th 8 Grade Mathematics / Science ILS—6A, 6B, 7A, 8A, 9A, 10B, 12E Shadows and Skyscrapers How do the sun’s position in the sky and our latitude affect the shadows created by tall buildings? Vocabulary latitude the distance north or south Theme of the equator measured in degrees; Using geometry, algebra, computation, and ratio skills, students will explore the latitude of Chicago is 41° north how the sun and shadows can be used to measure tall buildings. altitude the angle of the sun (0° - 90°) measured from the horizon; the Student Objectives higher the number, the higher the • explain the relationship between the position of the sun, a city’s latitude, sun is in the sky and shadow length horizon the visual line where it • use ratios, geometry, and algebra to determine shadow lengths and building heights appears that the sky meets the • use sun angle (altitude) charts to create and compare skyscraper shadows for land or sea four North American cities orientation the four points of a compass (north, south, east, west) Activities vernal equinox in the northern Grade Mathematics / Science / Mathematics Grade • sketch the outline of the school in plan to show its shadows hemisphere the first day of spring th 8 • trace the shadows of a simple object at intervals for 30 minutes as determined by the day when the • solve the height of buildings by measuring their shadows hours of daylight and night are equal; 329 • draw sun angles and determine shadow lengths for skyscrapers usually March 21 summer solstice in the northern Type hemisphere the first day of summer • indoor, desktop activities when the sun reaches its highest altitude and is in the sky for the • outdoor, around-the-school-building activities longest time in the year; usually June 21 Timeframe five class sessions of 30-40 minutes each autumnal equinox in the northern hemisphere the first day of autumn as determined by the day when the Materials hours of daylight and night are equal; Skyscrapersand Shadows • Handout A - photographs of buildings in the shadows of other buildings usually September 21 • Handout B - sun path diagrams winter solstice in the northern • Handout C - sun angle (altitude) chart for 24° N and 32° N latitude hemisphere the first day of winter • Handout D - sun angle (altitude) chart for 41° N and 52° N latitude when the sun is in the sky for the • Handout E - map of the U.S. showing four cities and their latitudes shortest time in the year; usually • Handout F - plotted shadow study for NBC Tower on graph paper December 21 • Handout G - top ten list of Chicago’s tallest skyscrapers equator an imaginary line around • large pieces of 1/4" graph paper (several sheets of 11" x 17" paper per student) the earth that is equal distant from • paper and pencils for quick sketches the north pole and the south pole • directional compass (one per class or student pair) • 50-foot or 50-meter tape measure (one per class or student pair) • small plastic protractor and straightedge (one per student or student pair) • optional but helpful: clipboards Schoolyards to Skylines © 2002 Chicago Architecture Foundation Teacher Prep • photocopy or scan Handouts A–G for display or projection • photocopy Handouts B–G (one per student) Background Information for Teacher Architects must consider latitude and orientation to maximize the positive impact and minimize the negative impact of sunlight on a building. In addition, architects must determine how the building’s shadow will affect the buildings around it. Such considerations take place throughout the design process; they are not afterthoughts. Vocabulary (continued) northern hemisphere the area Your students possess the math skills needed to solve many of these real-world of the earth north of the equator design problems. parallel lines lines that go in the same direction and are always the In 1893, Chicago enacted the first of many ordinances that set the maximum height same distance apart and never touch for tall buildings. The first ordinance limited building height to 130 feet, which is transverse line a line that bisects approximately 13 stories. (In contrast, the Sears Tower stands 1,450 feet tall.) In Shadows and Skyscrapersand Shadows parallel lines subsequent years, the height limit was raised and lowered several times. In 1923, corresponding angles of two a report cited problems that were a result of the dark caverns created by shadows parallel lines the equal-size angles of tall buildings: depressed workers due to lack of sun, an increase in “winter formed when a transverse line bisects diseases,” congestion in the streets, and the possibility of fires. parallel lines; the angles are on the same side on the transverse line Zoning laws today are much more complex than those early Chicago ordinances that simply imposed a height limitation. Chicago, like many major cities, requires every proposal for a new skyscraper to include a “Sunlight Access and Shadow Impact Study.” The study guidelines state that new buildings should not cast 330 significant shadows on public parks, plazas, waterways, beaches, or playgrounds. Each building is evaluated on its shape, height, and orientation. 