FISHING FOR CONICSHow can the mathematics concepts of conic sections be understood through our marine environment? What are some other applications of mathematics that can be applied to our physical world?

by Brennan Quiambo

High School (10th-12th grade)

Standard Benchmarks and Values: Mathematics Common Core State Standards (CCSS):

• Expressing Geometric Properties with Equations (G-GPE): Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. Derive the equation of a parabola given a focus and directrix. Derive the equation of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant.

Next Generation Science Standards (NGSS):

• Interdependent Relationships in Ecosystems: Animals, Plants, and their environment (K-ESS3-a). Collect, analyze, and use data to describe patterns of what plants and animals (including humans) need to survive.

Nā Honua Mauli Ola (NHMO) Cultural Pathways:

• ‘Ike Na’auao (Intellectual Pathway): We envision generations fostering the cycle of joyous learning through curiosity, inquiry, experience and mentorship. Fostering lifelong learning, curiosity and inquiry to nurture the innate desire to share knowledge and wisdom with others.

Waters of Hawai‘i Institute of Marine Biology (Photo credit:

Quiambao, 2013).

Fishing for Conics

• ‘Ike Honua (Sense of Place Pathway): We envision generations who accept kuleana for our honua. Demonstrating a strong sense of place, including a commitment to preserve the delicate balance of life and protect it for generations to come.

• ‘Ike Pilina – Relationship Pathway

-We envision generations that have, responsible and strong relationships in service to akua, ‘äina and each other.

-Nurturing respectful and responsible relationships that connect us to akua, ‘äina and each other through the sharing of history, genealogy, language and culture.

Enduring Understandings:

• Protecting our island’s marine life is important.• There are many things that we can do to protect not just our marine life but everything that makes Hawai‘i special.• Different types of marine life that exist in Hawai‘i.• Mathematics is everywhere.• Understanding of parabola, ellipse, and hyperbolas.

Background/Historical Context:

Students tend to think of mathematics as, “that difficult, boring subject” that only exists within the walls of a classroom. Many students do not realize that mathematics is all around us occurring in our daily lives. Mathematics can be found easily in everyday activities such as cooking, driving, and even walking.

Ubiratan D’Ambrosio, a Brazilian educator and mathematician, introduced the term “Ethnomathematics”, which is the relationship between culture and mathematics. D’Ambrosio argues that relationship has existed since the beginning of time. By incorporating the concept of ethnomathematics into lesson plans, students will not only have an easier time learning difficult math concepts but they will also gain a better understanding of different cultures and environments. By taking students out of the classroom and allowing them to experience something real and relevant, students will take more interest and become more engaged in the subject matter, in turn, making it much more enjoyable for both teacher and student (D’Ambrosio, 2001).

Brennan Quiambao

When one stumbles across Gorilla Ogo, it’s likely that it’ll be disregarded as seaweed and nothing more. At a glance, one may not even think that such a thing has a microhabitat existing inside it. If we’re ignorant to the little things that are around us, who’s to say that more animals and/or marine life won’t end up like the Laysan Albatross? (The Laysan Albatross is a seabird that is found primarily in the Northwestern part of the Hawaiian Islands. A large population of Albatross die every year because of all the plastic they intake. They mistake the plastic for food.)

Dead Laysan Albatross chick remains with ingested plastic (Photo credit: Wikipedia Commons, 2013).

Authentic Performance Task:• Write a short answer to answer the essential question using the information and lessons learned.

Authentic Audience:• Community, School Community, Parents

Hawai‘i is known for its tropical climates making it the ideal home for all sorts of animals—including marine life. Hawai‘i is not only home to such a diverse population but holds a strong connection to culture through the many plants, animals, and historical sites that inhabit these tiny islands. Just the Hawaiian Islands itself is a story on its own. With that said, this lesson plan will be primarily focused on the marine life located in Hawai‘i—particularly the microhabitat that exists in the Gorilla Ogo (can be found in Kaneohe Bay).

Gorilla Ogo (Photo credit: Quiambao, 2013).

Fishing for Conics

Learning Plan:• In the first activity, the teacher will review conics with the students.

