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 ID: 660105
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Slide1
Geometric Analysis of Suction FeedingSlide2
The universe is written in the language of mathematicsGalileo Galilei, 1623Quantitative analysis of natural phenomena is at the heart of scientific inquiry
Nature provides a tangible context for mathematics instruction
Math & NatureSlide3
ContextThe 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 ContextSlide4
Context-Specific Learning Facilitates experiential and associative learningDemonstration, activation, application, task-centered, and integration principles (Merrill 2002)
Facilitates generalization of principles to other contexts
The Importance of ContextSlide5
Geometry & BiologyBiological structures vary greatly in geometry and therefore represent a platform for geometric educationGeometric variability functional variability ecological variability
Mechanism for illustrating the consequences of geometry
Math & NatureSlide6
Vertebrate skulls vary greatly in form & function
Math & Nature
www.digimorph.orgSlide7
Vertebrate skulls vary greatly in form & function
Math & Nature
csi.whoi.edu
www.digimorph.org
22 bones
1 moving part
~50 bones
~7 moving partsSlide8
Vertebrate skulls vary greatly in form & function
Math & Nature
Liem
et al. (2001)Slide9
Vertebrate skulls vary greatly in form & functionMoveable parts of the fish skull are responsible for the diversity of feeding mechanisms in fishJaw protrusion in the sand tiger shark Carcharias
taurus
Math & Nature
D. HuberSlide10
Vertebrate skulls vary greatly in form & functionMoveable parts of the fish skull are responsible for the diversity of feeding mechanisms in fishJaw protrusion in the sling-jaw wrasse Epibulus
insidiator
Math & Nature
P. WainwrightSlide11
Fish feeding mechanismsFilter BitingRam Suction
Math & Nature
www.true-wildlife.blogspot.com
C. Fallows
www.z00n.netSlide12
Fish feeding mechanismsFilter feedingMath & Nature
W.
Mischler
2013
Motta et al. (
2010)Slide13
Fish feeding mechanisms
Filter feeding
Math & Nature
Motta et al. (
2010)Slide14
Fish feeding mechanismsFilter feedingMath & Nature
P. MottaSlide15
Fish feeding mechanismsBitingMath & NatureSlide16
Fish feeding mechanismsBitingMath & Nature
www.digimorph.orgSlide17
Fish feeding mechanismsRam feedingMath & Nature
D. Huber
C. FallowsSlide18
Fish feeding mechanisms
Ram feeding
Math & Nature
S.
Huskey
www.tennesseeaquarium.comSlide19
Fish feeding mechanisms
Ram feeding
Math & Nature
D. HuberSlide20
Fish feeding mechanisms
Suction feeding
Most common fish feeding mechanism
Water cohesion
Suction performance
Math & Nature
D. HuberSlide21
Fish feeding mechanisms
Suction feeding
http://
www.youtube.com/user/Wainwrightlab
Math & Nature
Wainwright et al (2006)Slide22
Fish feeding mechanisms
Suction feeding
http://www.youtube.com/user/Wainwrightlab
Math & Nature
Svanback
et al (2002)Slide23
Fish feeding mechanisms
Suction feeding
http://www.youtube.com/user/Wainwrightlab
Math & Nature
Grubich
(2001)Slide24
Fish feeding mechanisms
Suction feeding
Anterior
posterior expansion
Math & Nature
Gibb & Ferry-Graham (2005)
Wainwright et al (
2006)Slide25
Fish feeding mechanisms
Suction feeding
Fluid flow
Math & Nature
Holzman
et al (
2008)Slide26
Fish feeding mechanismsSuction feedingFluid pressure and movement speed
Math & Nature
Svanback
et al (2002)Slide27
Fish feeding mechanismsSuction feedingFeeding ecology
Math & Nature
Motta et al (2008)Slide28
Fish feeding mechanisms
Suction feeding
Geometric modeling
Math & Nature
Van
Wassenbergh
et al
(2007)
Bishop et al (2008
)Slide29
Fish feeding mechanisms
Suction feeding
Goliath grouper
Epinephelus
itajara
Math & NatureSlide30
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 & NatureSlide31
Geometry & BiologyNGSSSMA.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 & NatureSlide32
Geometry & BiologyNGSSSMA.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 & NatureSlide33
Geometry & BiologyCCSSMACC.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 & NatureSlide34
Geometry & BiologyCCSSMACC.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 & NatureSlide35
Goliath grouper modelObjectiveDetermine the velocity of water flow into the mouth
Procedure
Determine the volume of components A and B at rest (
t
0) and at maximum expansion (t
1
)
t
0
= time at
rest
t
1
= time at maximum expansion Determine the volume change during feedingMath & Nature
B
A
B
ASlide36
Goliath grouper modelObjectiveDetermine the velocity of water flow into the mouth
Procedure
Determine the area of the mouth
at maximum expansion (t
1)t
1
= time at maximum expansion
Math & Nature
B
A
B
ASlide37
Goliath grouper modelObjectiveDetermine the velocity of water flow into the
mouth
Procedure
Math & Nature
B
A
B
ASlide38
Suction feeding in the goliath grouperGivenDimensions of cones A and B at rest (t0)
Find
the
volume of the goliath grouper feeding mechanism at rest
(t
0
)
.
