Getting Started 
Misconception/Error The student can only complete the calculations with numbers in standard notation. 
Examples of Student Work at this Level The student attempts to convert each number to standard notation and then multiply or divide, as needed. The student may or may not be successful.

Questions Eliciting Thinking Why did you convert these numbers to standard notation? Could you have completed the multiplication with the numbers in scientific notation? Could you have completed the division with the numbers in scientific notation? 
Instructional Implications Provide instruction on converting between standard and scientific notation. Include converting numbers written in the form Â which are not in scientific notation (e.g., ) to scientific notation.
Review the Product Property of Exponents and the Quotient Property of Exponents. Using a few examples, demonstrate the ease and efficiency of calculating with numbers in scientific notation. Ask the student to redo his or her calculations withÂ the numbers in scientific notation. Then have the student determine if the final answer is in scientific notation and rewrite it if it is not.
Provide a set of practice problems in which the student must perform operations with numbers in scientific notation. Include problems where both standard and scientific notation are used. 
Moving Forward 
Misconception/Error The student makes errors working with numbers expressed in scientific notation. 
Examples of Student Work at this Level The student attempts to multiply numbers in scientific notation but:
 Multiplies coefficients and chooses one of the given powers of 10, (e.g., producing an answer of 17.6 x or 17.6 x ).
 Multiplies coefficients and multiplies exponents, producing an answer of 17.6 x .
 Rewrites the expressions so that they contain the same power of 10 and then applies that power of 10 to the quotient.
The student attempts to divide numbers in scientific notation but:
 Divides the coefficients and adds exponents, producing an answer of 1.6 x or 1.6 x .
 Rewrites the expressions so that they contain the same power of 10 and then applies that power of 10 to the quotient.

Questions Eliciting Thinking Can you explain how you found your answer?
What does mean? What does mean? If you multiply these powers of 10, how many factors of 10 will the product contain? If you divide by , how many factors of 10 will be in the quotient?
What does the Product Property of Exponents say?
What does the Quotient Property of Exponents say? 
Instructional Implications Provide feedback to the student concerning his or her specific error. As needed, review the Product Property of Exponents and the Quotient Property of Exponents. Ask the student to correct his or her error. Then provide additional problems in which the student must perform operations with numbers expressed in scientific notation. Include problems in which the results are not in scientific notation [e.g., (6.2 x ) x (4.3 x ) = 26.66 x ] and must be converted to scientific notation. 
Almost There 
Misconception/Error The student makes a minor error. 
Examples of Student Work at this Level The student:
 Rewrites 17.6 x as 1.76 x or as 1.76 x .
 Divides the two values given in problem #2 in the wrong order but divides correctly.Â

Questions Eliciting Thinking Can you show me how you converted 17.6 x to scientific notation?
Which number is larger, 1.66 x or 2.656 x ? Is an atom of hydrogen or an atom of oxygen heavier? 
Instructional Implications Provide feedback to the student regarding any errors made and allow the student to revise his or her work. If needed, provide opportunities for the student to rewrite numbers of the form a x in which a < 1 or a10 in scientific notation. Provide the student with a set of numbers written in the form a x only some of which are in scientific notation. Ask the student to identify those that are not in scientific notation and to convert them to scientific notation. 
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 multiplies and divides each pair of numbers in scientific notation getting answers of (1.76 x ) gallons per year and 16 times.

Questions Eliciting Thinking What do negative exponents mean? Can you rewrite the values in problem #2 using positive exponents? 
Instructional Implications Challenge the student to estimate very large and very small numbers using scientific notation and to use these estimates to calculate quantities in context. For example, given that the earth is 238,900 miles from the moon and 92,960,000 miles from the sun, ask the student to approximate each distance by writing it in scientific notation and rounding the coefficient to the nearest tenth. Then ask the student to determine how many times further the sun is than the moon. 