TITLE: Parallel Lines Cut by a Transversal

TASK DEVELOPER: Uriel Zea

CONTENT AREA AND GRADE: Basic Geometry for Limited English Proficiency

TARGET TEACHING DATE: April 26, 27, 28 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. Use geometric models to represent real-world situations and objects and to solve problems using those models.

2. Apply the properties of geometric shapes.

  • Parallel lines - transversal, alternate interior angles, corresponding angles
  • Triangles
    a. Conditions for congruence
    b. Segment joining midpoints of two sides is parallel to and half the length of the third side
    c. Triangle Inequality
  • Minimal conditions for a shape to be a special quadrilateral
  • Circles - arcs, central and inscribed angles, chords, tangents
  • Self-similarity

3. 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
  • Counterexamples to incorrect conjectures

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PERFORMANCES:

The students will understand the angle theorems related to parallel lines, then they will apply the theorems to find angle measurements when giving the measurement of some angle. Therefore, the students will discover learning by comparing angle measurements and types of angles working cooperatively in groups.

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SETTING:

Real World Setting: Home improvement

As a home owner, you need to improve the attic of your house. You need to install four windows in it. The design of the window includes parallel lines intersected by a transversal (see graph). You are faced to find angle measurements when you know the measure of some angle. Then you will go to Home Depot Store in order to buy the windows with the same measurement to be installed in your attic. For that purpose you must apply the theorems learned in the classroom.

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SMARTSKILLS:

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

  • Vocabulary/key Words (bilingual Setting): Parallel Lines, Transversal, Alternate Interior Angles, Alternate Exterior Angles, Same-Side Interior Angles, Corresponding Angles, and Vertical Angles.
  • Numbers: With the use of a Protractor students will find the measure of angles.
  • Images: Students will draw two parallel lines intersected by a transversal
  • Creating meaning:After analyzing the data (angles) they will pair angles with the same measures.

Level II: Visualizing Information - Data from Level I that are visualized as information in this standards-based task: Graphs of two parallel lines cut by a transversal.

  • Organizing: Students will organize their data (angles) and then they will apply the theorems involved.

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

  • Solving problems: Based on the graphs, solve problems involving these theorems and properties based on the facts they already know.

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PREFERENCES:

Student Involvement - The students will complete the graph and discover learning by comparing angle measurements and types of angles as they work cooperatively in groups.

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 the following electronic devices: Calculator, dictionary, work sheets, tracing paper, rulers, and protractors.

Special Education Accommodations - Students with special needs will require extra processing and response time.

Use of Resources - The school will provide calculators, work sheets, tracing paper, rules, and protractors as needed.

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

Customer for Student Work - The student will present their work as evidence of task completion to the teacher and to their 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 letter grade class work and a number grade for the quiz, and test.

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

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ACTIVITIES:

Teaching for Understanding in Mathematics

Activity 1: The teacher reviews yesterday's lesson and assigns a problem that was not finished
(Estimated time: 10 minutes)

  • Step 1: I will check the homework for the day before.
  • Step 2: I will review the the lesson about angles and classification of them.
  • Step 3: I will check the new vocabulary.

Activity 2: The students will construct a graph that includes two parallel lines cut by a transversal. The teacher will guide the students each step to complete the graph.
(Estimated time: 15 minutes)

  • Step 1: Use ruler to draw a pair of parallel lines 2 inches apart.
  • Step 2: Use the ruler to draw a transversal. It will form a 30 degree angle with one of the parallel lines.
  • Step 3: Number the angles 1 through 8 counterclockwise ensuring that all students' graphs are alike.

Activity 3: I will tell students they are going to discover some special properties of these angles formed by two parallel lines cut by a transversal.
(Estimated time: 25 minutes)

  • Step 1: Using worksheet and protractor.
  • Step 2: Tell students they are going to discover these properties by measuring and comparing angles.
  • Step 3: While class works on the activity, I will check their work.

Activity 4: The students will prove this theorem based on the theorems we have already learned.

(Estimated time: 25 minutes)

  • Step 1: This proof will be assigned as homework or done in class if time permits.
  • Step 2: Do another example: If one alternate exterior angle = 20x 10 and another = 15x - 40, solve for x. Explain the answer.

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BENCHMARKING:

Student Performance One: Graphing.

1. Assessment Benchmarking Example: Students will be able to construct a graph with two parallel lines cut by transversal and they will be able to find the measure of each angle.

Student Performance Two: Proof

1. Assessment Benchmarking Example: Students will prove this theorem based on the theorems we have already proved. This proof will be assigned as homework, or done in class if time permits.

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SCORING:

High School Mathematics Rubric: Brief Constructed Response Items

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 0The 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 when groups seem to be finishing, and refocus them as a whole for the explanation.

  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 by asking students in class and summarize their verbal responses.

  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?

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RESULTS:

Reflect: As I relate my students' results with my lesson activities, I noticed that in general 65% of them performed at or above the level 3 on my scoring mathematics rubric, 15% performed at level 2, and the rest at level 1. Also, I noticed that the students performed better and enjoy working cooperatively in groups. When they work in groups, they can share the translation from Spanish to English in order to understand the main idea of the activity. Groups will have students from different levels of ESL.

Analyze: As a result of examining the results and observing the performance of their activities, I can conclude that:

  • The geometric concepts had the most influence on student achievement.
  • The English language had the least influence on student achievement because: I was explaining the class in Spanish and the activities were made in cooperative groups.
  • The application of the theorems has the most promise for becoming a best practice in my classroom because: They know the basic concepts of algebra, especially how to solve equations.

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

  • Plan - My next standards-based task will focus on: Construction and Proving Lines Parallel
    Title: Construction and Proving Lines Parallel.
    Content Area: Basic Geometry - Bilingual Setting.
    Learning Standard(s): NJ CCCS 4.2 4.4
    Intent: Based on the concepts learned in this class, I will extend methods of how to make proofs with Parallel Lines.

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