Report

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 = 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. −1.24 + 76.26 if 0 ≤ < 61.5 = 0 if ≥ 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 -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 -intercept): = −1.24 ⋅ 0 + 76.26 = 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 -intercept): 0 = −1.24 + 76.26 −76.26 = −1.24 = 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 (), the recursive system of equations below should fit reasonably well for some value of . = 94 = ⋅ How are we going to determine a value for ? 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 value for the data. Alternatively, we can look at ratios of successive satiation values to obtain an estimate for the value of . Finding a Model = 0.50 = 0.75 = 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 value increases. Finding a Model = 0.95 = 0.96 = 0.97 A value of 0.96 seems to be a good fit: = 0.96 ⋅ Finding a Model We now want an explicit function for that gives the same values as the recursive model found. = . ⋅ , where is the number of hours since the mantid has filled its stomach and 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 = 40 hours is given by = . ⋅ = . . Now we have a value for satiation level that we can use in the linear function (): 18.36 = −1.24 ⋅ 18.36 + 76.26 = . 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 () = 47 and solve for to get = 23.60. Now set = 23.60. This gives us 23.60 = 0.96 ⋅ 94. This can be solved using logarithms or we can use the intersect feature of a graphing calculator or other technology to find = 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: −1.24 + 76.26 0 ≤ < 61.5 = 0 ≥ 61.5 = 0.96 ⋅ 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: = −1.24 ⋅ 0.96 ⋅ 94 + 76.26 It may be tricky for students to recognize that since is decreasing over time, in the vs. 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 [email protected] Talk materials available at: http://courses.ncssm.edu/math/talks/conferences/ Special thanks to Donita Robinson for creation of the online Mantid lesson materials.