Getting Started 
Misconception/Error The student is unable to correctly complete a table of functional values. 
Examples of Student Work at this Level The student miscalculates most of the functional values for the two functions and is unsure how to continue.

Questions Eliciting Thinking Do you know what it means to find the corresponding values of f and g? Can you calculate f(2) or g(1)?
Can you find the values of f and g for all of the integer values of x from 2 to 3? 
Instructional Implications Assist the student in completing the table. Indicate that there needs to be a column (or a row) for each of x, f(x), and g(x). Be sure the student understands the entries for x should be 2, 1, 0, 1, 2, 3. Explain how the prompt (e.g., â€śinteger values of x for â€ť) results in these values. Ask the student to carefully evaluate both f and g at each of these values. Then model how to use the table to find the solutions of the equation. Explain that since both f(2) = 8 and g(2) = 8 then f(x) = g(x) when x = 2. Explain that any value of x for which f(x) = g(x) is a solution of the equation f(x) = g(x) or Â = 3x+2. Make clear that the solutions of the equation f(x) = g(x) are the xcoordinates of the solutions common to the two equations. Ask the student to verify that both x = 1 and x = 2 satisfy the equation Â = 3xÂ + 2. Provide additional opportunities to make tables of values for two functions in order to find common solutions. Assist the student in understanding the implications of this kind of exercise for solving a system of equations.
If necessary, review function notation. Provide frequent opportunities for the student to use function notation to develop an understanding of its meaning and the conventions surrounding its use. 
Moving Forward 
Misconception/Error The student does not understand how to use the table to find solutions of the equation f(x) = g(x). 
Examples of Student Work at this Level The student creates a table inputting values of 2, 1, 0, 1, 2, 3 for x and correctly calculates most or all of the values of f and g but the student is unable to use the table to identify solutions of the equation f(x) = g(x). The student identifies:
 All values in the table as solutions of the equation f(x) = g(x).
 Values that are not solutions.
 Only x = 1 (since each entry in this row is 1).
 Only x = 2 [or (2, 8)] and upon questioning indicates that he or she thought there could be only one solution.
 The functional values, 1 and 8, as solutions rather than the corresponding values of x.
The student:
 Attempts to solve the equation Â rather than using the table.
 Indicates that he or she does not understand what it means to find solutions of the equationÂ f(x) = g(x).

Questions Eliciting Thinking What does it mean for f(x) to equal g(x)? Can you write out the equation f(x) = g(x) using the functions in this problem? Where can you find the solutions of this equation in the table?
How did you determine which values from the table are solutions?
What makes x = 1 (or x = 2) a solution? Are there any other solutions?
If you substitute 8Â for x, will you get a true statement? 
Instructional Implications Be sure the student understands the meaning of function notation such as f(2). Use the functions given in this task to show the student that f(x) = g(x) becomes the equation Â = 3xÂ + 2. Then model how to use the table to find the solutions of the equation. Explain that since both f(2) = 8 and g(2) = 8 then f(x) = g(x) when x = 2. Explain that any value of x for which f(x) = g(x) is a solution of the equation f(x) = g(x) or Â = 3xÂ + 2. Make clear that the solutions of the equation f(x) = g(x) are the xcoordinates of the solutions common to the two equations and there can be more than one solution. Ask the student to verify that both x = 1 and x = 2 satisfy the equation Â = 3x + 2. Provide additional opportunities to make tables of values for two functions in order to find common solutions. Assist the student in understanding the implications of this kind of exercise for solving a system of equations.
If necessary, review function notation. Provide frequent opportunities for the student to use function notation to develop an understanding of its meaning and the conventions surrounding its use. 
Almost There 
Misconception/Error The student makes a computational error when calculating values of f and g. 
Examples of Student Work at this Level The student creates a table inputting values of 2, 1, 0, 1, 2, 3 for x and correctly calculates most of the values of f and g but determines f (2) = 8, f (2) = 6, or g(1) = 4. The student correctly identifies the solutions of the equation f(x) = g(x) given the error.

Questions Eliciting Thinking I think you may have made a mistake when you calculated f (2) (or any other incorrectly calculated value). Can you review your work and see if you can find your error?
What does it mean to cube a value? Can you show me how you cubed 2?
What kind of function is g? What should be true of the differences between successive values of g(x) for consecutive values of x?
What does this exercise tell you about the solutions of the system of equations that include functions f and g? 
Instructional Implications Provide feedback to the student concerning any errors made and allow the student to revise his or her work. Be sure the student readdresses the issue of finding solutions of the equation f(x) = g(x). Provide additional opportunities to evaluate algebraic expressions that contain exponents, radicals, absolute value, and rational coefficients. 
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 creates a table inputting values of 2, 1, 0, 1, 2, 3 for x and correctly computes the values of both f and g for these inputs. In particular, the student determines that f(1) = 1 = g(1) and f(2) = 8 = g(2). The student indicates that x = 1 and x = 2 are solutions of the equation f(x) = g(x).

Questions Eliciting Thinking Do you think it is possible there are other solutions of the equation f(x) = g(x)?
When would this approach to finding common solutions be impractical? What other strategy might you use to find the solutions of this system?
What does this exercise tell you about the solutions of the system of equations that include functions f and g? 
Instructional Implications Allow the student to use a graphing calculator or other graphing technology to further explore systems of equations and their graphs. Have the student use graphing technology to find approximate solutions of systems of equations. 