Geometry in the Constellations: The ER-2


Author: Robin A. Ward, California Polytechnic State University-San Luis Obispo


Audience: Grades 2 - 4


Mathematical Topics: identification of polygons


Rationale: For grades 2 - 4, the NCTM Standardsrecommend that students be able to:

  • describe, model, draw, and classify shapes;

  • recognize and appreciate geometry in their world.

    • When learning geometry, students should investigate, explore, and experiment with everyday objects and other physical materials. Activities that encourage children to draw, visualize, and compare shapes in various positions will help develop their spatial sense.

    The NCTM Standardsalso emphasize making links across the curriculum. This activity integrates astronomy and mathematics in a creative way, providing students experience with exploring their night sky and seeing the geometry that exists in the various constellations.


    Materials:

    overhead map of the night sky
    worksheet
    posterboard
    crayons or magic markers
    Styrofoam cups
    pushpins


    Background: NASA is currently using an aircraft known as the ER-2 to collect information about our atmosphere and environment. The aircraft, based at NASA Dryden Flight Research Center, also makes celestial observations and is used for satellite calibration and satellite data validation.

    View several photos of the ER-2.
    Learn some facts about the ER-2.

    NASA is also currently collaborating with the German Space Agency (DLR) to build SOFIA, a Boeing 747-SP aircraft which holds a 2.5 meter telescope. SOFIA, which is expected to make its maiden voyage in the year 2001, will be the largest airborne telescope in the world and will make celestial observations beyond the capability of current ground-based telescopes.

    SOPHIA will replace the retired Kuiper Airborne Observatory (KAO) which was the world's only airborne telescope. Devoted exclusively to astronomical research, this converted C-141 military cargo plane, which carried a 36-inch reflecting telescope, was the scene of many major discoveries, including the first sightings of the rings of Uranus and a definitive identification of an atmosphere on Pluto.


    The Activity:

  • Promote the following discussion with students: Why might we study and record the location of stars? Knowing the location of certain stars serves as a frame of reference and guide to navigators such as sea captains. The study of the stars dates back many, many centuries. For example, long ago, before maps of the oceans and the world were developed, sea captains used the stars above to guide them to their destination. Provide students the opportunity to learn more about the history of the constellations and their importance in navigation by surfing the web. Some possible web sites are listed below in the "Enrichment Activity" section.

  • Present a picture of the night sky to students. Point out a few of the constellations and ask students if their shapes are indicative of their names. For example, Ursa Major (The Big Dipper) certainly looks like an ice cream dipper. Similarly, the constellation Draco (the Dragon) appears reptile-like, as it winds its way between Ursa Major and Ursa Minor (the Little Dipper). Notice how Sagittarius looks like a teapot. Promote a classroom discussion.

  • Next, the students will work individually and identify and record on the attached worksheet all of the various shapes (triangles, quadrilaterals, pentagons, etc.) they see in each of the constellations. Sketching the constellation will also help students in developing spatial skills. Place students in teams of four to share their findings. Facilitate a classroom discussion in which the students explain and justify their responses. Ask questions such as, Which constellation contains the most shapes? Which contains the least?

    For example, notice that the constellation, Sagittarius, is comprised of two triangles, one quadrilateral, and one isosceles trapezoid (shown in red).

  • For upper level students, the teacher can introduce and discuss such terminology as regular (meaning congruent angles and congruent sides) vs. non-regular polygons, convex versus concave polygons, simple figures versus closed figures, etc.

    For example, the constellation, Bootes, is a non-regularhexagon. Bootes is a hexagon because it is comprised of six line segments.

    A regularhexagon, shown in blue, is one whose angles are all congruent and whose sides are all congruent. Recall that congruent means "having the same measure." Clearly, Bootes is NOT regular.

    The constellation, Capricon, is a non-regular concave11-gon. Capricorn is non-regular since all of its angles (and sides) are not congruent. It is an 11-gon because it is made up of eleven line segments.

    A polygon is concaveif a line segment connecting any two interior points is not fully contained within the polygon. You can think of the polygon as being "caved in." Notice that the blue line semgent connecting two arbitrary interior points is NOT fully contained within Capricorn. Thus, Capricorn is concave,as opposed to being convex.

  • In the next activity, group students into pairs and provide each group with a piece of posterboard and crayons/markers. Students will develop their own (fictional) night sky, using the shapes they just discussed as well as any others they may have already learned. Encourage creativity on the part of students. See which teams can incorporate the most shapes in their night sky. Bring the class together as a whole and allow different pairs to present and discuss their night sky.

  • Finally, students can pick their favorite constellation in their night sky (or from the actual night sky map) and "create" that constellation using a pushpin and a Styrofoam cup. Provide each student with one Styrofoam cup and ask them to create the shape of the constellation by punching holes (representing the individual stars in the constellation) in the bottom of the cup using a pushpin. After the constellation has been created, students can then close one eye and hold their cup in front of their other eye. Students should hold the cup such that they are looking into the bottom of the cup and the bottom of the cup is pointed towards a light source. As students peer through the cup, the light will illuminate the stars (which are the holes in the cup) that makeup up their constellation.

  • Ask each student to place his or her name on their cup and then pass the cups around the classroom, allowing the students to view their classmates' constellations.


    Enrichment Activity: Allow students to learn more about astronomy by visiting the following web sites:

    Descriptions of particular constellations
    Some basic facts about the classical constellations
    The Northern Hemisphere constellations
    The Southern Hemisphere constellations

    As a writing activity, students can write a descriptive paragraph or story about various constellations in their journals and explain how mathematics can be found in the constellations.


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    • Funded by the NASA Dryden Flight Research Center


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