Getting Started 
Misconception/Error The student is unable to consistently use the standard algorithm for division. 
Examples of Student Work at this Level The student:
 Attempts to find the quotient by repeatedly multiplying the divisor until a number close to the dividend is found.
 Attempts to use a partial quotients strategy without success.
 Is unable to begin two or three of the problems.

Questions Eliciting Thinking Do you think trial and error is an efficient strategy? Do you know any other method you could use to find the quotient?
What do each of these products mean? How are you keeping track of how many multiples of the divisor you are pulling out?
Can you tell me what you know about solving a long division problem? 
Instructional Implications Have the student write familiar quotients such as 54 ÷ 9 and divisions involving a remainder such as 65 ÷ 4 using a long division format. It is also helpful to provide contexts for division problems that involve a given number of objects that must be divided into a given number of groups.
Given a division problem, ask the student to first estimate a quotient and use multiplication to check the estimate. Emphasize the relationship between division and multiplication. Have the student write the results of divisions in terms of multiplication (and addition, if there is a remainder). For example, after dividing 175 by 12, ask the student to summarize the result as 175 = (12 x 14) + 7. Model interpreting the result by saying, “The number 175 can be divided into 14 groups of 12 with seven leftover.”
Provide the student with direct instruction on the use of the standard algorithm for division. Explain and justify the steps in the process, so the student can develop a useful understanding of the process. Pay particular attention to the first step in each repeated cycle of steps in which a quotient is estimated. Provide focused practice with this step. Remind the student of the actual meaning of each digit in the quotient throughout the division process. For example, when dividing 9580 by 47, the first digit written above the division box is two, but this digit actually represents 200. Characterize the number 200 as an estimate of the quotient. Then explain that multiplying back and subtracting is just a means of finding what is “left over” or the remainder. If this amount is larger than the divisor, the process should be repeated in order to make the estimate more precise. Describe each cycle of the process as an attempt to find the quotient more precisely.
Show the student a division problem along with an associated visual model. Ask the student questions like:
 What are you dividing into? How many did you start with?
 What are you dividing by? How many in each group?
 What is the difference? How many do you have left over?

Moving Forward 
Misconception/Error The student makes systematic errors implementing some steps of the standard division algorithm. 
Examples of Student Work at this Level The student appears to understand the major steps of the standard division algorithm but:
 Brings down digits from the dividend incorrectly. For example, the student:
 Skips bringing down a digit.
 Does not bring down the final digit.
 Brings down the same digit twice.
 Brings down two digits at once while only putting one digit in the quotient – forgetting to hold one place in the quotient with a zero.
 Does not notice that a difference is greater than or equal to the divisor and creates an extra place value in the quotient.
 Adds an extra zero after each digit in the quotient (“because the divisor didn’t go into the difference”) before bringing down the next digit of the dividend.
 Adds a decimal point after the first digit in the quotient before bringing down any digits or does not add a decimal point at all, giving the quotient extra place value digits.

Questions Eliciting Thinking What numbers from the dividend have you already used? Are there any numbers in the dividend you have not used?
Should you ever use a digit greater than nine in the quotient? What should you do when the difference is greater than the dividend?
Can you tell me why you put a zero after each number in the quotient? Why is the difference smaller than the divisor? Should it always be smaller?
What is the next step in the division process? What will happen when you bring down the next digit from the dividend?
When should you add a decimal point to the quotient? What does it represent?
Where did you first have trouble with this problem? Can you work the problem again to see if you get the same answer? 
Instructional Implications As with the Getting Started student, provide the student with direct instruction on the use of the standard algorithm for division. Explain and justify the steps in the process, so the student can develop a useful understanding of long division. Pay particular attention to the first step in each repeated cycle of steps in which a quotient is estimated. Provide focused practice with this step. Remind the student of the actual meaning of each digit in the quotient throughout the division process. For example, when dividing 9580 by 47, the first digit written above the division box is two, but this digit actually represents 200. Characterize the number 200 as an estimate of the quotient. Then multiplying back and subtracting is just a means of finding what is “left over” or the remainder. If this leftover amount is larger than the divisor, the process should be repeated in order to make the estimate more precise. Describe each cycle of the process as an attempt to find the quotient more precisely.
Provide the student with error analysis practice. Give the student several division problems with all work shown including some that contain common student errors. Ask the student to decide if each problem is correct and if not, to find the error, describe it, and then correct it.
Ask the student to analyze his or her own work and describe the kinds of errors he or she typically makes. 
Making Progress 
Misconception/Error The student does not know how to appropriately write remainders as either a fraction or a decimal. 
Examples of Student Work at this Level The student correctly completes each division but:
 Writes the remainder by adding a decimal point followed by the digits of the remainder.
 Incorrectly writes the remainder as a fraction by writing the remainder:
 Over ten (or a multiple of 10).
 Over the dividend rather than the divisor.
 As the first digit over the second digit (for a two digit remainder).

Questions Eliciting Thinking Why does this problem have a remainder? How can you represent the remainder?
When should you include a decimal point in the quotient? What does the number after the decimal point mean?
When you write the remainder as a fraction, what goes in the numerator? What number is used for the denominator? 
Instructional Implications Emphasize the relationship between division and multiplication. Have the student write the results of divisions in terms of multiplication and addition. For example, after dividing 1963 by 27, ask the student to summarize the result as 1963 = (27 x 72) + 19. Model interpreting the result by saying, “The number 1963 can be divided into 72 groups of 27 with 19 leftover.”
Transition the student to writing remainders as both fractions and decimals. Help the student understand the meaning of the remainder written as either a fraction or decimal by considering divisions of smaller numbers such as 5 ÷ 2 which can be written as 2 with a remainder of 1, 2 , or 2.5. Have the student check his or her quotients with multiplication. Help the student to see that the remainder, whether written as a fraction or a decimal, improves the precision of the quotient.
Give the student additional practice with completed division problems that show the remainder as an amount leftover (e.g, 83 ÷ 12 = 6 R11). Ask the student to rewrite the quotient with the remainder written as a fraction or decimal. Additionally, ask the student to show how the decimal remainder is equivalent to the fraction remainder. 
Almost There 
Misconception/Error The student makes a minor or computational error. 
Examples of Student Work at this Level The student appears to understand the standard division algorithm but:
 Makes a multiplication or subtraction error.
 Transposes numbers within one step of the problem.
 Gets the correct decimal remainder (11.05555…) but writes it incorrectly as or 11.055.
 Neglects to bring down the final digit to finish the long division.

Questions Eliciting Thinking Can you check your work here? I think you may have made an error.
Do you know how you could check your final answer? 
Instructional Implications Provide the student with error analysis practice. Give the student several division problems with all work shown including some that contain common student errors. Ask the student to decide if each problem is correct and if not, to find the error, describe it, and then correct it.
Ask the student to analyze his or her own work and describe the kinds of errors he or she typically makes.
Consider using MFAS task Long Division  3 (6.NS.2.2). 
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 uses the standard division algorithm to correctly find each quotient:
 or
 170
 508
A student at this level will show fluency by completing the problems correctly in 35 minutes.

Questions Eliciting Thinking How would you begin a problem that has a threedigit or larger divisor?
How can you check your answer to be sure it is correct?
What does the remainder mean?
Do you also know how to write the remainder as a fraction (or decimal)?
Have you ever thought about why long division works? How would you explain it to someone else? 
Instructional Implications Have the student begin to address division with multidigit decimal numbers. 