Math & Nature
b
a
c
d
e
aSlide39
Suction feeding in the goliath grouperGivenDimensions of cones A and B at rest (t0)
Find
the
volume of the goliath grouper feeding mechanism at rest
(t
0
)
.
b
a
c
d
e
a
Math & NatureSlide40
Goliath Grouper Suction Feeding
Time 0
Cone A
Length (mm)
Area (
mm
2
)
Volume (
mm
3
)
a
34.9
N/A
b
153.6
Cone B
Length (mm)
Area (
mm
2
)
Volume (
mm
3
)
a
34.9
N/A
c
54.3
d
6.4
e
Volume of feeding mechanism before
expansion (t
0
)
Time 1
Cone A
Length (mm)
Area (
mm
2
)
Volume (
mm
3
)
a
N/A
b
Cone B
Length (mm)
Area (
mm
2
)
Volume (
mm
3
)
a
c
d
e
Volume of feeding mechanism at maximum expansion (t
1
)
Volume change during feeding event (
mm
3
)
Duration of
feeding event (sec)
0.132
Area of mouth at
maximum expansion (
t
1
)
(
mm
2
)
Velocity of water flow into mouth (mm/sec)
Slide41
Goliath Grouper Suction Feeding
Time 0
Cone A
Length (mm)
Area (
mm
2
)
Volume (
mm
3
)
a
34.9
N/A
195916.8
b
153.6
Cone B
Length (mm)
Area (
mm
2
)
Volume (
mm
3
)
a
34.9
N/A
84289.7
c
54.3
d
6.4
e
12.2
Volume of feeding mechanism before
expansion (t
0
)
180206.5
Time 1
Cone A
Length (mm)
Area (
mm
2
)
Volume (
mm
3
)
a
N/A
b
Cone B
Length (mm)
Area (
mm
2
)
Volume (
mm
3
)
a
c
d
e
Volume of feeding mechanism at maximum expansion (t
1
)
Volume change during feeding event (
mm
3
)
Duration of
feeding event (sec)
0.132
Area of mouth at
maximum expansion (
t
1
)
(
mm
2
)
Velocity of water flow into mouth (mm/sec)
Slide42
Suction feeding in the goliath grouper
Given
Dimensions of cones A and B at maximum expansion (t
1
)
Find
the
volume of the goliath grouper feeding mechanism at maximum expansion
(t
1
)
.
Math & NatureSlide43
Goliath Grouper Suction Feeding
Time 0
Cone A
Length (mm)
Area (
mm
2
)
Volume (
mm
3
)
a
34.9
N/A
195916.8
b
153.6
Cone B
Length (mm)
Area (
mm
2
)
Volume (
mm
3
)
a
34.9
N/A
84289.7
c
54.3
d
6.4
e
12.2
Volume of feeding mechanism before
expansion (t
0
)
280206.5
Time 1
Cone A
Length (mm)
Area (
mm
2
)
Volume (
mm
3
)
a
39.5
N/A
b
161.3
Cone B
Length (mm)
Area (
mm
2
)
Volume (
mm
3
)
a
39.5
c
56.4
d
32.6
e
Volume of feeding mechanism at maximum expansion (t
1
)
Volume change during feeding event (
mm
3
)
Duration of feeding event (sec)
0.132
Area of mouth at
maximum expansion (
t
1
)
(
mm
2
)
Velocity of water flow into mouth (mm/sec)
Slide44
Goliath Grouper Suction Feeding
Time 0
Cone A
Length (mm)
Area (
mm
2
)
Volume (
mm
3
)
a
34.9
N/A
195916.8
b
153.6
Cone B
Length (mm)
Area (
mm
2
)
Volume (
mm
3
)
a
34.9
N/A
84289.7
c
54.3
d
6.4
e
12.2
Volume of feeding mechanism before
expansion (t
0
)
280206.5
Time 1
Cone A
Length (mm)
Area (
mm
2
)
Volume (
mm
3
)
a
39.5
N/A
263547.1
b
161.3
Cone B
Length (mm)
Area (
mm
2
)
Volume (
mm
3
)
a
39.5
230974.7
c
56.4
d
32.6
e
266.5
Volume of feeding mechanism at maximum expansion (t
1
)
494521.7
Volume change during feeding event (
mm
3
)
214315.3
Duration of feeding event (sec)
0.132
Area of mouth at
maximum expansion (
t
1
)
(
mm
2
)
Velocity of water flow into mouth (mm/sec)
Slide45
Suction feeding in the goliath grouperGivenDimensions of cone B at maximum expansion (t1)
Find
the
area of the goliath grouper mouth at maximum expansion
(t
1
)
.
