Mathematics Grade 8
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)
Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., π2). For example, by truncating the decimal expansion of √2, show that √2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations.
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.
Building a Number Line
This lesson and activity will examine different number sets on a number line. Students will identify and plot integers, counting numbers, rational and irrational numbers on a number line. 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.
Comparing Rational and Irrational Numbers
This task can be used to either build or assess initial understandings related to rational approximations of irrational numbers. It allows students to construct viable arguments by identifying and justifying the greater of two expressions in each part.
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.
Irrational Numbers on the Number Line
In this task students plot irrational numbers on the number line in order to reinforce the idea that they fit into a number system that includes the more familiar integer and rational numbers. This task could be used for assessment, or if elaborated a bit, could be used in an instructional setting.
Placing a square root on the number line
The purpose of the task is to make connections between the definition and properties of squares and square roots and ordering on the number line, as prescribed by standard 8.NS.2.
The Number System (8.NS) - 8th Grade Core Guide
The Utah State Board of Education (USBE) and educators around the state of Utah developed these guides for Mathematics Grade 8 - The Number System.
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
and see the Mathematics - Secondary website. For
general questions about Utah's Core Standards contact the Director
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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
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