# MAFS.912.N-RN.2.3

Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.
General Information
Subject Area: Mathematics
Domain-Subdomain: Number & Quantity: The Real Number System
Cluster: Level 2: Basic Application of Skills & Concepts
Cluster: Use properties of rational and irrational numbers. (Algebra 1 - Additional Cluster) -

Clusters should not be sorted from Major to Supporting and then taught in that order. To do so would strip the coherence of the mathematical ideas and miss the opportunity to enhance the major work of the grade with the supporting clusters.

Date of Last Rating: 02/14
Status: State Board Approved
Assessed: Yes
Test Item Specifications
Assessed with:

MAFS.912.N-RN.1.2

## Related Courses

This benchmark is part of these courses.
1200310: Algebra 1 (Specifically in versions: 2014 - 2015, 2015 - 2022 (current), 2022 and beyond)
1200320: Algebra 1 Honors (Specifically in versions: 2014 - 2015, 2015 - 2022 (current), 2022 and beyond)
1200380: Algebra 1-B (Specifically in versions: 2014 - 2015, 2015 - 2022 (current), 2022 and beyond)
1200400: Foundational Skills in Mathematics 9-12 (Specifically in versions: 2014 - 2015, 2015 - 2022 (current), 2022 and beyond)
1207310: Liberal Arts Mathematics (Specifically in versions: 2014 - 2015, 2015 - 2022 (course terminated))
1206330: Analytic Geometry (Specifically in versions: 2014 - 2015 (course terminated))
1200700: Mathematics for College Algebra (Specifically in versions: 2014 - 2015, 2015 - 2022 (current), 2022 and beyond)
7912090: Access Algebra 1B (Specifically in versions: 2014 - 2015, 2015 - 2018, 2018 - 2019, 2019 - 2022 (current), 2022 and beyond)
1200315: Algebra 1 for Credit Recovery (Specifically in versions: 2014 - 2015, 2015 - 2022 (current), 2022 and beyond)
1200385: Algebra 1-B for Credit Recovery (Specifically in versions: 2014 - 2015, 2015 - 2022 (current), 2022 and beyond)
7912100: Fundamental Algebraic Skills (Specifically in versions: 2013 - 2015, 2015 - 2017 (course terminated))
7912075: Access Algebra 1 (Specifically in versions: 2014 - 2015, 2015 - 2018, 2018 - 2019, 2019 - 2022 (current), 2022 and beyond)

## Related Access Points

Alternate version of this benchmark for students with significant cognitive disabilities.
MAFS.912.N-RN.2.AP.3a: Know and justify that when adding or multiplying two rational numbers the result is a rational number.
MAFS.912.N-RN.2.AP.3b: Know and justify that when adding a rational number and an irrational number the result is irrational.
MAFS.912.N-RN.2.AP.3c: Know and justify that when multiplying of a nonzero rational number and an irrational number the result is irrational.

## Related Resources

Vetted resources educators can use to teach the concepts and skills in this benchmark.

## Formative Assessments

Sum of Rational Numbers:

Students are asked to define a rational number and then explain why the sum of two rational numbers is rational.

Type: Formative Assessment

Sum of Rational and Irrational Numbers:

Students are asked to describe the difference between rational and irrational numbers and then explain why the sum of a rational and an irrational number is irrational.

Type: Formative Assessment

Product of Rational Numbers:

Students are asked to define a rational number and then explain why the product of two rational numbers is rational.

Type: Formative Assessment

Product of Non-Zero Rational and Irrational Numbers:

Students are asked to describe the difference between rational and irrational numbers, and then explain why the product of a non-zero rational and an irrational number is irrational.

Type: Formative Assessment

## Lesson Plans

Rational and Irrational Numbers 2:

This lesson unit is intended to help you assess how well students reason about the properties of rational and irrational numbers. In particular, this unit aims to help you identify and assist students who have difficulties with:

• Finding irrational and rational numbers to exemplify general statements.
• Reasoning with properties of rational and irrational numbers.

Type: Lesson Plan

Rational and Irrational Numbers 1:

This lesson is intended to help you assess how well students are able to distinguish between rational and irrational numbers. In particular, it aims to help you identify and assist students who have difficulties in:

• Classifying numbers as rational or irrational.
• Moving between different representations of rational and irrational numbers.

Type: Lesson Plan

Calculating the Square Root of 2:

This task is intended for instructional purposes so that students can become familiar and confident with using a calculator and understanding what it can and cannot do. This task gives an opportunity to work on the notion of place value (in parts [b] and [c]) and also to understand part of an argument for why the square root of 2 is not a rational number.

Operations with Rational and Irrational Numbers:

This task has students experiment with the operations of addition and multiplication, as they relate to the notions of rationality and irrationality.

## Unit/Lesson Sequence

Sample Algebra 1 Curriculum Plan Using CMAP:

This sample Algebra 1 CMAP is a fully customizable resource and curriculum-planning tool that provides a framework for the Algebra 1 Course. The units and standards are customizable and the CMAP allows instructors to add lessons, worksheets, and other resources as needed. This CMAP also includes rows that automatically filter and display Math Formative Assessments System tasks, E-Learning Original Student Tutorials and Perspectives Videos that are aligned to the standards, available on CPALMS.

Learn more about the sample Algebra 1 CMAP, its features and customizability by watching the following video:

### Using this CMAP

To view an introduction on the CMAP tool, please .

To view the CMAP, click on the "Open Resource Page" button above; be sure you are logged in to your iCPALMS account.

To use this CMAP, click on the "Clone" button once the CMAP opens in the "Open Resource Page." Once the CMAP is cloned, you will be able to see it as a class inside your iCPALMS My Planner (CMAPs) app.

To access your My Planner App and the cloned CMAP, click on the iCPALMS tab in the top menu.

All CMAP tutorials can be found within the iCPALMS Planner App or at the following URL: http://www.cpalms.org/support/tutorials_and_informational_videos.aspx

Type: Unit/Lesson Sequence

## MFAS Formative Assessments

Product of Non-Zero Rational and Irrational Numbers:

Students are asked to describe the difference between rational and irrational numbers, and then explain why the product of a non-zero rational and an irrational number is irrational.

Product of Rational Numbers:

Students are asked to define a rational number and then explain why the product of two rational numbers is rational.

Sum of Rational and Irrational Numbers:

Students are asked to describe the difference between rational and irrational numbers and then explain why the sum of a rational and an irrational number is irrational.

Sum of Rational Numbers:

Students are asked to define a rational number and then explain why the sum of two rational numbers is rational.

## Student Resources

Vetted resources students can use to learn the concepts and skills in this benchmark.

Calculating the Square Root of 2:

This task is intended for instructional purposes so that students can become familiar and confident with using a calculator and understanding what it can and cannot do. This task gives an opportunity to work on the notion of place value (in parts [b] and [c]) and also to understand part of an argument for why the square root of 2 is not a rational number.

Operations with Rational and Irrational Numbers:

This task has students experiment with the operations of addition and multiplication, as they relate to the notions of rationality and irrationality.

## Parent Resources

Vetted resources caregivers can use to help students learn the concepts and skills in this benchmark.