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Glossary of Key Terms
Rational Number
A number that can always be written as a fraction of two integers (p/q, where q is not zero). Its decimal representation is either terminating or repeating.
Irrational Number
A number that cannot be written as a fraction of two integers. Its decimal representation is non-terminating and non-repeating.
Formative Assessment Lesson
A lesson unit designed to help teachers assess student understanding and reasoning, identify common difficulties, and provide targeted assistance to improve learning, rather than assigning a summative score.
Properties of Rational and Irrational Numbers
The rules governing how rational and irrational numbers behave under arithmetic operations (addition, subtraction, multiplication, division), determining whether the result of such operations is rational or irrational.
Always True
A classification for a mathematical statement that holds true for all possible cases and requires a general proof to be definitively established.
Sometimes True
A classification for a mathematical statement that holds true for at least one case but is false for at least one other case. This is established by providing one true example and one false example.
Never True
A classification for a mathematical statement that does not hold true for any possible case and requires a general proof to be definitively established.
Conjecture
An educated guess or statement formed based on observations and examples, but not yet rigorously proven.
Proof
A rigorous, logical argument that demonstrates the truth or falsity of a mathematical statement for all possible cases.
Example
A specific instance or set of values used to illustrate or support a mathematical statement or conjecture.
Counterexample
A specific instance that disproves a general mathematical statement; finding a single counterexample is sufficient to show a statement is not ‘Always True’.
Common Core State Standards (CCSS) N-RN
A specific content standard in the Common Core State Standards for Mathematics that focuses on students’ ability to use the properties of rational and irrational numbers.
Mathematical Practice Standards
A set of eight standards within the Common Core State Standards that describe varieties of expertise that mathematics educators should seek to develop in their students, including reasoning, precision, and argument construction.
Empirical Reasoning
A type of reasoning that relies on observations and specific examples to form conclusions, which can lead to false generalizations if not rigorously supported by proof.
Mini-whiteboards
A teaching tool used for quick, informal student responses and immediate feedback during a lesson.
Rational or Irrational? Task
An individual assessment task given before the lesson to evaluate students’ initial understanding of rational and irrational numbers, including definitions and application to geometric properties.
Always, Sometimes or Never True Task
A collaborative group activity where students classify mathematical statements about rational and irrational numbers into categories based on whether they are always, sometimes, or never true, providing examples and justifications.
See also: 01_Summary