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 1. The part of a text or statement that surrounds a particular word or passage and determines its meaning. 2. The circumstances in which an event occurs; a setting.
The Importance of Context Context-Specific Learning ◦ Facilitates experiential and associative learning Demonstration, activation, application, taskcentered, 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 ◦ 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 Epibulus insidiator P. Wainwright
Math & Nature Fish feeding mechanisms ◦ Suction feeding Goliath grouper Epinephelus itajara 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 twodimensional 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 A B Determine the volume change during feeding 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 A B B
Math & Nature Suction feeding in the goliath grouper ◦ Given Dimensions of cones A and B at rest (t0) 1) Find the volume of the goliath grouper feeding mechanism at rest (t0). b c a a e d
Math & Nature Suction feeding in the goliath grouper ◦ Given Dimensions of cones A and B at rest (t0) 1) Find the volume of the goliath grouper feeding mechanism at rest (t0). c b a a e d
Goliath Grouper Suction Feeding Cone A Length (mm) Area (mm2) a 34.9 N/A b 153.6 Cone B Length (mm) Area (mm2) a 34.9 Time 0 c 54.3 N/A d 6.4 e Volume of feeding mechanism before expansion (t0) Cone A Length (mm) Area (mm2) a N/A b Cone B Length (mm) Area (mm2) a Time 1 c d e Volume of feeding mechanism at maximum expansion (t1) Volume change during feeding event (mm3) Duration of feeding event (sec) Area of mouth at maximum expansion (t1) (mm2) Velocity of water flow into mouth (mm/sec) Volume (mm3) Volume (mm3) Volume (mm3) Volume (mm3) 0.132
Goliath Grouper Suction Feeding Cone A Length (mm) Area (mm2) a 34.9 N/A b 153.6 Cone B Length (mm) Area (mm2) a 34.9 Time 0 c 54.3 N/A d 6.4 e 12.2 Volume of feeding mechanism before expansion (t0) Cone A Length (mm) Area (mm2) a N/A b Cone B Length (mm) Area (mm2) a Time 1 c d e Volume of feeding mechanism at maximum expansion (t1) Volume change during feeding event (mm3) Duration of feeding event (sec) Area of mouth at maximum expansion (t1) (mm2) Velocity of water flow into mouth (mm/sec) Volume (mm3) 195916.8 Volume (mm3) 84289.7 180206.5 Volume (mm3) Volume (mm3) 0.132
Math & Nature Suction feeding in the goliath grouper ◦ Given Dimensions of cones A and B at maximum expansion (t1) 2) Find the volume of the goliath grouper feeding mechanism at maximum expansion (t1).
Goliath Grouper Suction Feeding Cone A Length (mm) Area (mm2) a 34.9 N/A b 153.6 Cone B Length (mm) Area (mm2) a 34.9 Time 0 c 54.3 N/A d 6.4 e 12.2 Volume of feeding mechanism before expansion (t0) Cone A Length (mm) Area (mm2) a 39.5 N/A b 161.3 Cone B Length (mm) Area (mm2) a 39.5 Time 1 c 56.4 d 32.6 e Volume of feeding mechanism at maximum expansion (t1) Volume change during feeding event (mm3) Duration of feeding event (sec) Area of mouth at maximum expansion (t1) (mm2) Velocity of water flow into mouth (mm/sec) Volume (mm3) 195916.8 Volume (mm3) 84289.7 280206.5 Volume (mm3) Volume (mm3) 0.132
