Getting Started 
Misconception/Error The student does not understand the concept of a rational number. 
Examples of Student Work at this Level The student describes rational numbers as whole numbers or negative numbers.

Questions Eliciting Thinking Do you know what the counting numbers are? The whole numbers? The integers?
Have you ever seen a definition of rational numbers? 
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. Guide the student to use this definition to test whether or not each number in the list is rational. That is, ask the student to try to convert each number to a fraction of integers to determine whether or not a number is rational. Then, assist the student in exploring and characterizing the decimal expansions of rational numbers. Be sure the student understands the distinction between the definition of a rational number and the characterization of rational numbers when written as decimals. Model using the definition to determine if a number is rational or not. 
Moving Forward 
Misconception/Error The student has a partial understanding of rational numbers. 
Examples of Student Work at this Level The student identifies some of the numbers that are rational (and possibly some that are not). Additionally, the student is unable to completely and correctly describe how to identify rational numbers. For example, the student:
 Describes rational numbers as ratios (or fractions) but does not indicate that the ratios must be comprised of integers.
 Attempts to describe rational numbers in terms of their decimal representations.
 Describes rational numbers as fractions, integers, or whole numbers.

Questions Eliciting Thinking You said that rational numbers are ratios? What kind of numbers are in the ratios?
Can you pick out a number from the list that is not rational? Suppose I form a ratio with this number? Does that make it a rational number?
Do you know what the integers are? 
Instructional Implications Review the definition of rational numbers and make explicit that rational numbers are ratios (or fractions) of integers (excluding zero from the denominator). Provide feedback to the student on his or her identifications of the rational numbers in the list, referring to the definition of rational numbers when necessary. Make the student aware of the existence of the irrational numbers and describe taking the square root of a nonperfect square as a way of generating examples of some irrational numbers.
Make sure the student understands how to identify rational numbers when written in decimal form. 
Almost There 
Misconception/Error The student can correctly define rational numbers but has trouble applying the definition to some of the examples in the list. 
Examples of Student Work at this Level The student:
 Is uncertain about .
 Does not have a complete understanding of the decimal expansions of rational numbers.
 Does not realize that integers are also rational.

Questions Eliciting Thinking What happens when you try to convert to a decimal? What happens when you try to convert to a decimal? Why are some square roots considered rational while others are irrational?
What can happen when you write a rational number as a decimal? What is true of their decimal expansions?
Can you write 12 in the form of a fraction? Could you do this for any integer? Does this mean that integers are also rational numbers? 
Instructional Implications Provide feedback to the student on his or her identifications of the rational numbers in the list, referring to the definition of rational numbers when necessary. Provide the student with another set of real numbers and ask the student to identify those that are rational. Be sure the student understands that the numbers in the list that are not rational are called irrational. 
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 identifies all of the rational numbers in the list, describes rational numbers as fractions of integers, and identifies the irrational number system as the one to which the remaining numbers in the list belong.
The student may need to be prompted to describe the restriction on the denominator. 
Questions Eliciting Thinking You described rational numbers as ratios (or fractions) of integers. Are there any restrictions? Can the denominator be zero?
How are the integers and the rational numbers related? Are integers also rational? Are all rational numbers integers? 
Instructional Implications 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.
Make the student aware that there are numbers that are neither rational nor irrational that will be studied in later mathematics courses. 