Student Actions

Teacher Actions

• Develop accurate and appropriate procedural fluency by tending to accuracy in operations and utilizing efficient procedures as they manipulate expressions with like terms and exponents to simplify polynomial expressions.
• Develop a deep and flexible conceptual understanding when applying previous knowledge of operations with integers and exponents to develop strategies for how and when to simplify polynomial expressions, including the connection between the concept of area and multiplying polynomials.
• Develop the ability to make conjectures, model, and generalize when exploring patterns in operations of numerical and algebraic expressions to develop conclusions of how to effectively generate equivalent polynomial expressions and make sense of processes.
• Develop the ability to communicate mathematically when justifying strategies for simplifying polynomials with appropriate mathematical notation and utilizing a variety of models (verbal, written, and/or visual) to demonstrate reasoning. 

• Elicit and use evidence of student thinking by providing opportunities for students to explore a variety of processes, rather than memorization of rules or series of steps, as well as flexibility in how students express their thinking with verbal, written, and visual models.

• Pose purposeful questions to support students in reaching conclusions of effective strategies for simplifying polynomial expressions.

• Implement tasks that promote reasoning and problem-solving by providing tasks and/or examples that incorporate the use of manipulatives or visual models that allow students to explore processes and encourage students to communicate their understanding in a variety of ways. 

Key Understandings

Misconceptions 

• Adding/Subtracting polynomial expressions involves adding/subtracting the coefficients of terms that have the same variable and exponent to produce an equivalent polynomial expression.

• Multiplying polynomial expressions involves multiplying the coefficients and adding exponents on like variables to produce an equivalent polynomial expression.

• Multiplying polynomial expressions can also be thought of as creating an area.

Ex: (2x + 3)(x - 4) = 2x² - 8x + 3x - 12 = 2x² - 5x - 12

 Didax Virtual Manipulatives

• Students may misuse exponent properties, often confusing multiplication with powers. 

• i.e. thinking 2 * x² = (x²)² 

• Students may incorrectly manipulate the base when using powers.  

• Students may raise the variable by the power and forget to raise the coefficient by the same power. 

• i.e. thinking (2x²)²  = 2x⁴

• Students may incorrectly add terms by adding exponents of like terms instead of maintaining the exponent or by combining terms of different variables and/or exponents.

• Students may not recognize that multiplying polynomial expressions is creating an area and will neglect partial products to only multiply the first and last terms.

• i.e. thinking (x - 3)² = x²- 3²

• When subtracting polynomial expressions, students may not fully distribute the negative to all terms of the second polynomial expression leading to only subtracting the first term of the second expression and then adding the remaining terms of the second polynomial expression. 

  Knowledge Connections

Prior Knowledge

Leads to 

• Use substitution to simplify and evaluate algebraic expressions. (PA.A.3.1)

• Develop and apply the properties of integer exponents. (PA.N.1.1)

• Justify steps in generating equivalent expressions by combining like terms and using order of operations (to include grouping symbols). Identify the properties used, including the properties of operations (associative, commutative, and distributive). (PA.A.3.2) 

• Add, subtract, multiply, divide, and simplify polynomial expressions. (A2.A.2.2)

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