Getting Started 
Misconception/Error The student converts the numbers to standard form in order to compare them. 
Examples of Student Work at this Level Note: Although the student may be able to correctly compare the numbers when written in standard form, standard 8.EE.1.3 requires that the student be able to compare numbers multiplicatively when “expressed in the form of a single digit times an integer power of 10.”
The student writes as 40,000,000 and as 1,000,000 before attempting to compare. The student writes as 0.06 and as 0.02 before attempting to compare. The student then compares appropriately using division.
The student writes as 40,000,000 and as 1,000,000 before attempting to compare. The student writes as 0.06 and as 0.02 before attempting to compare. The student then compares inappropriately using subtraction.
Note: Some students may convert to standard form incorrectly before attempting to compare.

Questions Eliciting Thinking Why did you write your numbers in standard form in order to compare them?
Do you think you can compare these numbers without converting them to standard notation?
What do you remember about the laws of exponents?
I see that you used subtraction to compare the numbers. What does this comparison tell you? 
Instructional Implications Provide instruction on the laws of exponents. Using manipulatives or visuals, model how powers of 10 increase when the exponent is increased (e.g., compare to each other). Encourage the student to recognize a pattern that directly relates powers of 10 to exponents. Consider using MFAS task Exponents Tabled (8.EE.1.1).
In addition, demonstrate how and is, therefore, . Also demonstrate how and is, therefore, . Encourage the student to explore efficient methods of multiplying and dividing powers of ten. Consider using MFAS task Multiplying and Dividing Integer Exponents (8.EE.1.1).
Address any issues with converting numbers from scientific notation to standard form. Provide instruction on the use of integer exponents and powers of ten. Make explicit the difference between and .
Be sure the student understands the difference between determining how much greater one number is than another (by finding the difference) and how many times greater one number is than another. 
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 writes is four times larger than . The student compares the single digits and does not consider the powers of ten.
The student writes is three times larger than . The student compares the single digits using subtraction and does not consider the powers of ten.
The student writes is 10 times larger than . The student compares the powers of ten and does not consider the single digit factors. 
Questions Eliciting Thinking How did you determine how many times larger one number is than the other number?
If you use only the single digits to compare, what happens to the powers of 10?
If you use only the powers of 10 to compare, what happens to the single digits?
How are the single digits related to the powers of 10?
Tell me what you remember about the laws of exponents. 
Instructional Implications Provide direct instruction on the laws of exponents and make explicit how the laws of exponents apply to numbers written in the form . Demonstrate how and is, therefore, . Also demonstrate how and is, therefore, . Encourage the student to explore efficient methods of multiplying and dividing powers of ten. Consider using MFAS task Multiplying and Dividing Integer Exponents (8.EE.1.1).
Provide additional opportunities for the student to practice applying the laws of exponents when comparing very large and small numbers. Consider using MFAS task Compare Numbers (8.EE.1.3). 
Almost There 
Misconception/Error The student attempts to compare the numbers in exponential form but makes minor errors. 
Examples of Student Work at this Level The student writes equals and , then writes instead of 3 x 1.
The student writes equals , then writes equals 30 instead of three.
The student writes equals . When subtracting the exponents, the student computed “2  2” instead of “2  (2).”

Questions Eliciting Thinking How did you get ?
Do you remember the laws of exponents?
When you divide an expression having the same base, what operation should you perform on the exponents?
Can you find your own mistake?
What is the value of any expression raised to the power of zero? For example, or ? 
Instructional Implications Review the laws of exponents with the student as needed. Clarify the value of any expression raised to the power of zero.
Pair the student with a Got It partner and give the pair more problems to complete individually and then have the pair compare answers and reconcile any differences.
Consider using MFAS task Multiplying and Dividing Integer Exponents (8.EE.1.1) and MFAS task Compare Numbers (8.EE.1.3) for further assessment. 
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 answers of (1) 40 times larger and (2) three times larger.
For number one, the student writes equals and then writes 40 as the final answer. For number two, the student writes equals and then writes three as the final answer.

Questions Eliciting Thinking Why did you divide?
How did you end up with when you divided?
How did you end up with when you divided?
Why did you subtract the exponents?
How does equal three?
What does mean? 
Instructional Implications Challenge the student with another MFAS task, Order Matters (8.EE.1.3).
Provide instruction on scientific notation and converting between standard and scientific notation. Using other 8.EE.1.3 and 8EE.1.4 MFAS tasks. 