Math & Nature
A. Collins
mouthSlide46
Goliath Grouper Suction Feeding
Time 0
Cone A
Length (mm)
Area (
mm
2
)
Volume (
mm
3
)
a
34.9
N/A
195916.8
b
153.6
Cone B
Length (mm)
Area (
mm
2
)
Volume (
mm
3
)
a
34.9
N/A
84289.7
c
54.3
d
6.4
e
12.2
Volume of feeding mechanism before
expansion (t
0
)
280206.5
Time 1
Cone A
Length (mm)
Area (
mm
2
)
Volume (
mm
3
)
a
39.5
N/A
263547.1
b
161.3
Cone B
Length (mm)
Area (
mm
2
)
Volume (
mm
3
)
a
39.5
230974.7
c
56.4
d
32.6
e
266.5
Volume of feeding mechanism at maximum expansion (t
1
)
494521.7
Volume change during feeding event (
mm
3
)
214315.3
Duration of feeding event (sec)
0.132
Area of mouth at
maximum expansion (
t
1
)
(
mm
2
)
Velocity of water flow into mouth (mm/sec)
Slide47
Goliath Grouper Suction Feeding
Time 0
Cone A
Length (mm)
Area (
mm
2
)
Volume (
mm
3
)
a
34.9
N/A
195916.8
b
153.6
Cone B
Length (mm)
Area (
mm
2
)
Volume (
mm
3
)
a
34.9
N/A
84289.7
c
54.3
d
6.4
e
12.2
Volume of feeding mechanism before
expansion (t
0
)
280206.5
Time 1
Cone A
Length (mm)
Area (
mm
2
)
Volume (
mm
3
)
a
39.5
N/A
263547.1
b
161.3
Cone B
Length (mm)
Area (
mm
2
)
Volume (
mm
3
)
a
39.5
3338.8
230974.7
c
56.4
d
32.6
e
266.5
Volume of feeding mechanism at maximum expansion (t
1
)
494521.7
Volume change during feeding event (
mm
3
)
214315.3
Duration of feeding event (sec)
0.132
Area of mouth at
maximum expansion (
t
1
)
(
mm
2
)
3338.8
Velocity of water flow into mouth (mm/sec)
Slide48
Suction feeding in the goliath grouperGivenVolume of the goliath grouper feeding mechanism at rest (t
0
) and at maximum expansion (t1
)
Duration of the feeding event (t
1
- t
0
)
Area of the mouth opening at maximum
expansion (t
1
)
Find
the
velocity of water flow into the mouth of the goliath grouper during suction feeding.
Math & NatureSlide49
Goliath Grouper Suction Feeding
Time 0
Cone A
Length (mm)
Area (
mm
2
)
Volume (
mm
3
)
a
34.9
N/A
195916.8
b
153.6
Cone B
Length (mm)
Area (
mm
2
)
Volume (
mm
3
)
a
34.9
N/A
84289.7
c
54.3
d
6.4
e
12.2
Volume of feeding mechanism before
expansion (t
0
)
280206.5
Time 1
Cone A
Length (mm)
Area (
mm
2
)
Volume (
mm
3
)
a
39.5
N/A
263547.1
b
161.3
Cone B
Length (mm)
Area (
mm
2
)
Volume (
mm
3
)
a
39.5
3338.8
230974.7
c
56.4
d
32.6
e
266.5
Volume of feeding mechanism at maximum expansion (t
1
)
494521.7
Volume change during feeding event (
mm
3
)
214315.3
Duration of feeding event (sec)
0.132
Area of mouth at
maximum expansion (
t
1
)
(
mm
2
)
3338.8
Velocity of water flow into mouth (mm/sec)
486Slide50
Wainwright et al (2006)
Suction feeding in the
snook
Centropomus
undecimalis
Math & Nature
Korhnak
JJ PhotoSlide51
Suction feeding in the snook Centropomus undecimalis
Given
Dimensions of
cones A and
B at rest (
t
0
) and at maximum
expansion of the feeding mechanism (t
1
)
Duration of the feeding event (t
1
- t0)
Find the velocity of water flow into the mouth of the
snook
during suction
feeding.
