UNIT PLAN

STUDENT TEACHING – GEOMETRY

During the spring of 2002, I completed my student teaching with the mathematics department at Blacksburg High School. I spent ten weeks teaching over 140 hours. A majority of those hours were spent teaching four 9th and 10th grade geometry classes. Of the four, two were Honors Geometry and two were General College Prep Geometry. All four geometry classes worked out of the Glencoe Geometry: Integration, Application, Connections textbook. I was required to create and implement a unit plan for these classes. My unit plan covered material from the first seven sections of chapter 9 entitled Analyzing Circles of their textbook. The chapter was the first of two chapters that were covered during the 5th grading period (5th six weeks) of the school year that ran from March 8 to April 25. It immediately followed a chapter on right triangles and trigonometry. Material and concepts from chapter 8, such as the Pythagorean Theorem and angles measures, are often used or applied in problems throughout my unit.

The material in my unit falls under only one of the Geometry Standards of Learning in Virginia :

G.10: The student will investigate and use the properties of angles, arcs, chords, tangents, and secants to solve problems involving circles. Problems will include finding the area of a sector and applications of architecture, art and construction.

The following NCTM geometry standards for grades 9-12 apply to my unit plan:

draw and construct representations of two- and three-dimensional geometric objects using a variety of tools

analyze properties and determine attributes of two- and three-dimensional objects;

explore relationships (including congruence and similarity) among classes of two- and three-dimensional geometric objects, make and test conjectures about them, and solve problems involving them;

establish the validity of geometric conjectures using deduction, prove theorems, and critique arguments made by others;

use trigonometric relationships to determine lengths and angle measures.

The following table briefly outlines the seven sections of my unit. To see a detailed lesson plans and reflections from my teaching for each of the seven sections, click on the section title. Each lesson plan contains links to relevant materials, handouts, overheads, and sample work from my students.

Section 9.1: Exploring Circles
Objectives	Identify and use parts of circles Solve problems involving the circumference of a circle
Concepts	Circle, center, radius, diameter, chord, circumference, pi, distance around a circle
Activities	Puzzle of the Day Construction of circles Class discussion Discovering Pi Activity Four Color problem (long-term project)
Assessment	Formal: Discovering Pi Activity Four Color Problem Informal: Homework problems Class observations
Section 9.2: Angles and Arcs
Objectives	Recognize major and minor arcs, semicircles, and central angles. Find measures of arcs and central angles. Solve problems by making circle graphs.
Theorems/ Postulates	Postulate 9-1: Arc Addition Postulate: The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs.
Concepts	Central angles, sum of central angles, arc measure, minor and major arcs, semicircle, adjacent arcs, arc length, concentric circles, similar circles, congruent circles, congruent arcs
Activities	Puzzle of the Day Review/Discussion on homework Use pi charts to explore central angles Exploratory activity with constructed circles
Assessment	Formal: 9-2 Practice sheet Informal: Homework problems Observations
Section 9.3: Arcs and Chords
Objectives	Recognize and use relationships among arcs, chords, and diameters.
Theorems	Thm 9-1: In a circle or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent. Thm 9-2: In a circle, if a diameter is perpendicular to a chord, then it bisects the chord and its arc. Thm 9-3: In a circle or in congruent circles, two chords are congruent if and only if they are equidistant from the center.
Concepts	Arc of the chord, inscribed polygons
Activities	Puzzle of the Day Group construction explorations Group discussions
Assessment	Formal: 5-minute Quiz (Intro Activity) Group work
Section 9.4: Inscribed Angles
Objectives	Recognize and find measures of inscribed angles. Apply properties of inscribed figures
Theorems	Thm 9-4: If an angle is inscribed in a circle, then the measure of the angle equals one-half the measure of its intercepted arc. Thm 9-5: If two inscribed angles of a circle or congruent circles intercept congruent arcs or the same arc, then the angles are congruent. Thm 9-6: If an inscribed angle of a circle intercepts a semicircle, then the angle is a right angle. Thm 9-7: If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary.
Concepts	Inscribed angles, intercepted arcs
Activities	Puzzle of the day Review of previous 3 sections Group exploratory activity Class discussion
Assessment	Formal: Group work Homework worksheet Informal: Observations Class review
Section 9-5: Tangents
Objectives	Recognize tangents and use the properties of tangents
Theorems	Thm 9-8: If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency. Thm 9-9: In a plane, if a line is perpendicular to a radius of a circle at the endpoint on the circle, then the line is a tangent of the circle. Thm 9-10: If two segments from the same exterior point are tangent to a circle, then they are congruent.
Concepts	Tangent, point of tangency, interior and exterior of a circle, common tangents, common external tangent, common internal tangent, tangent segment, circumscribed polygons.
Activities	Puzzle of the day Review homework Introduction to tangents Construction of a tangent
Assessment	Formal: Quiz Construction of tangent Informal: Observations
Chapter 9 – Test PART I (College)
Section 9-6: Secants, Tangents, and Angle Measures
Objectives	Find the measures of angles formed by intersecting secants and tangents in relation to intercepted arcs.
Theorems	Thm 9-11: If a secant and a tangent intersect at the point of tangency, then the measure of each angle formed is one-half the measure of its intercepted arc. Thm 9-12: If two secants intersect in the interior of a circle, then the measure of an angle formed is one-half the sum of the measures of the arcs intercepted by the angle and its vertical angle. Thm 9-13: If two secants, a secant and a tangent, or two tangents intersect in the exterior of a circle, then the measure of the angle formed is one-half the positive difference of the measures of the intercepted arcs.
Concepts	Secant
Activities	Puzzle of the Day Review of homework Class discussion with examples of three theorems
Assessment	Formal: Quiz Homework worksheet
Section 9-7: Special Segments in a Circle
Objectives	Use the properties of chords, secants, and tangents to solve segment measure problems
Theorems	Thm 9-14: If two chords intersect in a circle, then the products of the measures of the segments of the chords are equal. Thm 9-15: If two secant segments are drawn to a circle from an exterior point, then the product of the measures of one secant segment and its external secant segment is equal to the product of the measures of the other secant segment and its external secant segment. Thm 9-16: If a tangent segment and a secant segment are drawn to a circle from an exterior point, then the square of the measure of the tangent segment is equal to the product of the measures of the secant segment and its external secant segment.
Concepts	Secant segments, external secant segments
Activities	Puzzle of the Day Class discussion of theorems
Assessment		Formal: In-class problems (graded)
Chapter 9 – Reviews for Tests
Chapter 9 – Test PART II (College)
Chapter 9 – Test (Honors)

Along with teaching the unit plan outlined above, I was involved in the planning and teaching of most of chapters 8 and 10 in the Geometry text. Out of all the lessons that I planned and implemented, my most favorite lesson dealt with the proofs of the pythagorean theorem. As I read the section from chapter 8 that taught the pythagorean theorem, I became very interested in the geometric proofs of the theorem. The textbook presented one geometric proof in its Modeling Mathematics activity. After reading it, I began to look in other books and online resources to find other proofs. I read about numerous proofs used by different cultures and various mathematicians over the years. I decided to present three of these proofs to my geometry classes. In one of my honors geometry classes, I had the students work in small groups to construct, explore, and discover the three proofs. In my other classes, I did not have the time for the students to do this, but I was able to take the time to share the proofs with them. I do not know if the students found the proofs quite as exciting as I did, but I had a great time presenting them! Take a look at the three proofs! Hopefully you will find them just as interesting as I have.

PROOF #1
PROOF #2
PROOF #3

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