## Flowing Through Mathematics

9-12
Standards:
Math Content:
Data Analysis and Probability

Simulate water flowing from a tube through a hole in the bottom. The diameter of the hole can be adjusted and data can be gathered for the height or volume of water in the tube at any time.

Choose a Flow Measurement of centimeters (cm) if you want to focus on the height of the remaining water, or choose mililiters (ml) if you want to focus on the volume.
• Adjust the diameter of the hole on the tube with the "Adjust Diameter" arrow buttons. The range is 0.1 to 0.9 in increments of 0.05.
• Press the Start button for water to begin flowing from the tube; press the Pause button to stop the flow before the tube empties.
• Use "Enter A Time" or "Enter A Height" to place the pointer back on the graph at the desired location. Read the time or height on the left side of the screen.
• Press the Refill button to refill the tube with water.
• Press the Clear Graph button to clear the graph of previous results.

Run the simulation several times for different size diameters for the hole. Then try to answer the following questions based on your observation or by gathering data using the simulation.
1. What do you observe about the speed at which the water is flowing at the beginning compared to at the end? What might explain the difference?
2. What do you think is the shape of the graph? Does the graph ever reach the x-axis? Explain your thinking.
3. Why does the shape for the volume (ml) vs. time graph resemble the shape for the height (cm) vs. time graph?
4. Choose a diameter and gather data about the height and time. Find a formula that predicts the height given the time. Test your formula using the simulation.
5. Gather data on the total time required to empty the tube for various diameters. Explain why the graph makes sense. Be sure to discuss what should happen to the graph of the data for very small diameters and for diameters near the width of the tube.
6. Find a relationship between the diameter of the hole and the total time required to empty the tube.
7. Can you combine your findings from Questions 4 and 6 to predict the height of the water given the time and diameter of the hole?