Project Overview

Student Learning Objectives

Procedure

Content Material

Assessment

Links to Course Competencies

Supplementary Resources

Recommendations

 

 

Real World Learning Objects: Mathematics

Rate of Change in Temperature

Procedure

 Time:

Day 1: Approximately 10 minutes.

Day 2: Approximately 50 - 60 minutes.

 

Materials:

 

1.      Each student needs:

a.      A TI or equivalent graphing calculator with statistical capabilities.

b.      A sheet of graph paper. 

2.      The instructor could have:

a.      A TI or equivalent graphing calculator with statistical capabilities.

b.      A View Screen attachment for the overhead projector.  

3.      The instructor needs small prizes for correct predictions.  

 

Prerequisites:

 

This RWLO assumes that the student knows:

1.      Trigonometric differentiation.

2.      How to use the following features of a graphing calculator:

a.      Statistical features such as:  STAT, CALC, SinReg, and STAT PLOT

b.      Graphical features such as: Y=, GRAPH, ZOOM, ZoomFit, and TRACE.   

 

Implementation:

 

This RWLO can be used either in the classroom or initiated in the classroom and completed as a homework assignment in order to demonstrate the learning objectives.  Students will form groups and work collaboratively on an instantaneous rate of change word problem.  Each student will submit his/her own work that includes a chart with information from the Web-site, equations for a regression function and its derivative, graphs and plotted points on graph paper, and answers to calculus questions with mathematical reasoning to support their answers.  Students will make predictions at the onset and compare their predictions with actual results.  Prizes will be awarded to the students who correctly predicted all the months.

A graphing utility is needed for the RWLO.  There are many options for a graphing utility.  A computer with symbolic software or any graphing calculator with statistical features is appropriate.  Since more students have access to a graphing calculator, the RWLO is written with some suggested features for those students using a TI – graphing calculator with statistical features.  This can easily be modified by any instructor according to their classroom situation.           

 

 

Steps:

 

Day 1:

 

1.      Have students form groups of three or four.

2.      Distribute the “Content Material” section of this project to each student.

3.      Direct each group to complete the first two steps.    

4.      Collect the predictions and rationale before the students leave class.

5.      Assign step # 3 in the “Content Material” for homework.

 

Day 2:

 

6.      Direct students to use the information from the internet and their graphing calculator to complete step # 4 of the “Content Material”.  Inform students to use teamwork by working collaboratively to help each other with the calculator.  After a team had made this effort, assist the students with the use of their calculators. 

7.      Groups should proceed through steps # 5 and # 6 of the “Content Material”.  Assist students to insure that they have both the regression function and the derivative function visible on the calculator screen.

8.      Allow ample time for steps # 7, # 8, and # 9 of the “Content Material”.  Don’t rush the analysis.  This is the most crucial step in the process.  Have students support their answers with proper mathematical reasoning.  This should include a discussion about the derivative.

9.      Direct students to compare the predictions they made in the beginning with the actual real results found through mathematics.  Ask the students “How accurate were your predictions when using non-mathematical reasoning?  

10. Collect each student’s work.  Emphasize the mathematical reasoning used to interpret the graph of the derivative to find the actual results.

11. Give a small reward to all students who predicted all the months accurately.  It is necessary to verify the students’ answers.  This can be done quickly by checking:

a.      The largest y-value of the first derivative function.  This is where the temperature is increasing most rapidly, the greatest rate of change in the positive direction.  Use the Trace feature to find this point and read the x-value.   The x-value represents the month.  Example: If x = 4, then April is the month where the temperature increases most rapidly.  This is true for Mt. Laurel, New Jersey.  

b.      The smallest y-value of the first derivative function.  This is where the temperature is decreasing most rapidly, greatest rate of change in the negative direction.  For Mt. Laurel, New Jersey, x = 10.  Hence, October is the month where the temperature decreases most rapidly. 

c.      The zeros of the first derivative function.  This is where the y-value is 0 which indicates the temperature is the slowest, little or no rate of change. 

For Mt. Laurel, New Jersey, January and July are the months where the temperature changes most slowly.