Lesson 2: Applications of Recursion
Grade Levels: Appropriate for grades 9-12 in courses such as Discrete Mathematics, Advanced Functions and Modeling (AFM) and Algebra II. This lesson is the second in a series that covers the topic of recursion. It builds on the concepts introduced and practiced in the first lesson on this topic and includes some larger group projects for students to apply their understanding of recursion. Throughout the activities in this lesson, students are encouraged to think independently and build their problem-solving skills through discussion with their classmates. Teachers are encouraged to promote an environment that facilitates this type of discussion and to allow time for students to develop their thoughts and ideas during class.
A lesson plan for Applying Recursion to Larger Problems
Learning Outcomes
Students will further apply their understanding of recursion to real-world problems. They will answer questions using recursive systems and use technology to obtain visual representations of these systems.
Teacher planning
The resources listed below include a teacher handout that provides step-by-step instructions on how to develop explicit models from recursive systems.
TIME REQUIRED FOR LESSON
5-6 days
MATERIALS/RESOURCES
- Loan Problems Sheet Download Loan Problems Sheet
- Loan Problem Solutions Download Loan Problem Solutions
- Tim & Tom Problem Download Tim & Tom Problem
- Tim & Tom Problem Solution Download Tim & Tom Problem Solution
- Tim & Tom Rubric Download Tim & Tom Rubric
- Great Lakes Problem Download Great Lakes Problem
- Great Lakes Problem Solution Download Great Lakes Problem Solution
- Great Lakes Problem Rubric Download Great Lakes Problem Rubric
- Tips for Conducting Open-Ended Problems Sheet Download Tips for Conducting Open-Ended Problems Sheet
- Developing Explicit Models Download Developing Explicit Models
- Creating Exponential Models Problems Sheet Download Creating Exponential Models Problems Sheet
- Creating Exponential Models Problems Solutions Download Creating Exponential Models Problems Solutions
TECHNOLOGY RESOURCES
- TI-83 or TI-84 calculator
- Excel spreadsheet program
Pre-activities
Teacher will have covered the concepts and problems in Lesson 1: Introduction to Recursion of this unit.
Activities
DAY 1
- Teacher begins the lesson by presenting #1 on the Loan Problems sheet. The teacher leads the students through the problem in a whole-class discussion. (30 minutes)
- Teacher has the students work in groups on #2 on the Loan Problems sheet. (30 minutes)
- The class reconvenes to discuss their solutions to the Loan Problems sheet. (30 minutes)
DAY 2
- Teacher discusses the concept of Geometric Growth using the problems they worked in Lesson 1. (30 minutes)
- Teacher presents students with the Tim and Tom Problem, and they begin to work on it in pairs. Students turn in their solutions at the end of class. (60 minutes)
DAYS 3-4
- Teacher presents students with the Great Lakes Problem, and students work together in pairs for the next two days. They will turn in their projects at the conclusion of Day 4. (2 class days)
DAY 5
- Teacher begins class by revisiting the problems worked in Lesson 1 that were examples of exponential growth. (20 minutes)
- Teacher leads students through a lesson on developing exponential functions from recursive systems. (For teachers’ reference, Developing Explicit Models provides step-by-step instructions on how recursive models can be represented by explicit, continuous models.) They cover #1 and 2 on the Creating Exponential Models Problems Sheet together. (40 minutes)
- Students work on the remaining problems on the Creating Exponential Models Problems Sheet together in groups of 3 and 4. (30 minutes)
DAY 6
- Teacher leads a class discussion on student solutions to the remaining problems on the Creating Exponential Models Problems Sheet. (20 minutes)
Assessments
The following assessments are designed to be worked on in groups of 2-3 students. Solutions and rubrics for each of the problems are provided on this site for teachers.
Common Core Standards alignment
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A-SSE.1. Interpret expressions that represent a quantity in terms of its context.
- Interpret parts of an expression, such as terms, factors, and coefficients.
- Interpret complicated expressions by viewing one or more of their parts as a single entity.
- A-CED.1. Create equations and inequalities in one variable and use them to solve problems.
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F-BF.1. Write a function that describes a relationship between two quantities.
- Determine an explicit expression, a recursive process, or steps for calculation from a context.
- F-BF.2. Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.
- F-IF.1. Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range.
- F-IF.2. Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
- F-IF.3. Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers.
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F-LE.1. Distinguish between situations that can be modeled with linear functions and with exponential functions.
- Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.
- Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.
- Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.
- F-LE.2. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).
- F-LE.5. Interpret the parameters in a linear or exponential function in terms of a context.