Chords and Inscribed Angles
Developed by Paul Hartzer, Spring 2012
Topic: Honors Geometry (10th Grade)
Purpose: This is the fourth lesson in a unit on the parts and relationships within a circle. The basic textbook is Glencoe’s Geometry, Michigan Edition.
Objectives and Materials
Learn the basic properties of chords, particularly the relationship between chords and arcs and that a radius bisects a chord iff it is perpendicular to it.
Learn the basic property of inscribed angles, i.e., that the measure of an inscribed angle is always half the measure of the central angle with the same arc.
G1.6.3 Solve problems and justify arguments about central angles, inscribed angles, and triangles in circles.
G1.6.4 Know and use properties of arcs and sectors and find lengths of arcs and areas of sectors.
G1.3.2 Know and use the Law of Sines and the Law of Cosines and use them to solve problems. Find the area of a triangle with sides a and b and included angle q using the formula Area = (1/2) ab sin q.
Resources and Materials Needed
GeoGebra (for individual use or demonstration)
Basic or graphing calculators, one for each student
Introduction and Lesson
Begin by distributing the worksheet. This is a scaffolded document that explains the formula they used on an earlier worksheet, √(2(1-cos(θ))). This relies on prior knowledge of the Law of Cosines.
This lesson will cover two sections in the book, 10.3 and 10.4.
Once students have been given time to complete the worksheet, discuss it as a group. Discuss how it relies on the notion that any central angle of the same measure will cut an arc that is also congruent. For an Honors Geometry class, students will generally find this self-evident.
Complete some examples of arcs as a class, and then proceed to inscribed angles. For this, demonstrate through GeoGebra (using the “Inscribed Angles” file that we can change the end points of the arc or the placement of the vertex of the inscribed angle, and the relationship of 1:2 between the inscribed angle and the central angle will always be the same. Again, go through some examples.
Practice and Closure
If time allows, allow for some classtime group practice of exercises from both sections.
Again reiterate the three basic take-aways from this lesson. Assign homework.
Accommodations and Assessments
Accommodations and Adaptations
As with the other lessons in this section, balancing the dynamic presentation of GeoGebra with more traditional lecture allows students greater ability to visualize the information.
The worksheet is more scaffolded in order to try to direct students without giving information away. This is more scaffolded than I had originally visualized in response to the comments and complaints I’d gotten on the skeeball exercise.
Outcomes, Assessments, and Extensions
The students should be able to commuicate the three basic take-aways, as well as some other information about chords, radii, and inscribed and central angles. Monitor students during work for understanding or confusion; clarify as needed. Review the homework as a formative assessment, and the test as a summative assessment.
This lesson seemed on the one hand rushed and on the other hand to be too obvious.
On the first part: At this point, my mentor was pressuring me to go faster. The goal is to finish the entire text by the end of the year, and while we were on pace to finish this chapter by the week before spring break (that is, so that the test would be on the Tuesday before break), my mentor wanted it even faster than that. We agreed to put these two sections into one day. Due in part to my inexperience, the transition between the two sections felt rough and appeared to confuse the students.
On the second part: Little of the material in this section isn't blatantly obvious to an honors student. There is, obviously, a place in mathematics for “prove everything”, and the level of detail in these sections are more needed in a standard level mathematics class, but this late in the year, the students seemed to feel condescended to by the content. My challenge the next time I teach this material from this book is to better clarify the purpose of learning this in a way that doesn't bore the students or make them feel like their intelligence is being insulted.
Meanwhile, though, they struggled with the worksheet because most of them had forgotten the Law of Cosines. This is student habit tied to school expectations: They’ve told me that once a chapter test has been completed, they forget the information until final exams. This (unfortunately) may work in some subjects, but for mathematics, this is dangerous and counterproductive.
However, in future teaching, I’ll work on stressing which information that students get is more important for remembering beyond the chapter border.