Introduction to Circles
Developed by Paul Hartzer, Spring 2012
Topic: Honors Geometry (10th Grade)
Purpose: This is the first lesson in a unit on the parts and relationships within a circle. The basic textbook is Glencoe’s Geometry, Michigan Edition.
Objectives and Materials
Identify the pattern of the perimeters of polygons of increasing numbers of sides.
Get a basic understanding of limits.
Learn the formula for the circumference of a circle.
L1.1.6 Explain the importance of the irrational numbers and in basic right triangle trigonometry, and the importance of pi because of its role in circle relationships.
L2.2.3 Use iterative processes in such examples as computing compound interest or applying approximation procedures.
Resources and Materials Needed
Microsoft Excel and GeoGebra (for individual use or demonstration)
Basic or graphing calculators, one for each student
Introduction and Lesson
Begin with the handout. Guide students minimally. The purpose of this exercise is for them to notice that the number in the fourth column is converging on pi. Then discuss the importance of circles in general, tying the topic to circular objects in their life (wheels, cans), and prepare them for learning new vocabulary.
Once the students have been given time to explore the worksheet alone or in groups, bring them together to discuss their findings. Refer to the “Polygon half perimeter” Excel file (“Inscribed” tab), as well as to the “Inscribed Polygons” GeoGebra file. These demonstrate how, as polygons have increasingly high numbers of sides, they get closer to resembling circles. If possible, allow the students to explore the GeoGebra file on their own.
In the GeoGebra file, the square and hexagon have been marked to allow for a discussion of trigonometry. Students will likely have the most difficulty with the specific formula being used; the formula for circumscribed polygons (day 5) is simpler, but relies on the notion of tangents. Also, the inscribed polygon formula is used first because it more closely represents the historic development.
Practice and Closure
The practice comes in the form of the constructive worksheet.
Review the lesson. Point out that they’ve now learned an important building block of calculus.
Accommodations and Assessments
Accommodations and Adaptations
The same data is presented in different ways to allow for all students to see how different representations can mean the same thing. This differentiation is particularly useful for specific sorts of learners: Some will prefer the hands-on process of creating the data, some the more tabular nature of Excel, some the more visual nature of GeoGebra.
Outcomes, Assessments, and Extensions
Monitor students as they work on the worksheet; identify areas on which specific or multiple students are struggling. This is an introductory presentation, and isn’t intended to be part of the final chapter summative assessment: It’s meant to whet their creativity.
I felt like this exercise went particularly well. While students are often indifferent to their warm-ups, most of them dug into this worksheet and took up the challenge of finding the pattern. Students were also generally more interested in the GeoGebra portion of the demonstration than in the Excel portion. I would have prefered to have done this on a Smartboard; the set up in our classroom is to project from the computer onto the white board, which is generally adequate but inferior to a Smartboard. It’s also a limitation of GeoGebra that polygons take a long time to set up, and therefore relied on my having set up the file ahead of time. This limited my ability to show features dynamically; one student, for instance, wanted to see the radii on the polygons with more sides. I’m not sure if there is commercial software that is more flexible in this regard; the next time I teach this, I’ll explore perhaps using some sort of dynamic manipulation in Mathematica.