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3.3: Parallel and Perpendicular Lines

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Lines and Angles

I. Section Objectives

  • Identify parallel lines, skew lines, and perpendicular lines
  • Know the statement of and use the Parallel Line Postulate.
  • Know the statement of and use the Perpendicular Line Postulate.
  • Identify angles made by transversals.

II. Cross- curricular-Architecture

  • Use this image from Wikipedia for a discussion on lines and angles.
  • This is Figure 03.01.02
  • www.en.wikipedia.org/wiki/English_Gothic_architecture
  • Discuss the lines and angles in the picture.
  • Show students how perpendicular angles are a major feature in the structure.
  • This can be a fun lively discussion on how architecture and geometry come together.
  • Write student points on the board.

III. Technology Integration-Artist Todd Hoover

  • Use the following painting by Todd Hoover titled “Coming Together.”
  • It can be found at this website. This is Figure 03.01.01.
  • www.fineartamerica.com/featured/coming-together-todd-hoover.html
  • Have the students discuss the different lines and angles in this painting.
  • Then use this image as a springboard to have the students create their own painting.
  • Simplicity is key.
  • Allow time for the students to create their work.
  • Display art in the classroom.

IV. Notes on Assessment

  • Create a rubric for grading student art.
  • Establish how many types of lines need to be in the painting.
  • Be sure to include creativity when grading.
  • How well did the students take Todd Hoover’s simplicity and make it their own?
  • Provide feedback to students on their work.

Parallel Lines and Transversals

I. Section Objectives

  • Identify angles formed by two parallel lines and a non- perpendicular transversal.
  • Identify and use the Corresponding Angles Postulate.
  • Identify and use the Alternate Interior Angles Theorem.
  • Identify and use the Alternate Exterior Angles Theorem.
  • Identify and use the Consecutive Interior Angles Theorem.

II. Cross- curricular-Tube Map in London

  • Use the following image from Wikipedia. This is Figure 03.02.01.
  • www.en.wikipedia.org/wiki/File:Tube_map_thumbnail.png
  • Be sure that each student has a copy of the map.
  • Students are going to use this map to find an example of each of the postulates/theorems in this lesson.
  • 1. Corresponding Angles Postulate
  • 2. Alternate Interior Angles Theorem
  • 3. Alternative Exterior Angles Theorem
  • 4. Consecutive Interior Angles Theorem
  • Students will need to prove that each example in the map is accurate.
  • Have them use a protractor to measure and provide a list of statements and proof for each postulate/theorem.
  • Then allow time for students to share their work.

III. Technology Integration- Transportation Search

  • Have students use this map of the Tube and compare it with the map of the subway in NYC and the map of the “L” in Chicago.
  • Compare and contrast each map and the use of angles, parallel lines and transversals.
  • Have students write a few concluding statements to describe each in mathematical terms.
  • Then allow time for students to share their work.

IV. Notes on Assessment

  • Collect student tube maps and statements.
  • Did the students justify each theorem/postulate correctly and accurately?
  • Did they use angle measures in their justifications?
  • Provide students with feedback/correction when needed.

Proving Lines Parallel

I. Section Objectives

  • Identify and use the Converse of the Corresponding Angles Postulate.
  • Identify and use the Converse of Alternate Interior Angles Theorem.
  • Identify and use the Converse of Alternate Exterior Angles Theorem.
  • Identify and use the Converse of Consecutive Interior Angles Theorem.
  • Identify and use the Parallel Lines Property.

II. Cross- curricular-Washington DC

  • Use the following map of the mall in Washington DC.
  • This is Figure 03.03.01
  • www.visitingdc.com/images/national-mall-map.jpg
  • There are several different examples of parallel lines and transversals in this map.
  • Students are going to write a series of directions to take someone on a tour of the mall.
  • Ask them to start their directions at the American History Museum and write a list of directions in a mathematical way.
  • Students can work in small groups on this assignment.
  • When finished, have the students swap directions with a neighboring group and check to be sure that the directions work.
  • Have the groups provide each other with feedback on their directions.
  • Make corrections as needed.

III. Technology Integration

  • Have the students explore proving lines parallel by watching the video.
  • Use the following website.
  • www.yourteacher.com/geometry/provinglinesparallel.php
  • When finished, use this as a discussion starter.

IV. Notes on Assessment

  • Collect student maps and directions.
  • Check work for accuracy.
  • Provide students with feedback/correction as needed.

Slopes of Lines

I. Section Objectives

  • Identify and compute slope in the coordinate plane.
  • Use the relationship between slopes of parallel lines.
  • Use the relationship between slopes of perpendicular lines.
  • Plot a line on a coordinate plane using different methods.

II. Cross- curricular-Construction/Architecture

  • Ask a roof designer, architect or contractor to visit the class and present information on designing a roof.
  • Prepare the presenter that he/she needs to be able to talk about slope or pitch and how mathematics plays an important role in construction.
  • Ask the students to write questions for the presenter to answer.
  • Conduct a follow- up discussion with the students on career connections between architecture and slope.

III. Technology Integration- Roof Design

  • Have the students use the following website to explore slopes and roof design.
  • Given that the pitch of the roof is connected to the slope of the roof, students can see and explore the real life application of how slope is used in construction.
  • www.roofgenius.com/roofpitch.htm
  • There are several different websites that do a great job at this.
  • Ask the students to begin with this one, and then explore further with other websites.
  • A possible extension is for students to design their own roof plan.