Goliath Grouper Suction Feeding Cone A Length (mm) Area (mm2) a 34.9 N/A b 153.6 Cone B Length (mm) Area (mm2) a 34.9 Time 0 c 54.3 N/A d 6.4 e 12.2 Volume of feeding mechanism before expansion (t0) Cone A Length (mm) Area (mm2) a 39.5 N/A b 161.3 Cone B Length (mm) Area (mm2) a 39.5 Time 1 c 56.4 d 32.6 e 266.5 Volume of feeding mechanism at maximum expansion (t1) Volume change during feeding event (mm3) Duration of feeding event (sec) Area of mouth at maximum expansion (t1) (mm2) Velocity of water flow into mouth (mm/sec) Volume (mm3) 195916.8 Volume (mm3) 84289.7 280206.5 Volume (mm3) 263547.1 Volume (mm3) 230974.7 494521.7 214315.3 0.132
Math & Nature Suction feeding in the goliath grouper ◦ Given Dimensions of cone B at maximum expansion (t1) 3) Find the area of the goliath grouper mouth at maximum expansion (t1). A. Collins mouth
Goliath Grouper Suction Feeding Cone A Length (mm) Area (mm2) a 34.9 N/A b 153.6 Cone B Length (mm) Area (mm2) a 34.9 Time 0 c 54.3 N/A d 6.4 e 12.2 Volume of feeding mechanism before expansion (t0) Cone A Length (mm) Area (mm2) a 39.5 N/A b 161.3 Cone B Length (mm) Area (mm2) a 39.5 Time 1 c 56.4 d 32.6 e 266.5 Volume of feeding mechanism at maximum expansion (t1) Volume change during feeding event (mm3) Duration of feeding event (sec) Area of mouth at maximum expansion (t1) (mm2) Velocity of water flow into mouth (mm/sec) Volume (mm3) 195916.8 Volume (mm3) 84289.7 280206.5 Volume (mm3) 263547.1 Volume (mm3) 230974.7 494521.7 214315.3 0.132
Goliath Grouper Suction Feeding Cone A Length (mm) Area (mm2) a 34.9 N/A b 153.6 Cone B Length (mm) Area (mm2) a 34.9 Time 0 c 54.3 N/A d 6.4 e 12.2 Volume of feeding mechanism before expansion (t0) Cone A Length (mm) Area (mm2) a 39.5 N/A b 161.3 Cone B Length (mm) Area (mm2) a 39.5 Time 1 c 56.4 3338.8 d 32.6 e 266.5 Volume of feeding mechanism at maximum expansion (t1) Volume change during feeding event (mm3) Duration of feeding event (sec) Area of mouth at maximum expansion (t1) (mm2) Velocity of water flow into mouth (mm/sec) Volume (mm3) 195916.8 Volume (mm3) 84289.7 280206.5 Volume (mm3) 263547.1 Volume (mm3) 230974.7 494521.7 214315.3 0.132 3338.8
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) 4) Find the velocity of water flow into the mouth of
Goliath Grouper Suction Feeding Cone A Length (mm) Area (mm2) a 34.9 N/A b 153.6 Cone B Length (mm) Area (mm2) a 34.9 Time 0 c 54.3 N/A d 6.4 e 12.2 Volume of feeding mechanism before expansion (t0) Cone A Length (mm) Area (mm2) a 39.5 N/A b 161.3 Cone B Length (mm) Area (mm2) a 39.5 Time 1 c 56.4 3338.8 d 32.6 e 266.5 Volume of feeding mechanism at maximum expansion (t1) Volume change during feeding event (mm3) Duration of feeding event (sec) Area of mouth at maximum expansion (t1) (mm2) Velocity of water flow into mouth (mm/sec) Volume (mm3) 195916.8 Volume (mm3) 84289.7 280206.5 Volume (mm3) 263547.1 Volume (mm3) 230974.7 494521.7 214315.3 0.132 3338.8 486
Math & Nature Suction feeding in the snook Centropomus undecimalis ◦ Given Dimensions of cones A and B at rest (t0) and at maximum expansion of the feeding mechanism (t1) Duration of the feeding event (t1 - t0) 5) Find the velocity of water flow into the mouth of the snook during suction feeding.
