Student Actions

Teacher Actions

• Develop a deep and flexible conceptual understanding of how to simplify polynomials with mathematical operations by connecting prior knowledge of exponent rules, mathematical operations, and combining like terms to mathematical problems involving polynomials.

• Develop accurate and appropriate procedural fluency when modeling efficient strategies to solve problems involving addition, subtraction, multiplication division, and simplification of polynomial expressions. 

• Develop the ability to communicate mathematically when collaborating with peers about strategies and processes pertaining to operations on polynomials by discussion and also using written descriptions and/or visual models with attention to the use of correct terminology including term, coefficient, power, variable, etc. 

• Use and connect mathematical representations when connecting prior knowledge of arithmetic operations to simplifying polynomial expressions by engaging students in opportunities to model efficient strategies when adding, subtracting, multiplying, and dividing, including reviewing exponent rules.

• Pose purposeful questions that enable students to discuss efficient strategies for adding, subtracting, multiplying, dividing, and simplifying polynomials.

• Support productive struggle in learning mathematics by providing students with increasingly complex activities that allow them to expand their knowledge base about polynomial operations. Students can demonstrate their understanding in both written and verbal activities.

Key Understandings

Misconceptions 

• Polynomial expressions can be added, subtracted, multiplied, and divided in a similar manner to numerical expressions.

• Adding polynomial expressions involves combining like terms (same variable and exponent), maintaining the exponent, and combining any constants.

• Subtracting polynomial expressions involves distributing a negative throughout the second expression and then combining terms similar to adding polynomial expressions.

• Multiplying polynomial expressions involves distributing each term in the first expression to every term in the second expression, then multiplying the coefficients and adding the exponents by adhering to the exponent rules.

• Dividing polynomials could involve dividing each term of the first polynomial expression by a monomial or could involve an algorithm similar to long division with the second expression being used as the divisor. 

• Students try to combine terms with different powers or different variables when adding or subtracting polynomials.

• When subtracting polynomials, students may forget to distribute the negative and/or factor to all terms within the second expression.

• Students may forget to follow the rules of exponents when multiplying and/or dividing  polynomials.

• Students incorrectly apply the distributive property to multiply polynomials.

• When using long division, students may not consider both terms of a binomial divisor only utilize the first term, or may not be able to identify correct factors of the quotient.

• Students may incorrectly simplify factors of terms when dividing polynomial terms by a monomial term. 

  Knowledge Connections

Prior Knowledge

Leads to 

• Simplify polynomial expressions by adding, subtracting, or multiplying. (A1.A.3.2)

• Add, subtract, and multiply functions using function notation. (A1.F.3.3) 

• Add, subtract, multiply, and divide functions using function notation and recognize domain restrictions. (A2.F.2.1)

• Combine functions by composition.(A2.F.2.2)

• Add, subtract, multiply, divide, and simplify rational expressions. (A2.A.2.3)

• Model relationships through composition, and attend to the restrictions of the domain. (PC.F.2.1)

• Rewrite a function as a composition of functions. (PC.F.2.2)