Grade 8 Mathematics Module 7, Topic B, Overview

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In Topic B, students learn that to get the decimal expansion of a number (8.NS.A.1), they must develop a deeper understanding of the long division algorithm learned in Grades 6 and 7 (6.NS.B.2, 7.NS.A.2d).  Students show that the decimal expansion for rational numbers repeats eventually, in some cases with zeros, and they can convert the decimal form of a number into a fraction (8.NS.A.2).  Students learn a procedure to get the approximate decimal expansion of numbers like the square root of 2 and the square root of 5 and compare the size of these irrational numbers using their rational approximations.  At this point, students learn that the definition of an irrational number is a number that is not equal to a rational number (8.NS.A.1).  In the past, irrational numbers may have been described as numbers that are infinite decimals that cannot be expressed as a fraction, like the number pi.  This may have led to confusion about irrational numbers because until now, students did not know how to write repeating decimals as fractions and further, students frequently approximated pi using 22/7 leading to more confusion about the definition of irrational numbers.  Defining irrational numbers as those that are not equal to rational numbers provides an important guidepost for students’ knowledge of numbers.  Students learn that an irrational number is something quite different than other numbers they have studied before.  They are infinite decimals that can only be expressed by a decimal approximation.  Now that students know that irrational numbers can be approximated, they extend their knowledge of the number line gained in Grade 6 (6.NS.C.6) to include being able to position irrational numbers on a line diagram in their approximate locations (8.NS.A.2). 

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Common Core Learning Standards

CCLS State Standard
8.NS.1 Know that numbers that are not rational are called irrational. Understand informally that every...
8.NS.2 Use rational approximations of irrational numbers to compare the size of irrational numbers, locate...
8.EE.2 Use square root and cube root symbols to represent solutions to equations of the form x2 = p and x3...

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