Defining Functions Recursively

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Objective

SWBAT recognize a pattern in a situation involving bivariate data. SWBAT demonstrate understanding of recursive formulas.

Big Idea

A productive understanding of recursive definitions for functions provides a strong foundation for learning to write explicit formulas.

Do Now

10 minutes

Slide 1: This skill building time is strongly related to the work that students will be doing throughout the unit with evaluating functions at various values of x.  The exercises in this “do now” tie back to the order of operations but also can be used to survey student understanding of various mathematical symbols such as exponents, absolute value, and the operations.

Opening

10 minutes

Direct Instruction

5 minutes

Slide 4: There are two ways to approach this slide…you can go through the prompts one by one to help students build an understanding.  I like to put all four prompts up and see what students can come up with in their partnerships and then discuss possible solutions afterwards.  It is interesting for students to see that if different groups start at a different “NOW” spot that the “NEXT” will be the same regardless.  The term recursive is a vocabulary term that students should become familiar with.  However, the term recursive really doesn’t sink in for students until they have something to contrast it with, namely and explicit or general formula that will come in subsequent lessons.

Investigation

20 minutes

 Using some form of technology (laptops, computer lab, tablets, etc.) students can investigate the recursive online activity.  This activity walks students though several recursive tables and helps students develop an understanding of the notation involved in the recursive formula.  I like this applet because students can tell if they are filling in the answers correctly (cells turn green) and there is an opportunity for students to determine how much practice they need and continue to try more questions if they choose to.  During this time, the teacher role becomes one of a facilitator who is helping students navigate their way through the applet.  Success with this applet is very much dependent on student’s number sense and reasoning and so I am on the lookout for opportunities to pair up stronger and weaker students during this investigation so that a stronger student can help the weaker student see some of the patterns that they may be missing.  Often explaining the recursive formula to another student will help strengthen both students understanding of the concept.

Closure

5 minutes

 In this ticket out students will be asked to write at least one full sentence for each question.  The goal of this assessment is to determine if students can understand the meaning of the function in context (for example, the rule NEXT =  NOW + 9 would mean that Stephanie earns $9 every week.  Or that START = 36 means that after the first week Stephanie had $36).  In this table I also ask about how much she had to begin with before she started receiving the allowance $27 to plant the seed of an initial value f(0) which represents the y-intercept in functions that will be studied later.