Getting Started 
Misconception/Error The student is unable to correctly convert a fraction into a decimal. 
Examples of Student Work at this Level The student reverses the dividend and divisor when attempting the long division algorithm. The student puts the decimal point in the wrong place and puts the repeating bar over the as a “repeating whole number.” The student makes other major conceptual errors while performing the long division algorithm.

Questions Eliciting Thinking How did you decide which number to use as the dividend and which for the divisor?
What are the sequential steps that need to be followed in a division problem? Have you followed each one in this problem?
What digits actually repeat in the quotient? 
Instructional Implications Provide instruction and more practice converting fractions to decimals using the long division algorithm. Start with simple fractions whose decimal representations terminate. Then transition the student to fractions whose decimal representations contains only one repeating digit such as or . Make explicit what occurs in the process of division that gives rise to the repeating digit. Guide the student to write repeating decimals correctly. Eventually, introduce fractions whose decimal representations contain multiple repeating digits. Again, relate key features of the division process to repeating digits in the quotient. Model explaining how and why this occurs. 
Moving Forward 
Misconception/Error The student does not write the repeating decimal in a correct format. 
Examples of Student Work at this Level The student writes the answer as:
 A terminating decimal of 0.45 or 0.4545.
 A nonterminating decimal of 0.45... without referencing the repeating pattern in the answer.
 A repeating decimal but with the repeating bar over just one number (common answers are either or ).

Questions Eliciting Thinking Where did you put the decimal point? What digits actually repeated?
Why did you write 0.45 (or 0.45...)? What is happening in the division problem? Did you get a remainder? What does that remainder mean? How can you show that in your quotient?
Why did you write (or )? If you write out the next few digits of the number you wrote, it would show only the 4 (or 5) as repeating, but if you were to continue dividing, what would be the next number you would get? What numbers are repeating? 
Instructional Implications Provide the student with instruction regarding the use of proper notation in writing repeating decimals. Make explicit what occurs in the process of division that gives rise to repeating digits. Ensure the student has the opportunity to convert a variety of fractions to decimals, including those with one, two, and three repeating digits and those that have nonrepeating digits before the repeating digits. 
Almost There 
Misconception/Error The student cannot explain clearly why the decimal repeats. 
Examples of Student Work at this Level The student answers that it does repeat, but gives an unclear rationale. The student’s explanation may include why there will not be a zero remainder, but without explaining why the digits in the quotient repeat. Examples may include:
 It won’t go in evenly.
 There’s always going to be some number left over.
 You will have to keep annexing zeros.
 Eleven won’t go into a number with a 10, 100, 1000, etc.
 Eleven won’t go into 40 or 50.
 Eleven does not go into 5.
 You’ll get the same number every time.
The student gives a mathematically incorrect explanation for the repeating pattern.

Questions Eliciting Thinking If a number does not go into the new dividend evenly, will there always be the same repeating difference?
Does having a remainder again and again, even if it is different each time, mean that there is a repeating pattern in the quotient?
When you divide two numbers, is it possible for there to be a different remainder every time you annex a zero to make a new dividend?
What do you mean that “11 won’t go into a 10 number?”
What do you mean that the number will be the same every time – what number? 
Instructional Implications Have the student study a completed division problem that shows several steps that lead to the repeating pattern. Make explicit what occurs in the process of division that gives rise to the repeating digit. Model explaining how and why this occurs. Then have the student do a similar division problem and identify and explain any repeating pattern in the decimal representation. Also have the student work a division problem that will not have a repeating pattern and note the similarities and differences in each of the problems. 
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 converts the fraction to and explains, “Yes, the decimal repeats because you will continue to get the same two differences of 60 and 50 each time you bring down a zero and subtract from a new dividend, which will result in a repeating ‘45’ pattern in the quotient.”

Questions Eliciting Thinking Will all decimals end up repeating eventually once you start annexing zeros?
What will a repeating decimal look like in a calculator?
Can all fractions be written as decimals? 
Instructional Implications Have the student write a fraction that is equivalent to ; then have the student change the equivalent fraction into a decimal and explain the results.
Have the student convert mixed numbers and improper fractions into decimals that contain repeating digits.
Have the student convert repeating decimals back into fractions. Consider implementing MFAS task Decimal to Fraction Conversion. 