Equivalent Exponents

Getting Started

The student does not understand what it means for expressions to be equivalent.

The student does not demonstrate an understanding of the meaning of equivalent. The student states that:

• None of the expressions are equivalent to because they are all written differently.
• All expressions are equivalent to because they all use the variable x.
• 3x and x · 3 are equivalent to because they both use an x and a 3.
• 3x is equivalent to because x + x + x is equivalent to x · x · x since they both contain three x’s.

What does it mean for two expressions to be equivalent?

How can you show that two expressions are not equivalent?

How might you be able to show that two expressions are equivalent?

Explain what it means for two expressions to be equivalent (i.e., two expressions are equivalent if they always result in the same number when evaluated for the same values of the variables). Demonstrate the equivalence of two expressions such as x · x · x and by evaluating each for the same value of x. Explain to the student that it is not possible to evaluate each expression for every possible value of x to check for equivalence. However, definitions and properties of operations can be used to verify equivalence (i.e., x · x · x is equivalent to by the definition of exponents).

Review the definition of exponents, the properties of operations, and the associated vocabulary as needed (e.g., base, factor, exponent, coefficient). Emphasize the meaning of exponents when explaining their properties. For example, means there are three factors of x which can be written as x · x · x. Explain the difference between 3x and by expanding each (e.g., 3x = x + x + x while = x · x · x). Show that x + x + x is not equal to x · x · x by evaluating each for a particular value of x, such as five. Provide opportunities for the student to use the definition of exponents and properties of operations to generate equivalent expressions.

Consider administering other MFAS tasks.

Moving Forward

The student is unable to identify equivalent expressions using the definition of exponents and properties of operations.

The student “tests” the expressions by evaluating them for a particular value of the variable and concludes that only x · x · x is equivalent to .

If you had to explain to a classmate why x · x · x is equivalent to , what would you say?

Can you prove that something is true by testing only one value? Can you try testing another value? How many tests would you have to do to prove that the two expressions are equivalent?

Confirm that the student’s approach is consistent with the definition of equivalent expressions but explain that he or she only tested the expressions for one possible value of the variable. Clarify that expressions can be shown to not be equivalent by evaluating each for particular values of the variable and showing that they result in different numbers. Yet, showing that expressions are equivalent requires using definitions and properties. Show the student a pair of expressions such as 2x and 3x or 2x and . Evaluate the pairs of expressions for a particular value for which they are equal (i.e., x = 0 and x = 2, respectively). Then evaluate the pairs of expressions for another value to show that they are not always equal and are not, therefore, equivalent.

Review the definition of exponents, the properties of operations, and the associated vocabulary as needed (e.g., base, factor, exponent, coefficient). Explain that is equivalent to x · x · x since means there are three factors of x which can be written as x · x · x (by the definition of exponents). Explain the difference between 3x and by expanding each (i.e., 3x = x + x + x while = x · x · x). Show that x + x + x is not equal to x · x · x by evaluating each for a particular value of x, such as five. Provide opportunities for the student to use the definition of exponents and properties of operations to generate equivalent expressions.

Consider administering other MFAS tasks.

Almost There

The student holds a misconception about a property of exponents.

The student identifies only x · x · x as equivalent to and explains the equivalence in terms of the definition of exponents. The student explains why the other expressions are not equivalent to in terms of the definition of exponents and/or the properties of operations. However, the student identifies as equivalent to “since .”

Can you evaluate and for x = 2? What happens? Are they equal? What does this mean about the equivalence of the two expressions?

Explain to the student that, in general, is not equal to . Provide a specific example, such as, . Expand each exponential expression (e.g., rewrite the expression as 2 · 2 · 2 · 2 – 2 · 2, and explain that although addends can be subtracted, factors cannot be subtracted). Show the student that the only way to evaluate this expression is to first evaluate and and then subtract (e.g., = 16 – 4 = 12). Since the value of x is unknown in the expression , this expression cannot be written as a single power of x.

Consider implementing other MFAS tasks.

Got It

The student provides complete and correct responses to all components of the task.

The student identifies only x · x · x as equivalent to and explains the equivalence in terms of the definition of exponents. The student explains why the other expressions are not equivalent to in terms of the definition of exponents and/or the properties of operations. The student may also evaluate and the other expressions for a particular value of x to show that they are not equivalent.

Is equivalent to ? How can you tell?

Can you find an expression that is equivalent to · ?

Ask the student to identify every expression on the worksheet equivalent to 3x.

Ask the student to identify three different exponential expressions equivalent to (e.g.,  ). Guide the student to discover the Addition Property of Exponents (i.e., ).

Consider implementing other MFAS tasks.