Overview: Parts of a Circle

I taught this unit in Spring 2012, as part of an Honors Geometry course, to two classes of sophomores at Western International High School.
The purpose of the unit from the perspective of the curriculum and textbook (Geometry, Glencoe Mathematics, Michigan Edition) is to introduce students to some basic properties of circles. The sections covered in class include “Circles and Circumferences,” “Angles and Arcs,” “Arcs and Chords,” “Inscribed Angles,” “Tangents,” “Secants, Tangents, and Angle Measures,” and “Equations of a Circle.” There is an additional section of the book, “Special Segments in a Circle,” that is not covered in the class due to time considerations. The related HSCEs listed on the pacing chart are G1.6.1, G1.6.2, G1.6.3, G1.6.4, and G1.7.1.
I chose to go beyond the material in the book in several respects. Several times over the course of the year, students have asked me where π comes from, and I felt that this would be an excellent opportunity to introduce them to the concept of π in terms of circumference, both to address their curiosity and to introduce them to the calculus-based concept of limits. The book uses a method similar to the one I use later in the chapter on the area of circles, at which point I’ll remind students of their work here. The relevant HSCEs are L1.1.6 (“the importance of π because of its role in circle relationships”), L2.2.3 (“use iterative processes”), L2.4.2 (“describe and explain round-off error”), and L1.2.5 (“read and interpret representations from various technological sources”).
Because there is a good deal of vocabulary in this chapter, I also took the opportunity to discuss and present some basic morphology as it applies to mathematics; the book presents similar information as an optional exercise in the next chapter, albeit not as organized. I feel that tying morphology to their vocabulary words and including terms from outside of mathematics (circumvent, tangible, impending, and so on) allows students to build their vocabulary more quickly. The relevant ELA HSCE is CE4.1.2 (“use resources to determine word meanings”). I also exposed the students to the Greek alphabet; I am personally used to using the Greek characters (α, β, θ, etc.) to refer to angles, rather than the Latin characters (A, B, C, etc.) that the book uses. While the latter may increasingly be the standard, I informed them that having exposure to the former will be useful when dealing with certain sources (GeoGebra, for instance, uses Greek by default).

Lesson Structure and Assessment

In order to balance traditional district content expectations with my desire to encourage a deeper level of reasoning from my students, I structured most periods during this unit as follows: The “warm up” was a worksheet which encouraged some level of constructive reasoning; some class time was spent working on the previous day’s homework; lecture relevant to the textbook’s lesson was presented, with class exercises if time allowed; and homework was assigned.
For the final chapter assessment, in order to satisfy district and school expectations, I structured the summative assignment based on the textbook material. The summative exam included a mix of multiple choice and open ended questions. The multiple choice questions are largely due to student preference; my mentor teacher’s tests were all multiple choice photocopied out of Glencoe’s teacher support products, and so that’s what the students are used to. I strongly prefer open-ended questions so that I can get an understanding of students’ thought processes. As a compromise, I offer them a mix, with the additional understanding that students can get partial credit on multiple choice questions if they provide some rationale on the test for their answers.
At a minimum, the successful student will demonstrate knowledge of the basic relationships within a circle and an understanding of the relevant vocabulary.

Tools, Resources, and the Environment

I have two sections of Honors Geometry. The first section has 21 students; the second has 28 students. The default seating arrangement is in a traditional grid pattern, with five ranks across and six seats in each. This arrangement is a compromise based on having to share the room with two other teachers (my mentor, who teaches AP Calculus, and my practicum partner, who teaches Pre-Calculus).
The classroom has a whiteboard spanning most of the wall faced by the students, a projector which has a retractable screen (so images can be projected either onto the screen or onto a portion of the whiteboard), and a computer. However, because the computer is on the teacher’s desk and is difficult to get to while lecturing, I brought my own laptop and connected it to the projector via the alternate jacks. This way, I was able to manipulate images on the computer while still engaged with the class.
My primary technological tool for this and all units in this class is GeoGebra, a free geometry toolkit that allows for a good deal of flexibility in class demonstrations. I encourage my students to explore with GeoGebra on their own as well.
Students can keep track of assignments and get copies of missing worksheets via the Hero's Garden website (which I maintained at the time), as well as tracking their missing assignments via engrade.

Unit Schedule: Day by Day

All warm ups are handouts except day 2 (projected on the board) and days 8 and 9 (in the book).
Day Warm-up Topic
1 The perimeters of polygons Introduction to Circles (Chapter 10)
2 Skeeball diameters Parts of a Circle (Section 10.1)
3 Circle segments and pie charts (Section 10.2-.3) Arc Measurements
4 Chord lengths and polygon sides Chords and Inscribed Angles (Section 10.3-.4)
5 The perimeters of polygons revisited Tangents (Section 10.5)
6 Inscribed angles and tangents Tangents; Secants (Section 10.5-.6)
7 Circle vocabulary (front of handout) Secants (Section 10.6), Mathematical Morphology (back of handout)
8 Practice Quiz 2 (in the book) Equation of a Circle (Section 10.8)
9 Copy Standard Equation of a Circle into notes (in the book) Equation of a Circle
10 Chapter wrap-up with musical exploration: What Pi Sounds Like Chapter Review
11 (None) Chapter Test

Unit-Level Reflections

Because of the pacing, it was difficult for me to provide as much formatively assessed classwork as I would have liked. I did use the previous day’s homework review as an opportunity to do a formative assessment, as well as moving around the room during the warm up to monitor understanding of the exercise.
I also think that my novice status as a teacher led me to rely too much on “the book”; balancing those expectations with my own material often made the schedule seem overloaded, to the point that I gave one day that had been intended mostly for lecture (day 8) over to a breather day where students could catch up with their missing work over the course of the chapter.
This tendency to try to overfill a single class period is something I’ve noticed in myself in the past, such as with last semester’s lesson plan in Pre-Calculus. As I grow in confidence as a teacher, and feel less need to be tethered to the book’s content, I plan to balance out the pacing by better integrating the required school or district material into my own exercises, thus making more time for student-level work.
Finally, while much of the emphasis in this program has been on providing application-based, constructive exercises, I continue to find students giving me the most pushback when it comes to story problems and applied exercises. In this case, students overwhelmingly ignored the skee-ball exercise, despite my repeated prodding. They were excited to see a problem about skee-ball, which most of them reported playing, but they weren’t interested in actually trying the problem in question.
Based in part to some of their comments on that exercise, I worked to more explicitly scaffold for the “Chord lengths and polygon sides” worksheet, but they seemed overwhelmed with the amount of language on the sheet. It seems to be a delicate balance with these students: Too much language, and they get overwhelmed, too little in the instructions and they don’t feel confident in exploring. A good deal of this is school culture, but I do think some of it is due to the expectations set in the first half of the year by my mentor teacher. I hope that I will have more success in encouraging fuller inquiry when I can establish that behavior from the beginning of the school year.