Activities

  Measurement   Rotational Symmetry   Art and the Circle   Circle Theorems  

"For children, geometry begins with play,” writes Pierre van Hiele (1999). He goes on to say that for students to reach the higher levels of geometric thinking, their instruction should still begin “with an exploratory phase, gradually building concepts and related language, and culminating in summary activities that help students integrate what they have learned into what they already know” (p. 311). The resources in this section offer activities that can supplement your instruction as you move your students through exploring, building concepts, and integrating their learning through application. A wide range of play in the field of geometry!

The activities begin with measurement, including investigations of the number pi and projects on measuring the circumference of the Earth. Rotational symmetry is then studied through hands-on interactive simulations. The next activities involve constructions with compass and straightedge in which students create designs and study their symmetry. Appropriate for challenging middle school students, a final group of resources uses simulations to consider more advanced properties of circles.

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Measurement: Investigating Pi

Discovering the Value of Pi http://www.arcytech.org/java/pi/

Students measure the diameter and circumference of several circles, using a handy applet, record their data and reach conclusions about the ratio of circumference to diameter. A genuine guided exploration!

Infinite Secrets http://www.pbs.org/wgbh/nova/archimedes

How did Archimedes estimate the value of pi? A battered manuscript reveals Archimedes’ thinking, and an accompanying classroom activity allows students to duplicate the procedure using paper and pencil or an online applet. The lesson procedure is set out clearly and student handouts are included. Look in the right-hand margin for the link to the interactive applet and more resources. MSP full record

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Measurement: Circumference

Big Tree: Have You Ever Seen a Tree Big Enough to Drive a Car Through? http://www.figurethis.org/challenges/c15/challenge.htm

In this activity, students consider the girth and height of ten National Champion trees and determine which, if any, of the trees is large enough to drive a car through. MSP full record

The Noon Day Project: Measuring the Circumference of the Earth http://www.k12science.org/noonday

Here is a real-world project that will engage your class in measuring the circumference of the Earth! You will find all information you need to enable students to recreate the measurement as done by the Greek librarian Eratosthenes over 2000 years ago. The procedure is based on measurements of shadows taken at high noon local time on a designated day in March; results from several schools are posted online and used to calculate the circumference. Included are detailed explanations and illustrations of the mathematics involved.

Measuring the Globe: An Historical Activity http%3A//mathdl.maa.org/convergence/1/%3Fpa%3Dcontent%26sa%3DviewDocument%26nodeId%3D361

This activity introduces students to Eratosthenes' historical accomplishment in measuring the Earth's circumference. The mathematics involved is well explained. Only paper and pencil are used here, in contrast to the actual measurements of sun shadow needed in the Noon Day Project above. This activity is found on a site that highlights the interaction of history, mathematics, and teaching: Convergence.

The Great Circle http://nlvm.usu.edu/en/nav/frames_asid_308_g_3_t_3.html

By clicking on two cities on a world globe, students see two line segments connecting the cities, one showing the great circle route (the shortest) and the other showing the route on a flat map. A nice application of real-world math! MSP full record

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Measurement: Area

Circle, Square and Triangle http://www.math.nmsu.edu/~breakingaway/Lessons/CST/CST.html

Starting with a piece of clothesline cut into three equal pieces, students form a circle, a square, and a triangle, all having the same perimeter -- but do they all have the same area? An animation shows clearly the lesson procedure for comparing the areas. This activity is found in Breaking Away from the Mathbook. MSP full record

Windshield Wipers: It's Raining! Who Sees More? The Driver of the Car or the Truck? http://www.figurethis.org/challenges/c32/challenge.htm

In this activity, students compare the areas cleaned by different wiper designs. An animation shows the movement of the two windshield wipers, each cleaning off a different geometric shape on the window. Students are encouraged to draw the shape cleaned by each wiper and find its area. MSP full record

Three Piece Circle Puzzle http://www.math.nmsu.edu/~breakingaway/Lessons/puzzle3pc/puzzle3pc.html

