Getting Started 
Misconception/Error The student does not understand that the numbers cannot be added or subtracted in their given forms. 
Examples of Student Work at this Level The student attempts to add or subtract the numbers in scientific notation even though the powers of 10 are not the same. The student adds or subtracts coefficients and exponents.
 1.1 x Â as 1.100, 1100, or .
 8.4 x as 0.84, 840, 840, or 0.084.

Questions Eliciting Thinking How do you add and subtract numbers written in standard notation, for example, 1496 â€“ 72? How would you write this problem if you were preparing to subtract?
If written in standard notation, would the eight (inÂ ) and the one (in ) be in the same place and have the same place value? How could you tell? 
Instructional Implications If needed, provide instruction on converting between standard and scientific notation. Provide opportunities for the student to determine if a number is correctly written in scientific notation.
Demonstrate that it does not make mathematical sense to add or subtract numbers given in scientific notation by adding or subtracting their coefficients or exponents. Have the student add 120,000 and 23,000. Then ask the student to convert each to scientific notation. Next show the student that (1.2 x ) + (2.3 x ) Â 3.5 x since 3.5 x = 3,500,000,000 143,000 (the sum of 120,000 and 23,000).
Guide the student to determine when numbers of the form Â can be combined (i.e., when the powers of 10 are the same) and how to determine the result. Provide additional opportunities to add and subtract numbers written in scientific notation. 
Moving Forward 
Misconception/Error The student makes errors when converting numbers to a compatible form or combining compatible numbers. 
Examples of Student Work at this Level The student understands that each pair of numbers must be written in a compatible form before combining. However, the student:
 Makes errors when converting numbers to standard notation.
 Makes an error when converting numbers to a compatible exponential form, for example, writes Â as Â or Â as .
 Does not correctly align decimal points before adding or subtracting.
 Rewrites 1.07 x as 10.7 x and says the sum of 10.7 x and 1.5 x is 12.2 x + or 12.2 x .

Questions Eliciting Thinking How did you convert this number to standard notation (or to exponential form)?
Can you show me how you added (or subtracted) these two numbers? What must be true of the decimal points?
Your approach to the second problem is very clever, but I think you made an error in adding the two numbers. Can you factor out before adding? What will you get then? 
Instructional Implications As needed, review converting between scientific and standard notation.
Remind the student that the decimal points of the two numbers to be added or subtracted must be aligned. Indicate to the student that he or she made an addition or subtraction error and challenge the student to find and correct the error.
Model for the student how to add (1.07 x ) and (1.5 x ) by first rewriting 1.07 x as 1.07 x (10 x ) = 10.7 x . Then show the student that (10.7 x ) + (1.5 x ) can be rewritten as (10.7 + 1.5) x by applying the Distributive Property. Ask the student to complete the calculation and explain what must be true of the powers of 10 in order to use this strategy. 
Almost There 
Misconception/Error The student makes a minor error. 
Examples of Student Work at this Level The student:
 Subtracts another combination of masses (instead of the smallest from the largest) in the first problem but does so correctly.
 Correctly adds the two masses given in the second problem but makes an error in rewriting the sum in scientific notation. For example, the student writes the sum as 12.2 x .Â

Questions Eliciting Thinking I think you may have subtracted the wrong values in the first problem. Can you reread the problem and check your work?
Is 12.2 x Â written in scientific notation? What part of this expression violates the form of scientific notation? 
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 adds and subtracts in each problem, getting answers of 109.916 grams and 1.22 x kilograms.

Questions Eliciting Thinking One of the meteorites has a mass of 6.8 x . What does equal? What does 6.8 x equal? 
Instructional Implications If unfamiliar, show the student the following strategy for adding (or subtracting) numbers written in scientific notation: given (a x ) and (b x ) then (a x ) + (b x ) = (a + b) x (by an application of the Distributive Property). Provide opportunities for the student to rewrite numbers given in scientific notation so they involve equal powers of 10 and then apply this strategy. 