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 Website, 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 nonmathematical 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
yvalue 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 xvalue. The xvalue 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
yvalue 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 yvalue 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.
