TOPIC: Do rare males have a mating advantage? Using mathematical
modeling to explore sexual selection - Evolutionary Ecology Application
TUTOR GUIDE
MODULE CONTENT: This module contains simple exercises for biology majors
taking an introductory course in ecology and evolution to begin applying and
interpreting mathematical models of biological problems.
TABLE OF CONTENTS
Alignment to HHMI Competencies for Entering Medical Students………………...1
Outline of concepts covered, module activities, and implementation……..……....2
Module: Worksheet for completion in class......................................................3 - 7
Pre-laboratory Review Questions (optional) Preparation Questions..….……..8 - 9
Pre-laboratory Exercises (mandatory)..........................................................10 - 12
Suggested Questions for Formative Assessment................................................13
Guidelines for Implementation……………………………...............…............13 - 14
Contact Information for Module Developers........................................................15
Alignment to HHMI Competencies for Entering Medical Students:
Competency
E1. Apply quantitative reasoning
and appropriate mathematics to
describe or explain phenomena
in the natural world.
Learning Objective
E1.1 Demonstrate quantitative numeracy
and facility with the language of mathematics
Activity
1,5
E1.2. Interpret data sets and communicate
those interpretations using visual and other
appropriate tools.
E1.3. Make statistical inferences from data
sets (evaluating best fit linear relationships
based on calculating error sums of squares)
E1.5. Make inferences about natural
phenomena using mathematical models
1,2,3
1
2,3,5
7
Mathematical Concepts covered:
- mathematical modeling in a biological context
- linear models
- regression models
In class activities:
- group discussion
- graphing and interpreting data
- construction of linear models
- using regression approach calculations for determining “best fit” relationships
between two variables.
Components of module:
- preparatory assignment to complete and turn in as homework before class
- in class worksheet:
- discussion questions
- plotting and interpreting data
- calculations of sums of squared error values to quantitatively assess the
goodness of fit of lines to observed data.
- suggested assessment questions
- guidelines for implementation
Estimated time to complete in class worksheet
- 60 minutes
Targeted students:
- first year-biology majors in introductory biology course covering ecology and
evolution
Quantitative Skills Required:
- Basic arithmetic
- Logical reasoning
- Interpreting data from tables
- Graph/Data Interpretation
2
Worksheet: Introduction to Mathematical Modeling in Biology
Discussion Questions
To begin, form groups of 3 students. Discuss and write down the answers to the
following questions (individual answers should be turned in with your lab write
up):
1. What is a mathematical model? What is a mathematical model of a biological
system? What value would models have in these contexts?
2. How does a scientist know if a model is “good” at modelling a biological
phenomenon?
3
Lab Exercises – Part I
Students: To receive credit for this exercise, provide short answers and
supporting graphs and calculations for all questions below (except 6 instructors will do this).
In a species of cichlid fish, most males have yellow fins that they display when
attracting mates or defending their territories. However, in certain populations of
this species a small number of males have red fins instead. An evolutionary
biologist hypothesized that the red fins may be under frequency-dependent
sexual selection. According to this hypothesis, the fitness of fish with red fins is
greater when fewer males have this trait (i.e., when more males have yellow
fins). To test this hypothesis, the biologist set up several experimental pools with
cichlids in which the frequency of males with red fins was controlled. The
biologist then determined the number of offspring that each red male had.
Observation
(i)
1
2
3
4
5
6
7
8
9
10
Number of Red
Males per 100
Individual Fish
8
11
13
15
19
23
27
33
35
44
Average Number of
Offspring per Red
Male
325
380
337
248
147
103
152
175
153
79
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Instructions:
Open the Excel program and get a piece of graph paper.
1)
Individually: graph the data in the table on the graph paper provided.
a.
Draw a straight line that gets as close as you can to as many of the data
points as possible. This is your linear model of the data.
b.
Determine the equation for your model and write it here*:
*Remember that y = mx + b, where m is the slope and b is the y-intercept.
2)
As a group:
a.
Choose the “best” model from those within your group.
b.
Open the Excel spreadsheet and edit the formula in cell C5 to reflect your
equation (there is already an equation in that cell: =m*A5+b, referenced to the
value in cell A5 - change m and b to your calculated values of m and b).
c.
Copy cell C5 into the remaining cells in column C.
3)
As a group: Copy cell D5 into the remaining cells in column D. Examine
and discuss the Yi - Ŷi values, answering the following questions:
a.
Is the line generally too high or low at one end of the data range?
______________________________________________________________
______________________________________________________________
b.
What would happen to the values of Yi - Ŷi if you increased the slope of
the line?
______________________________________________________________
______________________________________________________________
c.
What if you increased the y-intercept?
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________________________________________________________________
__
______________________________________________________________
d.
Decide together what would improve your linear model.
