Inquiring Minds

Those pesky recursive functions

NC.M1.F-BF.1a has a nice table that shows the various options:

These functions can be written in function notation (linear or exponential) or as a sequence in explicit or recursive form. Students should recognize the explicit form of an arithmetic sequence as an equivalent structure to the slope-intercept form of a linear function and explicit form of a geometric sequence as an equivalent structure to the standard form of an exponential function. Using the concepts of rate of change, students should recognize that the forms of these sequences are one iteration forward from the y-intercept, which gives meaning to the 𝑛 − 1 notation. 

Students should be familiar with both function and sequence notations for defining a sequence recursively. 

NC.M1.F-BF.2 goes on to clarify the use of formal function and sequence notations:

Students should be able to use both the explicit and recursive forms of arithmetic and geometric sequences where the explicit form is a linear or exponential function, respectively. Students are expected to use formal notation (function or sequence notation, see NC.M1.F-BF.1a). Use of the informal notation, NEXT-NOW, can be used to build conceptual understanding of recursive functions, however the expectation is for students to know and use formal notation. Students should recognize explicit form of an arithmetic sequence as an equivalent structure to slope-intercept form of a linear function and explicit form of a geometric sequence as an equivalent structure to standard form of an exponential function. Using the concepts of rate of change, students should recognize that the forms of these sequences are one iteration forward from the y-intercept, which gives meaning to the n-1 notation. 

The unpacked standards use a n-1 exclusively. Joe Reaper invited us to submit other Checking for Understanding examples using f(n-1) for consideration, but said DPI could not include each and every possible iteration. 

It is curious that the resources I use most do not offer flexibility. They pick one method and stick with it. OpenUp Resources strictly uses f(n-1) while MathBits uses an-1 . While both representations are useful, Math 1 students (especially 8th grade Math 1 students) really struggle with understanding the function notation, especially the meaning of f(n-1). Each year I strive to improve my delivery of this concept to learners. This year I use a number line to help locate n as well as n+1 and n-1 and beyond. This helps some students, but is still far from perfect. 

How do you teach writing functions recursively to ensure your students have solid understanding? Please share!! 

Would you like to write an article for Math Matters? Please submit your article for consideration to ncctmcentral@gmail.com .