Math & NatureSlide52
Snook
Suction Feeding
Time 0
Cone A
Length (mm)
Area (
mm
2
)
Volume (
mm
3
)
a
2.1
N/A
b
27.6
Cone B
Length (mm)
Area (
mm
2
)
Volume (
mm
3
)
a
2.1
N/A
c
12.3
d
1.8
e
Volume of feeding mechanism before
expansion (t
0
)
Time 1
Cone A
Length (mm)
Area (
mm
2
)
Volume (
mm
3
)
a
7.0
N/A
b
28.9
Cone B
Length (mm)
Area (
mm
2
)
Volume (
mm
3
)
a
7.0
c
12.3
d
5.9
e
Volume of feeding mechanism at maximum expansion (t
1
)
Volume change during feeding event (
mm
3
)
Duration of feeding event (sec)
0.036
Area of mouth at
maximum expansion (
t
1
)
(
mm
2
)
Velocity of water flow into mouth (mm/sec)
Slide53
Snook
Suction Feeding
Time 0
Cone A
Length (mm)
Area (
mm
2
)
Volume (
mm
3
)
a
2.1
N/A
127.5
b
27.6
Cone B
Length (mm)
Area (
mm
2
)
Volume (
mm
3
)
a
2.1
N/A
147.2
c
12.3
d
1.8
e
73.8
Volume of feeding mechanism before
expansion (t
0
)
274.7
Time 1
Cone A
Length (mm)
Area (
mm
2
)
Volume (
mm
3
)
a
7.0
N/A
1482.9
b
28.9
Cone B
Length (mm)
Area (
mm
2
)
Volume (
mm
3
)
a
7.0
109.4
1611.5
c
12.3
d
5.9
e
66.0
Volume of feeding mechanism at maximum expansion (t
1
)
3094.4
Volume change during feeding event (
mm
3
)
2819.7
Duration of feeding event (sec)
0.036
Area of mouth at
maximum expansion (
t
1
)
(
mm
2
)
109.4
Velocity of water flow into mouth (mm/sec)
716Slide54
Suction feeding in the
longjaw
butterfly fish
Forcipiger
longirostris
Math & Nature
Ferry-Graham et al (2001)
S.
HuskeySlide55
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 (t
1
)
Duration of the feeding event (t
1
- t
0
)
Find the velocity of water flow into the mouth of the
longjaw
butterfly fish during
suction
feeding.
Math & NatureSlide56
Longjaw
Butterfly Fish Suction
Feeding
Time 0
Cone A
Length (mm)
Area (
mm
2
)
Volume (
mm
3
)
a
5.0
N/A
b
14.9
Cone B
Length (mm)
Area (
mm
2
)
Volume (
mm
3
)
a
5.0
N/A
c
31.2
d
1.1
e
Volume of feeding mechanism before
expansion (t
0
)
Time 1
Cone A
Length (mm)
Area (
mm
2
)
Volume (
mm
3
)
a
5.0
N/A
b
14.9
Cone B
Length (mm)
Area (
mm
2
)
Volume (
mm
3
)
a
5.0
c
31.6
d
1.1
e
Volume of feeding mechanism at maximum expansion (t
1
)
Volume change during feeding event (
mm
3
)
Duration of feeding event (sec)
0.022
Area of mouth at
maximum expansion (
t
1
)
(
mm
2
)
Velocity of water flow into mouth (mm/sec)
Slide57
Longjaw
Butterfly Fish Suction
Feeding
Time 0
Cone A
Length (mm)
Area (
mm
2
)
Volume (
mm
3
)
a
5.0
N/A
390.1
b
14.9
Cone B
Length (mm)
Area (
mm
2
)
Volume (
mm
3
)
a
5.0
N/A
1036.0
c
31.2
d
1.1
e
8.8
Volume of feeding mechanism before
expansion (t
0
)
1426.1
Time 1
Cone A
Length (mm)
Area (
mm
2
)
Volume (
mm
3
)
a
5.0
N/A
390.1
b
14.9
Cone B
Length (mm)
Area (
mm
2
)
Volume (
mm
3
)
a
5.0
3.8
1049.3
c
31.6
d
1.1
e
8.9
Volume of feeding mechanism at maximum expansion (t
1
)
1439.4
Volume change during feeding event (
mm
3
)
13.3
Duration of feeding event (sec)
0.022
Area of mouth at
maximum expansion (
t
1
)
(
mm
2
)
3.8
Velocity of water flow into mouth (mm/sec)
159Slide58
Suction feeding
Given
Velocities
of water flow into the
mouths of all three fish
Determine which fish is the best suction feeder.
Math & Nature
Korhnak
v(t) = 159 mm/sec
v(t) = 716 mm/sec
v(t) = 486 mm/secSlide59
ReferencesBishop, 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 & NatureSlide60
ReferencesLiem, 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
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Calculating the volume of a truncated cone
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