Getting Started 
Misconception/Error The student does not understand the meaning of exponents. 
Examples of Student Work at this Level The student:
 Interprets the meaning of the exponent,Â , asÂ Â or b + b + ... + bÂ including b as an addend x number of times.
Â Â Â
 Reverses the base and the exponent, writing .

Questions Eliciting Thinking How do you read Â out loud? What number is the base? What number is the exponent? What does the exponent mean?
If , why do we even need or use exponents? 
Instructional Implications Provide instruction on the meaning of exponents. Define the terms base and exponent and explicitly describe the exponent as indicating the number of factors of the base. To reinforce the meaning of the exponent, initially encourage the student to write exponential expressions in expanded form before calculating (e.g., ).
Demonstrate that Â can be rewritten as 10 â€“ (2 x 2 x 2) and Â can be rewritten as (2 x 2 x 2 x 2 x 2) â€“ (2 x 2). Provide additional opportunities for the student to evaluate numerical expressions involving whole number exponents. Ask the student to write powers in expanded form before evaluating them.
If necessary, review the order of operations rules. Guide the student through the evaluation of each expression in this task and model applying the order of operations rules. 
Moving Forward 
Misconception/Error The student makes an error in applying properties of exponents. 
Examples of Student Work at this Level The student demonstrates an understanding of the meaning of exponents by correctly calculating . But, to simplify , the student attempts to apply a rule such as:
 .
 Â writingÂ .
 Â writingÂ .

Questions Eliciting Thinking What does Â mean? What does Â mean?
What do the order of operations rules say about evaluating powers and about subtraction? Which takes priority and should be done first? 
Instructional Implications Review the order of operations rules. Guide the student through the evaluation of each expression in this task and model applying the order of operations rules.
Provide supplemental instruction in writing exponents within expressions in expanded form. Demonstrate that Â can be rewritten as (2 x 2 x 2 x 2 x 2) â€“ (2 x 2). Provide practice problems that require the student to write expressions in this form and evaluate.
Provide the student with examples of expressions containing whole number exponents that have been evaluated but include an error. Challenge the student to find and correct the error.
Ask the student to rework the problem by applying the strategy the student used. Challenge the student to identify his or her mistake and to explain how it affected the outcome. 
Almost There 
Misconception/Error The student makes a calculation error or provides an incomplete explanation. 
Examples of Student Work at this Level The student:
 Calculates Â as 16 or 64. All other work is correct.
 Provides correct computational work but does not include a complete explanation.

Questions Eliciting Thinking There is a computational error in your work. Can you find and correct it?
Can you explain why the second statement is false? 
Instructional Implications Provide feedback to the student concerning any errors or omissions and allow the student to revise his or her work. Provide additional opportunities to evaluate exponential expressions and compose justifications of mathematical work. 
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 evaluates Â writing . The student determines that Â is false and explains that .

Questions Eliciting Thinking Why might someone think that Â is equal to ?
Can you write 28 as a power of 2? Can you write 28 as a power of any whole number base? 
Instructional Implications Provide additional expressions for the student to evaluate that are more complex such as Â or incorporate grouping symbols around negative numbers such as .
Challenge the student to think of a realworld context in which these problems, or ones like them, might occur. Ask the student to create a context to accompany the expression.Â 