begin{alignat}{2} 2(1(x+3)) &= 2(1x3) &\qquad &\text{distribute the 1} \\ &= 2(x  2) &\qquad &\text{subtracting 3 from 1} \\ &= 2x  4 &\qquad &\text{distribute the 2} \\ \end{alignat} $$ 
Again, we add 3 to x:
$$ x+3 $$This time, next we double, meaning multiplying this expression by 2:
$$ 2(x + 3). $$Then we subtract 1 from the result and we have:
$$ 2(x+3)−1. $$If we choose to simplify this expression, we use the distributive and associative properties in the following way:
$$ \begin{alignat}{2} 2(x+3)1 &= (2x+6)1 &\qquad &\text{distribute the 2} \\ &=2x + 5 &\qquad &\text{subtracting 1 from 6} \end{alignat} $$Notice that the final expressions are very different, even though the instructions sounded very similar.
Writing Expressions
Write an expression for the sequence of operations.
 Add 3 to $x$, subtract the result from 1, then double what you have.
 Add 3 to $x$, double what you have, then subtract 1 from the result.