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Point of inflection of a logistic curve
Michael Cain
Mathematics




Unit created on 8/2/1999 EST.
Last modified 11/12/1999 3:06:19 PM EST.


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Fundamental Understandings (help)


Students will able to find the point of inflection of a logistic curve using the first and second derivative. The curve will be created using data gathered by the student and entered into the regression function or the CMBDERIV program of the CHEM/BIO application on the TI graphing calculator. One method of gathering data will involve an experiment using the CBL.


Technology ISTE Standards (info) 


Basic operations and concepts Identify capabilities and limitations of contemporary and emerging technology resources and assess the potential of these systems and services to address personal, lifelong learning, and workplace needs. Technology communication tools Select and apply technology tools for research, information analysis, problem-solving, and decision-making in content learning. Technology productivity tools Investigate and apply expert systems, intelligent agents, and simulations in real-world situations.

Information Literacy Standards (info) 


Category I: Information Literacy The student who is information literate: Standard 3: Uses information effectively and creatively, as described by the following indicators: 1.organizes information for practical application; 2.integrates new information into one's own knowledge; 3.applies information in critical thinking and problem solving; 4.produces and communication information and ideas in appropriate formats. Category II: Independent Learning The student who is an independent learner is information literate and Standard 6: Strives for excellence in information seeking and knowledge generation, as described by the following indicators: 1.assesses the quality of the process and products of one's own information seeking; 2.devises strategies for revising, improving, and updating self-generated knowledge. Category III: Social Responsibility The student who contributes positively to the teaming community and to society is information literate and: Standard 9: Participates effectively in groups to pursue and generate information, as described by the following indicators: 1.shares knowledge and information with others; 2.respects others' ideas and backgrounds and acknowledges their contributions; 3.collaborates with others, both in person and through technologies, to identify information problems and to seek their solutions; 4.collaborates with others, both in person and through technologies, to design, develop, and evaluate information products and solutions.

Relevance (help)


Graphs with S-shaped curves (logistic curves) are commonly found in chemistry and biology. The point of inflection of a logistic curve is called the equivalence point of the function. In biology, the logistic curves models experimental population data. Our experiment is in epidemiology. In chemistry, the change in concentration with time for certain autocatalytic reactrons is logistic. Our experiment is an acid/base titration. This unit is designed to help the student visualize the point of inflection (half-way point)

Assessment (help)


Students will be asked to enter data into the graphing calculator, use the correct regression function to created the best-fit graph, find the first and second derivative and estimate the point of inflection. Student will discuss the concavity around the point of inflection and describe the rate of the slope around that point.

Components (help)


Two days prior to this unit of study we will cover chapeter 3.3 THE FIRST DERIVATIVE TEST from the text Calculus of a Single Variable by Swokowski. One day prior to this unit of study we will cover chapter 3.4 Concavity and the Second Derivative test. Day one: Infectious disease experiment creates data to create a logistic curve. Point of inflection is found by the graphs of the derivatives. Day two: An acid/base titration experiment is done with the CBL and the Chem/Bio application to collect data to create a titration curve which is similar to a logistic curve Day three: Students will use data from the acid/base titration to find the point of inflection using the CMBDERIV program from the TI-83 and the nDeriv function.

URLs (help)


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Workforce Competencies (info) (help)


Standard Title: NUMERIC PROBLEM SOLVERS (3.3) Students use numeric operations and concepts to describe, analyze, disaggregate, communicate, and synthesize numeric data, and to identify and solve problems. Work Force Competencies Standard Title: INFORMATION MANAGERS (3.1) Students locate, comprehend, interpret, evaluate, maintain, and apply information, concepts, and ideas found in literature, the arts, symbols, recordings, video and other graphic displays, and computer files in order to perform tasks and/or for enjoyment. Standard Title: EFFECTIVE COMMUNICATORS (3.2) Students communicate in English and other languages using information, concepts, prose, symbols, reports, audio and video recordings, speeches, graphic displays, and computer-based programs. Standard Title: NUMERIC PROBLEM SOLVERS (3.3) Students use numeric operations and concepts to describe, analyze, disaggregate, communicate, and synthesize numeric data, and to identify and solve problems. Standard Title: CREATIVE AND CRITICAL THINKERS (3.4) Students use creative thinking skills to generate new ideas, make the best decisions, recognize and solve problems through reasoning, interpret symbolic data, and develop efficient techniques for lifelong learning. Standard Title: ETHICAL AND RESPONSIBLE WORKERS (3.5) Students display responsibility, self-esteem, sociability, self-management, integrity, and healthy decision-making. Standard Title: RESOURCE MANAGERS (3.6) Students appropriately allocate time, money, materials, and other resources. Standard Title: SYSTEMS MANAGERS (3.7) Students integrate their knowledge and understanding of how social, organizational, informational, and technological systems work with their abilities to analyze trends, design and improve systems, and use and maintain appropriate technology. Standard Title: COOPERATIVE WORKERS (3.8) Students work cooperatively to successfully complete a project or activity.
Lessons


Lesson 3 Find the Point of Inflection
Lesson 2 Acid/Base Titraton Experiment
lesson 1 Infectious disease experiment





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Career Connection to Teaching with Technology
USDOE Technology Innovation Challenge Grant
Marshall Ransom, Project Manager
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