Getting Started 
Misconception/Error The student is unable to correctly use long division to convert fractions to decimals. 
Examples of Student Work at this Level The student makes substantial errors when using the standard division algorithm. For example, the student:
 Interchanges the dividend and the divisor when setting up the long division problem.
 Puts the first number of the quotient in the wrong place value position.
 Begins the division process but is unable to continue.
 Does not notice that a difference is greater than or equal to the divisor.
 Does not place a decimal point in the answer at all, or places it in the wrong position.

Questions Eliciting Thinking Why did you set up the division problem the way you did?
Is the numerator the dividend or is the denominator the dividend? Which is the divisor?
What is the pattern of steps you use for division? What should you check for at each step? 
Instructional Implications Review the parts of a fraction and how to rewrite a fraction as an equivalent division problem. Then provide instruction on the use of the standard division algorithm.
Consider using MFAS tasks Long Division1, Long Division2, and Long Division3Â (6.NS.2.2) for review of long division and for additional instructional strategies. 
Moving Forward 
Misconception/Error The student is unable to explain how to determine whether a number is rational. 
Examples of Student Work at this Level The student correctly converts , , andÂ Â to decimal numbers, although the student may not understand that Â is undefined. The student is unable to identify , , and Â as rational. The student explains that:
 Only Â is rational because it repeats.
 Only Â is rational because of the zero.
 Only is rational because it does not repeat.

Questions Eliciting Thinking What is the definition of a rational number?
Do you know how to tell if a number written as a decimal is rational?
Do you know how to tell if a number is rational when it is written as a fraction? 
Instructional Implications If necessary, review the counting numbers, whole numbers, and integers. Then define rational numbers as numbers of the form Â where a and b are integers but b cannot be zero. Guide the student to use this definition to test whether or not each number is rational. If needed, explain to the student that division by zero is undefined. Consequently, Â is not defined as a number.
Have the student draw a diagram showing the relationship between each subset of real numbers, including examples of each.
Assist the student in exploring and characterizing the decimal expansions of rational numbers. 
Almost There 
Misconception/Error The student does not understand that division by zero is undefined. 
Examples of Student Work at this Level The student correctly convertsÂ ,Â , andÂ Â to decimal numbers and identifies each as rational. However, the student does not understand thatÂ Â is undefined. The student says both Â and Â are equal to zero.Â The student may identifyÂ as a rational number because â€śit is a fraction.â€ť 
Questions Eliciting Thinking If you start with zero objects, is it possible to divide them into 17 groups? How many would you have in each group?
Is a number divided by zero the same as dividing zero by a number? How could you demonstrate each of these situations with a picture or reallife example? 
Instructional Implications Use the relationship between multiplication and division to demonstrate the result of dividing into zero and by zero. Explain, for example, that ifÂ 0 Ă· 5 = n for some number n, then, because of the inverse relationship between multiplication and division, 5 x n = 0. But, in order for the product of five and n to be zero, n must be zero. Therefore, 0 Ă· 5 or Â = 0.Â Likewise, suppose 5 Ă· 0 = n for some number n. Then, because of the inverse relationship between multiplication and division, n x 0 = 5. But, the product of n and zero is always zero and can never be five. Therefore, 5 Ă· 0 or Â has no quotient and is said to be undefined. 
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 long division to convert each fraction to a decimal getting (A) 1.625, (B) undefined or â€śthere is no way to divide something by zero,â€ť (C)Â , and (D) 0. The student says that ,Â , andÂ Â are rational because they are fractions of integers without a zero denominator or because their decimal representations either terminate or repeat.Â 
Questions Eliciting Thinking Are all fractions rational numbers? If not, what is an example of a fraction that is not rational?
Can negative numbers be rational? If so, what is an example of a negative number that is rational? 
Instructional Implications Give the student a set of rational numbers and ask the student to order them from least to greatest. Include both positives and negatives, integers, repeating decimals, terminating decimals, fractions, improper fractions, mixed numbers, and zero. 