## A Perpendicular Pilgrimage

### Mason Clark

#### Description

Students examine the concept of perpendicularity both geometrically and algebraically. Students apply their knowledge by designing safe passage through a two-dimensional obstacle course using only perpendicular line segments.

#### Objectives

Understands geometric concepts such as perpendicularity, parallelism, tangency, congruency, similarity, reflections, symmetry, and transformations including flips, slides, turns, enlargements, and fractals.

#### Materials

-Brightly colored string or light rope, approximately five meters in length
-Overhead
-Overhead markers
-Blank transparencies (as many as necessary)
-Transparencies of the Blank Coordinate Axis and Practice with Perpendicular Lines (See Associated File)
-Optional: Copies of the Practice with Perpendicular Lines, one per student (See Associated File)
-Copies of Perpendicular Obstacle Course, one per student (See Associated File)
-Optional: Transparency of Perpendicular Obstacle Course and Obstacle Course Directions and Rubric (See Associated File)
-Protractors, one per student and teacher
-Tape
-Pencils with erasers for students

#### Preparations

1. Prepare a copy of each student handout ahead of time. (See Associated File)
2. Prepare teacher transparencies of the Blank Coordinate Axis, Practice with Perpendicular Lines and the optional Perpendicular Obstacle Course and Obstacle Course Directions and Rubric. (See Associated File)
3. Prepare the vertical for the introductory activity:
a. Tie one end of a piece onto one of the crossbars of the acoustical tile on your ceiling.
b. Allow the string to hang completely vertical, slightly taut.
c. Tape the bottom of your string to the floor.
d. Cut off the unused string and save for demonstration. (See Procedures, step #1)

#### Procedures

Note: This lesson examines the concept of perpendicularity only. Students should have prior knowledge and practice with - Using a rectangular coordinate system (graph), the student applies and algebraically verifies properties of two-and three- dimensional figures (including distance, midpoint, slope, parallelism, and perpendicularity). Students should also understand the following terms: line, intersection, and slope.

REVIEW OF PREREQUISITE CONCEPTS
1. PROMPT: “Examine the string extending from the floor to the ceiling. What geometric figure does it represent?” Ask for two volunteers. Have the volunteers come to the front of the room and take each end of the unused string. Ask them to hold the string taut. Point to the vertical string and the string held by the students. PROMPT: “Based on what we see with the first string, what does THIS string represent?” Students will likely indicate that it also represents a line. Note that some students may indicate that they are line segments, which is actually more correct, as our string models do indeed terminate!

2. Ask the students to intersect the two lines in any way they would like to do so. Make a sketch of their intersection on the overhead transparency and label it A. Leave space for four more sketches on the same overhead sheet.

3. Repeat Step 2 four more times so that the students have generated five different types of intersection. Label these lines B-E. Hopefully, they will have picked a perpendicular intersection. If not, continue on--it's no big deal!

4. Ask all of the students to get up and walk around the room. Give them about five minutes to examine the room and any real-world intersections that they can find.

5. Ask the students to return to their seats.

6. PROMPT: “What type of intersection did you observe most often in the real world?” If perpendicular lines are one of the five student-generated intersections, students will likely indicate that example. Otherwise, they may wish to describe or draw the perpendicular lines they found at the edges of the room, table, book, tile, or other right angles. PROMPT: “Remember that perpendicular lines are an important part of jobs like construction and design. They are kind of special in mathematics, too. Let's talk about what makes them special.”

DISCUSSION OF GEOMETRIC AND ALGEBRAIC CONCEPTS OF PERPENDICULARITY
7. Define perpendicular lines geometrically--they intersect forming a right angle. Draw two perpendicular lines on a clean overhead sheet by using a protractor and show that their resultant angle measures 90 degrees. Note: Do not assume that the students remember how to use a protractor. Your class may or may not require a quick refresher.

8. PROMPT: “So how do we know that two lines are perpendicular?” The students should include both conditions: The lines intersect and they form a right (or 90 degree) angle.

