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 2 nd Edition, Everyday Learning Corporation, The North Carolina School of Science and Mathematics Created by The North Carolina School of Science and MathThe North Carolina School of Science and Math

Satiation (cg) Reaction Distance (mm) Reaction Distance vs. Satiation We will now make a scatter plot of the data to determine what type of function will best fit the data.

Reaction Distance vs. Satiation

Finding a Model

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. License info: GFDL

Residual Plot

Assessing the Model

Interpreting the Model

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. License info: GFDL

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. License info: GFDL

Investigating the Eating Habits Of the Mantid Part II Created by The North Carolina School of Science and MathThe North Carolina School of Science and Math

Satiation vs. Time Time (hr) Satiation (cg) Time (hr) Satiation (cg) 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

Image of Mantid by Wikimedia User Fir0002, Licensed under the GNU Free Documentation License, via Wikimedia Commons. License info: GFDL

Finding a Model

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. License info: GFDL

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. License info: GFDL

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. License info: GFDL

Combining the Models

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: 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 Talk materials available at: Special thanks to Donita Robinson for creation of the online Mantid lesson materials.