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Geometric Analysis of Suction Feeding. Math & Nature. The universe is written in the language of mathematics Galileo Galilei , 1623 Quantitative analysis of natural phenomena is at the heart of scientific inquiry Nature provides a tangible context for mathematics instruction .
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Math & Nature • The universe is written in the language of mathematics • Galileo Galilei, 1623 • Quantitative analysis of natural phenomena is at the heart of scientific inquiry • Nature provides a tangible context for mathematics instruction
The Importance of Context • Context • The part of a text or statement that surrounds a particular word or passage and determines its meaning. • The circumstances in which an event occurs; a setting.
The Importance of Context • Context-Specific Learning • Facilitates experiential and associative learning • Demonstration, activation, application, task-centered, and integration principles (Merrill 2002) • Facilitates generalization of principles to other contexts
Math & Nature • Geometry & Biology • Biological structures vary greatly in geometry and therefore represent a platform for geometric education • Geometric variability functional variability ecological variability • Mechanism for illustrating the consequences of geometry
Math & Nature • Vertebrate skulls vary greatly in form & function www.digimorph.org
Math & Nature • Vertebrate skulls vary greatly in form & function ~50 bones ~7 moving parts 22 bones 1 moving part www.digimorph.org csi.whoi.edu
Math & Nature • Vertebrate skulls vary greatly in form & function Liem et al. (2001)
Math & Nature • Vertebrate skulls vary greatly in form & function • Moveable parts of the fish skull are responsible for the diversity of feeding mechanisms in fish • Jaw protrusion in the sand tiger shark Carcharias taurus D. Huber
Math & Nature • Vertebrate skulls vary greatly in form & function • Moveable parts of the fish skull are responsible for the diversity of feeding mechanisms in fish • Jaw protrusion in the sling-jaw wrasse Epibulusinsidiator P. Wainwright
Math & Nature • Fish feeding mechanisms • Filter • Biting • Ram • Suction www.true-wildlife.blogspot.com www.z00n.net C. Fallows
Math & Nature • Fish feeding mechanisms • Filter feeding W. Mischler 2013 Motta et al. (2010)
Math & Nature • Fish feeding mechanisms • Filter feeding Motta et al. (2010)
Math & Nature • Fish feeding mechanisms • Filter feeding P. Motta
Math & Nature • Fish feeding mechanisms • Biting
Math & Nature • Fish feeding mechanisms • Biting www.digimorph.org
Math & Nature • Fish feeding mechanisms • Ram feeding D. Huber C. Fallows
Math & Nature • Fish feeding mechanisms • Ram feeding S. Huskey www.tennesseeaquarium.com
Math & Nature • Fish feeding mechanisms • Ram feeding D. Huber
Math & Nature • Fish feeding mechanisms • Suction feeding • Most common fish feeding mechanism • Water cohesion • Suction performance D. Huber
Math & Nature • Fish feeding mechanisms • Suction feeding • http://www.youtube.com/user/Wainwrightlab Wainwright et al (2006)
Math & Nature • Fish feeding mechanisms • Suction feeding • http://www.youtube.com/user/Wainwrightlab Svanback et al (2002)
Math & Nature • Fish feeding mechanisms • Suction feeding • http://www.youtube.com/user/Wainwrightlab Grubich (2001)
Math & Nature • Fish feeding mechanisms • Suction feeding • Anterior posterior expansion Gibb & Ferry-Graham (2005) Wainwright et al (2006)
Math & Nature • Fish feeding mechanisms • Suction feeding • Fluid flow Holzmanet al (2008)
Math & Nature • Fish feeding mechanisms • Suction feeding • Fluid pressure and movement speed Svanback et al (2002)
Math & Nature • Fish feeding mechanisms • Suction feeding • Feeding ecology Motta et al (2008)
Math & Nature • Fish feeding mechanisms • Suction feeding • Geometric modeling Van Wassenbergh et al (2007) Bishop et al (2008)
Math & Nature • Fish feeding mechanisms • Suction feeding • Goliath grouper Epinephelusitajara
Math & Nature • Fish feeding mechanisms • Suction feeding • Goliath grouper Epinephelusitajara • Questions • What fluid velocity can the goliath grouper generate during suction feeding? • How does suction feeding by the goliath grouper compare to other fish?
Math & Nature • Geometry & Biology • NGSSS • MA.912.G.4.4: Use properties of congruent and similar triangles to solve problems involving lengths and area. • MA.912.G.5.4: Solve real-world problems involving right triangles. • MA.912.G.7.5: Explain and use formulas for lateral area, surface area, and volume of solids.
Math & Nature • Geometry & Biology • NGSSS • MA.912.G.7.7: Determine how changes in dimension affect the surface area and volume of common geometric solids. • MA.912.G.8.2: Use a variety of problem solving strategies, such as drawing a diagram, making a chart, guess-and-check, solving a simpler problem, writing an equation, and working backwards.
Math & Nature • Geometry & Biology • CCSS • MACC.912.G-GMD.1.3: Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. • MACC.912.G-GMD.2.4: Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.
Math & Nature • Geometry & Biology • CCSS • MACC.912.G-MG.1.1: Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). • MACC.K12.MP.1.1: Make sense of problems and persevere in solving them. • MACC.K12.MP.4.1: Model with mathematics
Math & Nature • Goliath grouper model • Objective • Determine the velocity of water flow into the mouth • Procedure • Determine the volume of components A and B at rest (t0) and at maximum expansion (t1) • t0 = time at rest • t1 = time at maximum expansion • Determine the volume change during feeding A B A B
Math & Nature • Goliath grouper model • Objective • Determine the velocity of water flow into the mouth • Procedure • Determine the area of the mouth at maximum expansion (t1) • t1 = time at maximum expansion A B A B
Math & Nature • Goliath grouper model • Objective • Determine the velocity of water flow into the mouth • Procedure A B A B
Math & Nature • Suction feeding in the goliath grouper • Given • Dimensions of cones A and B at rest (t0) • Find the volume of the goliath grouper feeding mechanism at rest (t0). b c e d a a
Math & Nature • Suction feeding in the goliath grouper • Given • Dimensions of cones A and B at rest (t0) • Find the volume of the goliath grouper feeding mechanism at rest (t0). e c b d a a
Math & Nature • Suction feeding in the goliath grouper • Given • Dimensions of cones A and B at maximum expansion (t1) • Find the volume of the goliath grouper feeding mechanism at maximum expansion (t1).
Math & Nature • Suction feeding in the goliath grouper • Given • Dimensions of cone B at maximum expansion (t1) • Find the area of the goliath grouper mouth at maximum expansion (t1). A. Collins mouth
Math & Nature • Suction feeding in the goliath grouper • Given • Volume of the goliath grouper feeding mechanism at rest (t0) and at maximum expansion (t1) • Duration of the feeding event (t1 - t0) • Area of the mouth opening at maximum expansion (t1) • Find the velocity of water flow into the mouth of the goliath grouper during suction feeding.
Math & Nature • Suction feeding in the snookCentropomusundecimalis JJ Photo Korhnak Wainwright et al (2006)