Lesson Plan

“Go with the Flow: Using Derivatives to Describe Storm Water Runoff Rates”
Grade Level &
Grade 12
Two 50 minutes
Field Trip
AB Calculus
Prepared By:
Regina Lamendella
Analyze Learners
Overview & Purpose (STEMcinnati theme)
Education Standards Addressed (Ohio).
The purpose of this unit is for students to discover how engineers use derivatives to solve realworld engineering problems (Careers). Students will utilize urban, sub-urban, and rural storm
water runoff data, to generate three different mathematical functions. Using the derivative
students will discover differences in instantaneous rates of change of water runoff within these
different settings (Application to the real-world). Students should be able to correlate water
runoff rate data to watershed characteristics, identify challenges associated with increased runoff
rates in an urban setting such as Cincinnati (Societal Impacts), and suggest appropriate best
management practices for its control. Finally, this unit will conclude with a trip to the national,
award-wining Sanitation District 1, for a tour of storm water management technologies utilized
for reduction of urban runoff. At the end of this unit, students should be able to feel comfortable
using derivatives to model change within environmental systems.
(Understanding Technology) Predict how decisions regarding the
implementation of technologies involve the weighing of trade-offs between
predicted positive and negative effects on the environment and/or humans.
Science: Students will be exposed to hypothesis driven research, data collection, and data
Understand the meaning of the derivative in terms of a rate of change and
local linear approximation and they should be able to use derivatives to solve
a variety of problems.
Technology: Students will be using Microsoft Excel to generate graphical representations of
their collected storm water data and to generated the slope of the tangent line.
Engineering: Students will learn how environmental and civil engineers design best
management practices to reduce stormwater runoff. Students will also learn how engineers use
derivatives to answer questions about environmental phenomena.
Mathematics: Students will apply the derivative to describe differences in storm water runoff
rates in urban, sub-urban, and rural watersheds.
(Scientifics Inquiry) Derive simple mathematical relationships that have
predictive power from experimental data (e.g., derive an equation from a
graph and vice versa, determine whether
a linear or exponential relationship exists among the data
in a table)
College Board AP Calculus
Select Goals and Objectives
Goals and
(Specify skills/information that
will be learned.)
Teacher Guide
Students should understand the meaning of the derivative in
terms of a rate of change and they should be able to use
derivatives to solve a variety of real-world environmental
engineering problems.
Objectives: Students will be able to:
Explain how engineers use derivatives to solve engineering
Generate a graph representing storm water runoff data to
create a graph representing the derivative of that function.
Compare instantaneous rates of change in runoff across
three different graphs, representing different watershed
scenarios and relate these different values to watershed
Recommend appropriate storm water management
technologies for reduction of urban runoff in a city such as
Cincinnati, and how these management practices might
impact the hydrographs they have created.
Select Instructional Strategies –
(Give and/or demonstrate
necessary information)
Direct Instruction: Powerpoint to show engineering
disciplines that use derivatives to solve everyday problems.
Presentation of storm water runoff and combined sewer
overflows in the greater Cincinnati area.
Indirect Instruction: Inquiry learning using three different
watershed models for discovery of how derivatives are
useful in understanding environmental phenomena, such as
storm water runoff rates.
Student Guide
Materials Needed
 Microsoft Excel
 Microsoft Powerpoint
 Sharpie markers (3)
 Three Aluminum Baking Pans
 Topsoil, Gravel, Sand (<1
cubic foot)
 Roofing Shingles (2)
 Graduated cylinders (3, 500
 Trash bags (3)
 Glue
 Acrylic paint
 Balsa Wood
 Drill of sharp puncturing tool
 Water (4 Liters)
 Watering cans (3)
 Packaging boxes (3)
Utilize Technology
Microsoft Excel- for plotting storm water runoff data and
derivative of this graph
Other Resources
(e.g. Web, books, etc.)
Microsoft Powerpoint- Direct instruction on how combined
sewer overflows are a problem in the greater Cincinnati area
and what type of engineers and scientists use derivatives to
solve environmental problems
Calculus: Graphical Numerical, and
Algebraic. 3rd edition. AP edition.
