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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.
Comments
Log in to commentAnne says:
over 1 yearI'm going to present this problem to my students tomorrow! I can't wait to see how they react to it.
leannef says:
over 5 yearsIn 6th grade, are we just introducing negative numbers, not asking students to add or subtract them. So the simplifying of this example would be above the content of the grade level.
Bill says:
over 5 yearsThanks Leanne for pointing this out. I think this task is misclassified and will reclassify it to Grade 7.
Bill McCallum