Building on the informal language students have used so far in the unit, the purpose of this lesson is for students to understand that sequences are functions and to use function notation when defining them with equations.

For example, in previous lessons students described the arithmetic sequence 30, 40, 50, . . . as a sequence that starts at 30 and in which each term is 10 more than the previous term. Now they think of it as a function of the position, starting at position 1, and write and for , where is an integer. This is called a recursive definition for because it describes a repeated, or recurring, process for getting the values of , namely, the process of subtracting 3 each time. Students will use recursive definitions to describe functions in both abstract and real-world contexts throughout the remainder of this unit. It is not necessary that students use the term "recursive definition," however.

In the Warm-up, students make sense of how a dot pattern is changing (MP1) in preparation for seeing the pattern as a function in which the number of dots in each step depends on the step number. Next, students learn to write a recursive definition for the function by expressing regularity in repeated reasoning (MP8) while using a table. This lesson also invites students to consider what values make sense for the domain of the functions, which leads to expanding their definition of a sequence to a function whose domain is a subset of the integers. Lastly, students return to sequences they have seen previously in the unit and practice defining them recursively using function notation.

Technology isn't required for this lesson, but there are opportunities for students to choose to use appropriate technology to solve problems, such as by using a spreadsheet to generate sequences. We recommend making technology available.

Math Community
Today’s community building centers on the teacher sharing their draft commitments as part of the mathematical community. At the end of the lesson, students are invited to suggest additions to the teacher sections of the chart.