TITLE: Angle Measures in Polygons
TASK DEVELOPER: Robert Kehoe
CONTENT AREA AND GRADE: 11th Grade Geometry
SCOPE AND SEQUENCE: Section 11.1 Geometry Text
TARGET TEACHING DATE: May 15, 2007
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:
1. Apply the properties of geometric shapes.

  • Parallel lines - transversal, alternate interior angles, corresponding angles.
  • Triangles

2. 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

Back to top

PERFORMANCES:

1.  The students will use basic tools (pencil, straightedge, paper, protractor) to create various polygons.

2.  Students will analyze drawings in order to determine a hypothesis about the angles of a polygon.

Back to top

SETTING:

You are a designer of sports equipment.  A home plate marker for a baseball field is a pentagon.  Three of the interior angles of the pentagon are right angles.  The remaining angles are congruent.  What is the measure of each angle?

You are an architect asked to draw up plans for a particular stage structure.  An octagonal floor plan for a theater stage has two right angles and two that measure 135 degrees each.  The remaining interior angles are congruent.  Find the measures of these angles.

Back to top

SMARTSKILLS:

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

  • Vocabulary: Polygon, diagonal, quadrilateral, pentagon, hexagon
  • Concepts: Students will be able to dissect a diagram to judge its parts.

Level II: Visualizing Information - Data from Level I that are visualized as information in this standards-based task: Students will use everyday implements and realize the significance of the lesson.

Back to top

PREFERENCES:

Student Involvement - The students will complete the task:

  • Individually
  • In a whole class group setting

Instruction - Activities will be organized and delivered:

  • By differentiating the content students are expected to learn based on assessment results.
    As a teacher-facilitated set of hands-on activities.

Special Education Accommodations - Students with special needs will require the following electronic devices:

  • Calculators

Special Education Accommodations - Students with special needs will require the following presentation of information:

Extra processing and response time

Use of Resources - The school will provide:

  • classroom time to complete the task

Use of Resources - The students will provide:

  • classroom materials such as pencils, paper, notebooks
  • homework time

Customer for Student Work - The student will present their work as evidence of task completion to:

  • peers

Assessment of Student Work - The following people will be involved in assessing student work generated to complete the task:

  • The student's teacher

Reporting Results - The assessment results will be reported:

  • as a proficiency profile over time
  • as a letter grade

Timeline - The estimated time needed to plan, teach, and score this task is

  • one to three class periods

Back to top

ACTIVITIES:

Quality-Driven Teaching

Activity 1: Setting High Expectations
Estimated time for this activity: 10 minutes

  • Step 1. Ask students what is the sum of the measures of the interior angles of a triangle.
  • Step 2. Then ask the students to do the same about rectangles.
  • Step 3. See if students can make a connection between the 3-sided triangle and the 4-sided rectangle.
  • Step 4. Students will conclude that there are two triangles in a rectangle when all possible diagonals from one vertex are drawn.

Materials for this activity: Paper, pencil, straightedge
Student product or performance for this activity: Realizing the significance of the number of sides in a polygon and its relationship to the amount of diagonals.
Scoring tool for this activity: Teacher assessment

Activity 2: Activating Prior Knowledge
Estimated time for this activity: 20 minutes

  • Step 1. Students will draw various polygons and apply the same principles.
  • Step 2. Students will interact with each other.
  • Step 3. An assessment quiz will follow.

Materials for this activity: Calculator, paper, pencil, straightedge
Scoring tool for this activity: Evaluation of quiz

Activity 3: Acquiring Data
Estimated time for this activity: 20 minutes

  • Step 1. Students will hone their skills by doing additional practice and reteaching.

  • Step 2. Students will use extra examples and checkpoint exercises involving pentagons, hexagons, octagons, etc.

  • Step 3. Students will share their own ideas and thoughts.

Materials for this activity: Various problems throughout section
Student product or performance for this activity: This activity leads students to discover the Polygon Interior Angles Theorem

Activity 4: Visualizing Information
Estimated time for this activity: 15 minutes

  • Step 1. Point out to students how the activity on page 661 uses inductive reasoning to arrive at the conjecture that forms Theorem 11.1.
  • Step 2. Remind students that for this conjecture to become a theorem it must be proven.
  • Step 3. Students will provide a deductive proof for a case of Theorem 11.1 in exercise 43.
  • Step 4. Students will realize the architectural benefits of this lesson.

Materials for this activity: Paper, notes, pencil, computer

Activity 5: Applying Knowledge
Estimated time for this activity: 20 minutes

  • Step 1. Students can find extra problems at www.mcdougallittell.com.
  • Step 2. Students will be given various projects regarding structures in everyday settings.
  • Step 3. Closure question.

Materials for this activity: Text and computer (internet)
Scoring tool for this activity: Teacher graded

Activity 6: Reporting Results
Estimated time for this activity: 20 minutes

  • Step 1. Students will be encouraged to visit the internet for added problems.
  • Step 2. Students will work in small groups to exchange ideas and methodology.

Materials for this activity: Internet, paper, pencil

Source: Mcdougal Littell

Back to top

BENCHMARKING:

Student Performances:

Students will be able to sketch many types of polygons and apply the Polygon Interior Angles Theorem.

Students will also understand the constant sum regarding the exterior angles of various convex polygons.

Students will be able to answer the challenge of related problems using the formula from the corollary of the above theorem.

Back to top

SCORING:

New Jersey High School Mathematics Rubric: Brief Constructed Response Items

New Jersey High School Mathematics Rubric: Brief Constructed Response
Level 3

The response indicates application of a reasonable strategy that leads to a correct solution in the context of the problem. The representations are essentially 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 2

The response indicates application of a reasonable strategy that may be incomplete or undeveloped. It 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

The response indicates little or no attempt to apply a reasonable strategy or applies an inappropriate 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 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

Back to top

METACOGNITION:

Cognitive Information: I will collect the following information by asking the questions to students in class and analyzing their responses.

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 gauge the following information based on their responses to my questions.

1. Do you feel that you are good in mathematics?
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?

Back to top

RESULTS:

Reflect: As I relate my students' results with my lesson activities, I noticed that the students were responsive to the theme of the lesson.  They enjoyed the freedom to investigate various situations.  They were able to follow the complexities of more difficult problems.  Students were assisting other students and it was evident that all students were involved.

Summarize: I assessed and scored the content standard geometry and measurement and realized that 83% of my students performed at or above the proficient level on my scoring rubric.  I was pleased with way the students accepted the challenges and dealt with the situations.  

The homework assignment and opening quiz were valuable to the lesson because they required students to address possible concerns prior to the class period. With most students clear on content expectations, they could then demonstrate their comprehension of the project. This assignment would not be as successful if students were not aggressively encouraged to complete the "guided practice."

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: Areas of Regular Polygons
    Content Area: Mathematics, Geometry    
    Learning Standard(s): 4.2,4.5
    Intent:
    • Find the area of an equilateral triangle.
    • Find the area of a regular polygon.

Back to top