Finney, Demana, Waits, and
Kennedy. 2006 Prentice Hall.
Boston, MA.
Storm water management technology-Students will be
observing best management technologies (i.e. green roofs,
porous pavement, riparian zones etc.) that Sanitation District
1 uses to reduce storm water runoff
For detailed “Catch” see attached
“Catch Activity for AP Calculus”
For Powerpoint presentation please
see “Storm Water Presentation.ppt”
For detailed description of
constructing watershed models,
please see attached “Constructing
Watershed Models”
For Student worksheet see attached
“Student Storm Water Handout”
For methods of storm water activity
see attached “Methods for Storm
Water Activity”
Require Learner
(Describe the independent
activity to reinforce this
Lesson 1
CATCH: Begin class with a “rainfall observation”.
Have two prepared models. One model is blacktop
and one model is turf. (5 minutes)
Discuss observations. (2-3 minutes)
PRE-ASSESSMENT: See below (5 minutes)
Powerpoint presentation 1) How engineers need to
use derivatives. 2) Why storm water runoff is a
problem, especially in the Cincinnati area.
Presentation of the class problem/activity (10
Three model Activity: Separate class into three
groups. Each group works on a different watershed
(urban, sub-urban, and rural) Students simulated
rainstorm event and measure cumulative volume of
runoff over time. See stormwater procedure for
details (20 minutes).
REVIEW: Make sure each group has water runoff
data for their model. Refer to essential questions
below. (i.e How will they graph the data they have
collected? (dependent/independent variable) How
will they determine which watershed scenario had
highest rate of runoff? (5 minutes)
Two students will be invited to
come up and pour 500 mL of
water out of a watering can
onto two different models.
See attached document for detailed description of
CATCH. Lamendella_R_08_Major 1_Catch
Students should be writing
down observations.
Students should observe that
rates of runoff are different
between the two models.
Students have chance to ask
questions about what engineers
do and help formulate the
methods for experimental
design of class activity.
Need a timer, a marker, a
recorder, an overseer, and a
rain-pourer. Collect water
runoff data. Students record
runoff water volume for each
time interval in student
See attached document to find pre-assessment for
this lesson.
See attached for the Powerpoint presentation.
See attached for description of how to make models.
See attached for a description of the procedure for
the activity.
See attached for runoff volume data sheet
Lesson 2
CATCH: Short Video “Troubled waters”. This video
will help students understand how combined sewer
overflows impact human health and the environment
(5 mins).
Review of problem and data collected during Lesson
1. Discuss how to approach our objective (i.e
graphing the runoff, curve fitting, graphing
derivative, and identifying differences in
instantaneous rates of change of runoff among three
different watershed models (5 mins).
Plot cumulative volume (y-axis) vs. time (x axis),
plot values of slopes of tangent lines to create graph
of the derivative in Excel Spreadsheet (15 mins)
Combine storm water graphs and derivative graphs
from each of the watershed scenarios (10 mins).
Review (5-10 minutes): Discuss conclusions and
environmental implications of water runoff in these
Each group will plot their
respective water runoff graph
and derivative graph. Then one
representative will bring their
data to me and we will plot all
three storm water graphs and
derivative of those graphs
Students will answer questions
on their student handouts,
comparing these graphs from
the different watershed
See attached for storm water runoff student
See attached for Excel spreadsheet of storm water
cumulative volume graph and derivative graph.
Require Learner Participation
Lesson 2 (Cntinued)
(Describe the independent activity
to reinforce this lesson)
POST-ASSESSMENT (See below) (5 mins)
REVIEW: Present the trip to Sanitation District 1 and
some of the stormwater management practices they use.
How will these practices impact stormwater runoff rates?
(5 mins)
Have students come to chalk board
and draw the graphs.
Essential Questions:
1. Name at least two differences in the derivative of the
storm water runoff function for each of the three watershed
2. What watershed characteristics result in the differences
rate of change of storm water runoff within each of these
different watershed scenarios?
