Getting Started 
Misconception/Error The student does not simplify expressions correctly. 
Examples of Student Work at this Level The student:
 Completes operations in order from left to right rather than using the order of operations rules.
 Combines unlike terms.
 Imposes an equal sign somewhere in the expression and attempts to â€śsolveâ€ť it (e.g., 8  10xÂ  6 becomes 8  10xÂ = 6).
 Drops the variable from the expression and then attempts to evaluate the expression.
 Makes sign errors (e.g., 8  10xÂ + 6 becomes (8  6) + 10x).
 Makes equivalency decisions based on how the expression â€ślooksâ€ť saying [e.g., 10xÂ + 14 is not equivalent â€śbecause it canâ€™t have a 14â€ť or 8  10xÂ  6 is equivalent to 8  (10x  6) because â€śitâ€™s the same, except that it has parenthesesâ€ť].
Note: Beware of relying on the â€śyes/noâ€ť answers to determine correctness of a studentâ€™s response. As evidenced by this paper, the student â€śdropped the variableâ€ť from the expressions essentially evaluating each one for x = 1. The studentâ€™s equivalency decisions were correct but not necessarily for the right reason.

Questions Eliciting Thinking What are the order of operations rules? According to these rules, which should be done first â€“ multiplication or subtraction?
What does like term mean? In the given expression, which terms are like terms?
What is the difference between an expression and an equation?
What does it mean for expressions to be equivalent? Is it possible to have two expressions that look different from each other still be equivalent? 
Instructional Implications Explain what it means for expressions to be equivalent and provide instruction on using the Commutative and Associative Properties to rewrite the terms of an expression in a more â€śconvenientâ€ť order. Review the Distributive Property and explain how it can be used to combine variable terms such as 5x and 2xÂ [e.g., 5x + 2x = (5 + 2)x = 3x]. Eventually introduce the concept of â€ślike termsâ€ť and transition the student to simplifying expressions by distributing (when necessary) and combining like terms. Next use the Distributive Property to introduce the concept of factoring. Emphasize that the Distributive Property can be used to both expand (e.g., multiply) and factor expressions. Provide additional opportunities for the student to rewrite linear expressions with rational coefficients in factored form using the fewest number of terms. Guide the student to check factorizations by expanding and comparing to the original expression.
Explain the distinction between an expression and an equation. Provide examples of each in both realworld and mathematical contexts. Ask the student to identify examples of variables, constants, and coefficients in expressions, and to identify expressions within equations. 
Moving Forward 
Misconception/Error The student does not apply the Distributive Property correctly when simplifying expressions. 
Examples of Student Work at this Level The student:
 Does not distribute the negative on the outside of parentheses, writing:
 8  2(5x  3) = 8  10x  6
 8  (10x  6) = 8  10x  6
 Uses the Distributive property incorrectly by multiplying by the wrong numbers, writing:
 8  2(5x  3) = 24  10x having multiplied the eight times the (3) and the (2) times the 5x.
 8  (10x  6) = 80x  48 having distributed the eight rather than a 1.
 8  2(5x  3) = 10x + 5 having multiplied (2) times 5x but then subtracted three from eight.Â
 Ignores the parentheses when attempting to rewrite a subtraction as an addition {e.g., the student changes: 8  2(5x  3) to 8 + (2)[5x + (3)] and 8  (10x  6) to 8 + [10x + (6)]}.

Questions Eliciting Thinking What do you remember about the Distributive Property? How is it used? Once you change the subtraction to addition and the two to a 2, what number will you actually be distributing?
When there is a subtraction sign before a parenthesis, what does that mean? How can you show that you will be subtracting each term in the parentheses? 
Instructional Implications Focus instruction on applications of the Distributive Property. Review how the Distributive Property can be used to combine variable terms such as 5x and 2xÂ [i.e., 5x + 2xÂ = (5 + 2)x = 3x]. Next review how the Distributive Property can be used to expand expressions of the form a(b + c) where a, b, and c contain rational coefficients, both positive and negative. Then introduce more complex expressions such as 2x + 3(x â€“ 8), 12 â€“ 2(y â€“ 9), or 5z â€“ (7 + 4z) + 3z. Guide the student to rewrite subtractions in terms of addition. Be sure the student understands what number is actually multiplying the expression in parentheses. Provide the student with additional opportunities to simplify expressions by distributing and combining like terms. 
Almost There 
Misconception/Error The student makes computational errors or provides insufficient explanations to justify answers. 
Examples of Student Work at this Level The student makes a minor computational error.
The student does not show adequate work to support an answer or gives an unclear explanation such as:
 6(5x  3): No, because you shouldnâ€™t subtract the eight and two.
 8  10x + 6: Yes, because it looks the same.
 8  (10x  6): Yes, because you can multiply by two.
 8  10x  6: No, because if â€śplus sixâ€ť is right, then â€śminus sixâ€ť is wrong.
 10x + 14: Yes, because eight and six makes 14.Â

Questions Eliciting Thinking I think you may have made an error. Can you check your work or rework the problem to see if you get the same expression?
How did you determine your answers? Can you explain in more detail? Do you know which properties you can use to justify your answers? 
Instructional Implications Provide direct feedback to the student regarding his or her error and allow the student to correct it. Provide additional opportunities for the student to rewrite linear expressions with rational coefficients by applying the Distributive Property to expand expressions and combine like terms.
Assist the student in developing a more formal understanding of the properties of operations and model using the properties to justify the steps in rewriting expressions in equivalent forms. Allow Got It students to present their explanations and justifications to the class to serve as a model for the student who is having difficulty explaining and justifying. 
Got It 
Misconception/Error The student provides complete and correct responses to all components of the task. 
Examples of Student Work at this Level The student identifies the equivalent expressions and provides clear explanations or shows sufficient work to support answers:
 6(5x  3): No, because you canâ€™t subtract the 8  2 first. You have to use the Distributive Property first.
 8  10x + 6: Yes, because you distribute the negative two and drop the parentheses.
 8  (10xÂ  6): Yes, because you can distribute a positive two but you have to keep the parentheses because you still have to distribute a negative.
 8 10xÂ  6: No, because it wonâ€™t be minus six because I distributed a negative two times a negative three, which gives me positive or plus six.
 10x + 14: Yes, because after you distribute, you can combine or add eight and six to make 14.

Questions Eliciting Thinking What would you do differently if the expression was 2(5xÂ  3) + 8? Would that change how you start the problem? Would it change your answer?
How would you rewrite the expression if the first thing you wanted to do was rewrite all the subtractions as additions? 
Instructional Implications Provide the student with an expression involving fractional coefficients and constants, both positive and negative, and challenge the student to rewrite the expression in an equivalent form. Ask the student to justify each step of work. 