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Lesson 5: Recursive De nitions
• Interpret and understand sequences de ned recursively using function notation. • Use a recursive de nition to generate the terms of a sequence.
Lesson Narrative
The purpose of this lesson is to learn what recursive notation is and how to interpret a recursively-de ned function. First, it is important for students to understand the meaning of
as it relates to This understanding is the focus of the warm-up, where students have an opportunity to express regularity in repeated reasoning (MP8) to extend their reasoning for speci c examples to any value of the input. In the rst activity, students are presented with a visual pattern and representations of it using function notation. This includes a recursively-de ned function which they interpret. Finally, students match sequences with their recursive de nitions.
Technology isn't required for this lesson, but there are opportunities for students to choose to use appropriate technology to solve problems. We recommend making technology available.
Student Learning Goals
• Let’s write recursive de nitions of sequences. 5.1 A Shrinking Function
Warm Up: 10 minutes
The purpose of this task is to understand to mean that the input to is 1 less than it is for . To accomplish this, students evaluate two pairs of such inputs at speci c values, and then express generality in repeated reasoning (MP8) to explain what happens with and for
any value of
It is not expected here, but it is possible that some students may reason this way to show that
is always 3 more than . If any students take this approach, ask them to demonstrate.
Instructional Routines
• Think pair share
What: Students have quiet time to think about a problem and work on it individually, and then time to share their response or their progress with a partner. Once these partner conversations have taken place, some students are selected to share their thoughts with the class.
Unit 1
Lesson 5: Recursive De nitions 49