Tasks that ask students to label rational number points on a number line are common not only in curricula in the upper elementary school grades but also on state assessments. Such tasks target foundational rational number concepts: A fraction (or a decimal) is more than a shaded part of an area, a part of a pizza, or a representation using base-ten blocks; a fraction (or a decimal) is also a number with a specific location on a number line. These concepts are described as core content in "Curriculum Focal Points" (NCTM 2006) and in the Common Core State Standards (2010). To explore the nature of students' difficulties when labeling rational number points on a number line, the author interviewed students in an urban school district in Northern California. The protocol she designed included a series of number line tasks asking students to label marked points on a number line as fractions and decimals. More students appropriately labeled points on the number line as decimals than as fractions, and more students appropriately labeled points when the interval from zero to one was equally partitioned than when the interval was unequally partitioned. An analysis of students' incorrect answers and their verbal reasoning revealed four common error types characterizing' incorrect answers: (1) Using unconventional notation; (2) Redefining the unit; (3) A two-count strategy focusing on discrete tick marks (or parts) rather than distances; and (4) A one-count strategy focusing on discrete tick marks (or parts) rather than distances. Instructional implications of this study are discussed. (Contains 13 figures.)