Snook Suction Feeding Cone A Length (mm) Area (mm2) a 2.1 N/A b 27.6 Cone B Length (mm) Area (mm2) a 2.1 Time 0 c 12.3 N/A d 1.8 e Volume of feeding mechanism before expansion (t0) Cone A Length (mm) Area (mm2) a 7.0 N/A b 28.9 Cone B Length (mm) Area (mm2) a 7.0 Time 1 c 12.3 d 5.9 e Volume of feeding mechanism at maximum expansion (t1) Volume change during feeding event (mm3) Duration of feeding event (sec) Area of mouth at maximum expansion (t1) (mm2) Velocity of water flow into mouth (mm/sec) Volume (mm3) Volume (mm3) Volume (mm3) Volume (mm3) 0.036
Snook Suction Feeding Cone A Length (mm) Area (mm2) a 2.1 N/A b 27.6 Cone B Length (mm) Area (mm2) a 2.1 Time 0 c 12.3 N/A d 1.8 e 73.8 Volume of feeding mechanism before expansion (t0) Cone A Length (mm) Area (mm2) a 7.0 N/A b 28.9 Cone B Length (mm) Area (mm2) a 7.0 Time 1 c 12.3 109.4 d 5.9 e 66.0 Volume of feeding mechanism at maximum expansion (t1) Volume change during feeding event (mm3) Duration of feeding event (sec) Area of mouth at maximum expansion (t1) (mm2) Velocity of water flow into mouth (mm/sec) Volume (mm3) 127.5 Volume (mm3) 147.2 274.7 Volume (mm3) 1482.9 Volume (mm3) 1611.5 3094.4 2819.7 0.036 109.4 716
Math & Nature Suction feeding in the longjaw butterfly fish Forcipiger longirostris ◦ Given Dimensions of cones A and B at rest (t0) and at maximum expansion of the feeding mechanism (t1) Duration of the feeding event (t1 - t0) 6) Find the velocity of water flow into the mouth of the longjaw butterfly fish during suction feeding.
Longjaw Butterfly Fish Suction Feeding Cone A Length (mm) Area (mm2) a 5.0 N/A b 14.9 Cone B Length (mm) Area (mm2) a 5.0 Time 0 c 31.2 N/A d 1.1 e Volume of feeding mechanism before expansion (t0) Cone A Length (mm) Area (mm2) a 5.0 N/A b 14.9 Cone B Length (mm) Area (mm2) a 5.0 Time 1 c 31.6 d 1.1 e Volume of feeding mechanism at maximum expansion (t1) Volume change during feeding event (mm3) Duration of feeding event (sec) Area of mouth at maximum expansion (t1) (mm2) Velocity of water flow into mouth (mm/sec) Volume (mm3) Volume (mm3) Volume (mm3) Volume (mm3) 0.022
Longjaw Butterfly Fish Suction Feeding Cone A Length (mm) Area (mm2) a 5.0 N/A b 14.9 Cone B Length (mm) Area (mm2) a 5.0 Time 0 c 31.2 N/A d 1.1 e 8.8 Volume of feeding mechanism before expansion (t0) Cone A Length (mm) Area (mm2) a 5.0 N/A b 14.9 Cone B Length (mm) Area (mm2) a 5.0 Time 1 c 31.6 3.8 d 1.1 e 8.9 Volume of feeding mechanism at maximum expansion (t1) Volume change during feeding event (mm3) Duration of feeding event (sec) Area of mouth at maximum expansion (t1) (mm2) Velocity of water flow into mouth (mm/sec) Volume (mm3) 390.1 Volume (mm3) 1036.0 1426.1 Volume (mm3) 390.1 Volume (mm3) 1049.3 1439.4 13.3 0.022 3.8 159
Math & Nature Suction feeding ◦ Given Velocities of water flow into the mouths of all three fish 7) Determine which fish is the best suction feeder. Korhnak v(t) = 486 mm/sec v(t) = 716 mm/sec v(t) = 159 mm/sec
