Strand: NUMBER SYSTEM (8.NS)

Know that there are numbers that are not rational, and approximate them by rational numbers (Standards 8.NS.1-3).

Standard 8.NS.1

Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.

• Approximating pi
The goal of this task is to explore some important aspects of approximating an irrational number with rational numbers. The irrational number chosen here is pi because it is one of the most interesting, well known, and grade appropriate irrational numbers.
• Approximating Square Roots of Nonperfect Squares
Students will learn a strategy for how to approximate the square root of a nonperfect square in this video and classroom activity. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
• Calculating and Rounding Numbers
This task is intended for instructional (rather than assessment) purposes, providing an opportunity to discuss technology as it relates to irrational numbers and calculations in general. The task gives a concrete example where rounding and then multiplying does not yield the same answer as multiplying and then rounding.
• Calculating the square root of 2
This Illustrative Mathematics 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.
• Chapter 7 - Mathematical Foundation (UMSMP)
This is Chapter 7 of the Utah Middle School Math: Grade 8 textbook. It provides a Mathematical Foundation for Rational and Irrational Numbers.
• Chapter 7 - Student Workbook (UMSMP)
This is Chapter 7 of the Utah Middle School Math: Grade 8 student workbook. It focuses on Rational and Irrational Numbers.
• Converting Decimal Representations of Rational Numbers to Fraction Representations
This task requires students to represent several rational numbers in fraction form.
• Converting Repeating Decimals to Fractions
The purpose of this task is to study some concrete examples of repeating decimals and find a way to convert them to fractions.
• Estimating Square Roots
The purpose of this task is to have students use the meaning of a square root to find a decimal approximation of a square root of a non-square integer.
• Grade 8 Math Module 7: Introduction to Irrational Numbers Using Geometry (EngageNY)
Module 7 begins with work related to the Pythagorean Theorem and right triangles. Before the lessons of this module are presented to students, it is important that the lessons in Modules 2 and 3 related to the Pythagorean Theorem are taught (M2: Lessons 15 and 16, M3: Lessons 13 and 14). In Modules 2 and 3, students used the Pythagorean Theorem to determine the unknown length of a right triangle. In cases where the side length was an integer, students computed the length. When the side length was not an integer, students left the answer in the form ofx2=c, where c was not a perfect square number. Those solutions are revisited and are the motivation for learning about square roots and irrational numbers in general.
• Grade 8 Unit 2: Exponents and Equations (Georgia Standards)
In this unit student will distinguish between rational and irrational numbers and show the relationship between the subsets of the real number system; recognize that every rational number has a decimal representation that either terminates or repeats; recognize that irrational numbers must have decimal representations that neither terminate nor repeat; understand that the value of a square root can be approximated between integers and that nonperfect square roots are irrational; locate rational and irrational numbers on a number line diagram; use the properties of exponents to extend the meaning beyond counting-number exponents; recognize perfect squares and cubes, and understanding that non-perfect squares and non- perfect cubes are irrational.
• Identifying Rational Numbers
Given a set of numbers students must decide whether each number is rational or irrational.
• Repeating Decimal Rings
In this interactive activity you will explore the patterns that occur when expanding seventh and thirteenth fractions into decimals. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
• Repeating Decimals
This lesson unit is intended to help educators assess how well students are able to translate between decimal and fraction notation, particularly when the decimals are repeating, create and solve simple linear equations to find the fractional equivalent of a repeating decimal, and understand the effect of multiplying a decimal by a power of 10.
• Repeating or Terminating?
The purpose of this task is to understand, in some concrete cases, why terminating decimal numbers can also be written as repeating decimals where the repeating part is all 9's.

http://www.uen.org - in partnership with Utah State Board of Education (USBE) and Utah System of Higher Education (USHE).  Send questions or comments to USBE Specialist - Lindsey  Henderson and see the Mathematics - Secondary website. For general questions about Utah's Core Standards contact the Director - Jennifer  Throndsen.

These materials have been produced by and for the teachers of the State of Utah. Copies of these materials may be freely reproduced for teacher and classroom use. When distributing these materials, credit should be given to Utah State Board of Education. These materials may not be published, in whole or part, or in any other format, without the written permission of the Utah State Board of Education, 250 East 500 South, PO Box 144200, Salt Lake City, Utah 84114-4200.