Math & Mantids: A Data Exploration
Cheryl Gann
NC School of Science and Math
One of Nature’s Perfect Predators
Investigating the Eating Habits
Of the Mantid
Adapted from Contemporary Precalculus Through Applications 2nd Edition, Everyday
Learning Corporation, The North Carolina School of Science and Mathematics
Created by The North Carolina School of Science and Math
Reaction Distance vs. Satiation
Satiation
(cg)
11 18 23 31 35 40 46 53 59 66 70 72 75 86 90
Reaction
65 52 44 42 34 23 23 8
Distance (mm)
4
0
0
0
0
0
We will now make a scatter plot of the data to
determine what type of function will best fit
the data.
0
Reaction Distance vs. Satiation
Finding a Model
Since the first nine data points seem to follow a
linear trend and the last six data points all have
R-values of zero, a piecewise-function would be
the best type of function for this data set.
The second part of the function will be 𝑅 = 0.
For the first part, technology can be used to find
the least squares line.
Finding a Model
The function below models the relationship
between Satiation (S) and the Reaction Distance
(R) a mantid will travel for food.
−1.24𝑆 + 76.26 if 0 ≤ 𝑆 < 61.5
𝑅 𝑆 =
0
if 𝑆 ≥ 61.5
Reaction Distance vs. Satiation
Is this a good model? We can determine this by
checking the residuals.
Image of Mantid by Wikimedia User Fir0002, Licensed under the GNU Free Documentation License, via Wikimedia Commons.
http://commons.wikimedia.org/wiki/File%3ALarge_brown_mantid_close_up_nohair.jpg
License info: GFDL 1.2 - http://www.gnu.org/licenses/old-licenses/fdl-1.2.html
Residual Plot
Assessing the Model
Since the residuals are
• small relative to the Reaction Distance values,
• scattered about the 𝑥-axis
we can assume the model is a good fit.
Then what information does the slope of our
model gives us in the context of our problem?
Interpreting the Model
The (non-zero) slope for our model is −1.24.
This tells us that according to our model, for
every additional centigram of food in the
mantid’s stomach, it is willing to travel
approximately 1.24 millimeters less to get to
food.
According to our model, what is the greatest
distance that a mantid will move for food?
Image of Mantid by Wikimedia User Fir0002, Licensed under the GNU Free Documentation License, via Wikimedia Commons.
http://commons.wikimedia.org/wiki/File%3ALarge_brown_mantid_close_up_nohair.jpg
License info: GFDL 1.2 - http://www.gnu.org/licenses/old-licenses/fdl-1.2.html
The mantid will move the greatest distance
toward food when its stomach is empty.
This is equivalent to the satiation being zero
(i.e. the 𝑅-intercept):
𝑅 = −1.24 ⋅ 0 + 76.26
𝑅 = 76.26
So, the greatest distance a mantid will move for
food is approximately 76.26 millimeters.
What is the hunger threshold for the mantid?
Image of Mantid by Wikimedia User Fir0002, Licensed under the GNU Free Documentation License, via Wikimedia Commons.
http://commons.wikimedia.org/wiki/File%3ALarge_brown_mantid_close_up_nohair.jpg
License info: GFDL 1.2 - http://www.gnu.org/licenses/old-licenses/fdl-1.2.html
The mantid will no longer move toward food
when it is full.
This is equivalent to the reaction distance being
zero (i.e. the 𝑆-intercept):
0 = −1.24𝑆 + 76.26
−76.26 = −1.24𝑆
𝑆 = 61.5
So, the hunger threshold for the mantid is
approximately 61.5 centigrams of food.
Investigating the Eating Habits
Of the Mantid
Part II
Created by The North Carolina School of Science and Math
Satiation vs. Time
Time
(hr)
Satiation
(cg)
Time
(hr)
Satiation
(cg)
0
1
2
3
4
5
6
8
10
94 90 85
82 88 83
70
66
68
12 16 19
20 24 28
36
48
72
50 46 51
41 32 29
14
17
8
We will again make a scatter plot of the data
to determine what type of function will fit
best.
Satiation vs. Time
Finding a Model
The biologists assume that the mantid will
digest a fixed percentage of the food in its
stomach each hour. That information, together
with the graph, tells us that an exponential
function should be a good fit.
Finding a Model
Since the amount of food decreases by a fixed
percentage each hour (𝑝), the recursive system
of equations below should fit reasonably well
for some value of 𝑝.
