Standard: MACC.912.N-RN.2.3 Depth of Knowledge Explain why the sum or product of two rational numbers is rational; Level 2: Basic Application that the sum of a rational number and an irrational number is of Skills & Concept irrational; and that the product of a nonzero rational number and irrational number is irrational. Explanations and Ideas to Support: Sample Test/Task Item(s): Know and justify that when adding or multiplying Given a right triangle whose hypotenuse is two rational numbers the result is a rational irrational, find measures for legs where: number. Know and justify that when adding a rational number and an irrational number the result is irrational. • Both legs are rational. • Both legs are irrational. Know and justify that when multiplying of a nonzero rational number and an irrational number the result is irrational. • One leg is irrational and one leg is rational. Since every difference is a sum and every quotient is a product, this includes differences and quotients as well. Explaining why the four operations on rational numbers produce rational numbers can be a review of students understanding of fractions and negative numbers. Explaining why the sum of a rational and an irrational number is irrational, or why the product is irrational, includes reasoning about the inverse relationship between addition and subtraction (or between multiplication and addition). What type of number is the product of 3 and √ ? Connections: SMPs to be Emphasized MP2- Reason abstractly and quantitatively. MP3- Construct viable arguments and critique the reasoning of others. _________________________________________ Algebra I Connection Topic 5: Quadratic Functions and Modeling. In high school, practicing operations with rational and irrational numbers helps students to understand the properties of real numbers and the relationships between number sets. Algebraic manipulations and reasoning become a powerful tool for transferring students’ experience in proofs from geometry to proofs in algebra. EOC Connections: Related NGSSS Standard(s) None related. Common Misconceptions:    Some students may believe that both terminating and repeating decimals are rational numbers, without considering nonrepeating and nonterminating decimals as irrational numbers. Students may also confuse irrational numbers and complex numbers, and therefore mix their properties. In this case, students should encounter examples that support or contradict properties and relationships between number sets (i.e., irrational numbers are real numbers and complex numbers are non-real numbers. The set of real numbers is a subset of the set of complex numbers). By using false extensions of properties of rational numbers, some students may assume that the sum of any two irrational numbers is also irrational. This statement is not always true (e.g., ( √ ) ( √ ) 4, a rational number), and therefore, cannot be considered as a property.