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

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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