• (Optional: Take the students to the Institute of Marine Biology at Coconut Island and have them do the Bioacoustics Lab. They will learn about all the creatures and get to do all the hands-on activities. Information can be found at http://www.hawaii.edu/himb/index.html.)

• Next, the teacher will engage the students by doing a small lesson showing them the different types of marine life present in our Hawaiian “backyard”. Remind them to try and see if they can identify the different types of conics that marine life exhibit.

• The teacher will then make the students think about the possible applications of math (based on what they just went over) and how it can be applied to marine life.

• In the next segment, the teacher will present silhouette-like pictures of marine life to help facilitate the process in identifying what conics belong with which sea-creature.

• Once identified, students will be given a worksheet and will be given the task of finding the missing information on the worksheet (i.e. focus, foci, major axis, etc.)

• Based on the information that they found, they will be given questions based on the information that just obtained from the worksheet.

Pool where Gorilla Ogo was collected at the Hawai‘i Institute of Marine Biology (Photo credit: Quiambao, 2013).

Brennan Quiambao

Standard Benchmarks, GLOs, or Nā Honua

Mauli Ola Skills Concepts Assessment

Interdependent Relationships in Ecosystems: Animals, Plants, and their Environment (K-ESS3-a)

Obtain information to describe the relationship between the needs of different plants and animals (including humans) and where they live on the land or in the water.

Understanding the relationship between the needs of different plants and animals (including humans) and where they live on land or in the water.

-Short Answer questions

Expressing Geometric Properties with Equations G-GPE: Translate between the geometric description and the equation for a conic section

-Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.

- Derive the equation of a parabola given a focus and directrix.

-(+) Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant.

-Conics

-Hyperbola

-Parabola

-Ellipse

Questions

‘Ike Na‘auao (Intellectual Pathway)

We envision generations fostering the cycle of joyous learning through curiosity, inquiry, experience and mentorship.

Fostering lifelong learning, curiosity and inquiry to nurture the innate desire to share knowledge and wisdom with others.

Essential Question

‘Ike Honua (Sense of Place Pathway)

We envision generations who accept kuleana for our honua.

Demonstrating a strong sense of place, including a commitment to preserve the delicate balance of life and protect it for generations to come.

Essential Question

‘Ike Pilina (Relationship Pathway)

We envision generations that have respectful, responsible and strong relationships in service to akua, ‘äina and each other

Nurturing respectful and responsible relationships that connect us to akua, ‘äina and each other through the sharing of history, genealogy, language and culture.

Essential Question

Fishing for Conics

References:

Common Core State Standards (CCSS). (2013). Common Core State Standards for Mathematics. Retrieved from http://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf

D’Ambrosio, U. (2001). What Is Ethnomathematics, and How Can It Help Children in Schools? Teaching children mathematics, 7(6), 308-10

Derbyshire, D. (2008). Deadly pistol shrimp that stuns prey with sound as loud as concorde found in uk waters. Mail Online. Retrieved from http://www.dailymail.co.uk/sciencetech/article-1085398/Deadly- pistol-shrimp-stuns-prey-sound-loud-Concorde-UK-waters.html

Hawai‘i Institute of Marine Biology (2010). Welcome to the Hawai‘i Institute of Marine Biology. Retrieved from http://www.hawaii.edu/himb/

Monterey Bay Aquarium Foundation (n.d.). Laysan albatross and plastics. Retrieved from http://www.montereybayaquarium.org/cr/oceanissues/plastics_albatross/

Nā Honua Mauli Ola (NHMO). (2010). Cultural Pathways for Culturally Healthy and Responsive Learning Environments. Retrieved from http://www.olelo.hawaii.edu/olelo/nhmo.php

Wikipedia Commons. (1999). Dead Laysan Albatross chick remains with plastic. Retrieved from http://en.wikipedia.org/wiki/File:Laysan_albatross_chick_remains.jpg

10 Search Terms:1. Fishing with Conics2. Conics3. Parabolas4. Ellipses5. Hyperbolas

6. Review of Conics7. Conics Review8. Real-world Applications of Math9. Brennan Quiambao10. Ethnomathematics

Snapping Shrimp (Photo credit: Quiambao, 2013).

