In this lesson, students explore finding decimal expansions of rational numbers as well as irrational numbers. Students begin by using long division to find the decimal expansion of , which starts to repeat after the seventh decimal place. Repeated reasoning allows students to stop the long division process and express the decimal expansion following the same steps (MP8).

Next, students learn how to take a repeating decimal expansion and rewrite it in fraction form by ordering a set of cards that show the steps and an explanation of the process. They then practice with different decimal expansions following the same steps (MP7).

In the last activity of this lesson, students investigate how to approximate decimal expansions of irrational numbers. Since values like and cannot be written as a fraction, students use successive approximations to find more and more digits of their decimal expansions. Students see that there is no easy way to keep zooming in on these irrational numbers since they are not predictable like repeating decimals, so we use symbols to name them. However, in practice, using approximations is usually good enough for a given purpose.