IV. Notes on Assessment

  • Observe students during the presentation.
  • Listen to student questions and answers.
  • Be sure that the students understand how roofing and slopes are connected.

Equations of Lines

I. Section Objectives

  • Identify and write equations in slope- intercept form.
  • Identify equations of parallel lines.
  • Identify equations of perpendicular lines.
  • Identify and write equations in standard form.

II. Cross- curricular-Ramp Design

  • Have a presenter from the local skate shop come in to explain ramps and how they are constructed.
  • Be sure that the person that you are having as a speaker is knowledgeable about skateboard ramps and how the ramps are designed.
  • You could also have someone come in who is an expert in snowboarding and ramps too.
  • Ask the person to bring in some designs or ramps and compare the slope to the equation of the line.
  • You can expand this after the presentation by asking the students to draw a diagram representing a skateboard ramp and demonstrate the slope and equation of the line in the diagram.

III. Technology Integration

  • Have students complete a websearch on parallel and perpendicular lines.
  • Students will find several different websites to explore about parallel lines and the equations of a line.
  • Also, they can search for perpendicular lines and equations of a line.
  • Use these websites to expand student understanding and prompt discussion.

IV. Notes on Assessment

  • Assessment is done through observation in this lesson.
  • You want to be sure that the students are engaging in exploring the concepts of the lesson.
  • There is not a specific measureable content piece for this lesson.

Perpendicular Lines

I. Section Objectives

  • Identify congruent linear pairs of angles.
  • Identify the angles formed by perpendicular intersecting lines.
  • Identify complementary adjacent angles.

II. Cross- curricular-Gymnastics

  • Use the following image from Wikipedia.
  • This is Figure 03.06.01
  • www.en.wikipedia.org/wiki/Parallel_bars
  • Use this image as a discussion point about perpendicular lines of the gymnast and the high bar.
  • One of the ways that gymnasts are scored is on their ability to reach a perfectly perpendicular point.
  • This is the basis for the discussion.
  • Ask the students to identify a linear pair of angles.
  • Also ask the students to find the angles formed by the perpendicular lines.
  • Begin this conversation as a springboard to extend into the technology integration.

III. Technology Integration

  • Have students continue to search gymnastics through Wikipedia.
  • There are several different examples of angles and geometric components of gymnastics.
  • Ask the students to make a list of the ways that geometry is integrated into gymnastics.
  • Allow time for a class discussion.

IV. Notes on Assessment

  • Assess student understanding through discussion.
  • Ask the students to point out different examples of geometric terms as they are illustrated in gymnastics.
  • Then participate with the students during discussion.

Perpendicular Transversals

I. Section Objectives

  • Identify the implications of perpendicular transversals on parallel lines.
  • Identify the converse theorems involving perpendicular transversals and parallel lines.
  • Understand and use the distance between parallel lines.

II. Cross- curricular-Airport Map

  • For this activity, download the map of the main terminal from the following website for the Atlanta International Airport.
  • Use this website, and consider this Figure 03.07.01
  • www.atlanta-airport.com/forms/passenger/frmPassengerInformation_terminallayout.aspx
  • Then use this map to show the main terminal and each of the concourses A- E at the bottom of the map to show a perpendicular transversal and the angles formed by the perpendicular transversal.
  • Have the students explore the rest of the map and discover ways that the perpendicular transversals are represented in the other places on the map.
  • Allow students to discuss this in small groups first.
  • Then bring the students back together and have them share in a large group.

III. Technology Integration

  • Have the students google perpendicular transversals.
  • Then after doing this, allow students the time to work through some of the websites and explore the information.
  • You can ask them to search through particular sites or allow this to be a general investigation time.
  • Ask students to make a list of the sites that they explore and at least three things that are presented on the site.

IV. Notes on Assessment

  • Look at the student maps and at the notes that the students make about the map.
  • Is their work accurate?
  • Are there patterns that they can find in the map having to do with perpendicular transversals?
  • How can they be sure that the lines are perpendicular?
  • What does this do to the angles formed?

Non- Euclidean Geometry

I. Section Objectives

  • Understand non- Euclidean geometry concepts.
  • Find taxicab distances.
  • Identify and understand taxicab circles.
  • Identify and understand taxicab midpoints.

II. Cross- curricular-Game Time

  • Review the basics of taxicab geometry, distances, circles and midpoints with the students.
  • Have them work in small groups.
  • Their task is to create a board game that uses the concepts of taxicab geometry.
  • You can provide students with a piece of cardboard for a game board, index cards, dice or number cubes, and small colored circle pieces.
  • Then set them to work.
  • The students will need to create a grid for the “taxis” to move on.
  • When finished, let the students play each other’s games.
  • This can be very in depth and take several days for the students to work on.

III. Technology Integration

  • This is a very fun website that has the students go on a treasure hunt while using taxicab geometry.
  • www.learner.org/teacherslab/math/geometry/shape/taxicab/
  • Allow time for students to explore this website and hunt for the treasure.
  • Then allow them time to play and then discuss what they have learned about taxicab geometry while hunting for treasure.

IV. Notes on Assessment

  • Create a rubric that gives the students guidelines on how their game will be graded.
  • Then walk around and observe students as they work.
  • When assessing the game, be sure to play it yourself or observe students playing it so that you can assess whether the game works or not.
  • Provide feedback/correction as needed.

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