Objective:
This
databased demo illustrates how linear functions can arise from real problems.
This demo makes use of a graphing calculator for data analysis.
Level: College
Algebra, Precalculus or any course in which modeling is incorporated.
Prerequisites:
It
would be helpful for students to have a knowledge of linear equations,
however, the demo could be used to motivate the study of linear functions.
Students should have basic graphing calculator
skills.
Materials:

paper cups

a sharp instrument for making a hole in the cup (needle, pin,
sharp pencil)

graduated cylinder, calibrated in mm or ounces

watch or clock with second hand

a blank table to record data

graphing calculator
Instructor's Notes:

Instructor selects two student assistants.

One assistant will hold the cup over the graduated
cylinder, fill the cup with water, makes a hole in the bottom of the cup,
and holds her finger over the hole until the instructor starts the experiment.

When the student starts the drip, the instructor
calls five second intervals and the second assistant records the amount
of water collected in the cylinder.

After the cup is empty, the data that was
collected is analyzed graphically.

In this sample data, the TI83 was used to analyze the
data. Instructions for calculator input/analysis may be different if a
different calculator is utilized.
Sample Data:
Time 
0

5

10

15

20

25

30

35

40

45

50

55

60

Amount(mm) 
0

7

15

21

28

34 
42

50

56

63

70

75

81

Calculator Input (TI83) for Sample Data:
In the STAT menu, choose EDIT
and enter the data in L1 and L2.
Scatterplot of Sample Data:
In the STATPLOT menu, turn on Plot
1, choose a scatterplot using L1 and L2. The window settings for
this data should be
Pressing GRAPH will show the following scatterplot.

Many students are surprised to see that the
data is linear. Questions about why it is linear are appropriate.

Discuss how the model can be used to predict
how much water a leaky faucet might waste in 1 hour, 2 hours, 24 hours.

Depending on the data collected, the scatterplot
may give the appearance of being "perfectly" linear. This scatterplot
does give that appearance. One way to demonstrate that not all points
lie on a line is to have the calculator graph the regression line on top
of the scatterplot. Even in a class where linear regression isn’t
a topic, you can use the command LinReg L1,L2, Y1 to get a regression
line and discuss the attributes of the line and the data without going
into the details of linear regression.

If this demo is used as a lab project students will probably
observe that different experiments produce different mathematical models because
the slope of the data line is dependent upon the size of the hole. Before
telling the students why the slopes are different, ask them to speculate on why
results from different experiments are not the same. You may instruct one
group to compare their experiment with that from another group.
This graph shows that not all points are
included on the line and how our model isn’t a 100% guarantee of accuracy.
Credits:
This demo was submitted by
Dr.
Kathleen Cage Mittag
Department of Mathematics and Statistics
University of Texas at San Antonio
Dr.
Sharon Taylor
Mathematics and Computer Science Department
Georgia Southern University
and is included in Demos
with Positive Impact with their permission.