9. To illustrate the algebraic concept of perpendicular lines, take out the Practice with Perpendicular Lines transparency and optional copies for the students. Write the slopes down on the table below the coordinate axis.

10. PROMPT: “Look for a relationship between the slopes of these perpendicular lines. What pattern do you notice?” The students should note that the slopes are negative reciprocals of each other.

11. On the overhead, demonstrate how to build the equation of a line in point-slope form IF you are given one point on the line and the slope.

12: PROMPT: “Any questions?” You may informally assess their understanding here or continue with the student activity, based on your judgment of their progress.

STUDENT ACTIVITY
13. Distribute the two student handouts--the Perpendicular Obstacle Course and the Obstacle Course Directions and Rubric. (See Associated File)

14. Review with the students the directions. Make sure that they understand the goal of the activity (to plan a route to the finish line in as few moves as possible). Also ensure that they understand the constraints of the activity (all turns are perpendicular to the previous path and the entire route may never touch an obstacle or a wall).

15. Explain to the students how they will be assessed. Their geometric solutions will be assessed by using a protractor to verify that their route indeed uses only perpendicular lines. Their algebraic solutions will be assessed by verifying that they have indeed transferred their geometric solutions to algebraic equations (i.e., each change-in-y and change-in-x indeed results in the distance of their path).

16. Break the students into groups of three and ask them to discuss possible travel plans. At this point, they should discuss ideas only. Ask the students to NOT begin work on the obstacle course in groups. This group work is to share ideas only!

17. After five minutes or so, ask the students to return to their seats and begin working on their individual solution to the activity.

18. Circulate through the room and answer any questions that the students may have about the process. This is particularly important as they transfer their geometric solution to an algebraic one. Students tend to intuit a correct geometric route right away, but they may make errors in representing that solution algebraically. Pay attention to their questions about the slope. Each path's slope should be the negative reciprocal of the previous slope. Also, students sometimes confuse the positioning in the x- and y- positions in the slope. Finally, make sure that they are accurately relating the distance traveled by each path in the route in terms of the x- and y-distances traveled.

19. Ask the students to place the optional Practice with Perpendicular Lines sheet into their notebook or portfolio, if applicable. Collect their Perpendicular Obstacle Course and formatively assess their solutions with the Obstacle Course Rubric.

#### Assessments

Students are given the Perpendicular Obstacle Course which challenges them to navigate an obstacle course using only perpendicular line segments. The students then represent their geometric route into algebraic equations. Perpendicularity is the primary assessed concept, although the concepts of slope and the point-slope form of a line are indirectly assessed.

The Obstacle Course Rubric is used to assess the students' routes and solutions. (See Associated File)

#### Extensions

1.Support may be required in forming the equation for some students with special needs. You may wish to create a document in large type that provides the point-slope form with blanks for the slope and point coordinates and allow the students to actually cut out the slope and coordinates of the point and help them to move them into their correct places on the formula until they are comfortable with constructing the formulae on their own.
2. See Weblinks for ESOL support.

#### Web Links

This site offers both parallel and perpendicular algebraic practice. The instructor may wish to use this site as an enrichment activity for students who pick up rather quickly the idea of basic perpendicularity.
Practice with Parallel and Perpendicular

This site allows you to translate blocks of text (150 words at a time) or a Web page from English into several common languages to assist your ESOL students. Keep in mind that the site is a machine translator and makes mistakes in meaning sometimes. To translate sections of the associated file, open the associated file in a word processing program. Highlight the text you wish to translate, right click and choose “Copy.” Then go to the translation site and right click in the “Translate a block of text” box. Choose “Paste” and the text from the associated file will appear. Click translate and you should receive a translated copy of the text. Highlight the translated version and then copy and paste it into a new document in the word processing program. Piece by piece, this tool enables you to translate text to assist your ESOL students.
The Altavista Translation Site