Getting Started 
Misconception/Error The student cannot correctly define a rational number. 
Examples of Student Work at this Level The student may be able to provide an example of a rational number. However, the student is unable to write a complete and correct definition. The student says:
 A rational number is not a whole number.
 A rational number is a fraction.
 A rational number â€śdoesnâ€™t go on forever.â€ť
 A rational number does not have an infinite number of random numbers following the decimals.
 A rational number is a real number that can be written as a fraction.

Questions Eliciting Thinking You said that a rational number can be written as a fraction. Can you be more specific about the numerator and denominator? What type of numbers must they be? Are there any restrictions on the denominator?
What happens when you try to write a rational number as a decimal?
Can you explain what you mean by â€śdoesnâ€™t go on foreverâ€ť?
Can you explain what you mean by â€ścan be written as a fractionâ€ť? Can you provide examples?
Can you give me an example of a number that is not rational? 
Instructional Implications Remind the student that the integers consist of the set {â€¦3, 2, 1, 0, 1, 2, 3, â€¦}. Then, review the definition of a rational number as a number that can be written in the form Â where a and b are integers but . Use the definition as a way to â€śbuildâ€ť rational numbers by substituting integers for a and b to form a variety of rational numbers. Then use the definition as a way to show a number (e.g., 0, 8, 12, , ) is rational by rewriting it as a fraction of integers. Finally, ask the student to convert a variety of rational numbers written in fraction form to decimals and to observe that the decimal representation of a rational number will either terminate or repeat.
Challenge the student to review his or her initial explanation of a rational number. Ask the student to correct any errors and revise the explanation. Also, ask the student to provide additional examples of rational numbers. 
Making Progress 
Misconception/Error The student is unable to explain why the sum of two rational numbers must be rational. 
Examples of Student Work at this Level The student can define and provide examples of rational numbers. However, the student is unable to explain why the sum of two rational numbers must be rational. The student:
 Indicates that he or she does not understand why the sum of two rational numbers must be rational.
 Offers an incorrect explanation such as:
 Since a rational number can be written as a fraction, then combining two fractions requires a common denominator and the result is another fraction. Therefore, the result is another rational number.
 Since a rational number has a repeating or a terminating decimal, when the decimals are combined, the result is another (larger) repeating or terminating decimal. Therefore, the sum is a rational number.

Questions Eliciting Thinking How do you know that the common denominator is an integer?
How could you use variables to model rational numbers and then model the sum of the two rational numbers? 
Instructional Implications Review the fact that the integers are closed for addition, subtraction, and multiplication. Guide the student to understand that the rational numbers are closed for addition, subtraction, multiplication, and division. Show that the sum of two rational numbers must be rational by reasoning that ifÂ Â andÂ Â are rational then their sum, , must be rational since both ac and bd are integers. Ask the student to reason in a similar fashion to show that the rational numbers are closed for subtraction, multiplication, and division.
Consider implementing other MFAS tasks for NRN.2.3. 
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 explains that rational numbers can be written as a fraction of two integers with a nonzero denominator. As a result, rational numbers written as decimals are repeating or terminating. The student is able to provide a variety of examples of rational numbers (e.g., whole numbers, integers, fractions, repeating and terminating decimals, and radicals that reduce to a rational number). To show that the sum of two rational numbers must be rational, the student provides a proof such as: Suppose p and q are rational numbers. Since p and q are rational, they can be represented as fractions of integers with nonzero denominators, for example, as Â andÂ Â where a, b, c, d, are integers such that ,Â . Then . Since integers are closed under multiplication and addition then ad + cb and bd are also integers. Additionally, sinceÂ ,Â Â thenÂ . So, Â is a fraction of integers withÂ ,Â which means it is a rational number. Therefore, the sum of two rational numbers must be rational. 
Questions Eliciting Thinking To what number system do both the rational numbers and irrational numbers belong?
What is an irrational number? Are there any rational numbers that are also irrational? 
Instructional Implications Challenge the student to determine whether each of the following statements is always true, sometimes true, or never true:
 The product of two rational numbers is rational.
 The difference between two rational numbers is rational.
 The quotient of two rational numbers is rational.
For statements that are sometimes or never true, ask the student to provide a counterexample.
Consider implementing other MFAS tasks for NRN.2.3. 