Writing Expressions

Task

Write an expression for the sequence of operations.

1. Add 3 to $x$, subtract the result from 1, then double what you have.
2. 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

1. This problem can be done step-by-step. 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:

   $$ \begin{alignat}{2} 2(1-(x+3)) &= 2(1-x-3) &\qquad &\text{distribute the -1} \\ &= 2(-x - 2) &\qquad &\text{subtracting 3 from 1} \\ &= -2x - 4 &\qquad &\text{distribute the 2} \\ \end{alignat} $$

2. 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.