TITLE: Inscribed Angles in a Circle
TASK DEVELOPER: Blair Seidler
CONTENT AREA AND GRADE: 11th Grade Basic Geometry
SCOPE AND SEQUENCE: Section 10.5 Basic Geometry Text
TARGET TEACHING DATE: March 26, 27, 28
SCHOOL: John F. Kennedy High School

STANDARDS:

GEOMETRY AND MEASUREMENT - GRADE 9-12

STANDARD 4.2 GEOMETRY AND MEASUREMENT: All students will develop spatial sense and the ability to use geometric properties, relationships, and measurement to model, describe and analyze phenomena

Strand A. Geometric Properties: Building upon knowledge and skills gained in preceding grades, by the end of Grade 12, students will:

3. Apply the properties of geometric shapes.

  • Circles - arcs, central and inscribed angles, chords, tangents

4. Use reasoning and some form of proof to verify or refute conjectures and theorems.

  • Verification or refutation of proposed proofs
  • Simple proofs involving congruent triangles

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PERFORMANCES:
  1. The students will use standard tools (compass, protractor, ruler) to perform a geometric construction.
  2. The students will collect data from their constructions, and use this data to form a hypothesis
  3. The students will use the results of this exercise and the beginning of the proof that the measure of the Inscribed Angle is half the measure of the corresponding Central Angle to complete the proof.

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SETTING:

Real World Setting: Architecture/Planning

You are an architect. You have been asked to help design the seating for a movie theater. You must design the seating so that all of the seats will provide an acceptable view of the screen. You will be given parameters for what constitutes an acceptable view. How can you make sure that every seat in the theater is correctly placed?

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SMARTSKILLS:

Level I: Acquiring Data - Data students will acquire in this standards-based task:

Vocabulary: Students will learn the definition of inscribed angles.
Skills: Students will use protractor and compass to construct and measure circles and angles.
Processes: Students will follow a multi-step procedure for the construction.

Level II: Visualizing Information - Data from Level I that are visualized as information in this standards-based task:

Organizing: Students will organize their measurements into a table of values.
Creating meaning: Students will analyze the data to determine the relationship between inscribed angles and central angles in a circle.

Level III: Applying Knowledge - Visualized information from Level II that is applied knowledge in this standards-based task:

Solving problems: Students will generalize from their specific measurements to a hypothesis about why inscribed angles have half the measure of the corresponding central angle.
Creating solutions: Students will complete a proof of the theorem, verifying their hypothesis.

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PREFERENCES:

Student Involvement - The students will complete the construction of the circle and angles and the proof of the theorem individually. The students will analyze the data collected in a whole class group setting. The students will verify or refute each other's proofs in a whole class group setting.

Instruction - Activities will be organized and delivered as a teacher-facilitated set of hands-on activities.

Special Education Accommodations - Students with special needs will require a calculator

Special Education Accommodations - Students with special needs may require extra processing and response time, or
written or photocopied notes of orally presented instruction or assessment materials.

Use of Resources - The school will provide protractors, compasses, rulers, calculators as needed.

Use of Resources - The students will provide classroom materials such as pencils, paper, notebooks

Customer for Student Work - The students will present their work on the construction to the teacher. The students will present their completion of the proof to the entire class.

Assessment of Student Work - The students' teacher will be involved in assessing student work generated to complete the task.

Assessment of Student Work - The following forms of assessment will be used to determine progress and results:

The completed construction and proof will be assessed. Each student's participation in the group discussions will be assessed. There will be a test at the end of this unit to assess student mastery of the underlying concepts.

Reporting Results - The assessment results will be reported as a letter grade.

Timeline - The estimated time needed to plan, teach, and score this task is three to four class periods

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ACTIVITIES:

Activity 1: Review of yesterday's lesson and new definition (Est. time: 3 minutes)

  • Ask students to define central angle and the measure of an arc
  • Define inscribed angle: An inscribed angle has its vertex on a circle and rays which intersect the circle at points other than the vertex.

Materials: notebooks, pencils. Students will produce yesterday's homework, provide definitions, and write the new definition in their notebooks.