3. Explain how you might use derivatives to solve other
watershed related problems?
Lesson 3
Field Trip to Sanitation District 1.
Leave Norwood HS at 8:30am.
Arrive to SD#1 at 9:30am.
9:30 am: Begin Guided tour of SD #1 to see storm water
management practices (2 hours).
Students will observe impacts of
storm water management practices
(i.e green roofs, porous pavement,
detention ponds, wetlans, riparian
Lunch 11:30-12:15 (Donated by Kroger)
12:15 pm: Meeting with scientists and engineers that work
at SD 1 to learn their educational goals, training,
experience, and professional development. Students will
also gain a better understanding of what environmental
scientists and engineers do for the county.
Students will gain and understanding
of everyday problems that scientist
and engineers solve within the
environmental field.
Evaluate (Assessment)
See pre/post assessment sheet.
(Steps to check for student
See Student Handout sheet.
Please see attached sheet for rubric for grading.
Additional Notes
Discuss some
assumptions/limitations of the
models Measuring runoff at one
point (real-world multiple points)
No pipelines in models. In the realworld what would we measure?
Student have trouble understanding
the difference between discrete data
(data point they collected) and a
continuous function. Stress that we
can only take derivatives of
continuous functions. This is why
we generate a best-fit curve, that
generate a continuous function.
Bell: ______
Pre/Post assessment on Applications of Derivatives
1. Define a derivative in your own words.
2. Each of the following professionals uses derivatives to solve real-world problems in their respective discipline. Match the professional (on
left) with how they might use derivatives to solve a real-world problem (on right).
A. Chemical Engineer
B. Environmental Engineer
C. Civil Engineer
Can use a Derivative to:
i. Describe how fast a reactor can
Make a new petroleum product
ii. Describe how efficiently their
Global Position System works
iii. Describe how a fluid moves
inside a reactor
iv. Describe concentration of
Chlorine in water distribution
v. Describe strength of material
D. Electrical Engineer
3. You are a hydrologist who has been calculating storm water runoff data within the Mill Creek Watershed before, during, and after a
significant rainstorm. Using storm water runoff data how you could calculate when the maximum water runoff rate occurred during this
storm event?
4. List three different storm water management practices that could be used to reduce storm water runoff in the greater Cincinnati area.
Reflection for Lesson #1. The three watershed models held up fine for two class periods and can be reused in subsequent classes. However, I
would suggest using plastic tuberware stands in place of the cardboard stands, which became wet and subsequently weakened. Students were
very excited to perform the group work, in which they measured runoff from the models. If I were to do this lesson again, I would use smaller
groups, (i.e. 3-4 people instead of 5-6 people) as I noticed a few people standing around without a task. The additional people could measure
storm water runoff from a fourth model, which could represent a modified Cincinnati with green roofs, porous pavement, etc. I would shorten the
lecture on calculus and engineering to no more than 10 minutes, as I saw I was losing some of my audience. It might be more effective to ask the
students their current understanding of what engineers do and how they use calculus, in order to break them of any misconceptions. Make each
student record volume, so as to have multiple copies of data and to keep the students engaged in the twenty minute activity.
Reflection for Lesson #2. Instead of working in teams for the Microsoft excel, make each student plot the data. I noticed that students who were
more proficient at using excel took over the exercise and the less experienced simply watched. Instead of plotting the discrete data and taking the
slope between two successive points on the graph, and using that to represent the derivative, I would fit the runoff data to a continuous function,
generate an equation for the function, and take the derivative of the equation to produce a graph of the derivative.
Reflection for Lesson #3. While the field trip was useful in giving the students a clearer understanding of what environmental engineers and
scientist do, the trip could have been improved in the following ways. Since SD#1 is actually collecting data on storm water runoff from different
scenarios, this lesson could be repeated with the students actually collecting data from their test sites, plotting it, and using the derivative to
describe differential rates of runoff in a real-world scenario. Additionally, it would be beneficial to describe some of the limitations of the watershed
design, and how scientist and engineers use computer models to estimate runoff in various watersheds (i.e SWIM 5.0)