Title III Technology Literacy Challenge Grant

Learning Experience

Learning Context | Procedure | Instructional/Environmental Modifications | Time Required | Resources | Assessment Plan | Student Work | Reflection

LEARNING CONTEXT

Purpose or Focus of Experience

The purpose of this learning experience is to develop mathematical power, through knowledge gained about angles and triangles, and to further discover the circumference of the earth. This power is also enhanced, with the Internet, through accessing data base sites and through communication with other students via an on-line project site.

        Time of Year Required For Completion

This project has a window for completion, which is by March 21. Please check the www.kencole.org  website to register your class so as to participate with other classrooms and share in the data base.

WebQuest to Accompany this Project at www.geocities.com/goodapple_2000_2001/noonshadow.html

Connection to Standards

Standard 2: Information Systems - Intermediate

Information technology is used to retrieve process, and communicate information, and as a tool to enhance learning. Students in this learning experience will acquire data about the latitude and longitude of Colton, NY, to find the distance from Colton to the equator, along the same lines of longitude. Once longitude and latitude are found, students will access an Internet site to find the miles to the equator. Students will share their findings with other students around the globe to compare their findings of the distance around the earth.

Standard 3: Mathematics – Key Idea:  Patterns and Functions

Students use patterns and functions to develop mathematical power, appreciate the true beauty of mathematics and construct generalizations that describe patterns simply and efficiently. Students in this learning experience will explore relationships involving point, lines, angles and planes to find the circumference of the earth.

ESSENTIAL QUESTIONS

Is it possible to find the circumference of an object as big as the earth without measuring it?

Has anyone else ever discovered a way of finding the circumference of the earth?

What are the sources of proof to findings about circumference of the earth?

CONTENT KNOWLEDGE

PROCEDURE
(Chronologically ordered description of all teacher & student activities and interactions.)

This learning experience takes place in the middle of a geometry unit where students move from one-dimensional linear measurements to three-dimensional measurements. To build upon the history of mathematics and informal methods used to solve problems, students are asked, "Is it possible to find the circumference of the earth? If so, show do you think it can be calculated?"

Students made suggestions about how this could be accomplished. Some thought about the use of maps and the scale on the map.

Day 1 Two parallel lines cut by a transversal were drawn on the board and students were asked, "What angles do you think are equal?" Students then responded that the vertical angles were equal and then another students said, "There are four equal angles." Students then concluded that alternate interior angles were equal, this verified with knowledge about straight angles and supplementary angles. Students were then given time to practice finding the measure of angles when two given parallel lines were cut by a transversal. Students then wrote definitions for vertical angles and alternate interior angles.

Day 2 Students read together the book, The Librarian Who Measured the Earth, by Kathryn Lasky and published by Little Brown and Company (1994). Students took notes about Erastosthenes and his constant questioning about the earth and the sky. Students learned that Greek children went to a school called a gymnasium, where there were no desks, pencils and paper, in school as there are today. They learned that Eratosthenes was knowledgeable about all subjects, but "his favorite subject was geography." After he had learned all that he could in the gymnasium, he was sent to Athens where he studied mathematics and science.

Students learned Eratosthenes enjoyed making lists and made the first chronology of the Olympic Games. He became famous for his writing of books, which covered many topics such as astronomy, history, and comedy.

Next, students learned Eratosthenes worked for Ptolemy, who was then the ruler of Egypt, where he worked as a tutor of Ptolemy’s son. Being in Alexandria, was a dream come true for Eratosthenes. Alexandria was considered the "center of all learning." Here there were museums, not as we know them, but as laboratories for learning. Great minds came to the places where they could work and eat as they pleased. There were dissecting laboratories where it was first learned of the relationship, which exists between a pulse and a heartbeat. Eratosthenes was nicknamed Pentatholos, which meant the "Greek all-rounder." It was here in Alexandria that he became the chief librarian. Questions were now abound in his mind, the biggest question had to do with geography.

