Getting Started 
Misconception/Error The student does not demonstrate an understanding of the definitions and properties of exponents. 
Examples of Student Work at this Level The student attempts calculations:
 That are not consistent with the definition of negative exponents. For example, the student:
 Applies the negative in an exponent to the base.
 Disregards negative exponents, treating them as positive exponents.
 That are not consistent with one or more properties of exponents. For example, the student:
 Multiplies bases by their exponents.
 Multiplies or divides bases and multiplies or divides exponents.
The student does not use properties of exponents.

Questions Eliciting Thinking What do you know about negative exponents? What do negative exponents mean? Can you rewrite these expressions in an equivalent form with positive exponents?
What do you know about the property for multiplying two exponential expressions with the same base?
What do you know about the property for dividing two exponential expressions with the same base?
What happens if you expand each of these exponential expressions before multiplying (or dividing) them? Would you get the answer shown on your paper? 
Instructional Implications Review the meaning of positive integer exponents. Using bases with whole number exponents less than ten, model rewriting the expression in an expanded form (e.g., rewrite Â as 3 x 3 x 3 x 3). Provide the student with sample problems of this type and ask the student to rewrite in expanded form and evaluate. Then, provide examples in expanded form and ask the student to rewrite in exponential form.
Model the process of writing numbers with negative integer exponents in an equivalent form using only positive exponents, expanding, and then evaluating (e.g., ). Ask the student to use this approach to evaluate other expressions with negative integer exponents.
Model the multiplication of two numbers with the same base and whole number exponents by first writing in expanded form. For example, given , write (3 x 3 x 3 x 3) x (3 x 3). Demonstrate rewriting this as 3 x 3 x 3 x 3 x 3 x 3 and then applying the definition of exponents to rewrite the six repeated factors of three as . Provide the student with other problems of this type and ask the student to first expand the exponential expressions and count factors to determine the exponent when rewriting as a single power of the base. Remind the student of the Product Property of Exponents (i.e., ) and relate this property to the previous numerical examples. Provide examples of expressions to which this property does not apply, such as , and challenge the student to explain why this property cannot be used. Then, discuss how to correctly evaluate the expression.
In a similar manner, explain division of two numbers with the same base and whole number exponents. For example, ask the student to rewrite Â as , â€ścancelâ€ť two of the threes, and rewrite the result as . Again, provide additional problems of this type and relate them to the Quotient Property of Exponents (i.e., ). 
Moving Forward 
Misconception/Error The student makes computational errors in applying the properties of exponents or in simplifying exponents. 
Examples of Student Work at this Level The student:
 Makes computational errors when adding or subtracting exponents.
 Makes an error when applying an exponent or is unsure how to simplify a base raised to a negative power.
Additionally, the student is unable to simplify exponential expressions with different bases (as in problem #4).

Questions Eliciting Thinking What do you know about adding and subtracting integers? Can you show me how you added (or subtracted) these exponents?
What is the same or different about Â and ? 
Instructional Implications Provide feedback to the student concerning the specific error made. As needed, review addition and subtraction of integers. Allow the student to correct his or her errors.
Model the process of writing numbers with negative integer exponents in an equivalent form using only positive exponents, expanding, and then evaluating (e.g., ). Ask the student to use this approach to evaluate other expressions with negative integer exponents.
Explain why the Product Property of Exponents (i.e., ) does not apply to the expression . Assist the student in finding a strategy to evaluate the expression. Suggest the student evaluate this expression by first evaluating each factor and then multiplying. Provide additional numerical problems involving exponential expressions and include some with different bases. Ask the student to evaluate the expressions and provide feedback. 
Almost There 
Misconception/Error The student is unable to evaluate the product of two exponential expressions with different bases. 
Examples of Student Work at this Level The student correctly applies the properties of integer exponents to generate equivalent numerical expressions for the expressions in the first three problems. For the fourth problem,Â , the student:
 Multiplies the bases and adds the exponents.
 Treats the base of 4 as if it were a 2.
 Adds the bases and adds the powers, generating an answer of or .
Additionally, the student may not initially evaluate completely each expression in the first three problems leaving final answers of , , and . If so, ask the student to complete the evaluation of each expression.

Questions Eliciting Thinking Does the property Â apply to this expression?
What is ? What is ? If you multiply these numbers, will you get Â (or another incorrect answer the student might have determined)? 
Instructional Implications Explain why the Product Property of Exponents (i.e.,Â )Â does not apply to the expression . Assist the student in finding a strategy to evaluate the expression. Suggest the student evaluate this expression by first evaluating each expression and then multiplying. Provide additional numerical problems involving exponential expressions and include some with different bases. Ask the student to evaluate the expressions and provide feedback. 
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 correctly applies the properties of integer exponents to perform calculations. The student may rewrite exponents in expanded form before correctly evaluating the expressions.

Questions Eliciting Thinking One property of integer exponents is . Why canâ€™t this property be applied to ?
Why are exponents added when multiplying two exponential expressions with the same base?
Why are exponents subtracted when dividing two exponential expressions with the same base? 
Instructional Implications Review the properties of integer exponents including the Product Property of Exponents (), the Quotient Property of Exponents (), and the Power Properties of Exponents [Â and ]. Ask the student to use the definition of exponents to justify each property. Then, ask the student to complete problems involving multiplication and division of exponential expressions that include some expressions to which these properties do not apply.
Introduce the student to exponential expressions in which the bases contain variables. Ask the student to simplify the expressions by applying the properties of exponents. 