### Lesson Review

ETP Title: | Designing a Solar Array- An Area Optimization Problem |

Organization: | Lockheed |

ETP Type: | Enhance Existing Curriculum |

Grade Level(s): | 9,10,11,12 |

Subject Area(s): | Science,Mathematics |

### Lesson Abstract:

Upon completion of this project, students will be able to determine the most efficient shape of a silicon photovoltaic cell to use in order to build a solar array. Students will arrive at this shape through

- calculating the areas of certain polygons including squares, rectangles, trapezoids, hexagons, and circles;
- determining which of the polygons most efficiently fills a large rectangle (the solar array)
- determining which of the polygons makes the most efficient use of a circular silicon wafer
- determining the effective cost of a solar array given the shape of the wafer chosen and the size of the rectangular array said wafer is intended to fill.

This project is most suitable for high school geometry classes. It could be also be used as an enrichment exercise in a science class after a lesson on renewable energy.

### California Standards

Geometry 8.0 Students know, derive, and solve problems involving the perimeter, circumference, area, volume, lateral area, and surface area of common geometric figures.10.0 Students compute areas of polygons, including rectangles, scalene triangles, equilateral triangles, rhombi, parallelograms, and trapezoids.

### Measurable Objectives:

Students will gain competence/mastery in the calculation of areas of polygons and circles. In addition, they will evaluate the appropriateness of design choices with respect to product function of solar cells and arrays.

### Assessment:

Solar Array Rubric (attachment 5)

### The connection between the ETP and Fellowship. :

My responsibilities are in the Solar Array Manufacturing Division of Lockheed where we assemble Solar Arrays for a number of satellites. The process involves laying groups of cells in a specific arrangement onto a surface and bonding them in place. I have observed differing geometries of photovoltaic cells on the arrays for different satellites and found myself wondering "why are they using those shapes" of cells? This question is a good optimization problem and represents a concept math students encounter in second year algebra. Fitting the various shapes to fill a larger area is a good geometry exercise. Put the two exercises together and students can get experience with a real life exercise requiring the direct use of algebra and geometry.

### Instructional Plan:

__Content__

In this exercise students will:

- Find the area of a given rectangle, trapezoid, hexagon, circle, and square.
- Determine what percentage of a circular “cell” each of the above shapes takes up.
- Produce a physical model of a solar “array” based upon a trapezoid, hexagon, circle or square cell shape.
- Calculate the efficiency of an arrangement of cell shapes.
- Calculate a corresponding cost and electrical output of the final solar “array”

__Prerequisite Knowledge__

Ideally students should already have familiarity with area formulas for various polygons and circles.

__Materials__

- Scissors – 1 pair/student
- Glue or tape
- 1 piece of 8 ½ x 11 cardstock (this is the solar “array”) per student
- Photocopies of solar “cells” as follows:
- 2 sheets each of hexagon cells, circular cells, and square cells;
- 3 sheets of trapezoid cells
- Parts I and II Question Handouts
- Parts III -> V Question Handouts

__Outline__

- Start the class with images of a variety of solar cells and arrays – pictures taken from various links provided – to include cells on space stations, homes, businesses. Show both individual cells and arrays of cells.
- Ask students to notice that not all cell shapes are the same. Also show students pictures of cylindrical ingots of silicon, the main ingredient of solar cells. Ask them if they can visualize what shape results when the cylinder is sliced like a loaf of bread into very thin pieces: Answer: circles.
- Explain that many (but not all) solar cells come from this beginning shape.
- Ask: “What must be done to a circle to produce a square, a hexagon, a trapezoid?” Answer: remove part of it.
- Ask: “Which of the given shapes—circles, hexagons, squares, trapezoids –would make the most use of a circular piece of silicon?” Answer: Circle, followed by hexagon, and two trapezoids.
- Ask students to compare the shapes of each of the arrays pictured and notice they all have the same basic shape. “A rectangle”.
- Relate that solar
**arrays**are almost always rectangular but solar**cells**vary in shape, performance and cost. Given these variables, we wish to build an “array” using different shaped cells and determine the performance and cost of each.

