Standard: MACC.912.N-RN.2.3 Depth of Knowledge Level 2: Basic Application

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.
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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:
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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.