NAVIGATING THE HAWAIIAN CANOE

High School:
10th-12th Grade
NAVIGATING THE
HAWAIIAN CANOE
THROUGH THE OCEAN
Why does the shape of the Hawaiian Outrigger Canoe varies?
How do we differentiate between rounded and “V-shaped” hulls?
How do we differentiate between speed and carrying capacity of canoes?
by Milena Boritz
Standard Benchmarks and Values:
Mathematics Common Core State Standards
(CCSS):
• Geometric Definition of Parabola.
• Equations of Parabola.
• Analyze Parabola with Vertex at the Origin (0,0).
• Differentiate Horizontal and Vertical Axis of Symmetry.
• Focus of Parabola; Latus Rectum; Directrix.
Nā Honua Mauli Ola (NHMO) Cultural
Pathways:
• ‘Ike Mauli Lāhui (Cultural Identity Pathway):
Perpetuating Native Hawaiian cultural identity through
practices that strengthen knowledge of language,
culture and genealogical connections to akua, ‘āina and
kanaka.
• ‘Ike Honua (Sense of Place Pathway):
Demonstrating a strong sense of place,
including a commitment to preserve the
delicate balance of life and protect it for
generations to come.
• ‘Ike Na‘auao (Intellectual Pathway):
Fostering lifelong learning, curiosity and
inquiry to nurture an innate desire to share
knowledge and wisdom with others.
• ‘Ike Ola Pono (Wellness Pathway): Caring
for the wellbeing of the spirit, na‘au and
body through culturally respectful ways that
strengthen one’s mauli and build responsibility
for healthy lifestyles.
• ‘Ike Piko‘u (Personal Connection Pathway):
Promoting personal growth, development
and self-worth to support a greater sense of
belonging compassion and service toward
oneself, family and community.
Enduring Understandings:
• Recognize parabolic shapes in everyday life.
• Identify horizontal or vertical axis of
symmetry.
• Be able to apply equations of parabola.
• Find the vertex, focus, directrix.
Background/Historical Context:
Polynesian voyaging canoes were built
centuries ago. The voyaging canoes were
sailed from Western Polynesia, Samoa,
towards Eastern Polynesia, Hawai‘i, between
BC 500 – AD 600. The Polynesian voyaging
canoes were designed to last long distances
and transport people, food, plants, animals,
culture and traditions.
Canoe designs vary extensively from
island to island. Each island group had made
unique improvements to the canoe design to
meet the challenges of local sailing conditions
and timber resources. Major differences are
observed in hull shape, paddle shape, sail
shape, number of hulls, and number of floating
outrigger. The main differences in the shape
of a canoe hull originate from the practical
purpose of the vessel – sailing or paddling.
Sailing canoes require deeper keel - rounded
“V”-shape, to provide a greater carrying
capacity and ocean stability for longer
voyages, and in various wind conditions.
In contrast, paddling canoes are built with
rounded keels that have less depth than the
sailing “V”-shaped hull. Paddling canoes are
built for greater maneuverability that offers
quick and easy turns.
Authentic Performance Task:
Task #1: The Hawaiian paddling outrigger canoe has parabolic-shape hull (bottom). The
waterline of a canoe is at focus, or at 0.125 feet height (waterline-to-bottom). The width at seat
level is 1.6 feet. Find the canoe height (in inches) at seat level (seat-to-bottom).
Navigating the Hawaiian Canoe
Solution:
Equation:
Vertex: (0,0)
Axis of Symmetry: Y-axis
Focus: (0,0.125)
Parabola: Opens Up
Directrix: y = -0.125
Latus Rectum: F (0, 0.125)
Width at Seat level: 1.6 feet
Height at Waterline: 0.125 feet
The height of the canoe is: 1.28 feet = 15.36 inches
Task #2: The Hawaiian paddling outrigger canoe has its hull (bottom) shaped in a parabolic form.
The waterline of a canoe is at focus, or at 0.1 feet height (waterline-to-bottom). The height of the canoe
is (top-to-bottom) is 1.5 feet. Find the canoe width (in inches) on top.
Solution:
Equation:
Vertex: (0,0)
Axis of Symmetry: Y-axis
Focus: (0,0.1)
Parabola: Opens Up
Directrix: y = -0.1
Latus Rectum: F (0, 0.1)
Width at Top: 1.5 feet
Height at Waterline: 0.1 feet
The canoe width on top is: 2(x), or 2(0.7746) = 1.5492 feet = 18.6 inches.
