Bath Time with Archimedes
Project Overview
Understanding the connection between functions and their graphs is one of the
fundamental concepts a developmental Algebra student needs to comprehend.
Students will explore the connections between certain events and the level of water in a
bathtub. This project uses a bathtub simulation that allows students to control the level
of water in a bathtub by manipulating the faucet, plug and whether or not Archimedes
sits in the bathtub. Students watch the graph as well as a chart of the depth of the
water at certain time increments being drawn on the screen in real-time. Students are
then asked to identify the events that caused each change in the graph.
After analyzing the bathtub, students are given a brief description and asked to sketch
graphs for other functions. This is designed to determine whether the student is able to
transfer their knowledge from the bathtub exercise to other situations.
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Student Learning Objectives
Upon completing this RWLO, the student will be able to:

Describe the connections between functions and their graphical representations.

Understand the connections between a function’s graph and a chart containing
data points.

Sketch a reasonable graph of a linear function in two variables given a brief
description.
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Procedure
Time: Approximately 60 minutes
Materials: Internet Access, Student Worksheet and Writing Utensil
Prerequisites: Student understanding of how to navigate the Internet is needed to
complete this RWLO.
Implementation: This RWLO is designed to supplement classroom discussion and
activities during the introduction of functions. This activity is intended to be used as
either an individual or group assignment to be completed outside of class. It is
recommended that each student complete the worksheet individually.
Steps:
1. Describe the goal and the desired final result (students can correctly interpret the
events which affect the water level and/or the rate at which the water level
changes).
2. Provide instructions on how to access and begin the Bath Time with Archimedes
Activity. If you have the technological ability, you may wish to demonstrate.
3. Distribute the Student Worksheets and provide instructions to students for
completing and submitting their work.
4. Collect the Student Instruction Worksheets and evaluate according to
assessment procedures.
5. Discuss other events that might affect the level of water in the bathtub, as well as
the differences between the five rates of tap flow.
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Content Material
Background Information:
Through this RWLO activity, students will explore the connections between real world
actions and the graphs that represent these actions. Utilizing an online simulation
program, students will explore different factors which affect the level of water in a
bathtub. Students will complete a minimum of three trials, each time examining how
each change they make affects the graph. After completing the trials, students will be
asked to make conclusions about the graphs for other functions.
Student Directions:
 Obtain a Student Worksheet for the Bath Time with Archimedes Activity.
 Experiment with the Online Simulation and then Complete Steps 1-3 on the
Worksheet.
 Using the information you learned during this activity, answer the questions on
Step 4.
 Relate the information you learned during this activity to other situations by
sketching graphs for the functions described in Step 5.
Referenced URLs:

