Getting Started 
Misconception/Error The student is unable to apply the properties of integer exponents to generate equivalent numerical expressions. 
Examples of Student Work at this Level The student:
 Uses the wrong integer exponent rule for the given situation.
 Uses a contrived rule (e.g., multiplying the base by the exponent or multiplying the bases while adding the exponents).
 Does not attempt to use any exponent properties but instead tries to do the actual calculations.

Questions Eliciting Thinking How do you know when to add exponents and when to multiply them?
What rule or strategy did you use to find equivalent expressions?
What does an exponent mean? Is there another way you can try this problem rather than multiplying all the factors?
Can you explain the work you did for this problem? 
Instructional Implications Review the meaning of integer exponents and the properties of integer exponents. Demonstrate why each property is stated the way that it is [e.g., show the student the following: Â x Â = (4 x 4 x 4) x (4 x 4) = (4 x 4 x 4 x 4 x 4) = Â or ]. Remind the student that the exponent simply describes the number of factors of the base. Provide similar examples as explanations of other exponent properties using both positive and negative exponents.
Provide additional practice using properties of exponents to write exponential expressions in equivalent forms. 
Moving Forward 
Misconception/Error The student is unable to apply the properties of integer exponents to generate equivalent numerical expressions requiring more than one step. 
Examples of Student Work at this Level The student has work to support the selection of:
 One correct answer for each question with nothing else marked.
 Some correct answers for each problem, but the student misses the choices that require multiple steps to evaluate (e.g., #1: ; #2: Â ; #3: ).

Questions Eliciting Thinking Are there other answer choices that are also equivalent?
Are there other combinations of exponents that will also give you the same product? 
Instructional Implications Have the student evaluate the given expressions and each of their answer alternatives on the worksheet. Ask the student to find every possible expression that is equivalent to the given one.
Give the student an exponential expression such as Â and ask the student to use properties of exponents to rewrite the expression in equivalent forms. Challenge the student to find as many ways as possible to rewrite Â using integer exponents.
Provide additional practice using properties of exponents to write exponential expressions in equivalent forms. 
Almost There 
Misconception/Error The student is unable to apply the properties of integer exponents to numerical expressions involving fractions. 
Examples of Student Work at this Level The student:
 Does not recognize the answer of Â as equivalent to .
 Chooses both Â and Â as equivalent to .
 Only recognizes Â as equivalent to .
 Does not know how to evaluate .

Questions Eliciting Thinking Is there a way to simplify the fraction in parentheses? How do you write that as a decimal?
What does equivalent mean? Can you show me how Â and Â are equivalent?
How can you use other exponent properties to find equivalent expressions to this one?
What does the negative exponent mean? 
Instructional Implications Review with the student how to write fractions in lowest terms and convert fractions to decimals. Have the student write three new fractions that are equivalent to each of 0.5 and 2. Have the student describe how the fractions are alike and different.
Provide additional instruction on evaluating exponential expressions with fractional bases and integer exponents. Give the student additional practice and pair the student with another student to compare answers and reconcile differences.
Consider using the MFAS task Negative Exponential Expressions. 
Got It 
Misconception/Error The student provides complete and correct responses to all components of the task. 
Examples of Student Work at this Level The student chooses all equivalent expressions for each problem.
 Â
 Â

Questions Eliciting Thinking Why isnâ€™t Â equivalent to the original expression?
Can you use the same original numbers but rewrite the expression so that it is equivalent to ? 
Instructional Implications Provide the student with additional examples of expressions that contain fractional bases and negative exponents to evaluate (e.g., ask the student to evaluate expressions such as ). 