Geometry in Art, Architecture, and Nature
INTRODUCTION: Our world is a collection of geometric figures in many shapes and sizes. Artists, architects, and naturalists have long been inspired by these geometric relationships and they discovered early in their lives that our world is rich in abundance of these geometric properties. Fortunately, their work has been (and can be) captured in photographs and images. It is your job to search the world around you for examples of this phenomenon. You may take your own photographs using a digital camera or you may search the web for appropriate images. Your photographs or images must highlight specific geometric properties and you will present your findings to the class in a collection of at least 10 slides that you created in SMART Notebook. Your imagination and creativity will be your greatest tools.
TASK: Your task is to explore the world around you and find occurrences of specific geometric relationships. Think of your mission as having to teach visitors from Mars about Earth’s geometric properties. Your mission is divided into three parts. The first part will require you to find ten photographs (using either a digital camera or images found on the web) that illustrate specific geometric properties. The second part will be to provide an analysis of the photographs or images and explain and discuss the geometric relationships. The third part will be to create and present a SMART Notebook presentation of your findings.
PROCESS:
Part 1: The Photographs and Images: You will take specific photographs or find images on the Web of the following geometric relationships. The pictures that you use must be appropriate for presentation in the classroom. You may use one slide for several of the properties but your entire project must have at least 10 slides in addition to an Introduction Slide and a Conclusion Slide.
1. Two parallel lines cut by a transversal, not right angles, which shows the relationship between the angles (alternate interior, corresponding, same side interior, alternate exterior etc.)
2. Triangles: vertices, angles, right, obtuse, scalene, isosceles, equilateral, and equiangular.
3. Angles: acute, obtuse, right, vertical, supplementary, complementary, and linear pair.
4. Lines: skew, parallel, perpendicular, and intersecting at various angles.
5. Congruent triangles: Find two or more congruent triangles which show that their corresponding parts are also congruent.
6. Find a triangle inscribed in a circle which shows that the center of the circle is also the circumcenter of the triangle.
7. Find a regular polygon with more than 4 sides; find the measure of one interior angle and the sum of all interior angles.
8. Find a (non-square) rectangle, and (non-square) rhombus and a square either in the same picture or separate pictures that shows the relationship between all three polygons.
9. Find a circle and its tangent which show the properties at the point of tangency.
10. Find two non-congruent polygons whose areas are equal.
11. Find a cylinder and a cone that display the relationship
between their volumes.
12. Circles: chord, radius, diameter, point of tangency,
Common external tangent or common internal tangent.
13. Figures and relationships: sphere, hemisphere, right rectangular prism, right cylinder, oblique square prism, oblique cylinder, cube, rectangular prism, cylinder, regular pyramid, regular hexagonal pyramid, circular cone, right cone, parallelogram, rectangle, square, trapezoid, rhombus, and ellipse.
14. Your Choice: May be any geometric relationship that you find intriguing, interesting or perhaps puzzling. Explain why you chose this picture or image and its geometric significance.
PART II—THE ANALYSES
Choose three of your slides and provide some mathematical analysis to further enhance your conclusions. You may do your measurements by hand using a ruler or protractor. Think of the analysis as a way to verify the mathematical relationship. The analysis section involves taking actual measurements and verifying the mathematics behind the visual representations.
PART III—THE PRESENTATION
1. Make a title slide and choose an aesthetic image for it. Include your name, the title of the project, the name of the class, and the date.
2. You will need a total of 10 Slides showing geometric relationships. You must show a variety of relationships—in other words you don’t want the same things shown on different pictures.
3. Each photograph will be labeled (location and date).
4. On the same slide, add the relevant geometric information about the picture. You may want to include definitions or theorems that are represented.
5. The three mathematical analyses slides should each be on a separate slide and should follow your 10 pictures.
6. The conclusion slide will be the summary slide and will include some overall thoughts about the project: What was your method of finding pictures and images? Did you enlist any help? What was the most difficult picture or image to find? What was the easiest? Did you enjoy doing this project? Why or Why not?
EVALUATION:
You will be evaluated on the three parts of the project: the photographs and images, the analyses, and the presentation. You must have ten photographs or images that adequately show specific geometric relationships. The analyses should be convincing and use the mathematical properties (conjectures) that you have learned in this course. The information on the slides must be presented in a pleasing way and contain no errors in usage and mechanics. The information must be mathematically accurate.
STANDARDS:
S.D. Geometry Standards (9-12)
1. Justify properties of geometric figures
2. Apply properties of circles
NCTM Geometry Standards 9-12
1. Analyze properties and determine attributes of two- and three- dimensional objects.
2. Explore relationships (including congruence and similarity) among classes of two-and three-dimensional geometric objects, make and test conjectures about them, and solve problems involving them.
3. Use geometric models to gain insights into, and answer questions in, other areas of mathematics.
4. Use geometric ideas to solve problems in, and gain insights into, other disciplines and other areas of interest such as art and architecture.
CREDITS:
A project adapted from the work of Romeo Baldeviso, Bishop O’Dowd High School and Goodwin, A Photography Exploration
Geometry Summer Institute, Mills College summer 2005
Kathleen A. Fugate
Kadoka Area High School
PO Box 99
Kadoka, SD 57543
605-837-2172
Beginning4 / Developing
6 / Accomplished
8 / Exemplary
10 / Score
Photographs and Geometric Relationships / Photographs of at least two geometric relationships. / Photographs of at least four geometric relationships. / Photographs of at least eight geometric relationships. / Photographs of ten geometric relationships.
Photographs Exhibits a Unified Theme / Photographs do not adhere to a single unifying theme. / Photographs exhibited in each of the three unifying themes (Art, Architecture, Nature) / Photographs exhibited in two of the three unifying themes. / Photographs exhibited all belong to one of the three unifying themes.
Analyses - Quantity / Analyses of at least two geometric relationships. / Analyses of at least four geometric relationships. / Analyses of at least eight geometric relationships. / Analyses of ten geometric relationships.
Analyses - Quality / Analyses fails to demonstrate the geometric relationships. / Analyses attempts to demonstrate the geometric relationships, with several errors. / Analyses somewhat demonstrates the geometric relationships, with some errors. / Analyses clearly demonstrates the geometric relationships with no errors.
Presentation / Title slide, aesthetic template, summary included / All of previous plus: 10 photos included / All of previous plus: labels for each photo included / All of previous plus: geometric information for each photo included