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 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 â€śhas a patternâ€ť while an irrational â€śgoes on forever in a random sequence.â€ť
 A rational can be expressed as a fraction while an irrational cannot.

Questions Eliciting Thinking 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 formÂ Â whereÂ aÂ andÂ bÂ are integers butÂ . Use the definition as a way to â€śbuildâ€ť rational numbers by substituting integers forÂ aÂ andÂ bÂ to form a variety of rational numbers. Then use the definition as a way to show a number (e.g., 0, 8, 12,Â , ) 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 thanÂ . 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.,Â ). 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 sum of a rational and 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 sum of a rational number and an irrational number is irrational. The student:
 Indicates that he or she does not understand why the sum of a rational and an irrational must be irrational.
 Offers an incorrect explanation such as:
 Since an irrational number cannot be written as a fraction, when a nonfraction (an irrational number) is combined with a fraction (rational number), a common denominator cannot be determined, so the result cannot be a fraction. Therefore, the result is a nonfraction or an irrational number.
 Since an irrational number has a nonrepeating, nonterminating decimal, when this type of decimal is combined with a repeating or terminating decimal (rational number), the decimal becomes another nonrepeating, nonterminating (an irrational number) because the decimal of the irrational number does not reduce. Therefore, the sum is a larger irrational number.

Questions Eliciting Thinking Why canâ€™t an irrational number be written as a fraction? What if I rewriteÂ Â asÂ ? Does this mean thatÂ Â is rational?
What do you mean by â€śthe decimal does not reduceâ€ť? What happens when you add negative five to ? Is the sum larger than five? Is the sum larger thanÂ ?
What happens when you add two rational numbers? Will the sum be rational?
Can you use the fact that the sum of two rational numbers is always rational to explain why the sum of a 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 sum of a 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 the irrationals are not. Show that the sum of two rational numbers must be rational by reasoning that ifÂ and are rational, then their sum,Â , must be rational since ad + bcÂ andÂ bdÂ are integers. Ask the student to reason in a similar fashion to show that the rational numbers are closed for subtraction, multiplication, and division.
Next, ask the student to consider if a number such asÂ Â is rational or irrational. Guide the student to reason that (givenÂ Â is irrational and 5 is rational) Â cannot be rational. If it is rational, thenÂ Â is equal to some rational numberÂ xÂ which means thatÂ . But the rational numbers are closed for subtraction, so ifÂ xÂ is rational, x  5Â is rational, which contradicts the fact thatÂ Â is irrational. Ask the student to use similar reasoning to explain whyÂ Â is irrational. Then, ask the student to develop a general explanation for why the sum of a rational number and an irrational number must be irrational.
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 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 sum 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. Then let c = a + b and assume c is rational. Then b = a  c. Since a and c are rational numbers (and the rational numbers are closed for subtraction) then b must be rational which contradicts the assumption that b is irrational. Therefore, the sum 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 addition? 
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 irrational numbers is irrational.
 The product of two irrational numbers is irrational.
For statements that are sometimes or never true, ask the student to provide a counterexample.