It was in Alexandria as a librarian that Eratosthenes came up with the question, "How big around is the earth?" He knew he could not walk around the earth, so he began his research. He soon realized that he was not going to find his answer in one of the scrolls in the library. He then decided to write a geography book to collect all the information he need and put it into one place. He still wondered about the distance around the earth. He tried to develop ways of knowing, measuring, and describing. No one had considered measuring something as big as the earth. He imagined the earth as a grapefruit, and how many of the same size sections there were, and thought you would then be able to find the distance around by finding the shorter distance.

So began Eratosthenes’ idea, that if you knew one central angle of the earth, one would need only to divide 360 degrees by this angle to determine how many sections there were. You would then multiply the number times the arc distance for one section.

Day3 Using the Internet site Http://k12science.stevens-tech.edu/~ihor/noon1html students were shown an overhead of what Eratosthenes imagined. Students were then instructed to make point O in the center of the paper to represent the center of the earth. They then were to construct a circle, with point O as the center. Next, they were instructed to locate point A on the outside of the circle. Draw the line segment OA, connecting point O with point A. Students were asked, "What is this line segment called?" Students then measured line segment OA and were asked, "Could you now find the circumference of your circle without further measuring?" One student said, "This is like our PI discovery. If we know the radius, we can double that and then multiply this by approximately three to find the circumference." Using string they measured the diameter and then saw the circumference. Cutting one diameter from the length of the string, they found they could cut two more, which proved the circumference was about three times the diameter.

Next students were instructed with, "Construct an angle of 60 Degrees with point O as the vertex of the angle and line segment OA as one side of an angle. Students were then told that this is called a central angle. They were asked, "Why do you think this is called a central angle? One student responded, it is called a central angle because its vertex is at the center of the circle.

If the central angle is 60 degrees, how many sections are there?" Students then made their estimates based on prior knowledge and perceptions. (Answers were 4, 5, and 6. I left this as is until proven, then we discussed why people answered as they did.)

"Let us prove it through constructing as many central angles as we can. Using line segment OB as a ray, draw another angle of 60 degrees. Label the angle, BOC. Can we draw another? Label this one angle COD, etc. Continue until the circle is complete with central angles of 60 degrees."

Then ask the question again, "How many central angles of 60 degrees are there in the circle, or in the earth?" Students unanimously agreed there were six. We then discussed why there were the prior misconceptions. Students verbally shared their misconceptions as well as understandings. One student said, "We could have just divided 360 degrees by 60 degrees, everyone agreed.

Students were then instructed to measure, with string, the length of arc AB. They were then asked, "What do you think the circumference of the circle is now, using the number of central angles and the length of arc AB?" Students quickly said six time the number of the arc length. So all students multiplied their arc times six to find the circumference of their circles. In pairs, each group checked their partner’s measurements and calculations.

Students were then asked, "Can we could prove this with the formula, C = p x D?"

Day 4 Eratosthenes dilemma was that he could not cut the earth in half like a grapefruit. Eratosthenes then thought of how he could use the sun to help solve his problem. He knew that on the twenty-first day of June, the sun would shine directly into the bottom of a well in Syrene, but in Alexandria, at the same time, shadows were noted. He realized this was because the earth was round. He knew if he measured the shadow at Alexandria, when the sun was directly overhead in Syrene, he could find the angle of the sun. This angle would be the same as the central angle of the earth that could then be divided into 360 degrees to find the number of sections of the earth. Knowing the distance from Syrene to Alexandria, he was able to calculate the distance around the earth, or as we know it, the circumference of the earth.

To prepare the students for using the WebQuest titled, Eratosthenes in the Twenty-First Century, we continued with the following discourse.

The next question asked was, "When would the sun be directly over the equator so the sun would shine directly into the well as Eratosthenes had figured?" One student said, March 21. A diagram was drawn on the board of the earth, the sun, and the angle formed between Colton and the equator. The two parallel lines were then noted, which were cut by a transversal. The question was asked, "If we know the angle of the sun, do we know any other angles?" Some students said, "Yes."

