Students will conduct the infectious disease experiment as an activity to collect data. The data will be entered into the TI-83 calculator. Students will find the equation that best-fits the data, the derivatives and the point of inflection of the data.
Students prior to this have learned the first derivative test and the second derivative test to find the point of inflection.
This activity will relate a real-life situation to the equations discussed above. The cross-curricular connecton is biology.
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
Understand and apply basic notions of chance and probability.
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.
Students can find the first and second derivative mannually,
Students can enter data into a TI-83 calculator, find the best-fit graph using the regression functions and find the derivatives,
Students can do the first and second derivative tests.
Lesson 1 must be done after students have already mastered entering data into a TI-83 calculator and using the regression function to find the best-fit equation.
This lesson will also follow the lessons from the text introducing the first and second derivative tests.
The experiment in its entirety is in the CHEM/BIO 1999 participant packet.
Students results from the introductory activity and the experiment will be recorded on paper. These results will be graded on the students ability to enter the data, find the graph, derivatives and point of inflection.
Students will be asked to copy data from overhead and enter in calculator as they enter the room.
sample data: (3,7) (5,11) (6,13)
Students should use a linear regression formula to find the equation. The purpose of this activity is to review these skills.
Infectious disease experiment:from Chem/Bio packet from TI
Purpose: To simulate an outbreak of an infection disease using the randInt function of the TI-83. The number of people infected will be plotted vs. time to produce a logistic curve.
Procedure: Students will count off and remember their number.
The randInt button will generate integers between 1 and the total number of participants. The first time only one number will be generated. That person will stand. A second number is generated and two people stand. Two numbers are generated and two more are infected. There may be 4 at this time unless an earlier number appeared. The number of students standing determines how many numbers are generated or how many new people are infected. Continue until all are standing. Keep track of data as following example:
Round no of sick
Students will enter this data and enter into Stat Plot 1, press zoom 9, press stat calc and enter into the Logistic function. Students will look at the graph and determine what appears to be the point of inflection by determining where the graph appears to change from concave up to concave down.
Students can now press trace and follow the cursor to that point to estimate the x value. Students will use the key nDeriv to graph the first derivative. Students describe the relationship between this graph and the estimated point of inflection. Students find the graph of the second derivative and describe the relationship between it and the first and between it and the estimated point of inflection.
Students must be able to enter data into a TI-83 and use the regression function to find the best-fit graph and equation.
Students must be able to find the graphs of the derivatives on the TI-83.