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Primary Subject Area: Mathematics |
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Grade Level: 9 - 12
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Overview:
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Students will use the measurements of circular objects to discover the relationship between the circumference and diameter of circles using a spreadsheet application. They will also use the TI-92 graphing calculator to create circles and calculate the circumference diameter ratio. |
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Approximate Duration: 2 class periods |
Content Standards:
- Measurement
In problem-solving investigations, students demonstrate an understanding of the concepts, processes, and real-life applications of measurement. - Geometry
In problem-solving investigations, students demonstrate an understanding of geometric concepts and applications involving one-, two-, and three-dimensional geometry, and justify their findings. - Data analysis, Probability, and Discrete Math
In problem-solving investigations, students discover trends, formulate conjectures regarding cause-and-effect relationships, and demonstrate critical thinking skills in order to make informed decisions.
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Benchmarks:
- M-1-H
selecting and using appropriate units, techniques, and tools to measure quantities in order to achieve specified degrees of precision, accuracy, and error (or tolerance) of measurements; (1,3,4)- M-2-H
demonstrating an intuitive sense of measurement (e.g., estimating and determining reasonableness of results as related to area, volume, mass, rate, and distance); (1,2,3,4)- M-3-H
estimating, computing, and applying physical measurement using suitable units (e.g., calculate perimeter and area of plane figures, surface area and volume of solids presented in real-world situations); (1,2,3)- M-4-H
demonstrating the concept of measurement as it applies to real-world experiences. (1,2,4)- G-1-H
identifying, describing, comparing, constructing, and classifying geometric figures in two and three dimensions using technology where appropriate to explore and make conjectures about geometric concepts and figures; (1,2,4)- G-2-H
representing and solving problems using geometric models and the properties of those models (e.g., Pythagorean Theorem or formulas involving radius, diameter, and circumference); (3)- G-4-H
using inductive reasoning to predict, discover, and apply geometric properties and relationships (e.g., patty paper constructions, sum of the angles in a polygon); (1,2,4)- G-6-H
demonstrating deductive reasoning and mathematical justification (e.g., oral explanation, informal proof, and paragraph proof). (1,3,4)- D-7-H
making inferences from data that are organized in charts, tables, and graphs (e.g., pictograph; bar, line, or circle graph; stem-and-leaf plot or scatter plot); (1,2,3,4,5)
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Grade-Level Expectations (GLEs):
- Grade 9
17.
Distinguish between precision and accuracy
(M-1-H)
19.
Use significant digits in computational problems
(N-2-H) (M-1-H)
21.
Determine appropriate units and scales to use when solving measurement problems
(N-2-H) (M-1-H) (M-3-H)
22.
Solve problems using indirect measurement
(M-4-H) - Grade 10
10.
Form and test conjectures concerning geometric relationships including lines, angles, and polygons (i.e., triangles, quadrilaterals, and n-gons), with and without technology
(G-1-H) (G-4-H) (G-6-H)
13.
Solve problems and determine measurements involving chords, radii, arcs, angles, secants, and tangents of a circle
(G-2-H)
17.
Compare and contrast inductive and deductive reasoning approaches to justify conjectures and solve problems
(G-4-H) (G-6-H)
19.
Develop formal and informal proofs (e.g., Pythagorean theorem, flow charts, paragraphs)
(G-6-H)
22.
Interpret and summarize a set of experimental data presented in a table, bar graph, line graph, scatter plot, matrix, or circle graph
(D-7-H) - Grade 11-12
12.
Explain the unit circle basis for radian measure and show its relationship to degree measure of angles
(M-1-H)
13.
Identify and apply the unit circle definition to trigonometric functions and use this definition to solve real-life problems
(M-4-H)
22.
Explain the limitations of predictions based on organized sample sets of data
(D-7-H)
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Interdisciplinary Connections:
- English/Language Arts : Standard 7
Students apply reasoning and problem solving skills to reading, writing, speaking, listening, viewing, and visually representing.
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Educational Technology Standards:
- Demonstrate knowledge and skills of Internet use and other resources consistent with acceptable use policies including the legal consequences of plagiarism and the need for authenticity in student work through an understanding of copyright issues.
( 5 )- Refine knowledge and enhance skills in keyboarding, word processing, desktop publishing, spreadsheets, databases, multimedia, and telecommunications in preparing and presenting classroom projects.
( 3, 6 )- Use appropriate technology to locate, retrieve, organize, analyze, evaluate, and communicate information for problem solving and decision making.
( 1, 2, 4 )- Evaluate the usage of technology and the processes involved during and upon completion of individual and group projects.
( 2, 5 )
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Objectives:
The learner will:
1. discover a relationship between the circumference and diameter of round objects.
2. demonstrate the ability to measure given objects accurately.
3. utilize a spreadsheet program to organize data for individual groups. |
Lesson Materials and Resources:
1. Assorted circular objects such as tin cans, lids, garbage cans, pencils, clocks, plates, pans, bowls, etc.
2. Tape Measure
3. String and yardsticks
4. TI-92 Graphing Calculator
5. Computers with Microsoft Excel Software. |
Technology Tools and Materials:
Hardware:
Computer
TI-92 Graphing Calculator
Software:
Microsoft Excel
Cabri Geometry(TI-92)
Websites:
•PI Mathematics
ncsa.uiuc.edu/Edu/RSE/RSEorange/buttons
•The PI pages
www.cecm.sfu.ca/pi/pi.html
•Nine Planets
seds.lpl.arizona.edu/nineplanets/nineplanets/nineplanets
Other:
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Background Information:
Many students tend to memorize, without understanding, formulas that we use in geometry or other mathematic areas. This particular activity allows students to discover why PI works in solving problems dealing with finding circumference. Students should be familiar with the TI-92 graphing calculator and spreadsheet applications. This lesson corresponds to a unit involving circles, their area, circumference and diameter and how they are related. |
Lesson Procedures:
Introduction: Discuss with the students the history concerning the calculation of an irrational number known as “PI.“ Present the “PI Mathematics“ http://www.ncsa.uiuc.edu/edu/RSE/RSEorange/History.html web site to illustrate an interactive timeline.