__Procedure__

- With students working in pairs, give each student one 8 ½ by 11 inch piece of cardstock, scissors and tape/glue. Allow them to choose which shape of cell they wish to use for their array insuring that each student in the pair uses a different shape.
- Give each student the required number of cell handouts according to their shape choice -- 2 sheets for anything but the trapezoid—trapezoids get 3 sheets.
- With the question sheet as their guide, students are to first compute the area of the rectangle (showing all work) and the area of their cell shape using the dimensions given on each figure. Area formulas which students should recall are

A= bh rectangle/square;

A = πr^{2} circle (use 3.14 for π);

A = ½ san hexagon;

A= ½ (b_{1} + b_{2}) h trapezoid.

- By dividing the area of the rectangle by the area of their cell have students estimate how many cells should be able to fit onto the rectangle.
- Check the results of the computation. Once you are satisfied that their area answers are correct, then give them scissors and tape and charge them with the task of fitting as many cells as possible onto their rectangle. They are not allowed to alter the shape of any individual cell and no part of any cell can overlap another or extend past the edge of the rectangle. .
**NOTE: there may be more than one configuration of cells possible. The solutions included are guides. The object is for students to try to get whatever maximum they can find.** - Once they have arrived at a number, they are to record their results on their question handout and glue/tape their cells in place in whatever configuration they arrived at. Ask them if this number is higher, lower, or the same as the estimate they made in step 4 above (it will be lower). Ask them why this is? (the estimate computation assumes that the rectangle’s area can be completely covered by the cells with nothing left uncovered).
- Once they have completed the computations for their and their partner’s cells shapes, they then gather the same data for the other cell shapes (Part II on the handout).
- As a teaching/classroom management strategy you may require students to complete parts I and II correctly before proceeding on to Parts III- V. You may have them submit their results for check/verification, or you may choose to have Parts I – V assembled as one large packet.
- Be sure to carefully check and be ready to assist the students work relating to the hexagons. Students frequently encounter difficulty determining the apothem length. Remember Pythagoras! Also use care when checking students’ work to insure they have shown not only the results, but the correct formulas and substitutions.
- Assessment may be performed using the enclosed rubric.

### Supplies:

8.5 x 11" Cardstock Scissors Tape/Glue Photocopies of Cell Shapes (included) Photocopies of images of solar arraysTime required: (1=2 50 minute periods, or 1 90 minute block)

### Bibliographic or other resources you used in creating this curriculum:

"Solar Cell." *Wikipedia*. 26 July 2011. Web.

Fairley, Peter. "Ultraefficient Photovoltaics." *Technology Review* (15 June 2007): 1-2. Web. 8 Aug 2011.

Masi, G.G. "Solar Screening." *Vision Systems Design* 01 July 2009: n. pag. Web. 8 Aug 2011.

"Slicing solar power costs with new wafer-cutting method." *Reliable Plant* 1. Web. 8 Aug 2011.

"Wafering." *SolarWorld*. SolarWorld AG, n.d. Web. 8 Aug 2011.

Lenardic, Denis. "Photovoltaic Technologies." *Solar cell materials - production and features*. PV Resources, 22 November 2008. Web. 8 Aug 2011.

Teschler, Leland. "Next Big Challenge for PV Makers: Wafer Handling." *MachineDesign.com* 10 July 2008: 1. Web. 8 Aug 2011.

"Gaining on the Grid." *BP Frontiers* August 2007: 1. Web. 8 Aug 2011.

Secrest, Rose. "Solar Cell." *How Products are Made*. 1. Advameg, 2011. Web.

Chourdia, Prateek. "Solar Panel Installations are changing! How? Why?." *Solar Choice* 07 May 2010: 1. Web. 8 Aug 2011.

"Integrated Truss Structure." *Wikipedia*. 10 July 2010. Web.

"Assembly of the International Space Station." *Wikipedia*. 30 July 2011. Web.

Zordak, Samuel. "Circle Packing." *Illuminations* (2011): 1-4. Web. 8 Aug 2011.

"International Space Station after undocking of STS-132.jpg." Web. 8 Aug 2011. http://en.wikipedia.org/wiki/File:International_Space_Station_after_undocking_of_STS-132.jpg.

### The needs this ETP will fulfill in the classroom, teaching or school:

When studying math concepts involving a bunch of formulas (such as areas) it is easy for students simply to memorize those formulas, or try to memorize them, and then move on to the next topic. Frequently the practice exercises are a bit dry and students don't associate the proper formula with the appropriate shape. This ETP will provide an alternate way of teaching areas of polygons and sectors, and by asking students to work through a real life application which requires these formulas, I hope it will engage them more fully and result in better comprehension and retention.

### Keywords:

geometry, polygon areas,
**Attachments:**