MiddleSchoolPortal/Going Around in Circles: Around and About A Geometric Figure
From NSDLWiki
Going Around in Circles: Around and About A Geometric Figure - Introduction
Geometry can be an exercise arena for strengthening those logic muscles that middle school students need to flex. When we work with a geometric figure—a circle, for instance—and apply the ancient tools of compass and straightedge, geometry can become a rich ground for developing design. And a circle has size, so a unit on this topic necessarily brings in the mathematics of its measurement. Circles, then, is a geometric topic that can provide mental challenge, opportunity for artistic development, and connections to both the history of measurement and its everyday applications.
For this unit we have selected online activities that will enable your students to look at circles from these various viewpoints. The activities range from measurement to theorems about the circle to art and symmetry. Many will engage students in solving problems or in creating designs. Others will allow them to investigate the thinking behind the area and circumference formulas, either through virtual simulations or hands-on projects. We hope these activities will add to your repertoire of ways to present the "many-sided" topic of circles.
Contents |
If you would like to refresh your own understanding of the circle, the Teacher Background section offers online reviews of the properties of circles, their measurement, and rotational symmetry. A final section, NCTM Standards, considers thoughts offered on the teaching of this integrated topic in Principles and Standards for School Mathematics.
Background Information for Teachers
The circle holds few mysteries for the teacher—at least when we are teaching its measurement and general properties. But as you work through the Activities section with your class, you may find you are asking yourself, What are the underpinnings of the area and circumference formulas? How did pi appear on the scene? What is the value of rotation symmetry in the big mathematics picture? The resources below, developed for teachers, offer not only information but also online exercises that will involve you in answering your questions.
Circles and Pi Pi is central to the formulas for circumference and area. These activities, intended for K-8 teachers, explore the formulas.
Parallel Lines and Circles This is a chance to "play" online with dynamic software. You will construct figures and discover properties of parallel lines and circles. The lesson, created for K-8 teachers, includes the formal geometric definitions, but the emphasis is on investigating relationships among the geometric figures.
Rotation Symmetry What is rotational symmetry, and why is it important? In this session, developed for K-8 teachers, you can find the answers. Using this symmetry, you will create designs and consider their properties. A video segment shows teachers determining rotational symmetry.
[http%3A//mathdl.maa.org/convergence/1/%3Fpa%3Dcontent%26sa%3DviewDocument%26nodeId%3D646 Eratosthenes and the Mystery of the Stades] This article gives valuable background information about Eratosthenes' ingenious method for measuring the circumference of the Earth.
Activities
"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. Measurement: Investigating Pi
Discovering the Value of 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 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.
Measurement: Circumference
Big Tree: Have You Ever Seen a Tree Big Enough to Drive a Car Through? 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.
The Noon Day Project: Measuring the Circumference of the Earth 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.
[http%3A//mathdl.maa.org/convergence/1/%3Fpa%3Dcontent%26sa%3DviewDocument%26nodeId%3D361 Measuring the Globe: An Historical Activity] 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 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!
Measurement: Area
Circle, Square and Triangle 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.
Windshield Wipers: It's Raining! Who Sees More? The Driver of the Car or the Truck?
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.
Three Piece Circle Puzzle 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.
Measurement: Arcs
[www.amblesideprimary.com/ambleweb/mentalmaths/angleshapes.html Angle Activities] 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
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.
A Shard or Two: How Big Was the Plate? 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
Rotational Symmetry
Transformations - Rotation 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 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 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.
Art and the Circle
Designs with Circles 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.
Tessellating a Circle 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.
Native American Geometry 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!
Circle Theorems
Circles 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.
Projects
Here are digital teaching resources that demonstrate how data and statistics are a vital part of learning mathematics in a meaningful context. The resource activities are often interdisciplinary, which makes them time-consuming to prepare, as additional expertise is often needed. But the payoffs can be huge: student engagement, in-depth learning, and a real-world context for learning mathematics.
The starting point for developing an interdisciplinary activity can be a teacher's personal interests or a student's passion. Are you a birder or pool player? Check out the ideas in Classroom FeederWatch, Backyard Birding--Research Project, or Analyzing Numeric and Geometric Patterns of Paper Pool below. Or if you want to begin with a simpler data collection lesson, take a look at Junk Mail below or Backpack Project.