Math & Nature References ◦ Bishop, K.L., Wainwright, P.C., and Holzman, R. (2008). Anterior to posterior wave of buccal expansion in suction feeding fish is critical for optimizing fluid flow velocity profile. Journal of the Royal Society, Interface. 5:1309-1316. ◦ Ferry-Graham, L.A., Wainwright, P.C., and Bellwood, D.R. (2001). Prey capture in long-jawed butterflyfishes (Chaetodontidae): the functional basis of novel feeding habits. Journal of Experimental Marine Biology and Ecology. 256:167-184. ◦ Galileo Galilei, The Assayer, as translated by Stillman Drake (1957), Discoveries and Opinions of Galileo pp. 237 - 238. New York: Doubleday & Company. ◦ Gibb, A.C. and Ferry-Graham, L.A. (2005). Cranial movements during suction feeding in teleost fishes: Are they modified to enhance suction production? Zoology. 108(2): 141-153. ◦ Grubich, J.R. (2001). Prey Capture in Actinopterygian Fishes: A Review of Suction Feeding Motor Patterns with New Evidence from an Elopomorph Fish, Megalops atlanticus. Integrative and Comparative Biology. 41(6): 1258-1265. ◦ Holzman, R., Day, S.W., and Wainwright, P.C. (2007). Timing is everything: coordination of strike kinematics affects the force exerted by suction feeding fish on attached prey. Journal of Experimental Biology. 210: 3328-3336. ◦ Holzman, R., Day, S.W., Mehta, R.S., and Wainwright, P.C. (2008). Jaw protrusion enhances forces exerted on prey by suction feeding fishes. Journal of the Royal Society, Interface. 5(29): 1445-1457.
Math & Nature References ◦ Liem, K., Bemis, W., Walker, W.F., and Grande, L. (2001). Functional Anatomy of the Vertebrates: An Evolutionary Perspective. New York. Cengage Learning. ◦ Merrill, M.D. (2002). First principles of instruction. Educational Technology Research and Development. 50 (3): 43 – 59. ◦ Motta, P.J., Hueter, R.E., Tricas, T.C., Summers, A.P., Huber, D.R., Lowry, D., Mara, K.R., Matott, M.P., Whitenack, L.B., and Wintzer, A.P. (2008). Functional morphology of the feeding apparatus, feeding constraints, and suction performance in the nurse shark Ginglymostoma cirratum. Journal of Morphology. 269(9): 1041-1055. ◦ Motta, P.J., Maslanka, M., Hueter, R.E., Davis, R.L., de la Parra, R., Mulvany, S.L., Habegger, M.L., Strother, J.A., Mara, K.R., Gardiner, J.M., Tyminski, J.P., and Zeigler, L.D. (2010). Feeding anatomy, filter-feeding rate, and diet of whale sharks Rhincodon typus during surface ram filter feeding off the Yucatan Peninsula, Mexico. Zoology. 113: 199-212. ◦ Sanford, C.P.J. and Wainwright, P.C. (2002). Use of sonomicrometry demonstrates the link between prey capture kinematics and suction pressure in largemouth bass. Journal of Experimental Biology. 205: 3445-3457. ◦ Svanback, R., Wainwright, P.C., and Ferry-Graham, L.A. (2002). Linking cranial kinematics, buccal pressure, and suction feeding performance in largemouth bass. Physiological and Biochemical Zoology. 75(6): 532-543.
Math & Nature References ◦ Van Wassenbergh, S., Herrel, A., Adriaens, D., and Aerts, P. (2007). No trade-off between biting and suction feeding performance in clariid catfishes. Journal of Experimental Biology. 210: 27-36. ◦ Wainwright, P.C., Huskey, S.H., Turingan, R.G., and Carroll, A.M. (2006). Ontogeny of suction feeding capacity in snook, Centropomis undecimalis. Journal of Experimental Zoology. 305A: 246-252.
