Getting Started 
Misconception/Error The student is not able to correctly subtract the polynomials and is unable to determine or explain that the polynomials are closed for subtraction. 
Examples of Student Work at this Level The student:
 Does not distribute correctly across the subtraction.
 Combines terms that are not like.
 Combines terms incorrectly.Â

Questions Eliciting Thinking Can you explain how you distributed across this subtraction? What value are you actually distributing?
Is an expression such as a polynomial? How many terms must a polynomial have?
What does it mean for terms to be like? Are the terms you subtracted like terms? 
Instructional Implications Review the concept of like terms and remind the student that terms must have the same variable(s) raised to the same powers to be considered like terms. Provide practice for the student in determining whether or not terms are like terms and justifying his or her choices.
Review the process of adding and subtracting like terms. Relate this process to an application of the Distributive Property. Show the student that by applying the Distributive Property, + can be rewritten as (a + b). Consequently, when adding or subtracting like terms, the coefficients are added or subtracted while the variable portion of the term is unchanged.
Provide additional problems for the student involving the addition and subtraction of polynomials.
Once the student is proficient subtracting polynomials, review the concept of closure of the integers under addition, subtraction, and multiplication. Then introduce the concept of closure of the polynomials under addition, subtraction, and multiplication. 
Making Progress 
Misconception/Error The student correctly subtracts the polynomials but is unable to adequately explain that the polynomials are closed for subtraction. 
Examples of Student Work at this Level The student understands that polynomials are closed for subtraction but:
 Does not offer any explanation.
 Simply states that when you subtract polynomials, the difference is a polynomial.
 Does not recognize as a polynomial and determines that the polynomials are not closed for subtraction.Â

Questions Eliciting Thinking What is a polynomial?
What must be true of the terms of a polynomial?
What about the process of subtracting like terms ensures that the differences of like terms of polynomials are polynomials? 
Instructional Implications Review the definition of a polynomial as the sum or difference of terms of the form where a is a real number and n is a whole number. Provide additional examples and nonexamples of polynomials (e.g., 0, , , 3,Â yÂ  9, ). Ask the student to identify the examples of polynomials and explain why each nonexample fails the definition.
Model explaining that for any two terms and (where a and b are real numbers and n is a whole number), . Since the real numbers are closed for subtraction, (a  b) is a real number. Therefore, subtracting like terms of polynomials results in terms that fit the definition of a polynomial (i.e., a  b is a real number and n is a whole number).
Consider implementing MFAS tasks Adding Polynomials (AAPR.1.1), Multiplying Polynomials  1Â (AAPR.1.1), and Multiplying Polynomials  2 (AAPR.1.1) to assess the student's understanding of the closure of the polynomials under addition and multiplication. 
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 correctly subtracts each pair of polynomials and is able to explain:
 Â is a polynomial since it consists of a term in the form where a is a real number and n is a whole number. Likewise, is a polynomial since each of the four terms has the form of a real number multiplying a nonnegative power of t (since 4 = ).
 The difference of two polynomials will always be a polynomial because subtracting like terms of the form results in more terms of the form . The student may show that for any two terms and (where a and b are real numbers and n is a whole number), . Since the real numbers are closed for subtraction, (a  b) is a real number. Therefore, subtracting like terms of polynomials results in terms that fit the definition of a polynomial.Â

Questions Eliciting Thinking What are the coefficients of and x in the expression Â  x? To what power is x being raised?
When you add two polynomials, will the result always be a polynomial?
When you multiply two polynomials, will the result always be a polynomial? 
Instructional Implications Challenge the student to find an example of an operation with two polynomials that will result in an expression that is not a polynomial.
Consider implementing MFAS tasksÂ Adding PolynomialsÂ (AAPR.1.1),Â Multiplying Polynomials  1Â (AAPR.1.1), andÂ Multiplying Polynomials  2Â (AAPR.1.1) to assess the student's understanding of the closure of the polynomials under addition and multiplication. 