http://www.mathsonline.co.uk/nonmembers/gamesroom/sims/archi/archi4.html
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Student Worksheet: Bath Time with Archimedes
Goal:
To understand the connections between real world actions and their representations on a graph.
1.
Use this link to load the Bath Time with Archimedes Activity on your computer.
http://www.mathsonline.co.uk/nonmembers/gamesroom/sims/archi/archi4.html
During this activity, you will control the level of water in a bathtub by turning on and off
the tap, putting in and pulling out the plug, and by having Archimedes get in and out of
the tub. This activity will last 100 seconds. You will see a graph showing the level of
water as a function of time drawn as you manipulate the faucet and plug. You will also
see a chart which indicates the depth of the water in 10 second increments. You should
examine these carefully each time you complete the activity.
2.
First Trial: You will need to make a minimum of three changes during the 100 seconds.
a)
Record the tap strength you chose:
b)
In the space provided below, sketch the graph and chart.
__________
Time
Water
Level
10
20
30
40
50
60
70
80
90
100
c)
Write brief sentences to describe the event that caused each change in the function graph.
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2.
Second Trial: You will need to make a minimum of four changes during the 100 seconds.
a)
Record the tap strength you chose:
b)
In the space provided below, sketch the graph and chart.
__________
Time
Water
Level
10
20
30
40
50
60
70
80
90
100
c)
Write brief sentences to describe the event that caused each change in the function graph.
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3.
Third Trial: You will need to make a minimum of five changes during the 100 seconds.
a)
Record the tap strength you chose:
b)
In the space provided below, sketch the graph and chart.
__________
Time
Water
Level
10
20
30
40
50
60
70
80
90
100
c)
Write brief sentences to describe the event that caused each change in the function graph.
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4.
Answer the following questions pertaining to the Archimedes activity.
a)
Name another event that might result in an almost instantaneous increase in the level of
water in the bathtub.
b)
What would be the fastest way to overflow the tub?
c)
Describe the conditions under which there should be no change to the level of water in
the bathtub.
5.
For each function described below, sketch the graph.
a)
The height of a person depending on his/her age.
b)
The height of the top of your head as you bounce on a pogo stick.
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Assessment
This is a sample rubric awarding a total of 80 points for this activity.
1.
Trial One
_____/ 2
Tap Strength
2 if Recorded
_____/ 2
2 for 3 Events
_____/ 3
0 for No Events
1 if Partially Recorded
0 if Not Recorded
1 if Partially Recorded
0 if Not Recorded
Chart
3 if Recorded
_____/ 5
1 for 1-2 Events
Graph
3 if Recorded
_____/ 3
0 if Not Recorded
# of Events
Description Sentences
5 for Detailed Sentences which Accurately Describe All Events and Changes
3 for Sentences containing Inconsistent Explanations or only Addressing Some Events
0 if Not Attempted
2.
Trial Two
_____/ 2
Tap Strength
2 if Recorded
_____/ 2
2 for 4 Events
_____/ 3
0 for No Events
1 if Partially Recorded
0 if Not Recorded
1 if Partially Recorded
0 if Not Recorded
Chart
3 if Recorded
_____/ 5
1 for 1-3 Events
Graph
3 if Recorded
_____/ 3
0 if Not Recorded
# of Events
Description Sentences
5 for Detailed Sentences which Accurately Describe All Events and Changes
3 for Sentences containing Inconsistent Explanations or only Addressing Some Events
0 if Not Attempted
3.
Trial Three
_____/ 2
Tap Strength
2 if Recorded
_____/ 2
2 for 5 Events
_____/ 3
0 for No Events
1 if Partially Recorded
0 if Not Recorded
1 if Partially Recorded
0 if Not Recorded
Chart
3 if Recorded
_____/ 5
1 for 1-4 Events
Graph
3 if Recorded
_____/ 3
0 if Not Recorded
# of Events
Description Sentences
5 for Detailed Sentences which Accurately Describe All Events and Changes
3 for Sentences containing Inconsistent Explanations or only Addressing Some Events
0 if Not Attempted
4.
Questions
5 for Detailed Answer Indicating Thorough Grasp of Concepts
_____/ 5
_____/ 5
_____/ 5
5.
3 for Partial Understanding
Event for “Instantaneous” Increase
Fastest Way to Overflow
No Changes to Water Level
Function Sketches
10 for Detailed Sketch, Indicates Thorough Understanding, Axes Labeled
8 for Detailed Sketch, Indicates Thorough Understanding, No Labels on Axes
6 for Sketch Showing Good Grasp of Concepts
4 for Sketch Showing Partial Understanding
2 for Sketch Showing Little to No Understanding or for Partial Attempt
0 if Not Attempted
_____/ 10
_____/ 10
Lifetime Height
Pogo Height
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0 for No Answer
Links to Course Competencies
The RWLO is best suited for use in the developmental algebra course where functions
are introduced, but could easily be utilized in any algebra course that covers functions.
Specifically, this RWLO meets the following Intermediate Algebra course competency:

To identify functions and sketch linear equations and functions in two variables
and use a multivariate approach to solving application problems involving linear
equations and functions.
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Supplementary Resources

http://illuminations.nctm.org/LessonDetail.aspx?id=L375
Mathematics as Communication Activity that focuses on Creating and
Interpreting Graphs that are Functions of Time.

Mathematicians Are People, Too: Stories from the Lives of Great
Mathematicians. Dale Seymour Publications. December 1999. ISBN#
0866515097.

Identifying Qualitative Graphs Worksheet. From Mathematics Teacher.
September 1994.
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Recommendations
Recommendations for Integration:
This RWLO is suggested to be incorporated as an out-of-class assignment for students
during an Introduction to Functions unit. It can be adapted and given as a group activity
or an in-class activity, if time allows.
If this is administered as an out-of-class assignment, it is recommended that the Bath
Time with Archimedes Activity be demonstrated in class.
The emphasis for Developmental Algebra courses is to be able to identify the events
that affect the change of the water level. This activity can also be incorporated into
College Algebra courses, where a stronger emphasis is placed on determining the
different rates of the faucets.
Regardless of where and how this activity is incorporated, it is recommended that a
discussion of Archimedes and his discovery of the Law of Buoyancy be included either
as a springboard for this activity or an extension of the topics presented here.
Back-up:
It is highly recommended that the Instructor create several Bath Tub scenarios and save
or print these to be utilized during class discussion, in the event that the website for the
Bath Time with Archimedes Activity is inaccessible during the project.
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