Getting Started 
Misconception/Error The student does not understand the concept of a zero. 
Examples of Student Work at this Level The student substitutes zero for the independent variable and solves for the dependent variable. 
Questions Eliciting Thinking What is the definition of a zero of a polynomial?
How can you find the zeros of a polynomial? 
Instructional Implications Review the definition of a zero (e.g., a value of x that makes y equal to zero) and the process of finding the zeros of a polynomial. Remind the student that finding zeros is equivalent to finding xintercepts. Make clear that the process involves substituting zero for the dependent variable, y, and solving for the independent variable, x. Provide the student with additional linear and quadratic functions, and ask the student to find the zeros of each. Allow the student to selfcheck by using graphing technology to find the zeros (which will also reinforce the relationship between zeros and xintercepts). 
Moving Forward 
Misconception/Error The student attempts to find the zeros but makes errors. 
Examples of Student Work at this Level The student:
 Factors incorrectly.
 Correctly factors but says the zeros are the constants in the linear factors.
 Correctly factors and sets each linear factor equal to zero but makes an error solving one of the linear equations.

Questions Eliciting Thinking How can you check to see if you factored correctly?
Did you check your solutions to see if they were correct? 
Instructional Implications Provide feedback to the student regarding errors. Provide additional opportunities to solve linear and quadratic equations. Remind the student to always use the original equation when checking the solutions in case the error was in the factoring step. Remind the student to be diligent in checking for sign errors.
Pair the student with another Moving Forward student and have them work together to find the zeros of polynomials. Ask the students to compare solutions and resolve any differences. Allow the students to selfcheck by using graphing technology to find the zeros. 
Almost There 
Misconception/Error The student is unable to identify the zeros as representing the xintercepts of the graph. 
Examples of Student Work at this Level The student correctly finds the zeros of each function. However, when asked what the zeros of the polynomials indicate about its graph, the student:
 Leaves the question unanswered.
 States the graph starts there.
 States that it means the equation can be solved.
 States the zeros are the origin.

Questions Eliciting Thinking What does the graph of a quadratic function look like?
Can you write the zeros as ordered pairs? Where would the zeros be located if you graphed them? 
Instructional Implications Ask the student to write the zeros as ordered pairs and to graph each function clearly showing the locations of the zeros. Explain that the zeros coincide with the xintercepts and emphasize that the processes of finding zeros and xintercepts are the same. Give the student additional linear and quadratic functions to graph using graphing technology and ask the student to find the zeros of each.
Consider using MFAS task Zeros of a Cubic (AAPR.2.3). 
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 writes:
 0 = , then 0 = (xÂ  7) (xÂ  7). The student then sets each factor equal to zero and determines x = 7 is a zero of the polynomial.
 0 = Â  7xÂ  15, then 0 = (xÂ  5) (2x + 3). The student then sets each factor equal to zero and determines x =  and x = 5 are the zeros of the polynomial.
 The zeros of a polynomial are the same as the xintercepts of the graph of the polynomial.

Questions Eliciting Thinking What else do you know about the graph? What is the graph called? What is its shape? How is it oriented? Is it symmetric?
How can you find other points, so you can graph the function?
How can you determine the number of times the graph of a polynomial function changes direction?
What does it indicate about the graph if there are no zeros?
For quadratics, is there a way to determine the number and nature of the roots without finding the zeros of the polynomial? 
Instructional Implications Consider using MFAS task Zeros of a Cubic (AAPR.2.3).
Challenge the student to use the zeros of a function to sketch the graph of the function. Consider using MFAS task Use Zeros to Graph (AAPR.2.3).
Challenge the student to find zeros in context. Consider using MFAS task Jumping Dolphin (ASSE.2.3). 