Getting Started 
Misconception/Error The student is unable to identify the sequence of operations used in the algebraic solution. 
Examples of Student Work at this Level The student correctly identifies operations in order in the arithmetic solution but lists the operation signs in the order in which they appear in the algebraic solution rather than the actual operations used to solve the equation.

Questions Eliciting Thinking What is Haleyâ€™s approach to this problem? What does her solution method contain that Luisâ€™s does not?
Where in the problem can you see the operations that Haley used? What are they? 
Instructional Implications Review the procedure of solving an equation using the properties of equality. Emphasize how the properties of equality are used to transform an equation into a statement of the form x = a (where x is a variable and a is a real number). Give the student equations to solve and ask the student to clearly show operations completed as a result of applying the properties of equality. Ask the student to summarize, in order, which operations were used. Demonstrate comparing an algebraic solution to an arithmetic solution, identifying the sequence of operations used in each approach.
Provide the student with solved equations (similar to the algebraic solution given in the Algebra or Arithmetic? worksheet) and ask him or her to identify the properties of equality and the operations used to transform the equation at each step. 
Moving Forward 
Misconception/Error The student is unable to accurately compare the sequence of operations used in the two solution methods. 
Examples of Student Work at this Level The student correctly identifies the operations used in each solution method but does not compare them. Instead, the student compares other aspects of the two approaches. The student says:
 Haleyâ€™s solution is â€ślongerâ€ť or â€śmore complicated.â€ť
 Haleyâ€™s solution â€śused a variableâ€ť or â€śused an equation.â€ť
 Haley and Luis did the problem different ways but got the same answer.
 They both got an answer of seven.

Questions Eliciting Thinking What operations did Luis use? What operations did Haley use? How do these two sequences of operations compare? 
Instructional Implications Direct the studentâ€™s attention to the actual operations used in each approach. Ask the student to compare the specific operations. Guide the student to observe that the sequence of operations used in each approach is the same. Provide additional problems that can be solved either arithmetically or algebraically and ask the student to solve each problem using both approaches. Then ask the student to compare the sequence of operations used in each. 
Almost There 
Misconception/Error The student is unable to identify the significance of an expression or quantity given in an algebraic solution. 
Examples of Student Work at this Level The student correctly identifies the operations used in each solution method and compares them. However, the student is unable to identify the significance of 79.95x within the story context. The student:
 Provides a noncontextual response (e.g., â€ś79.95 times xâ€ť).
 Describes the meaning of x rather than the meaning of 79.95x.
 Provides a literal interpretation (e.g., â€śthe cost of a cartridge times the number of cartridgesâ€ť).

Questions Eliciting Thinking In the problem, what does 79.95 stand for? In Haleyâ€™s equation, what does x stand for?
How would you describe the quantity 79.95x? What does it equal in Haleyâ€™s equation? What does 559.65 represent in this problem?
Is the value of 79.95x the same as the value of 559.65 Ă· 79.95? 
Instructional Implications In the process of solving word problems with equations, emphasize the relationship between algebraic expressions and the quantities they represent in context. For example, 4x + 9 = 21 describes a situation in which $21 is the total cost of a $9 item and an unknown quantity of $4 items. Ask the student to describe the meaning of 4x and 4x + 9 as well as other quantities that arise in the course of solving the equation (e.g., the 12 in 4x = 12). Provide additional opportunities to write equations that model relationships among variables given in word problems. Ask the student to explain the meaning of the variable, variable expressions that appear in the equation, and quantities that arise in the course of solving the equation.
Provide additional problems that can be solved both arithmetically and algebraically and ask the student to solve each problem using both approaches. Guide the student to use the arithmetic solution to help identify the meaning of expressions and quantities in the algebraic solution. For example, to identify the meaning of 79.95x, the student can refer to the equation 79.95x = 559.65 to determine that it equals 559.65. Then the student can refer to the arithmetic approach to determine the meaning of 559.65 as the total cost of cartridges and conclude that 79.95x must represent the total cost of cartridges. 
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 states that subtraction then division are the operations used both in the arithmetic solution and the algebraic solution. The student states that 79.95x represents the total cost of the cartridges and this value is represented as 559.65 in the arithmetic solution.
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Questions Eliciting Thinking How did you determine the significance of 79.95x in the algebraic solution?
How did you know that 559.65 represents the total cost of the cartridges in the arithmetic solution? 
Instructional Implications Challenge the student to consider the advantages an algebraic approach might have over an arithmetic approach.
Have the student develop a word problem along with both an algebraic and an arithmetic solution. Ask the student to explain the meaning of variable(s) used, variable expressions that appear in the equation, and quantities that arise in the course of solving the equation. The word problem can be shared with other students for additional practice and experience. 