**LESSON 1 (**

**Approximately 2 Hours)**

**THE MATHEMATICAL UNDERPINNING OF PARABOLAS & PARABOLOIDS**

*or*THE PHYSICAL PROPERTIES BEHIND PARABOLIC SOLAR MELTING DEVICES

*Overview*- Juxtaposing ‘parabolas’ with ‘paraboloids’
- Investigating and devising methods by which to construct ‘paraboloids’
- Understanding the relationship between dilation value and focal length (i.e. distance between vertex and focus)
- Experimenting with the reflectivity, malleability, pliability, and elasticity of various materials (e.g., aluminum foil, aluminum flashing, wood, glass, and mirrors)
- Grouping students into teams of four to begin brainstorming designs for a parabolic solar melting device

*Materials*Large whiteboard Blank white paper

Individual whiteboards Professionally constructed parabolic satellite dish

Expo markers Non-parabolic solar oven

Aluminum foil 'Google Documents’

Aluminum tape ‘Popplet’

Aluminum flashing ‘Google SketchUp’

Wood ‘Rhinoceros’

Mirrors Digital projector

Glass Projector screen

Graph paper Student and teacher laptops

*Key Words*Conic | Parabola | Paraboloid | Focus/Focal length | Directrix | Axis of symmetry

Dilation value | Reflectivity | Malleability | Pliability | Elasticity

__TEACHING PLAN__

*Introduction*This class period represents a transition between a chalk-talk style class period dedicated to the basic mathematical abstractions of parabolas and a project-based learning unit dedicated to investigating one of the applications for parabolas: solar ovens. Prior to launching into engineering design, members of the class will have opportunities to explore how a two dimensional parabola can be extended into the three dimensional realm as a paraboloid. Some of the essential mathematical relationships embedded in parabolas (e.g., the relationship between dilation value, its graphical rendering, and focal length) are explored as are potential building materials.

__ASK Questions and discuss as a class__(5 minutes)

- In pairs, name as many conics as you can recall from last class.
- In your journal, brainstorm what the ‘oid’ ending signifies in conics given the images of a paraboloid, hyperboloid, and ellipsoid projected on screen.

*Task*- In pairs,
two or more ways to construct a three-dimensional paraboloid. Feel free to use individual whiteboards to draw your ideas or Google SketchUp. (10 minutes)*imagine* your ideas and drawings with a neighboring pair of students. (5 minutes)*Share*- Head outside the classroom. In teams of four (using aluminum flashing and a white piece of paper)
with how light reflects off the flashing and onto a white piece of paper.*experiment*what happens when you bend and straighten the aluminum flashing. How does the shape of the flashing influence the reflection of light onto the piece of paper? At what distance from the aluminum flashing does the reflected light appear sharpest and brightest on the white paper? (10 minutes)*Observe* - As a class,
your observations and ideas on how to construct a three-dimensional paraboloid. Challenge: Can we describe a mathematical relationship between the curvature of the aluminum flashing (i.e. ‘dilation value’) and the location where the reflected light appears sharpest and or brightest on the white paper (i.e. distance from vertex to focus; ‘focal length’)? (10 minutes)*share* - As a class,
the inverse relationship between ‘dilation value’ of a parabola and the ‘focal length’ (5 minutes) and the algorithm used to calculate focal length given dilation value and vise versa. (10 minutes)*discuss* - Head outside the classroom once more.
with the reflectivity, malleability, pliability, and elasticity of wood, aluminum foil, aluminum tape, aluminum flashing, glass, and reflective mirrors. Decide how you will communicate your findings to the rest of the class.*Experiment*how your findings will eventually influence your design choices as we will begin constructing parabolic solar melting devices/ovens at the beginning of next week. (15 minutes)*Consider*

*Wrap Up*__Discuss__

- How can one transform a two-dimensional conic section into a three-dimensional structure?
- How does the curvature of three-dimensional, curved structures influence the location of its focal point(s)?
- How do materials influence the reflectivity of light off of three-dimensional, curved structures?
- Review homework deliverables due next class:
- View all of the suggested online videos and websites on solar melting devices. (See
*Distribute*) - Complete Algebra 2 Worksheet: Vertex, Factored, and Standard Form of Quadratic Functions
- ‘NetLogo’ parabola interactive through Northwestern University: http://ccl.northwestern.edu/netlogo/models/ConicSections2
- Create a 3/4 to 1 page summary of what you learned from the class period and from your classmates.

- View all of the suggested online videos and websites on solar melting devices. (See

__Distribute__

- List of suggested websites and videos on solar melting devices and their uses:
- Solar Cookers International: http://www.solarcookers.org/involved/basics/
- How Much Solar Energy Hits Earth: http://www.ecoworld.com/energy-fuels/how-much-solar-energy-hits-earth.html
- BBC: Bang Goes The Theory: https://www.youtube.com/watch?v=z0_nuvPKIi8

- Algebra 2 Worksheet: Vertex, Factored, and Standard Form of Quadratic Functions
- Grading rubric for 3/4 to 1 page summary on what students learned from the class period and from classmates
- Link to ‘NetLogo’ parabola interactive through Northwestern University: http://ccl.northwestern.edu/netlogo/models/ConicSections2

__Web Resources__

- Solar Cookers International: http://www.solarcookers.org/involved/basics/
- How Much Solar Energy Hits Earth: http://www.ecoworld.com/energy-fuels/how-much-solar-energy-hits-earth.html
- BBC: Bang Goes The Theory: https://www.youtube.com/watch?v=z0_nuvPKIi8
- ‘NetLogo’ parabola interactive through Northwestern University: http://ccl.northwestern.edu/netlogo/models/ConicSections2

__Formative Assessment__

, assess that students are engaged in class discussions and actively involved in their pair discussions.*In class*: Algebra 2 Worksheet: Vertex, Factored, and Standard Form of Quadratic Functions (graphing parabolas and expressing quadratic functions in vertex, factored, and standard form)*Homework*

__Summative Assessment__

- 3/4 – 1 page summary of what students learned from the class period and from their classmates.