Milena Boritz
√
Task #3: The Hawaiian paddling outrigger canoe has its hull (bottom) shaped in a parabolic form.
The width at top of a canoe is 1.34 feet (top-to-bottom). The height of the canoe is (top-to-bottom) is one
foot. Find the focus of the parabolic canoe hull (in inches).
Solution:
Equation:
Vertex: (0,0)
Focus: (0,a)
Directrix: y = - a
Axis of Symmetry: Y-axis
Parabola: Opens Up
Latus Rectum: F (0, a)
Width at Top: 1.34 feet
Height Top-to-Bottom: 1 foot
The focus of a canoe is: 0.11225 feet = 1.3467 inches
Task #4: Akela (Happy) and Kahewai (Flowing Water) keep their canoes in the same storage hale.
Akela’s canoe is measurements are: width (beam) 1.5 feet and focus at 0.1 feet. Kahewai’s canoe is
measurements are: width (beam) 1.6 feet and focus at 0.13 feet.
a. If the waterline of Akela’s canoe is at focus, find the distance top-to-bottom of the canoe. Or,
what is the depth of the canoe?
b.If the waterline of Kahewai’s canoe is at focus, find the distance top-to-bottom of the canoe.
Or, what is the depth of the canoe?
c. Based on the above information, which of the two canoes has deeper keel and greater
carrying capacity for long voyages?
A. Solution:
Equation:
Vertex: (0,0)
Axis of Symmetry: Y-axis
Focus: (0,0.1)
Parabola: Opens Up
Directrix: y = - 0.1
Latus Rectum: F (0, 0.1)
Width at Top: 1.5 feet
Focus: 0.1 feet
The depth of Akela’s canoe is 1.4 feet.
Navigating the Hawaiian Canoe
B. Solution:
Equation:
Vertex: (0,0)
Axis of Symmetry: Y-axis
Focus: (0,0.13)
Parabola: Opens Up
Directrix: y = - 0.13
Latus Rectum: F (0, 0.13)
Width at Top: 1.6 feet
Focus: 0.13 feet
The depth of Kahewai’s canoe is 1.23 feet.
C. Solution:
Akela’s canoe is 1.4 feet deep; and Kahewai’s canoe is 1.23 feet deep.
Evidently, Akela’s canoe is deeper and it has a greater carrying capacity than Kahewai’s canoe.
Authentic Audience:
Students, parents, and community members.
The assigned tasks are designed to familiarize new paddlers with the canoe shape.
The background information is designed to educate both students and their ‘ohana.
Learning Plan:
1. Students visit a canoe club or a shipyard to
examine various canoe hulls. Field trip day for
credit.
6. Examine the parabolic shape and analyze
the Axis of Symmetry; Origin; Latus Rectum;
Directrix.
2. Listen to historical lecture on Hawaiian
voyaging and the cultural impact on the
Hawaiian people.
7. Draw conclusion of Parabolic Equation
type. Apply Analytic Geometry knowledge
and skills of Conics (Parabola) to the
assigned tasks.
3. Paddle out on a sailing or paddling canoe to
experience the outrigger Hawaiian canoe.
4. Write a reflection about the paddling or
sailing experience from the field trip day.
Students share their prior knowledge and skills
to reflect the new experience.
5. Draw a parabolic conic shape of a canoe
hull as if canoe body is bisected.
Milena Boritz
8. Visit the Bishop Museum Polynesian and
Hawaiian exhibits.
9. Perform research on Polynesian
voyaging canoes and present stories and
pictures to the classmates.
10. Prepare a group presentation on
popularity of Hawaiian canoes today.
References:
Dierking, G. (2007). Building Outrigger Sailing Canoes: Modern Construction Methods for Three Fast, Beautiful Boats. McGraw-Hill Professional Pub.
Davis, D. (1997). Build Your Own Canoe. Crowood Press, Ltd.
West, S. (2013). Outrigger Canoeing: The Art and skill of Steering. Batini Books Pub.
Holmes, T. (1993). The Hawaiian Canoe. Editions Ltd.
D’Ambrosio, U. (2001). Ethnomathematics: Link between traditions and Modernity. Amsterdam: Sense Publishers.
Navigating the Hawaiian Canoe