Getting Started 
Misconception/Error The student cannot distinguish between rational and irrational numbers. 
Examples of Student Work at this Level The student may be able to provide an example of a rational number and an irrational number. However, the student is unable to completely and correctly distinguish between the two types of numbers. The student says:
 A rational number is a decimal while an irrational number is a whole number.
 A rational â€śhas a patternâ€ť while an irrational â€śgoes on forever in a random sequence.â€ť
 A rational number is a fraction while an irrational number is a decimal.
 A rational â€śhas an endâ€ť while an irrational â€śgoes on forever.â€ť
 A rational can be expressed as a fraction while an irrational cannot.
 A rational number is a â€śpositive whole numberâ€ť while irrational numbers are fractions, decimals, and negative numbers.

Questions Eliciting Thinking You said that 0.333... is irrational. Is Â rational? What isÂ Â written as a decimal? So what type of number is 0.333â€¦?
What happens when you try to write a rational number as a fraction of integers? What happens when you try to write an irrational number as a fraction of integers?
What happens when you try to write a rational number as a decimal? What happens when you try to write an irrational number as a decimal?
Can you explain what you mean by â€śhas an endâ€ť and â€śgoes on foreverâ€ť?
Can you explain what you mean by â€śhas a patternâ€ť? 
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.
Introduce irrational numbers by describing them as numbers that are not rational (i.e., numbers that cannot be expressed as a quotient of integers). Provide examples of irrational numbers other than . Explain to the student that a way to generate examples of irrational numbers is by taking the square root of a nonperfect square (e.g., ). Next, compare the decimal representations of rational and irrational numbers by explaining that when an irrational number is written as a decimal, it neither terminates nor repeats.
Challenge the student to review his or her initial explanation of the difference between rational and irrational numbers. Ask the student to correct any errors and revise the explanation. Also, ask the student to provide additional examples of rational and irrational numbers. 
Making Progress 
Misconception/Error The student is unable to explain why the product of a nonzero rational and an irrational must be irrational. 
Examples of Student Work at this Level The student can explain the difference between a rational and an irrational number and provide examples of each. However, the student is unable to explain why the product of a nonzero rational and an irrational must be irrational. The student:
 Indicates that he or she does not understand why the product of a nonzero rational and an irrational must be irrational.
 Offers an incorrect explanation such as:
 The product of a rational and an irrational number is irrational because the rational â€śstateâ€ť of the first number is changed.
 A nonzero rational number multiplied by an irrational will not come out to be written as a fraction or ratio.
 If you have a rational number, three, and an irrational number, , the product will be irrational because no matter what you do toÂ , you cannot make it terminating.

Questions Eliciting Thinking What do you mean by the â€śstateâ€ť of a number?
Why canâ€™t an irrational number be written as a fraction? What if I rewrite Â as ? Does this mean thatÂ Â is rational?
What happens when you multiply two rational numbers? Will the product be rational?
Can you use the fact that the product of two rational numbers is always rational to explain why the product of a nonzero rational and irrational can never be rational? 
Instructional Implications Introduce the student to proof by contradiction. Explain that a proof by contradiction contains the following features: (1) the negation of the conclusion is assumed (e.g., assume that the product of a nonzero rational and an irrational is rational); and (2) then one shows how this assumption leads to the contradiction of something known or established. The contradiction of something known or established indicates the negation of the conclusion cannot be true. Therefore, the conclusion must be true.
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 but that the irrationals are not. Show that the product of two rational numbers must be rational by reasoning that if Â andÂ Â are rational then their product,Â Â , 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.
Next, ask the student to consider if a number such as Â is rational or irrational. Guide the student to reason that (given Â is irrational and 5 is rational) Â cannot be rational. If it is rational, thenÂ Â is equal to some rational number x which means that . But the rational numbers are closed for division, so if x is rational, Â is rational, which contradicts the fact thatÂ Â is irrational. Ask the student to use similar reasoning to explain why Â is irrational. Then, ask the student to develop a general explanation for why the product of a rational number and an irrational number must be 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 explains that rational numbers can be written as a ratio of two integers with a nonzero denominator but irrational numbers cannot. As a result, rational numbers written as decimals are repeating or terminating while irrational numbers are nonrepeating, nonterminating decimals. The student is able to provide examples of each type of number.
To show that the product of a rational and an irrational must be irrational, the student provides a proof by contradiction such as: Suppose a is a rational number and b is an irrational number. Let c = ab and assume c is rational. Then . Since a and c are rational numbers (and the rational numbers are closed for division) then b must be rational which contradicts the assumption that b is irrational. Therefore, the product of a rational number and an irrational number must be irrational. 
Questions Eliciting Thinking Are the irrational numbers closed for multiplication?
To what number system do both the rational numbers and irrational numbers belong? Do you think the real numbers are closed for multiplication? 
Instructional Implications Challenge the student to determine whether each of the following statements is always true, sometimes true, or never true:
 The difference between two irrational numbers is irrational.
 The quotient of two irrational numbers is irrational.
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 Rational Numbers (NRN.2.3) if not previously used. 