Students create the puzzle themselves, using compasses, and are challenged to find the area of each of the three pieces. You will need to guide your eighth- and ninth-grade students through the given solution. This activity is found in Breaking Away from the Mathbook. MSP full record

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Measurement: Arcs

Angle Activities www.amblesideprimary.com/ambleweb/mentalmaths/angleshapes.html

Measuring arcs of a circle with a virtual protractor, students engage in several activities on measuring angles. The protractor clearly shows how the arcs of a circle are measured in degrees from 0 to 180. This activity is from Ambleweb: Numeracy Hour.

Geoboard - Circular http://nlvm.usu.edu/en/nav/frames_asid_284_g_3_t_3.html?open=activities

Using an online circular geoboard, students work through five interactive activities that link the measure of the central angle to the measure of its arc. Even if your class doesn't have access to the Internet, the ideas here are worth transferring to paper. MSP full record

A Shard or Two: How Big Was the Plate? http://www.figurethis.org/challenges/c73/challenge.htm

Given only a few pieces of the edge of a circular plate, how can an archaeologist find the plate's original size? In a hands-on solution, students fold a sheet of paper to find the center and radius of the plate. MSP full record

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Rotational Symmetry

Transformations - Rotation http://nlvm.usu.edu/en/nav/frames_asid_299_g_3_t_3.html?open=activities

Two challenging activities push students to explore the connection between the angle of rotation and the image created. In the first, students see a rocket and its rotated image. They can move the vertex of the angle around the screen, widen or narrow the angle, or change the position of the original rocket and see what happens to the image. In the second activity, a game, they must create a rotation scheme that will move a rocket onto a target. These activities are from the National Library of Virtual Manipulatives.

Symmetries and Their Properties: Part I-Rotational Symmetry http://illuminations.nctm.org/LessonDetail.aspx?ID=U138

This unit is marked for grades 9-12, but don't be put off! It can be used profitably at lower levels as well. Each of the four lessons in the unit relies on an excellent interactive applet to explain a main idea of rotational symmetry.

Transmographer http://www.shodor.org/interactivate/activities/transform/index.html

This applet shows a triangle or square on a coordinate axis grid. Students may choose to rotate, reflect, or translate the figure, selecting all the parameters. For rotation, the center may be chosen and the number of degrees through which the figure is moved. Included are answers to the What? How? and Why? of the activity. MSP full record

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Art and the Circle

Designs with Circles http://mathforum.org/alejandre/circles.html

In addition to making designs using a compass, students also consider the patterns they make and the types of symmetry they can see in their designs. MSP full record

Tessellating a Circle http://www.math.nmsu.edu/~breakingaway/Lessons/TOAC/TOAC.htm

Beautiful designs based on the circle are shown here, along with clear directions on how to make the designs. This lesson is from Breaking Away from the Mathbook. MSP full record for Breaking Away

Native American Geometry http://www.earthmeasure.com/index_earth.html

This site describes Native American geometry as "a physical, proportional geometry that originates from the simple circle." You may use the material essentially as math concepts in art, or as a source of activities that connect the circle to polygons. It is a site worth investigating! MSP full record

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Circle Theorems

Circles http://www.ies.co.jp/math/java/geo/circles.html

If you are teaching a geometry course, or looking for challenging material, investigate these applets on circles from Manipula Math with Java. Working with these applets, students can visualize the theorems before they have to prove the theorems. In particular, they may like Eye Ball, Inscribed Angles, Inscribed Angle and Central Angle, and Flashlight. MSP full record

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Reference

van Hiele, Pierre M. (1999, February). Developing geometric thinking through activities that begin with play. Teaching Children Mathematics, 5(6), 310-316.

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Copyright June 2006 — The Ohio State University. This material is based upon work supported by the National Science Foundation under Grant No. 0424671. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

This work is licensed under a Creative Commons License.