In many of the standard first-year calculus books (e. g., Stewart, Larsen, Hughs-Hallett, Tan),
there is a common exercise that asks students to draw a rough sketch of the depth of water in a vase that is being filled
at a constant rate. We have given this exercise to our students as a class activity and also as a problem on a test.
Our students struggle to visualize the rate of change of the depth of the water and most of the time they
draw an incorrect graph. To help our students better visualize this situation, we decided to video a clear
glass vase being filled with water. By observing this video and its subsequent analysis, students will develop
an improved intuition and understanding of rates of change.
We also wanted our students to check how well their
hand-drawn graphs of the depth of water over time models the actual experiment. To do this, we used the software
package Graphic Converter to convert the video into a sequence of individual frames equally spaced in time.
This sequence of digital images was then imported into the TEMATH software package and the first image was positioned over
a pair of coordinate axes (red food coloring was added to the water to make the water level more visible).
Using TEMATH's Point tool, students can mark the depth of the water by
- Clicking a coordinate point at the top edge of the water level, measuring the depth of the water.
- Next, they can move frame by frame through all the digital images and mark the depth of the water
as a function of time.
- Placing the measured depths are then into a table and plotted as a function of time.
Students now have the opportunity to observe the plotted data points
and determine the important characterisitics of the plot. For example, they can easily relate the concavity of the
depth curve and the position of the points of inflection to the shape of the vase.
