Getting Started 
Misconception/Error The student is not able to correctly multiply the polynomials and is unable to determine or explain that the polynomials are closed for multiplication. 
Examples of Student Work at this Level The student:
 Does not correctly distribute. Answers may include:
 Multiplies the expressions partially or fully but then combines terms that are not like or adds exponents. Answers may include:

Questions Eliciting Thinking What does it mean to distribute?
How did you multiply the two binomials?
How do you square a binomial? Did you rewrite it as a multiplication first?
There is an error in your multiplication; can you find it? 
Instructional Implications Review the application of the Distributive Property with the student. Model the use of the Distributive Property when multiplying two binomials. Explain that the Distributive Property is used twice, so both terms of the second binomial are multiplied by each of the terms of the first binomial. When the student is proficient using this approach, guide the student to discover and use more efficient methods such as â€śFOIL.â€ť Provide problems for the student to complete.
When squaring a binomial, guide the student to first rewrite the binomial as the product of two factors, , and then multiply using the same approach as when multiplying two binomials. Eventually, guide the student to observe the relationship between the terms of the binomial base and the terms of its expansion [e.g., ]. Assist the student in using this relationship to square binomials. Provide additional problems for the student to complete.
Once the student is proficient multiplying 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 multiplies the polynomials but is unable to adequately explain that the polynomials are closed for multiplication. 
Examples of Student Work at this Level The student answers one question positively and one negatively or both negatively (questions 3 and 4). The student does not demonstrate an understanding of a polynomial.
The student answers both questions (3 and 4) positively, but cannot explain why the answers are polynomials. 
Questions Eliciting Thinking What is a polynomial?
What must be true of the terms of a polynomial? What do they look like?
Can you give me an example of an expression that is not a polynomial?
Is the expression 0 a polynomial? 
Instructional Implications Review the definition of a polynomial as the sum or difference of terms of the formÂ Â where a is a real number and n is a whole number. The prefix poly can be misleading since a polynomial may have only one term, which could be a constant including zero. 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 m and n are whole numbers),. Since the real numbers are closed for multiplication, (a Â· b) is a real number. Since the whole numbers are closed for addition, (m + n) is a whole number. So, multiplying terms of polynomials results in terms that fit the definition of a polynomial [i.e., (a Â· b) is a real number and (m + n) is a whole number].
Consider implementing MFAS tasksÂ Subtracting PolynomialsÂ (AAPR.1.1),Â Multiplying PolynomialsÂ Â 1Â (AAPR.1.1), andÂ Adding PolynomialsÂ (AAPR.1.1) to further assess the student's understanding of the closure of polynomials under the operations of addition, subtraction, 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 responds to the four questions as follows:
 The student explains that Â is a polynomial since it consists of the sum of two terms each of the form where a is a real number and n is a whole number. Likewise, is a polynomial since the coefficients are real numbers for each term and all of the exponents are whole numbers.
 The student further explains that the product of two polynomials will always be a polynomial because the result of multiplying 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 m andÂ nÂ are whole numbers), . Since the real numbers are closed for multiplication, (a Â· b) is a real number. Therefore, multiplying terms of polynomials results in terms that fit the definition of a polynomial.Â

Questions Eliciting Thinking When you add or subtract two polynomials, will the result always be a polynomial?
When you divide 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 tasks Subtracting Polynomials (AAPR.1.1), Multiplying PolynomialsÂ Â 1Â (AAPR.1.1), and Adding Polynomials (AAPR.1.1) to further assess the student's understanding of the closure of polynomials under the operations of addition, subtraction, and multiplication. 