Another approach is to look at situations in your community or larger world issues and have the students frame questions to investigate. Students may develop a passion for scientific inquiry when a topic can be analyzed with numbers. Requiring quality work and including a component about sharing results with the community will add value to an interdisciplinary contextual learning experience. Teachers may want to enlist a science teacher or community person to provide additional expertise. Whether thinking small activity or big project, be ready to be surprised at what the data analysis reveals!
Individual Lessons
Analyzing Numeric and Geometric Patterns of Paper Pool Look out, pool sharks! Begin the study of data and statistics with this super student exploration where data are collected and analyzed while students apply mathematical topics studied in grades 6 and 7: factors, multiples, rectangles, and the meaning of being relatively prime. In the Paper Pool applet, a ball is hit from the lower left-hand corner of a grid-lined pool table at a 45-degree angle. Students modify the size of the rectangular pool table and observe how the ball always travels on diagonals of the grid squares. After gathering and organizing data, students look for patterns to predict the corner pocket into which a ball will fall and the number of side hits the ball makes as it moves on the table to a corner pocket. The goal is to determine how the number of hits, final pocket, and number of squares crossed depend upon the relative lengths of the sides of the pool table. Sounds like fun, doesn't it?
Junk Mail (a mini project) No one is immune from receiving junk mail, but just how much of it is really finding its way to your address? In this simple activity, data collection and analysis are a key part of a project to learn about the importance of recycling. For one week, students count and record the number of pieces of junk mail received in their homes. The display and organization of the data can be modified to address the data and statistics topics the class is working on.
WWW--Wonderful Web Weather Just how on-target are those weather forecasters we watch and listen to? In this webquest, students work in groups to track online weather reports for several locations over the course of three days and determine the accuracy of forecasts. Students develop an understanding of how weather can be described by measurable quantities, such as temperature, wind, and precipitation as they find and compare weather data found on the Internet, chart and graph data, and present their conclusions about forecasting. This straight-forward activity is suitable for students who are just beginning their work with data and statistics.
Projects
Classroom FeederWatch Students can participate in an annual bird-feeder survey through this web site from the Cornell Lab of Ornithology. The project integrates science and math skills and concepts as students collect local information about the abundance and species composition of birds. Students observe birds at a local feeder and use the Internet to share the data with scientists and other students. There is an initial registration fee to participate in the program and a lesser renewal fee for previous participants.
The Global Sun Temperature Project With this free online collaborative project, students measure the temperature and record the minutes of sunlight for one week. Data are collected on the web site, and average daily temperatures and amount of sunlight are compared. Students draw conclusions about how the distance from the equator influences temperature. If you like this collaborative project, be sure to check out Down the Drain: How Much Water Do You Use?, another collaborative data project from the Center for Innovation in Engineering and Science Education (CIESE).
The Gulf Stream Voyage If ocean travel is your passion, this site offers a way to spend time at sea without ever leaving your classroom. Here is a science project that uses actual data to help students investigate the science and history of the Gulf Stream. Math students can greatly benefit from the opportunity to collect data and draw conclusions based on the data. In the lesson called Current Now, students use real-time data and satellite images to determine how the Gulf Stream moves in the course of a year. In another activity, students use data about water temperature obtained from ships and buoys to determine the course of the Gulf Stream.
Boil, Boil, Toil and Trouble: The International Boiling Point Project Be part of an annual event: Enroll your class in this free Internet-based collaborative project. Students discover which factors--room temperature, elevation, volume of water, or heating device--have the greatest influence on boiling point. Students boil water, record their data, and send it via email to be included in the site's database of results. Student activities focus on analyzing the compiled data to find answers to questions about how and why water boils.
Backyard Birding--Research Project Birds are everywhere, and here are ideas for creating a data collection project. Work with a science teacher and, possibly, an industrial tech teacher to expand this multiweek activity into a cross-curricular project to help students see how data analysis can support an understanding of nature.
Population Growth These nine online lesson/activities investigate population growth and its impacts. Students use archived census and demographic data from the U.S. Census Bureau to model population growth and examine how population change affects the environment. Teachers will want to carefully review this resource to choose the activities most appropriate for their students' mathematics background. Linear, quadratic, and exponential functions are used in some lessons.