4)
Now obtain data from another group. Write the equation for this model:
___________________________
5)
Whose group had the better fitting line? How do you know? Can you tell
easily by looking at lines or do you need error sum of squares calculations to
evaluate your two models? (to calculate error sum of squares, copy cell E5 into
the remaining cells in column E)
______________________________________________________________
______________________________________________________________
______________________________________________________________
______________________________________________________________
6)
Instructors: take example from one group & edit equations on their
computer, then graph data (Yi) and Ŷi for each model (scatter chart with line but
no markers). This will display data & the Student Group’s two models, which can
be projected for entire class to see. Can also show the SS for each of their
models.
7)
Explain your own results (from your own model) in the context of the
original hypothesis. Do the data support the idea that male color might be a
sexually selected trait (and under frequency dependent selection)? If so, how
would you expect the frequency of red and yellow color morphs to change across
generations?
MODULE FEEDBACK - Each year we work to improve the modules in the active
learning "discussion" sections. Please answer the following question with regard
to this module on this sheet and turn in your answer to the TA. You can do this
anonymously if you like by turning in this sheet separately from your module
answers.
How helpful was this module in helping you understand a fundamental
concept in mathematical modeling of biological data?
A = Extremely helpful
B= Very helpful
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C= Moderately helpful
D= A little bit helpful
E = Not helpful at all
Module Rating ____________
Thank you!
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Pre-laboratory Review Questions (OPTIONAL)
This homework is designed to review lines and linear models and to prepare you
for the upcoming module on mathematical modeling and subsequent models that
you will have to work with. If you encounter an unfamiliar term, please refer to a
textbook for high school algebra or pre-calculus.
On her way to her volleyball game, Joanne stops by the grocery store for a
healthy snack that will give her energy for the game. She decides to get a jar of
peanut butter and some bananas. She notices that the peanut butter costs $3.99
for a jar and that the bananas cost $0.49 per pound of bananas.
1.
How much will one pound of bananas and a jar of peanut butter cost?
2.
Joanne decides to bring a snack for each of the 6 girls on the volleyball
team. How much will it cost for six pounds of bananas and a jar of peanut
butter?
3.
On the grid provided below, draw a Cartesian coordinate system with
number of pounds of bananas on the x-axis and the total cost of the peanut
butter and bananas on the y-axis. Plot the two points that you found for the
previous two questions.
8
7
6
5
4
3
2
1
0
4. Plot the line on the grid above through the two points that you found. Now
compute the slope of the line that goes through these two points (remember,
“rise over run”: (y2-y1) = m (x2-x1) where m is the slope of the line).
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Now write the equation of the line in slope-intercept form, y= mx + b, where b is
the y-intercept (the place where the line crosses the y axis).
Congratulations! You have just made a mathematical model of peanut butter and
bananas!
5.
What does the slope correspond to in terms of the scenario above? What
is the y-intercept of the line? What does it correspond to in the scenario above?
6.
What is the independent variable?
What is the dependent variable? Respond in symbols as well as in words.
7.
How would the line change if the price of bananas increased or
decreased?
How would the line change if the cost of the jar of peanut butter increased or
decreased?
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Pre-laboratory Exercise: Evolutionary Ecology
(excel file that goes with this exercise: math modeling pre-laboratory
ecology data.xlsx)
An ecologist was studying a transition zone from grassland to conifer forest in
central Asia. The ecologist wanted to see if a linear model could describe the
relationship between latitude and the density of conifer trees. Using a small
amount of data from a study in North America (NA), the ecologist developed the
following model:
Ŷ = 36x -1767 where Ŷ is the predicted conifer density and x is the
latitude.
The ecologist then collected data from the first five study sites in Asia,
shown in Table 1.
In the following exercises you will evaluate how well the model developed
from data collected in North America fits the data collected from Asia. A
common method for evaluating a model involves comparing the actual
data with the predictions of the model.
Table 1. (KEY)
Study Site (i)
1
2
3
4
5
Latitude
(degrees)
48.97
49.35
49.55
49.70
50.24
Conifer
density
(number of
trees per
hectare; Yi)
2.2
25.3
17.5
47.8
46.9
Ŷi
(estimated
from the NA
model)
Yi – Ŷi
1. Begin by using a calculator to compute the conifer density that the North
American model predicts at each of the latitudes of the five study sites.
This will give you a series of Ŷi for each site, where i = 1, 2, 3, 4, or 5
corresponding to the study sites at each latitude. Record Ŷ in Table 1 of
your worksheet and in your excel spreadsheet.
2. The next column in the table will show the difference between the
measured conifer density in Asia (Yi) and the predicted density (from the
model; Ŷi) at each latitude. Use a calculator to fill in this column.
3. Plot the actual observed data (Asian). To do this, place a mark at each
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value of Yi at each value of i (1 through 5) on a piece of graph paper.
4. Next, graph the model Ŷ = 36x -1767 (Hint: To graph the model, you will
find it easier to plot two or three values of Ŷi and draw a line directly
through them, rather than drawing a line through the y-intercept).
5. At what latitude does this model predict that conifer density equals zero
trees per hectare (ha)?
__________________________________________
6. Look at where the model (the line you’ve just drawn) is relative to the data
points. How could the model be changed to improve the fit of the line to
the actual data? In one or two sentences, support your recommendation
for how to improve the model, referring to the values that you calculated.