Activity 2: Construction of an inscribed angle (Est. time: 20 minutes) The teacher will direct the students through this process, and will draw each step of the construction on the board while the students work at their seats.

  • Use the compass to draw a circle. Mark the center of the circle and label it as point O.
  • Choose 3 points on the circle. Label them A, B, and C.
  • Use the ruler to draw ray BA and ray BC. Use the protractor to measure angle ABC and record the measurement of the inscribed angle.
  • Use the ruler to draw ray OA and ray OC. Use the protractor to measure angle AOC and record the measurement of the central angle.

Materials: notebook, pencil, compass, protractor, ruler. The students will draw the required construction using the tools provided. The students will be able to visually observe the difference between central angles and inscribed angles.

Activity 3: Synthesis and analysis of data (Est. time: 15 minutes)

  • Draw a table on the board with two columns labeled central angle and inscribed angle.
  • Ask each student in turn to provide the measures of the central and inscribed angles from his or her construction.
  • The teacher will lead a group discussion starting with the question, "Do you see any relationship between the measures of the two angles?" The group should work towards the conclusion that the inscribed angle will have half the measure of the corresponding central angle. Once this relationship has been established, ask the students to hypothesize about why this relationship works the way it does.

Activity 4: Proof of the theorem m(inscribed)=1/2 m(central)   (Est. time 30 minutes) The teacher will present the first two cases of the proof.

  • Draw case I, in which one of the rays of the inscribed angle contains a diameter of the circle. Label the center O, and the other points A, B, and C (BC is the diameter). Draw angle ABC and segment OA.
  • Consider triangle AOB. m(ABO) m(BAO) m(AOB)=180°
  • segments OA and OB are radii, and therefore congruent, so OA=OB.
  • triangle AOB is isosceles, so angles ABO and BAO are congruent and m(ABO)=m(BAO).
  • Therefore, 2m(ABO) m(AOB)=180°
  • Note that angles AOC and AOB are supplementary, so m(AOC) m(AOB)=180°
  • By transitive, 2m(ABO) m(AOB)=m(AOC) m(AOB)
  • Subtracting m(AOB) from both sides gives us the desired result of 2m(ABO)=m(AOC)

At this point, the teacher should start a group discussion of how and why this proof works. The students should be able to understand the components and structure of the proof. The discussion needs to include the fact that this is only a partial proof because it only works if BC is a diameter. This should lead into the presentation of case II.

  • Draw case II, which is a circle with inscribed angle ABC, both of whose rays are on the same side of the diameter of the circle through B. Draw the diameter through B and label the point on the other side of the circle D.
  • According to the result of Case I, 2m(CBD)=m(COD) and 2m(ABD)=m(AOD)
  • Because angles AOC and COD are adjacent, m(AOD)=m(AOC) m(COD)
  • Therefore m(AOC) = m(AOD)-m(COD) = 2m(ABD)-2m(CBD) = 2 (m(ABD)-m(CBD))
  • Because angles ABC and BCD are adjacent, m(ABD)=m(ABC) m(BCD)
  • Therefore m(ABD)-m(CBD)=m(ABC)
  • Substituting yields m(AOC)=2m(ABC)

Materials: notebooks, pencils. The students will take notes on the proof and the explanation. Assign the third case (rays of inscribed angles on opposite sides of the diameter) for homework.

Activity 5: Discuss student proofs (Est. time 30 minutes)

  • Have several students put their proofs of case III (homework) on the board.
  • Have the class verify or refute each of the proofs with the teacher as moderator

Materials: notebooks, pencils, chalk.

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BENCHMARKING:

Student Performance One: Construction

Assessment Benchmarking Example: The student's construction will include a circle, an inscribed angle, and the corresponding central angle. All points will be correctly labeled. The student will correctly determine and record the measure of each angle.

Student Performance Two: Proof

Assessment Benchmarking Example: The student will produce a valid proof in two-column format. Each step will follow logically from previous steps and each step will correctly identify the theorem or postulate used.