Name: Date:

Fishing for Conics

Parabola:x² = ±4ayVertex: (0,0)Focus: (0,±a)Directrix: y =∓ a

y² = ±4axVertex: (0,0)Focus: (±a,0)Directrix: x =∓ a

(x – h)² = ±4a(y – k)Vertex: (h,k) Focus: (h, k±a)Directrix: y = k∓ a

(y – k)² = ±4a(x – h)Vertex: (h,k) Focus: (h±a,k)Directrix: x = h∓ a

Ellipse:

x2

a2+y2

b2 =1

Center: (0,0)Foci: (±c,0)Vertices: (±a,0)Major Axis: x-axisb² = a² − c²

x2

b2 +y2

a2 = 1

Center: (0,0)Foci: (0,±c)Vertices: (0,±a)Major Axis: y-axisb² = a² − c²

(x − h)2

a2+(y− k)2

b2= 1

Center: (h,k)Foci: (h±c,k)Vertices: (h±a,k)Major Axis: to x-axisb² = a² − c²

(x − h)2

b2 +(y− k)2

a2 = 1

Center: (h,k)Foci: (h,k±c)Vertices: (h,k±a)Major Axis: to y-axisb² = a² − c²

Hyperbola:

x2

a2− y2

b2= 1

Center: (0,0)Foci: (±c,0)Vertices: (±a,0)Transverse Axis: x-axisb² = c² − a²

Asymptote: y =±bax

y2

a2 −x2

b2 =1

Center: (0,0)Foci: (0,±c)Vertices: (0,±a)Transverse Axis: y-axisb² = c² − a²

Asymptote: y =±abx

(x − h)2

a2 −(y− k)2

b2 = 1

Center: (h,k)Foci: (h±c,k)Vertices: (h±a,k)Transverse Axis: to x-axisAsymptote:

(y−k) =±bax(x−h)

(x − h)2

b2 −(y− k)2

a2 = 1

Center: (h,k)Foci: (h,k±c)Vertices: (h,k±a)Transverse Axis: to y-axisAsymptote:

(y−k) =±abx(x−h)

Recall:

1.

a.) Math can be applied anywhere. You see below that Mr. Krabs is standing in a parabolic pose (The shape of crabs alone resemble a parabolic shape just like below). The equation of his pose is . Determine the Focus, Vertex, Directrix, and the Value of a.

Vertex: ( , )Focus: ( , )Directrix: y = a =

 

Name: Date:

b. Crabs are known to be detritivores meaning they eat decomposing plants and animal parts. Mr. Krabs is located at the origin. He sees some food floating nearby at the coordinates of (6,6). If his initial equation was , what will his new equation be if he wanted to go eat his fill? Determine the new Focus, Vertex, and Directrix.

Equation:

Vertex: ( , )Focus: ( , )Directrix: y =

c.) Based on the information you obtained from above, how many steps to the right and how many steps up did Mr. Krabs have to make in order to reach his destination?

2.

a.) When Snapping Shrimp make a snap, it shoots a cavitation bubble (the area of effect is an elliptical shape) up to 4cm away and about 3mm wide. The bubble travels at approximately 60mph. For a brief moment, the bubble reaches a temperature of approximately 5,000°F (that’s almost as hot as the sun!). Snapping Shrimp use this weapon to stun their prey and can also be used for communication.

Using the space below to graph the cavitation bubble’s area of effect based on the information above. Use the space below to find the Equation of the Ellipse, Center, Foci, Vertices, and the Major Axis of the graph centered at the origin.

 

Food

Name: Date:

3.a.) Iridescent Fireworms are found in the Gorilla Ogo of Kaneohe bay. They are part of the

microhabitat that exists within the Gorilla Ogo. There are two Iridescent Fireworms sleeping in a hyperbolic shape (an example of their sleeping formation is below). When put onto a graph, their shape can be described as x

2

16 − y2

16 = 1.

Graph their hyperbolic shape and find the Center, Foci, Vertices, and Asymptotes.

b.) If a wave were to push the left fireworm so that it would roll, how many times would it roll before it reached the exact position as the fireworm? (How many times does it have to be reflected before it reaches the right side?)