Getting Started 
Misconception/Error The student converts the numbers to standard form in order to compare them. 
Examples of Student Work at this Level The student converts each number to standard notation and then divides to compare. The student may or may not correctly interpret the result.

Questions Eliciting Thinking Can you compare these numbers without converting them to standard notation?
Do you know how many times larger Â is than Â (or 10)? Can you tell just by looking at the exponents?
What does Â mean? How many factors of 10 are in ? So how does Â compare to 10? 
Instructional Implications Have the student rewrite each number as a product of factors, (e.g., 6 Ă— Â = 6Â·10Â·10Â·10Â·10 and 6 Ă— Â = 6Â·10Â·10). Ask the student to compare the numbers in this form and determine how many more factors of 10 are in 6 Ă— Â than in 6 Ă— . Do the same for the other two problems. Then provide additional opportunities to compare numbers written as a single digit times a power of 10. Guide the student to observe the general rule for dividing exponential expressions with the same base. Help the student appreciate how easy it is to compare two numbers each written in the form a xÂ Â with equal values of a.
Provide the student with problems in which numbers must first be converted to scientific notation and then compared. 
Moving Forward 
Misconception/Error The student attempts to compare the numbers in exponential form but makes significant errors. 
Examples of Student Work at this Level The student:
 Compares exponents and indicates that 6 Ă— Â is two times larger than 6 Ă— Â since four is twice as large as two, and 2 Ă— Â is two times larger than 2 Ă— Â since 3 is twice as large as 6.
 Makes an error when subtracting exponents or when dividingÂ whole numbers.

Questions Eliciting Thinking What does Â mean? How many factors of 10 are in ? What does Â mean? How many factors of 10 are in ? So how does Â compare to ?
What does Â mean? What does Â mean? Can you write these numbers as fractions? 
Instructional Implications Have the student rewrite each number as a product of factors, (e.g., 6 Ă— Â = 6Â·10Â·10Â·10Â·10 and 6 Ă— Â = 6Â·10Â·10). Ask the student to compare the numbers in this form and determine how many more factors of 10 are in 6 Ă— Â than in 6 Ă— . Do the same for the other two problems. Then provide additional opportunities to compare numbers written as a single digit times a power of 10. Guide the student to observe the general rule for dividing exponential expressions with the same base.
Pair the student with a Getting Started partner and have the students work on problems similar to those in this task. Ask students to compare answers and reconcile any differences. 
Almost There 
Misconception/Error The student does not interpret his or her answer correctly. 
Examples of Student Work at this Level The student correctly divides each pair of expressions but is unable to correctly interpret the results. The student leaves the quotients in exponential form and:
 Says that 6Ă— is two times larger than 6Ă— Â since Â = 1Ă— and the exponent is two.
 Says only that one number is larger or smaller than the other.
 Provides no interpretation of the result.

Questions Eliciting Thinking Can you write your quotients in standard notation? What do these values indicate about how many times larger one number is than the other?
Can you be more specific? How many times larger is 6Ă— than 6Ă—? 
Instructional Implications Provide additional opportunities to compare numbers written in the form . Have the student work with a partner to compare answers and reconcile differences.
Provide guidance on how to show mathematical work appropriately. Have the student justify his or her work by stating a rule of exponents. 
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 compares the numbers in exponential form getting (1) Â or 100; (2) Â or 1000; (3) 3 x 10 = 30.

Questions Eliciting Thinking Can you explain why you subtracted the exponents when you divided Â by ?
How would you complete this problem: ? Can you put your answer in scientific notation? 
Instructional Implications Provide the student with additional problems involving all four operations with numbers written in scientific notation (see 8.EE.1.4). 