Student Performance Three: Group discussions

Assessment Benchmarking Example: The student will participate actively in the group discussions. The student's comments will be mathematically valid, relevant, and courteous to classmates and teacher.

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SCORING:

High School Mathematics Rubric: Extended Constructed Response Items

High School Mathematics Rubric: Extended Constructed Response
Level 4: A The response indicates application of a reasonable strategy that leads to a correct solution in the context of the problem. The representations are correct. The explanation and/or justification is logically sound, clearly presented, fully developed, supports the solution, and does not contain significant mathematical errors. The response demonstrates a complete understanding and analysis of the problem.
Level 3: B

The response indicates application of a reasonable strategy that may or may not lead to a correct solution. The representations are essentially correct. The explanation and/or justification is generally well developed, feasible, and supports the solution. The response demonstrates a clear understanding and analysis of the problem.
Level 2: C

The response indicates an incomplete application of a reasonable strategy that may or may not lead to a correct solution. The representations are fundamentally correct. The explanation and/or justification supports the solution and is plausible, although it may not be well developed or complete. The response demonstrates a conceptual understanding and analysis of the problem.
Level 1: D

The response indicates little or no application of a reasonable strategy. It may or may not have the correct answer. The representations are incomplete or missing. The explanation and/or justification reveals serious flaws in reasoning. The explanation and/or justification may be incomplete or missing. The response demonstrates a minimal understanding and analysis of the problem.
Level 0: F The response is completely incorrect or irrelevant. There may be no response, or the response may state, "I don't know."
Notes: Explanation refers to the student using the language of mathematics to communicate how the student arrived at the solution.

Justification refers to the student using mathematical principles to support the reasoning used to solve the problem or to demonstrate that the solution is correct. This could include the appropriate definitions, postulates and theorems.

Essentially correct representations may contain a few minor errors such as missing labels, reversed axes, or scales that are not uniform.

Fundamentally correct representations may contain several minor errors such as missing labels, reversed axes, or scales that are not uniform.
Source: http://www.mdk12.org/mspp/high_school/structure/algebra/index.html

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METACOGNITION:

Cognitive Information: I will collect the following information in a survey at the end of the unit.

  1. Describe what skills you needed to complete this task.
  2. Explain how you solved the goal, problem, or issue in this task.
  3. Draw a picture that shows how you solved the goal, problem, or issue in this task.
  4. Explain why you completed the task your way.

Attitude Information: I will collect the following information in a survey at the end of the unit.

  1. Do you feel that you are good in geometry?
  2. Did you find this task to be difficult?
  3. Did you see the usefulness of what you were asked to do in real life?
  4. Did you enjoy the task?

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RESULTS:

Analyze: I will examine the data in the chart to look for trends, contributing factors, and implications of student performance over a series of assessments of the same learning standard.

Trends: I will look for improvement relative to previous lessons which included construction and proof.

Reflect: I will consider two or more of the following stems to reflect on the results and instructional practices I used and others I might benchmark and apply in the future. Then, I'll write a brief summary about my findings, contributing factors, and implications for improvement.

As I relate my students' results with my lesson activities, I noticed that...

Having the students perform a construction has the most promise for becoming a best practice in my classroom because I find that students have a better retention of geometric concepts if they have a chance to see and measure shapes rather than just reading or hearing about them.

This connects to previous and subsequent lessons in the chapter on circles. The students are becoming familiar with the techniques of construction and proof and are using both correctly.

Action Plan: I will complete the following TaskBuilder Figure 8 Strategy Action Plan to prepare for my next standards-based task.

1. Plan - My next standards-based task will focus on:
Title: Square roots in algebra
Content Area: Algebra (10th grade)
Learning Standard(s): NJ CCCS 4.4
Intent: Define the concept of square root and apply it to the simplification of algebraic expressions.

5. Team or Grade Level Portfolio and School Web Site - I will insert the standards-based instruction or assessment task, results, samples of student work, and summary into my Team or Grade Level Portfolio and upload them to my School's Instructional Web Site on the following dates:

Target date for School Instructional Web Site: April 4

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