Getting Started 
Misconception/Error The student does not convert a terminating decimal into a fraction. 
Examples of Student Work at this Level The student does not use place value for the denominator, but instead writes a one over the given digits to make the fractions.
The student uses the wrong place value when writing the denominators of each fraction.
The student attempts to change the numbers to a percent.

Questions Eliciting Thinking How did you choose that numerator and that denominator?
How do you read the decimal number? How do you read your fraction? Does the place value you said for each one match?
Why did you change to a percent? What does percent mean? What do the instructions ask you to do? 
Instructional Implications Provide instruction to the student regarding converting terminating decimals to fractions and mixed numbers.
Have the student use mathematically precise vocabulary when saying decimal and fraction names. Emphasizing the place value words helps the student see the relationship between the two. 
Moving Forward 
Misconception/Error The student does not convert a decimal with all repeating digits into a fraction. 
Examples of Student Work at this Level The student does not convert the decimals 0.777…, 0.272727… or 0.27777… to fractions explaining, “They cannot be written as fractions because they are repeating decimals.”
The student writes the fraction as if it were a terminating decimal:
 Over a place value of ten, based on the number of digits shown.
 But adds a repeating bar over the numbers in the numerator to show the repeating digits continue.
 After adding a zero onto the end of the decimal to “finish” the number, before writing it over its new place value denominator.
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Questions Eliciting Thinking How is a repeating decimal different than a terminating decimal? What do the “dots” mean? How can you rewrite this decimal number using repeating bar notation?
Why did you add a zero onto the repeating digits of the decimal? Does that change the meaning of the number? If this were a terminating decimal, would adding a zero give you an equivalent decimal? 
Instructional Implications Provide the student with a calculator and a wide variety of fractions (repeating and nonrepeating). Have the student change each fraction to a decimal while looking for patterns. From this they can develop general rules and procedures regarding writing decimals as fractions. Show the student how the patterns they discovered come from algebraic solutions.
Review with the student why adding zeros onto the end of terminating decimals produces an equivalent decimal, but why this is not possible with a repeating decimal. 
Making Progress 
Misconception/Error The student does not convert a decimal that has nonrepeating digits before the repeating digits into a fraction. 
Examples of Student Work at this Level The student converts 0.27777… to i.e., as if it were 0.272727… (or with a repeating bar over the repeating seven as ). 
Questions Eliciting Thinking Why did you write the fraction for 0.27777… the same as you did for 0.272727… ? What is the difference between these two decimals? Should there be a difference between their fraction equivalents? 
Instructional Implications Provide direct instruction with the algebraic method of changing decimals that have nonrepeating digits before the repeating digits into fractions. 
Almost There 
Misconception/Error The student cannot explain that there are some decimals that cannot be written as fractions. 
Examples of Student Work at this Level There are no decimals that cannot be written as fractions because:
 You can always round.
 They all have a place value.
 A decimal is just a different form of a fraction.
 Everything is terminating or repeating.
 Decimals are rational numbers.
There are decimals that cannot be written as fractions because:
 There are decimals that go on and on.
 Some decimals are not tenths, hundredths, thousandths, etc.

Questions Eliciting Thinking Is it possible to have a decimal with digits that don’t repeat but keep on going? Can you think of an example? How would you read that number? 
Instructional Implications If necessary, review the counting numbers, whole numbers, and integers. Then define the rational numbers as numbers of the form where a and b are integers but b cannot be zero. Use this as the basis to discuss examples of irrational numbers.
Provide the student a wide range of differing decimal numbers of all types: terminating, nonterminating, repeating and nonrepeating. Have the student categorize these into rational and irrational numbers based on their characteristics, then write fraction equivalents for all rational numbers. 
Got It 
Misconception/Error The student provides complete and correct responses to all components of the task. 
Examples of Student Work at this Level (1)
(2)
(3)
(4) or
(5) or
(6) Yes, irrational numbers (or decimals that are nonterminating and nonrepeating) cannot be written as fractions.

Questions Eliciting Thinking If any of these decimals was greater than one, how would that change your fraction answer?
How do you write an integer as a decimal? How are integers and rational numbers related? Are integers also rational? Are all rational numbers integers?
What are some examples of irrational numbers? 
Instructional Implications Have the student convert decimals greater than one (with and without repeating decimals) into mixed numbers and improper fractions.
Ask the student to draw a diagram that shows the relationships among the following number systems: counting numbers, whole numbers, integers, rational numbers, irrational numbers, and real numbers. Provide feedback as needed.
Have the student practice converting fractions into repeating decimals and identifying rational numbers. Consider implementing MFAS tasks Fraction to Decimal Conversion and Rational Numbers (8.NS.1.1).
Make the student aware that there are numbers that are neither rational nor irrational that will be studied in later mathematics courses. 