MA.912.AR.1.4

• Instruction includes the connection to addition, subtraction and multiplication of polynomials to develop the understanding of closure, and the connection to properties of exponents. 
• Instruction includes proper vocabulary and terminology, keeping in mind that when dividing, the word “cancel” can become a misconception for students. A number or an expression divided by itself is equivalent to 1 and does not disappear (see Appendix D). 
• Instruction includes the use of manipulatives, like algebra tiles, and various strategies, like the area model, properties of exponents and decomposing fractions. 
  • Decomposing fractions 
  •  The expression can be written as   . Students can then   perform the division with  each fraction to determine the difference.

Common Misconceptions or Errors

• Students may not understand the meaning of closure (although not directly discussed in this benchmark, polynomials are not closed under division). 
• Students may not understand like terms.

Strategies to Support Tiered Instruction

• Teacher provides examples showing that polynomials are closed under the operations of addition, subtraction and multiplication, but not under division. 
  • For example, when subtracting or multiplying polynomials (as shown below), the result is always a polynomial.
    
     7$x$ +5−(3$x$ + 8) = 4$x$ − 3
    (7$x$ + 5)(3$x$ + 8) = 21$x$² + 71$x$ + 40

• For example, when dividing polynomials (as shown below), the result may or may not be a polynomial.  

 

Instructional Tasks

• Part A. Determine the quotient of $x$ + $x$² and $x$^-1. 
• Part B. What do you notice about your answer and the Laws of Exponents? 
• Part C. What happens when you divide $x$ + $x$² by the expression $x$^{$\frac{\text{1}}{\text{2}}$} (which is not a monomial)?

Instructional Items

• What is the quotient of the expression