Lesson 3 Find the Point of Inflection
Part of Curriculum Unit:Point of inflection of a logistic curve
Abstract help
Students will use the data gathered on the previous lesson and find the point of inflection first using the CMBDERIV program on the TI-83 and then using the best-fit curve method used in lesson one. Once again this will complete the cross-curricular connection with chemistry which was introduced in lesson two.
National Standards help
Mathematics instructional programs should include attention to data analysis, statistics, and probability so that all students
Pose questions and collect, organize, and represent data to answer those questions
Interpret data using methods of exploratory data analysis
Develop and evaluate inferences, predictions, and arguments that are based on data
Mathematics instructional programs should emphasize connections to foster understanding of mathematics so that all students
Recognize and use connections among different mathematical ideas
Understand how mathematical ideas build on one another to produce a coherent whole
Recognize, use and learn about mathematics in contests outside of mathematics.
Mathematics instructional programs should emphasize mathematical representations to foster understanding of mathematics so that all students
Create and use representations to organize, record, and communicate mathematical ideas
Develop a repertoire of mathematical representations that can be used purposefully, flexibly, and appropriately
Use representations to model and interpret physical, social, and mathematical phenomena.
Pose questions and collect, organize, and represent data to answer those questions
Interpret data using methods of exploratory data analysis
Develop and evaluate inferences, predictions, and arguments that are based on data
Mathematics instructional programs should emphasize connections to foster understanding of mathematics so that all students
Recognize and use connections among different mathematical ideas
Understand how mathematical ideas build on one another to produce a coherent whole
Recognize, use and learn about mathematics in contests outside of mathematics.
Mathematics instructional programs should emphasize mathematical representations to foster understanding of mathematics so that all students
Create and use representations to organize, record, and communicate mathematical ideas
Develop a repertoire of mathematical representations that can be used purposefully, flexibly, and appropriately
Use representations to model and interpret physical, social, and mathematical phenomena.
Pre-requisite Skills help
Use of TI-83 Calculator
How to find first and second derivatives on the TI-83
How to find first and second derivatives on the TI-83
Teacher Information help
CMBDERIV is a program which is part of the CHEM/BIO appication.
The entire procedure is on page 4-18 in the TI CHEM/BIO 1999 Participant Packet.
The entire procedure is on page 4-18 in the TI CHEM/BIO 1999 Participant Packet.
Student Activity help
Part 1:
Students will use the CBMDERIV program to determine from the sample titration data the equivalence point and concentration of a weak acid.
Students must have data from lesson 2 in lists 1 and 2 in the TI-83 calculator. Students enter the CBMDERIV program and press enter. They will be given the option to view the graph, first derivative or second derivative.
By selecting 1 students can trace along the titration graph and view the y values to estimate the point when the delta y changes from increasing to decreasing. (point of inflection)
By selecting 2 students can trace the derivative. The point of infleciton is where the derivative is at its maximum point.
By selecting 3 students can trace the second derivative. The point of inflection is where the derivative is a zero.
Part 2:
Students will find the best-fit curve and equation using the logistic regression function as in lesson one.
(note: this function wants the first y value to be near 0, therefore one must subtract the initial y value for when x=0 from all data, apply the regression and then ad the amount subtracted to the resulting equation.
Students will then discuss the point of inflection as in lesson one.
Students will use the CBMDERIV program to determine from the sample titration data the equivalence point and concentration of a weak acid.
Students must have data from lesson 2 in lists 1 and 2 in the TI-83 calculator. Students enter the CBMDERIV program and press enter. They will be given the option to view the graph, first derivative or second derivative.
By selecting 1 students can trace along the titration graph and view the y values to estimate the point when the delta y changes from increasing to decreasing. (point of inflection)
By selecting 2 students can trace the derivative. The point of infleciton is where the derivative is at its maximum point.
By selecting 3 students can trace the second derivative. The point of inflection is where the derivative is a zero.
Part 2:
Students will find the best-fit curve and equation using the logistic regression function as in lesson one.
(note: this function wants the first y value to be near 0, therefore one must subtract the initial y value for when x=0 from all data, apply the regression and then ad the amount subtracted to the resulting equation.
Students will then discuss the point of inflection as in lesson one.
Assessment help
Students will be able to use the data stored in their calculators to find the point of reflection by both methods.
At the end of the unit, students will be tested by taking a set of data, entering it on the calculator and find the point of inflection by both methods learned.
At the end of the unit, students will be tested by taking a set of data, entering it on the calculator and find the point of inflection by both methods learned.
Enrichment / Alternative Activity help
Students can use the slope formula to explore the behavior of the curve on both sides of the point of inflecton.
Technology Requirements/Integration help
Use of CBL
Use of TI-83 calculator (CHEM/BIO application, CMBDERIV program, nDeriv function and regression functions)
Use of TI-83 calculator (CHEM/BIO application, CMBDERIV program, nDeriv function and regression functions)