Student Actions

Teacher Actions

• Develop accurate and appropriate procedural fluency ​​by selecting the most efficient procedure when solving a variety of tasks involving square root and cube root equations of one variable and explaining the mathematical basis for chosen strategies and/or procedures.

• Develop a deep and flexible conceptual understanding of the concept that raising an equation to a higher power can introduce extraneous roots and knowing how and when to apply the mathematics to verify the validity of all possible solutions. 

• Pose purposeful questions that allow students to discuss and justify the procedures that they have chosen to solve square and cube root equations.

• Facilitate meaningful discourse as students make connections between their strategies and methods to more efficient and appropriate procedures.

• Implement tasks that promote reasoning and problem solving providing opportunities for students to explore various mathematical situations that incorporate the examination of extraneous solutions and common errors as well as allowing multiple entry points and varied solution strategies. 

Key Understandings

Misconceptions 

• Solve square root and cube root equations by first isolating the radical, then squaring or cubing both sides of the equation. If a square or cube root remains, this process must be repeated.

• Squaring or cubing an equation may introduce extraneous roots, so solutions must be verified.

• Extraneous solutions of a radical equation occur when replacing a value in the original equation makes the equation untrue.

• A square root equation has no solution when the radical is equal to a negative value. 

• Students may forget to square or cube both sides of the equation.

• Students may forget to isolate the square root before squaring both sides of the equation or isolate the cube root before cubing both sides of the equation.

• Students may incorrectly square or cube individual terms in the equation, instead of all terms of each side of the equation.

• Students may forget to check all answers to eliminate the possibility of extraneous solutions. 

  Knowledge Connections

Prior Knowledge

Leads to 

• Evaluate radical expressions (A1.A.3.4)

• Use knowledge of solving equations with rational values to represent, use and apply mathematical models and interpret the solutions in the original context. (A1.A.1.1)

• Also include since a student may need to multiply and/or factor polynomial expressions.

• Factor common monomial factors from polynomial expressions and factor quadratic expressions with a leading coefficient of 1. (A1.A.3.3)

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

• Solving equations of various types in higher-level mathematics courses.