𝐼𝑛𝑖𝑡𝑖𝑎𝑙 𝑆𝑎𝑡𝑖𝑎𝑡𝑖𝑜𝑛 = 94
𝑁𝑒𝑤 𝑆𝑎𝑡𝑖𝑎𝑡𝑖𝑜𝑛 = 𝑝 ⋅ 𝑂𝑙𝑑 𝑆𝑎𝑡𝑖𝑎𝑡𝑖𝑜𝑛
How are we going to determine a
value for 𝑝?
Image of Mantid by Wikimedia User Fir0002, Licensed under the GNU Free Documentation License, via Wikimedia Commons.
http://commons.wikimedia.org/wiki/File%3ALarge_brown_mantid_close_up_nohair.jpg
License info: GFDL 1.2 - http://www.gnu.org/licenses/old-licenses/fdl-1.2.html
Finding a Model
Using the sequence mode on a graphing
calculator or other technology, we can “guessand-check” to find the best 𝑝 value for the data.
Alternatively, we can look at ratios of successive
satiation values to obtain an estimate for the
value of 𝑝.
Finding a Model
𝑝 = 0.50
𝑝 = 0.75
𝑝 = 0.90
The actual data is shown as red squares. The
recursive values we are generating are shown as
blue dots. Notice how the recursive values are
tending toward the actual data as the 𝑝 value
increases.
Finding a Model
𝑝 = 0.95
𝑝 = 0.96
𝑝 = 0.97
A 𝑝 value of 0.96 seems to be a good fit:
𝑁𝑒𝑤 𝑆𝑎𝑡𝑖𝑎𝑡𝑖𝑜𝑛 = 0.96 ⋅ 𝑂𝑙𝑑 𝑆𝑎𝑡𝑖𝑎𝑡𝑖𝑜𝑛
Finding a Model
We now want an explicit function for 𝑆 that
gives the same values as the recursive model
found.
𝑺 = 𝟎. 𝟗𝟔𝑻 ⋅ 𝟗𝟒, where 𝑇 is the number of
hours since the mantid has filled its stomach
and 𝑆 is its satiation in cg. Notice that 0.96 is the
percentage of food in the mantid’s stomach and
94 represents the initial satiation.
Suppose a mantid has been without food for 40
hours. How far do you estimate it will travel
seeking food?
Image of Mantid by Wikimedia User Fir0002, Licensed under the GNU Free Documentation License, via Wikimedia Commons.
http://commons.wikimedia.org/wiki/File%3ALarge_brown_mantid_close_up_nohair.jpg
License info: GFDL 1.2 - http://www.gnu.org/licenses/old-licenses/fdl-1.2.html
The satiation level after 𝑇 = 40 hours is given by
𝑺 = 𝟎. 𝟗𝟔𝟒𝟎 ⋅ 𝟗𝟒 = 𝟏𝟖. 𝟑𝟔.
Now we have a value for satiation level that we can
use in the linear function 𝑅(𝑆):
𝑅 18.36 = −1.24 ⋅ 18.36 + 76.26 = 𝟓𝟑. 𝟒𝟗 mm
According to the models, a mantid that has been
without food for 40 hours will travel approximately
53.49 mm for food.
Suppose a mantid is willing to travel 47 mm for
food. Approximately how long has it gone
without eating?
Image of Mantid by Wikimedia User Fir0002, Licensed under the GNU Free Documentation License, via Wikimedia Commons.
http://commons.wikimedia.org/wiki/File%3ALarge_brown_mantid_close_up_nohair.jpg
License info: GFDL 1.2 - http://www.gnu.org/licenses/old-licenses/fdl-1.2.html
Set 𝑅(𝑆) = 47 and solve for 𝑆 to get 𝑆 = 23.60.
Now set 𝑆 𝑇 = 23.60.
This gives us 23.60 = 0.96𝑡 ⋅ 94.
This can be solved using logarithms or we can
use the intersect feature of a graphing calculator
or other technology to find 𝑇 = 33.86 hours.
Having looked at all this information, what do
we now know about the eating habits of
mantids?
Image of Mantid by Wikimedia User Fir0002, Licensed under the GNU Free Documentation License, via Wikimedia Commons.
http://commons.wikimedia.org/wiki/File%3ALarge_brown_mantid_close_up_nohair.jpg
License info: GFDL 1.2 - http://www.gnu.org/licenses/old-licenses/fdl-1.2.html
Combining the Models
Notice the relationship between our two
functions:
−1.24𝑆 + 76.26 𝑖𝑓 0 ≤ 𝑆 < 61.5
𝑅 𝑆 =
0
𝑖𝑓 𝑆 ≥ 61.5
𝑆 𝑇 = 0.96𝑇 ⋅ 94
The output for the second is the same quantity
as the input for the first.