RoadKill: Collision in the Wildlife Corridors For this online data collection project, students monitor and report the number and type of animals killed by motor vehicles in their locale, along with information about the environmental conditions that may have contributed to an animal's death. The RoadKill Monitoring Project's goal is to make students aware of the environmental issues related to the death of wildlife on roads--the statistics are considered evidence of the conflict between humans and wildlife. Teachers can use the data to teach ecological principles, to portray information graphically, and to develop elementary concepts in statistical analysis.
Lesson and Project Collections
http://www.ciese.org/currichome.html CIESE: K-12 Online Classroom Projects] The Center for Innovation in Engineering and Science Education at the Stevens Institute of Technology develops free standards-based projects that offer real-time data found on the Internet and online collaborative projects linking classrooms from around the world. There are opportunities for cross-curricular activities and for rich experiences with data collection, data display, and for drawing conclusions or inferences based on the data. The Global Sun Temperature Project and the Gulf Stream Voyage highlighted above are two examples of this site's materials.
New York Times Daily Lesson Plan: Mathematics What better way to make data and statistics lessons real than to use the daily newspaper? Here are interdisciplinary lesson materials based on New York Times articles. The stories offer ways to draw on real-world issues and statistics to develop lessons in mathematics. For example, in one lesson, "students convert statistics about gun injuries into visual presentations, then use these as the basis for a poster campaign to teach children about the dangers of guns in home," while another lesson idea involves analyzing the job market for older workers.
and Probability. Grades 6-8 This unit contains seven lessons with hands-on learning activities to help students explore statistics and probability. The lessons, designed for mentoring situations, may also be helpful to teachers new to teaching data and statistics. Each lesson features an overview of the mathematics, preparation guidelines, teaching tips, and suggestions for how to use each activity to develop specific mathematics concepts. Statistics activities focus on posing questions, gathering data and identifying bias, understanding measures of central tendency, representing data with graphs, and interpreting data.
NCTM Measurement Standard
The NCTM Principles and Standards for School Mathematics addresses the topic of circles in both the Measurement and the Geometry standards. It states that students should come to the middle grades with informal knowledge of several two-dimensional shapes, with experience in visualizing and drawing them, and "with intuitive notions about shapes built from years of interacting with objects in their daily lives" (NCTM, 2000, p. 233). Likewise, students bring experiences with measurement from prior instruction as well as from everyday use.
Building on this knowledge, middle school students "should explore a variety of geometric shapes and examine their characteristics" (NCTM, 2000, p. 233), learning to precisely define and describe the features of the shapes. Furthermore, at this level students should develop a strong understanding of transformational geometry, including rotational geometry. The NCTM document notes that one important aspect of measurement in the middle grades is "solving problems involving the perimeter and area of two-dimensional shapes" (p. 241). Students need to learn to use appropriate measuring tools and apply appropriate formulas to determine measurements and "whenever possible . . . develop formulas and procedures meaningfully through investigation rather than memorize them"(p. 244).
The resources highlighted in Going Around in Circles! can be used to focus attention on the underlying logic of circle formulas and rotational symmetry. In selecting these resources, our aim has been to invite students to develop intuitive understanding through working with actual materials or interacting with online simulations. Creating their own art with circles, students will come to an appreciation of the use of circles in designs found in many cultures.
Teachers will find resources that support their standards-based instruction. As they investigate the history of circle measurement, experiment with rotating figures, and question the formulas used, teachers will find opportunities to incorporate ideas from the Reasoning and Proof, Problem Solving, and Connections standards. Other valuable resources that feature NCTM Standards-based teaching ideas for geometry and measurement are the series Navigating Through Geometry and Navigating Through Measurement. (Ordering information can be found at the NCTM web site). Reference
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author.
Author and Copyright
Terese Herrera taught math several years at middle and high school levels, then earned a Ph.D. in mathematics education. She is a resource specialist for the Middle School Portal 2: Math & Science Pathways project.
Please email any comments to msp@msteacher.org.
Connect with colleagues at our social network for middle school math and science teachers at http://msteacher2.org .
Copyright June 2006 - The Ohio State University. This material is based upon work supported by the National Science Foundation under Grant No. 0424671 and since September 1, 2009 Grant No. 0840824. 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.
