Student Actions

Teacher Actions

• Develop Accurate and Appropriate Procedural Fluency when identifying numerical and variable expressions which are not simplified within a square root or cube root and use knowledge of operations with factors to accurately simplify square and cube roots. 

• Develop the ability to communicate mathematically when discussing, writing, interpreting, translating, and justifying their thinking to both teachers and peers using appropriate mathematical vocabulary and notation.

•  Develop mathematical reasoning by justifying the processes used to simplify radicals, and demonstrating and understanding of how roots will change appearance but are still equivalent to the original. 

• Elicit and use evidence of student thinking by providing opportunities for students to justify operations related to both whole numbers and variables within radicals, and allowing students to examine alternative approaches in finding their solutions beyond rote memorization of a process. 

• Pose purposeful questions and create examples that highlight how the expression changes when students compare equivalent representations of the same expression. 

• Facilitate meaningful mathematical discourse, as students and teachers discuss common errors various strategies for simplifying radicals and the use of proper mathematical notation.

• Implement tasks that promote reasoning and problem-solving when students compare alternate solutions, utilize flexibility in solving strategies, and examine error analysis of common mistakes. 

Key Understandings

Misconceptions 

• Square roots are simplified utilizing factors that are perfect squares and cube roots are simplified utilizing factors that are perfect cubes.

• Produce the simplest form of square and cube root expressions including both whole numbers and variables, using knowledge of prime factorization.

Examples:

• When taking the square root of a variable that results in an odd exponent, absolute value notation is necessary to ensure the simplified variable expression results in a positive value.

• Example:

• Students may square or cube a number, rather than finding its root. For example, students may incorrectly say  rather than rather than .

• Students may use division instead of factors and divide a square root by 2 or a cube root by 3 

• Students may not apply absolute value bars on the variable that is simplified to an odd exponent outside the radical of an even root.

• Students may incorrectly multiply the perfect square or perfect cube number outside of the radical instead of using the perfect square or cube factor. For example, students may think  instead of  

  Knowledge Connections

Prior Knowledge

Leads to 

•  Identify the square roots of perfect squares to 400 or, if it is not a perfect square root, locate it as an irrational number between two consecutive positive integers  (PA.N.1.4) 

• Apply the distance formula, the Pythagorean theorem, and the Pythagorean theorem converse (approximate and exact values, including Pythagorean triples) to solve problems.(G.RT.1.1)

• Use the distance and midpoint formula to recognize and write the radius r, center (h,k), and standard form of the equation of a circle (x − ℎ) 2+ (y − k) 2 = r2  (G.C.1.2)

• Understand and apply the relationship between rational exponents and radicals to solve problems. (A2.N.1.3)