______________________________________________________________
______________________________________________________________
______________________________________________________________
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Notice that the data points don’t actually all fall on the line. The distance that
an observation (Yi) at each value of i falls from the line (Yi – Ŷi) is a form of
"error". In a linear model, the line through the data that best explains, or fits,
the data is the one that the observed data points are closest to. We can
evaluate how well a model fits the data by adding up (summing) the errors for
all the data points. However, some errors may be negative while others are
positive (some points will be above the line while others will be below it). So
to evaluate how well a model fits the data we first we square the value of
each error (Yi – Ŷi) and then sum these values for all data points. This
procedure is formally described by the following:
or another way of putting it is = (Y1 - Ŷ)2 + (Y1 - Ŷ) 2 +.... (Yn - Ŷ) 2
where the "error sum of squares" is the summation of all of the (Yi – Ŷi)2
values for every observation, where n is the total number of observations. So
now we can say the model that best explains the data is the one with the least
sum of squares error.
7. Next you will use the Excel spreadsheet provided (math modeling prelaboratory ecology data.xlsx) to compute the sum of squares for the model.
This will give you a chance to practice the Excel skills you will need to
complete the exercises in class.
a. Double click on cell F3 and enter the formula ‘=E3^2’ (which tells
excel to square what’s in cell E3).
b. Hit enter to save this edit.
c. Copy cell F3 and paste into cells F4 through F7.
i. Excel will automatically adjust the formula to square the
values in cells E4 through E7.
d. Record the Error sum of squares (SS) value displayed in cell F8
here __________.
8. Model 2 in the spreadsheet is based on the equation computed by Excel
for this data set (y = 36.051x - 1758.8). Is Model 2 a better fit of the data?
How do you know?
__________________________ ___________________________
______________________________________________________________
______________________________________________________________
______________________________________________________________
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Suggested Questions for Formative Assessment
Learning Objective
E1.2. Interpret data sets and communicate those
interpretations using visual and other appropriate tools.
E1.3. Make statistical inferences from data sets
(evaluating best fit linear relationships based on
calculating error sums of squares)
E1.5. Make inferences about natural phenomena using
mathematical models
Activity
1
3,5
7
Guide for implementation: Discussion
Have students break up into groups of 3 to discuss and come up with answers to
the questions. Groups get together to talk about each question – work pauses
after 10 minutes or as soon as you think each group has come up with something
for both questions (no longer than 15 minutes). The TA should then pick a person
from 3 groups chosen at random to share with the class their group’s answer to
one of the questions. Tell them that everyone in the group should be prepared to
share the answer with the class, as you will choose who speaks randomly among
the group. Suggested ideas that should emerge from each question are listed
below.
An alternative way to run this discussion section of the class would be to run it
“Question Time” style where you let them discuss each question in turn for three
minutes, then ring a bell or use another method to cut off discussion, then have
the whole group report their answer. Then move on to the next question.
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Guide for Implementation: Lab Activity
Students work in groups. Each group has a computer.
Materials needed: Graph paper for each student
1) TA hands out graph paper to each member of a group and instructs students
to open excel spread sheet (Introduction to Mathematical Modeling.xlsx).
Each individual is instructed to plot the data points and construct a line through
the data points based on the two model equations. Each individual will then
decide which model (line) more accurately “fits” the data and provide rationale.
One or two groups share their decision with the class.
a) The graph paper should have x and y axes drawn in lower left when held
in landscape orientation.
b) If possible, project the data plot & regression lines for discussion –maybe
via instructor’s computer & projector or overhead projector.
c) Discussion of possible data outliers may come up – explain why the data
may not fall exactly along the best fit line.
2) Instructor provides brief explanation of least squares method of evaluating
regression lines, writing the equation for this calculation on the board.
Students then perform these calculations in their groups (can divide up who
does computation for which model).
a) Sample explanation: By plotting the data it looks like y varies with x. The
models describe a linear relationship where y varies with x. But the data
points don’t actually all fall on the line. The distance a point falls from the
line is a form of error. The model that best explains, or fits, the data is the
one that the data points are closest to, or has the least error (the smallest
error sum of squares).
n
ErrorSumOfSquares = å (Yi - Yˆi ) 2
i=1
Yi is a data point at Xi, and Yhat is the value of Y that falls on the line at Xi as
defined by the model.
3) Instructor-led discussion (as a class) about which group’s model is the “best
fit”.
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Module Developers:
Please contact us if you have comments/suggestions/corrections
Kathleen Hoffman
Department of Mathematics and Statistics
University of Maryland Baltimore County
khoffman@math.umbc.edu
Jeff Leips
Department of Biological Sciences
University of Maryland Baltimore County
leips@umbc.edu
Sarah Leupen
Department of Biological Sciences
University of Maryland Baltimore County
leupen@umbc.edu
Acknowledgments:
This module was developed as part of the National Experiment in Undergraduate
Science Education (NEXUS) through Grant No. 52007126 to the University of
Maryland, Baltimore County (UMBC) from the Howard Hughes Medical Institute.
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