The question asked by the teacher, "How do we measure the angle of the sun?" This was followed by another question, "Would the measure of the sun’s angle be the same all day?" Someone said, "No, because in the afternoon the shadow becomes shorter. It was decided we must measure the shadow of the sun. Using a meter stick, we recorded the shadow every half-hour. Students then graphed this data and found the shape to be that of a parabola opening upward. Sixth grade students just knew it was not the graph of a straight line. They found the shadow the shortest around noon, but not exactly noon, which surprised them. The shadow was thirty-seven inches. Students were guided by questioning into the type of triangle formed by the meter stick and the shadow. They said, "A right triangle." So we drew the meter stick to a scale of one inch equal to one centimeter, and thus 37 inched became 37 cm. Students drew the triangle and then using a protractor found the sun’s angle that they knew was also the dental angle of the earth. Student’s then divided 360, representing the entire earth by 42 degrees, and found there were 8.6 sections of 42 degrees in the earth.

Using Internet site for the Noon Shadow Project, http://www.kencole.com/ , and the WebQuest page www.geocities.com/goodapple_2000_2001/noonshadow.html, we typed in the town of Potsdam, which was the nearest town so we could find the distance from Colton, NY to the equator. This was 3, 074, which represents the arc length of one section, whose central angle in 42 degrees. Multiplying 8.6 times 3,074, the students calculated the circumference of the earth to be 26,436 miles. Others were close to the figure.

Student data was then recorded on the NOON SHADOW PROJECT site of Ken Cole.

After figuring the Circumference, students were assigned a project to be done in groups of two, as this was challenging, and yet motivating. As a hook the following introductory scenario was developed and the task was given. Students had two class days to use the computer lab or to work in their groups within the classroom. Students were given a graphic organizer to help them to break down their product into individual roles.

This is an except from the webquest developed to go with this unit www.geocities.com/goodapple_2000_2001/noonshadow.html_

Introduction

Congratulations! You have just been selected to serve on a search committee for an extraordinary mathematician. Your quest will be to convince this mathematician to travel forward in time to Colton, New York, in the year 2000, so he can benefit mankind. At the same time, you must convince the Administrators and teachers of Colton, that Eratosthenes is the man (or Librarian?) for the job.

Who is Eratosthenes?

What did he discover that made a significant contribution to the field of mathematics?

How can we use what he taught us in the year 2000 and beyond?

The Task

Your task is to report your findings about Eratosthenes to the search committee here at Colton-Pierrepont Central. This will be done in the form of a poster, to describe the mathematics pictorially, and will be accompanied by a written report.

Your written report to the committee should include the following:

1. A brief biographical sketch of the life of Eratosthenes.
2. How Eratosthenes contributed to the field of mathematics.
3. How his understandings of the world helped to create a better understanding of the world.

   Prepare this presentation with poster and paper. You will have two class periods to use the computer lab and to work within the classroom.

    

    

   INSTRUCTIONAL/ENVIRONMENTAL MODIFICATIONS

To allow for involvement from all students included the use of cooperative groups, and teams of two for the creative retelling project. The classroom was set up so that groups of four could work together to decide what must be done first. All students were successful with the graphing and enjoyed that. Others needed one-on-one assistance to understand the application of angles to find circumference. The material was presented over five class periods, which meant systematic questioning for understanding.

TIME REQUIRED

The project was first identified for use in the classroom when researching online projects, which correlate with the math curriculum, and which, employ the use of mathematics in a meaningful way. This learning experience in itself, takes six to seven class periods provided the prior knowledge suggested in the first part of the learning experience is understood. We also had to vie for the sun. School vacations made for a short window of completion of sun's shadow measurement, added to the fact that this is a departmentalized sixth grade class, who are not always with me to allow for measurement.

RESOURCES 

For the student: The Librarian Who Measured The Earth by Kathryn Lasky

Protractors

Meter Sticks

Graph paper

Paper folding activities to prove alternate interior angles are equal

Overheads from the internet site http://k12science.stevens-tech.edu/noonday/cosmos.html

Internet site for finding arc distance given longitude and of Colton, NY http://www.kencole.org/

This could be done with one computer but the webquest would need to be eliminated.

For the Teacher: Same as for students

Danielson, Charlotte. A Collection of Performance Tasks and Rubrics, Middle School Mathematics. Larchmont, NY: Eye On Education, 1997.

ASSESSMENT PLAN

(Include samples of rubrics, checklists, etc.)

Performance Assessment List for Graph

Noon Shadow Length