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PA-N-1-4

Page history last edited by Tashe Harris 6 years, 1 month ago

PA.N.1.4 Classify real numbers as rational or irrational. Explain why the rational number system is closed under addition and multiplication and why the irrational system is not. Explain why the sum of a rational number and an irrational number is irrational; and the product of a non-zero rational number and an irrational number is irrational.


In a Nutshell

Rational numbers occur as integers, fractions, and decimals that can be written in fractional form. Irrational numbers simplify to non-repeating, non-terminating decimals. In Pre-Algebra the most common irrational numbers are π and non-perfect square roots. All integers and rational numbers are considered closed if specific operations with numbers from a set always produces a number within the same set: (rational +rational = rational  AND  rational/rational=rational). Sets are open if an operation with numbers from the set sometimes produces numbers not in the set: (rational +irrational=irrational AND rational  x irrational=irrational). 

 

EX:

 

Student Actions

Teacher Actions

  • Develop Mathematical Reasoning by comparing and contrasting sets of rational, irrational and real numbers, students can identify characteristics of each set and make meaning of the classification of independent sets (rational and irrational) within the larger set (real).

  • Develop a Deep and Flexible Conceptual Understanding through creating or using visual models of our number classification system to explore the relationships between the independent sets of numbers within the larger set of real numbers with an optional extension/introduction of imaginary numbers.

  • Develop Accurate and Appropriate Procedural Fluency by discovering that the sum and products of two rational numbers will always be rational and the sum or product of two irrational numbers will not always be irrational.
  • Develop the Ability to Make Conjectures, Model, and Generalize when analyzing problems to determine if the property sets are closed with certain operations (Ex. Is the sum of an irrational number and a rational number always, sometimes, or never a rational number?)

  • Develop the Ability to Make Conjectures, Model and Generalize when generalizations about rational and irrational operations are able to be defended by providing evidence of examples, counterexamples and conceptual reasoning.

     

  • Build procedural fluency from conceptual understanding by using a variety of representations of rational and irrational numbers ( like a card sort with fractions, decimals, integers, square roots, etc…) to engage students to make comparisons and connections of their place in the number system.

  • Implement meaningful tasks where students create a detailed number classification system organizer or provide different examples of pre-made classification organizers for students to elaborate on and make connections between individual sets of numbers within the larger set of real numbers.

  • Pose purposeful questions to stimulate students’ curiosity and encourage them to investigate further by asking leading questions such as, “If you add a whole number and pi, would your answer be rational or irrational? How do you know? Does this result always happen when you find the sum of any rational and irrational number?”

  • Implement meaningful tasks that promote reasoning and problem solving by creating opportunities for students to come up with their own questions and examples as they move from an individual problem to a  generalization that can be defended.

  • Extend the properties of numbers by widening the possible types of number categories.  Students have been exposed to rational numbers in 6th grade (6.N.1.2) and this should be connected to their new knowledge of irrational numbers.

Key Understandings

Misconceptions

  • Classify numbers as rational or irrational;

  • Locate rational and irrational number on a number line;

  • Estimate the value of irrational numbers;

  • Understand that adding or multiplying a rational number by an irrational number will create an irrational numbers, and adding an irrational number by an irrational number will create and irrational number and adding only rational numbers will result in a rational sum.

  • Understand that adding opposite irrational numbers will result in the sum of zero which is rational (Ex. -2+2 = 0)
  • Students might assume all repeating numbers are considered irrational.

  • Students might think that irrational numbers are not real numbers.

  • Students might think repeating decimals can not be written as a fraction.

  • When a square root of a number is in a fraction students sometimes believe that it is a rational number since it is already in fraction form.

  • Students might think that negative numbers are always irrational.

  • Students might assume that improper fractions are irrational.

OK Math Framework Introduction

Pre-Algebra Introduction

 

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