Getting Started 
Misconception/Error The student is unable to consistently apply the definition of negative exponents to generate equivalent numerical expressions. 
Examples of Student Work at this Level The student may do any of the following:
 Confuse negative exponents with additive inverses.
 Not distinguish between expressions with the same base but with opposite exponents.
 Interpret the exponent as a factor.

Questions Eliciting Thinking What does a positive exponent mean?
What does the negative in an exponent mean?
Is different from ? 
Instructional Implications Provide the student with direct instruction regarding exponents in general and specifically the meaning of negative exponents. Guide the student to evaluate every exponential expression on the worksheet including the answer alternatives. Provide additional exponential expressions for the student to simplify. Be sure to include some with positive bases, negative bases, positive exponents, and negative exponents.
Review the meaning of opposite and reciprocal. Guide the student to observe that generally a number cannot be equal to either its opposite or its reciprocal. Challenge the student to find the exceptions. 
Moving Forward 
Misconception/Error The student is unable to explain the difference between –b and . 
Examples of Student Work at this Level The student completes the first two problems correctly. When asked to explain the difference between –b and , the student says:
 The negative is in front of the b in one but in the exponent in the other.
 One has an exponent but the other doesn’t.
 The b is positive in one but negative in the other.
 One is negative b and the other is b to the negative one power.
The student says there is no difference between –b and because:
 It doesn’t matter where the negative is.
 1 times b is –b.
 They are both negative numbers.

Questions Eliciting Thinking Can you explain the difference between and ?
Can you explain the difference between and ?
Does b have to stand for a positive number? 
Instructional Implications Ask the student to evaluate b and for several positive values of b. Then ask the student to evaluate b and for several negative values of b. Guide the student to generalize the difference between the two types of expressions using mathematically correct vocabulary. 
Almost There 
Misconception/Error The student does not understand that an expression such as –b can represent a positive number. 
Examples of Student Work at this Level The student completes the first two problems correctly. When asked to explain the difference between –b and , the student says that, “ b represents a negative number but means b itself is positive but the negative exponent tells you to rewrite it as a fraction .”

Questions Eliciting Thinking Is there another way to interpret the negative symbol in the expression –b?
Does b have to stand for a positive number?
In the expression can b represent a negative number? 
Instructional Implications Make it clear that b can represent both positive and negative numbers. Ask the student to evaluate each expression for several positive and negative values of b.
Challenge the student to evaluate each expression for several positive and negative fractional values of b. 
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 the correct answers of:
 and with work or an explanation to support the answer.
 32 and with work or an explanation to support the answer.
 The student explains that b means the opposite of b; but means , or the reciprocal of b.

Questions Eliciting Thinking Is it possible for b to represent a negative number?
If b is negative, what is the sign of –b? If b is negative, what is the sign of ? 
Instructional Implications Challenge the student to determine and explain the differences among: , , , , , , , .
Challenge the student to evaluate for several positive and negative fractional values of b. 