To demonstrate proficiency in generating mathematical expressions that are equal in value by applying the order of operations, including whole number exponents and utilizing prime factorization techniques, students can follow these steps:

Understand the order of operations: Ensure that students understand the correct order in which mathematical operations should be performed. The order of operations is typically represented by the acronym PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).

Practice simplifying expressions with parentheses: Provide students with expressions that contain parentheses and ask them to simplify the expressions by applying the order of operations. For example, given the expression 3 + (4 Ã— 2), students should first perform the multiplication inside the parentheses (4 Ã— 2 = 8) and then add 3 to the result (8 + 3 = 11).

Introduce whole number exponents: Explain to students that exponents represent repeated multiplication. For example, 2^3 means 2 multiplied by itself three times (2 Ã— 2 Ã— 2 = 8). Provide examples of expressions with whole number exponents and ask students to simplify them using the order of operations. For instance, given the expression 2^2 + 3 Ã— 4, students should first calculate the exponent (2^2 = 4), then perform the multiplication (3 Ã— 4 = 12), and finally add the results (4 + 12 = 16).

Teach prime factorization techniques: Introduce prime factorization as a method to break down numbers into their prime factors. Show students how to find the prime factors of a number by repeatedly dividing it by prime numbers until only prime factors remain. For example, the prime factorization of 24 is 2 Ã— 2 Ã— 2 Ã— 3. Encourage students to practice prime factorization on various numbers to become comfortable with the technique.

Combine order of operations and prime factorization: Provide students with expressions that involve both the order of operations and prime factorization. For instance, given the expression 2^3 Ã— 3^2, students should first calculate the exponents (2^3 = 8 and 3^2 = 9), then multiply the results (8 Ã— 9 = 72).

Provide ample practice opportunities: Offer students a variety of practice problems that require them to generate mathematical expressions equal in value by applying the order of operations, whole number exponents, and prime factorization techniques. Gradually increase the complexity of the problems to challenge their understanding and proficiency.

Assess student understanding: Evaluate studentsâ€™ proficiency by assigning assessments or quizzes that test their ability to generate equivalent mathematical expressions using the order of operations, whole number exponents, and prime factorization techniques. Provide feedback and additional support as needed.
By following these steps and providing sufficient practice opportunities, students can develop proficiency in generating mathematical expressions that are equal in value by applying the order of operations, including whole number exponents, and utilizing prime factorization techniques.