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 cannot be negative.
 A rational number is a number that can be written as a fraction (or a ratio).
 A rational number is a positive whole number.
 A rational number is any number that does not have a repeating decimal.

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?
Is Â a rational number? How do you writeÂ Â as a decimal number? So repeating decimal numbers must be what?
Can a rational number be negative?
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 and . 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 product of two rational numbers must be rational. 
Examples of Student Work at this Level The student can define a rational number and provide examples. However, the student is unable to explain why the product of two rational numbers must be rational. The student:
 Indicates that he or she does not understand why the product of two rational numbers must be rational.
 Offers an incorrect explanation such as:
 The product of two rational numbers is always rational because you are multiplying a rational number a rational amount of times.
 The product always has an ending and is not continuous.
 The product of two rational numbers is rational because nothing about their â€śstateâ€ť changes.

Questions Eliciting Thinking What do you mean by the â€śstateâ€ť of a number?
What happens when you multiply two integers? Will the product be an integer?
Can you use the fact that the product of two integers is always an integer to explain why the product of two rational numbers is always rational? 
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 product of two rational numbers must be rational by reasoning that if Â and Â are rational then their product, , 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, addition, and division.
Challenge the student to find an example that shows the irrational numbers are not closed for multiplication. 
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 ratio of two integers with a nonzero denominator. Rational numbers written as decimals are repeating or terminating. The student is able to provide correct examples.
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 and Â where a, b, c, d, are integers such that Â andÂ . Then . Since integers are closed under multiplication, both ac and bd are integers. Additionally, sinceÂ Â andÂ Â thenÂ . So, Â is a fraction of integers withÂ Â which means it is a rational number. Therefore, a rational number times a rational number will always be rational. 
Questions Eliciting Thinking To what number system do both the rational numbers and irrational numbers belong?
Do you think the real numbers are closed under multiplication? 
Instructional Implications Challenge the student to determine whether each of the following statements is always true, sometimes true, or never true:
 The sum of two rational numbers is rational.
 The difference between two integers is an integer.
 The quotient of two integers is an integer.
For statements that are sometimes or never true, ask the student to provide a counterexample.
Consider using the MFAS tasks Sum of Rational Numbers (NRN.2.3), Sum of Rational and Irrational Numbers (NRN.2.3), and Product of NonZero Rational and Irrational Numbers (NRN.2.3) if not previously used. 