Number Theory

In this chapter, we deepen our study of numbers and number systems. We first look into  how the counting numbers decompose through multiplication (or division). This leads us  to ideas about factors, multiples, and prime numbers. We then look into decimal representations of fractions. We’ll see that decimal representations of fractions are of a special  type, so that the numbers that have decimal representations form a larger system of numbers  than the fractions do.  We focus on the following topics and practices within the Common Core State Standards  for Mathematics.  Standards for Mathematical Content in the CCSSM  In the domain of Operations and Algebraic Thinking (Kindergarten–Grade 5), students  learn about even and odd numbers, about factors and multiples, and about prime numbers. In the domain of The Number System (Grades 6–8), students work with common  factors and common multiples, including greatest common factors and least common  multiples. They also learn that there are irrational numbers, that fractions have decimal  representations that eventually repeat (or terminate with repeating zeros), and they learn  how to express repeating decimals as fractions. 

Standards for Mathematical Content in the CCSSM  

In the domain of Operations and Algebraic Thinking (Kindergarten–Grade 5), students  learn about even and odd numbers, about factors and multiples, and about prime numbers. In the domain of The Number System (Grades 6–8), students work with common  factors and common multiples, including greatest common factors and least common  multiples. They also learn that there are irrational numbers, that fractions have decimal  representations that eventually repeat (or terminate with repeating zeros), and they learn  how to express repeating decimals as fractions. 

Standards for Mathematical Practice in the CCSSM  

Opportunities to engage in all eight of the Standards for Mathematical Practice described  in the Common Core State Standards occur throughout the study of number theory,  although the following standards may be especially appropriate for emphasis:  

• 3 Construct viable arguments and critique the reasoning of others. Students  engage in this practice when they explain how they know they have found all the  factors of a number or why a given number must be prime.  

• 7 Look for and make use of structure. Students engage in this practice when  they look for how least common multiples and greatest common factors apply to  contexts such as gears, spirograph flower designs, and musical rhythms.  

• 8 Look for an express regularity in repeated reasoning. Students engage in this  practice when they see that decimal representations of fractions repeat because of  the repeating remainders that occur in the standard division algorithm or when they  work through multiple explanations for why 0.9999... = 1. 

 (From Common Core Standards for Mathematical Practice. Published by Common  Core Standards Initiative.)