8 th Grade Mathematics / Science / Mathematics Grade Chicago requires the architect or developer to submit a drawing that shows the shadows the new tall building will create at 9:00 a.m., noon, 3:00 p.m., and 6:00 p.m. at four times during the year: the vernal equinox, summer solstice, autumnal equinox, and winter solstice. Chicago sits at 41° north latitude. The path of the sun in our sky is highest in the summer and lowest in the winter, as it is everywhere in the northern hemisphere. The specific point on the horizon of sunrise and sunset varies seasonally, but students should understand that the sun generally rises in the east and sets in the west. The path of the sun in spring and autumn falls in between the higher summertime sun path and the lower wintertime sun path. In the spring and fall, the paths of the sun and the sunrise and sunset times are almost the same. Summer Autumn/Spring Winter West South North East Schoolyards to Skylines © 2002 Chicago Architecture Foundation Equinoxes and solstices The vernal equinox marks the first day of spring. Due to the motion and tilt of the earth, the sun crosses over the equator on its way north. Typically, the vernal equinox occurs on March 21. (“Equinox” comes from the Latin words “equal” and “night.”) The length of daylight and night are equal. After this date, the day becomes longer than the night. The summer solstice marks the first day of summer. It usually occurs on June 21. In the northern hemisphere, this is when the sun is in the sky for the longest period of any day all year. The sun is also at its highest point in the sky on this day. The autumnal equinox marks the first day of autumn. Due to the motion and tilt Discussion Points of the earth, the sun crosses over the equator on its way south. Typically, the • In which direction does the sun rise? autumnal equinox occurs on September 21. After this date, the night becomes • In which direction does the sun set? longer than the day. • Is this true everywhere in the world? The winter solstice marks the first day of winter. It usually occurs on December 21. • Approximately what time did the sun In the northern hemisphere, this is when the sun is in the sky for the shortest period rise this morning? of any day all year. • Approximately what time will the sun set tonight? Altitude • Which direction do your classroom The angle of the sun, as measured from the horizon, is called altitude. The higher windows face? the number, the higher the sun is up in the sky. 0° = the sun is directly at the • Does the sun ever shine in your horizon. 90° = the sun is exactly overhead (which occurs only at the equator). classroom windows? • Is the direct sunlight welcome or Science / Mathematics Grade th annoying? 8 • At what times of the day or months of the year does the sun shine in 331 your classroom windows? altitude = 49° West • Do any nearby buildings block South North the sun’s light from coming in your classroom windows? East angle of the sun at noon at the vernal equinox (March 21) in Chicago Note: Both city latitudes and altitude numbers exist in a measurement of degrees Skyscrapersand Shadows and minutes. For this lesson, calculations have been simplified by dropping the minutes. In addition, the charts state when the sun is below the horizon, rather than displaying the more standard method of listing degrees as negative numbers. This lesson has also been simplified for your students by eliminating reference to azimuth, which is the other factor used to trace the sun’s path across our sky. Azimuth measures the angle of the sun, as seen in plan, in relation to due south. The azimuth clarifies that at 9:00 a.m., for example, the sun is 32° up from the horizon and in the eastern sky. At 3:00 p.m., the sun would be at the same altitude, but in the western sky. Due to the complex nature of azimuth, it has been omitted here. Schoolyards to Skylines © 2002 Chicago Architecture Foundation Reading sun angle (altitude) charts Handouts C and D contain charts, such as the example below. For four cities at different latitudes, they show the angle of the sun at four times during the year and at five times during the day. Example: Sun position - vernal equinox 41 degrees north latitude = Chicago, Illinois (altitude measures how many degrees the sun is above the horizon) date time altitude Interdisciplinary Connections 3/21 6:00 a.m.