Combining the Models
We can write a function for Reaction Distance in
terms of time as a composition of functions.
The non-zero part is:
𝑅 𝑆 𝑇 = −1.24 ⋅ 0.96𝑇 ⋅ 94 + 76.26
It may be tricky for students to recognize that
since 𝑆 is decreasing over time, in the 𝑅 vs. 𝑇
graph, the zero portion will be first…
Reaction Distance vs. Time
From the graph we
can see that if a
mantid has filled its
stomach within the last
10 hours, it is satisfied
and will not move toward food at all. Its
satiation decreases exponentially over time, and
the distance it will travel toward food increases
over time.
After about 10.4 hours the distance a mantid
will move toward food increases approaching a
limiting value of about 76 mm.
Implementation Suggestions
• Showing the video first is a great way to motivate and engage
the students.
• GeoGebra is a free online graphing application that would be
very useful on this problem if students have access to computers
during class.
• Working through both parts of the problem will likely take at
least 2 hours of class time.
• Having groups share their solutions with the class or write a
mock newspaper article is a nice opportunity to get the students
to explain their work.
• Full lesson plans, calculator tips, and other lesson details
available online at: http://betterlesson.com/unit/144785/math.
A free account is required for download.
Common Core Standards in 9-12
Mathematics
High School Algebra Mathematics Standards
Math.A-REI.1: Explain each step in solving equation as following from the
equality of numbers asserted at the previous step, starting from the
assumption that the original equation has a solution. Construct a viable
argument to justify a solution method.
Math.A-REI-11: Explain why the x-coordinates of the points where the
graphs of the equations of y=f(x) and y=g(x) intersect are the solutions of the
equation f(x)=g(x); find the solutions approximately, e.g., using technology to
graph the functions, make a table of values, or find successive
approximations. Include cases where f(x) and/or g(x) are linear, polynomial,
rational, absolute value, exponential, and logarithmic functions.*
Math.S-CED.2: Create equations in two or more variables to represent
relationships between quantities; graph equations on coordinate axes with
labels and scales.
Common Core Standards in 9-12
Mathematics
High School Functions Mathematics Standard
Math.F-BF.1a: Write a function that describes a relationship between
quantities.
Math.F-IF.1: Understand that a function from one set (called the domain) to
another set (called the range) assigns to each element of the domain exactly
one element of the range. If f is a function and x is an element of its domain,
then f(x) denotes the output of f corresponding to the input x. The graph of f
is the graph of the equation y=f(x).
Math.F-IF.2: Use function notation, evaluate functions for inputs in their
domains, and interpret statements that use function notation in terms of a
context.
Math.F-IF.4: For a function that models a relationship between two
quantities, interpret key features of graphs and tables in terms of the
quantities, and sketch graphs showing key features given a verbal description
of the relationship. Key features include: intercepts; intervals where the
function is increasing, decreasing, positive, or negative; relative maximums
and minimums; symmetries; end behavior; and periodicity.
Common Core Standards in 9-12
Mathematics
High School Functions Mathematics Standard
Math.F-IF.5: Relate the domain of a function to its graph and, where applicable,
to the quantitative relationship it describes.
Math.F-IF.7a: Graph linear and quadratic functions and show intercepts,
maxima, and minima.
Math.F-IF.7b: Graph square root, cube root, and piecewise-defined functions,
including step functions and absolute value functions.
Math.F-IF.7e: Graph exponential and logarithmic functions, showing intercepts
and end behavior, and trigonometric functions, showing period, midline, and
amplitude.
Math.F-IF.8b: Use the properties of exponents to interpret expressions for
exponential functions.
Math.F-LE.1c: Recognize situations in which a quantity grows or decays by a
constant percent rate per unit interval relative to another.
Common Core Standards in 9-12
Mathematics
High School Statistics and Probability Mathematics Standards
Math.S-ID.6: Represent data on two quantitative variables on a scatter plot,
and describe how the variables are related.
Math.S-ID.6b: Informally assess the fit of a function by plotting and analyzing
residuals.
Math.S-ID.6c: Fit a linear function for a scatter plot that suggests a linear
association.
Math.S-ID.7: Interpret the slope (rate of change) and the intercept (constant
term) of a linear model in the context of the data.
Thank You for Attending!
Cheryl Gann
NC School of Science and Math
gann@ncssm.edu
Talk materials available at:
http://courses.ncssm.edu/math/talks/conferences/
Special thanks to Donita Robinson for creation of
the online Mantid lesson materials.