1. Divide the class into groups of “2.“
2. Each group should gather at least 4 circular objects, string, tape measure, and TI-92 graphing calculators.
3. Discuss Laboratory guidelines.(Group norms should have been established in prior class periods)
4. The group members may use the tape measure and string to measure the circumference and diameter of each circular object. These measurements should be recorded in the appropriate places on an electronic spreadsheet such as MS Excel. Have the students save and print the spreadsheet.
5. Next, use the data in the spreadsheet to calculate the Circumference to Diameter (C/D) ratio.
6. The group members will use the calculator instruction sheet (see reproducible materials) to create, measure and calculate the C/D ratio. Each set of measurements and ratios should be recorded on the calculator generated measurement form.
7. The students will discuss the relationship between the circumference and diameter of a circle as a group and then as a whole class.
Closing Activity: Using Internet resources you can find the radius or diameter of the planets in our solar system and calculate the Circumference to Diameter (C/D) ratio. Choose the site “Nine Planets“ http://seds.lpl.arizona.edu/nineplanets/nineplanets/nineplanets to locate the necessary data. Students may use the “Data Recording form“ to record data. Discuss the results with the students. Other discussions may include:
“How do their circumferences compare?“
“What is the largest planet? Smallest?
“Are there any planets that are similar in measure?“ |
Assessment Procedures:
1. Group observation and whole class discussion.
2. Individual journal writing on What is Pi?
3. Give students practice applications involving
listing only diameters of each object and have
them find the circumference.
4. Groups will research and present other methods
for finding Pi. |
Accommodations/Modifications:
----- written by Ada Lawless
Teachers must follow special needs student's Individual Education Plan(IEP) or 504 Modifications Plan.
NON READER (BELOW GRADE-LEVEL READER)
1. Objective # 2: Allow the reluctant reader or non reader to collect the items for this assignment
2. Objective # 4, 5, 6: Pair reluctant reader with fluent reader to insure comprehension of directions, have a correct reading of the objects measured, and that the measurments are properly recorded. Allow non reader or reluctant reader to plug numbers into the Excel program. Pair reluctant or non reader with a fluent reading peer to insure that the material is read and understood by the non reader.
3. Objective # 7: Allow the reluctant reader to help the group prepare a PowerPoint presentation of the findings and operate the PowerPoint slide show as the other group members present.
4. Closing: Allow the reluctant or non reader, with supevision by fluent group leader, to copy and paste from the web sites to a Word document for the group's ending presentation.
5. Journal: Allow reluctant reader/writer to use a word prediction software program (CoWriter) to create his journal document. If no computer is available allow the reluctant reader/writer to tape record his journal or have a peer scribe record the special student’s journal writings.
STUDENTS WITH PHYSICAL IMPAIRMENTS
1. Objectives # 1, 2, 4, 5, Journal: Pair the non disabled peer for students with arm/ hand control difficulties. Other accommodations: Allow a clip board to be used to hold sheet for initial data collection, use a ball grip on student's pencil for better control, assign a notetaker as student dictates the information or use speech to text software to record students responses (ViaVoice, Dragon-Naturally Speaking, MS Windows accessibility feature).
2. Objective # 4: Allow student with arm/hand control difficulties to read the measurments as a non disabled peer places the tape measure at the directed spot for measurment.
3. Objective # 4, 5, 6-Journal: Adjust key repeat rate in MS Windows accessibility feature and/or use adaptive/enlarged keyboard (Intellikeys) in activities requiring computer responses.
4. Objective # 4, 5, 6: Use an overlay to restrict student striking more than one key at a time.
5. Closure: Allow the physically challenged student to report the group finding to the class.
Additional general classroom accommodations found in "helpful Links"
----- written by Barbara Reeves
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Reproducible Materials:
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Explorations and Extensions:
1. Have students to derive a formula to find the circumference of an object knowing only the diameter of that object. Students must prove their formula is accurate by demonstration and measuring to check their results.
2. Create a department wide “PI day“ in which all mathematics students can participate.
3. Have a Pi memorization contest in which the student who remembers the most number of places after the decimal is the winner. |
Lesson Development Resources:
Arnet, Bill. Nine Planets. [Online]. Available http://seds.lpl.arizona.edu/nineplanets/nineplanets/nineplanets February 2000. Description: Various information about the nine planets of our solar system.
Boyd, Cindy et al.(1998)Geometry Integration Applications Connections. New York, NY: Glencoe/McGraw-Hill Companies.
Kenda, Margaret.(1995). Math Wizardry for Kids. Hauppauge, NY: Barron's
Keyton, Michael.(1996) 92 Goemetric Explorations on the TI-92. Dallas, TX: Texas Instruments Inc.
Math Forum, The.Cabri Geometry Classroom Resources.[ONLINE]Available http://forum.swarthmore.edu/dynamic/cabri.links.html, December 1, 1999.Description:This site offers links on the web, and classroom resources for Cabri Geometry.
Serra, Michael.(1997). Discovering Geometry An Inductive Approach. Berkeley, CA: Key Curriculum Press. |
Reflections:
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Contact Information:
Ada Lawless
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Additional Contacts:
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Group: none |
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MarcoPolo Lesson: No
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