Introduction: This lesson plan uses the concept attainment strategy. It guides learners to the important property that the values of a rational function whose numerator and denominator have the same degree essentially is single value outside a large interval. In essence, the drama of these special rational functions is confined to the finite interval. Beyond that interval, the function is basically constant. A rational function is the quotient of two polynomials. So, this is not a trivial function nor is this an expected answer.
The content attainment strategy employs a two-column table which contains examples that help learners to understand the concept being studied. The first column is the YES column. It contains examples of the concept being studied. Learners master the concept by observing the characteristics of these YES examples. The second column is the NO column. It contains non-examples of the concept. These examples are helpful to refine and to distinguish incorrect features that a learner may harbor. In other words, they limit the scope of the concept.
Content attainment strategy requires learners to view an example of the concept. And to use the characteristics in the example to formulate a hypothesis of the concept. Additional examples are presented to give learners further opportunities to refine their hypotheses or adopt new hypotheses. After teacher confirms the correctness of a hypothesis, learners test their hypothesis through another set of examples. Then they explain how they arrived at the correct hypothesis. This may be a written or verbal explanation.
Subject Area: Pre-Calculus
Specific Content: Rational Functions
Grade Level: High school senior
Length of Lesson:2 class periods
Instructional Objective(s): Upon completion of the lesson, the learner will state and be able to graph the behavior of large values of the independent variable of a rational function when the numerator and denominator have the same degree.
Subsequent Lesson : Discuss the behavior of a rational fraction for large values of the independent variable when the degree of the denominator is greater than the degree of the numerator. Offer likely models for their graph.
Model of Teaching: Concept Attainment
Prerequisite Knowledge or Behaviors Needed:
Skills: Experience with Concept Attainment
Graphing interpretations of graphs
Concepts: Functions, independent variable, polynomials, rational functions, interval, outside of an interval on a line.
Raise hands to comment to the lesson.
Work with partners and in table groups
Why is the Content of Today’s Lesson Relevant for Your Students?
Understanding the behavior of a rational functions for large values of the independent variable means the function is known by its behavior within an interval. For this class of rational functions, the lesson teaches that the values of the function for large absolute values of the independent variable approximate one number. The lesson reveals how to obtain that number.
The table of YES/NO examples are posted here.
Slide 1. Title
Slide 2. First Example: Study then formulate a hypothesis about the behavior of the
function for large values of the independent variable.
Slide 2. Confirm your hypothesis. Comment?
Slide 3. Confirm your hypothesis. Comment? Change your hypothesis.
Slide 4. Confirm your hypothesis.
Slide 5. Confirm with the teacher.
Slide 8. Test your hypothesis.
Procedures: List of procedure according to stages of Concept Attainment
1. Review of rational functions
2. Objective of the lesson explained
3. First set of YES/NO examples – several offer
4. Student speak with a partner
5. First hypothesis offered in the group : comments?
6. Additional YES/NO examples
7. Additional comments on first hypothesis
8. Additional YES/NO examples
9. New hypotheses?
11. Additional YES/NO examples
12. Elicit thinking, comments on responses
13. Clarify vocabulary as needed based on student responses
Questions you plan to ask students in the appropriate place in your lesson plan. (Questions 1 through 6 are repeated throughout the lesson.)
1. What is your hypothesis?
2. Why do you think this is so
3. What do you think about what was just said
4. Do you see anything new in these examples?
5. Can you draw your path to help explain your thinking?
6. How can you explain your thinking?
7. Can you provide some new examples of YES/NO numbers?
Closure: 1. Tomorrow student must be prepared to:
1. Given five examples of the concept learned today and,
2. Discuss the behavior of rational functions whose denominator have larger degree than its numerator outside large intervals.
How are learning styles addresses during this lesson? Describe all that apply.
Visual Recording examples on board
Auditory Verbal discussion and presentation
Kinesthetic Scratch paper available for thoughts and notes
What tangible evidence will demonstrate your students’ learning today?
1. Write how the lesson it unfolded to them.
2. Give answers to question about the property
3. Draw partial graphs for large values of rational functions whose numerator and denominator have the same degree.
As homework, they will use the YES/NO examples used in class and add 5-6 more.
What will be considered quality work?
1. Students give an accurate recounting of today’s class process and provide correct responses to questions about the property.
2. Students working in groups will provide opportunities for partners to best learn material during discussions.
3. Lesson assessment will also be done in partners as students backtrack their thinking during the lesson.
1. Students with special needs shall have
2. Examples with large print,
3. A reader if necessary,
4. Examples in colors that aid learning.
ESL student in Spanish will be encouraged to contribute and to explain his mathematical reasoning in English-with the help of a bilingual person if needed.