316 MatheMatics teaching in the Middle school ● Vol. 13, No. 5, December 2007/January 2008

mathematical explorations

mMiddle school teachers in Clark County School District, Nevada, have found the following geometry activities effective with sixth-, sev-enth-, and eighth-grade students when they study translations, rota-tions, and reflections of points in the plane. Students use a local map, in this case a university campus map, to find particular points under a transla-tion, rotation, reflection, or composite of these symmetries. Teachers might choose to customize the lesson for their students with a map from their community, preparing clues similar to those in this activity using familiar locations.

In activity 1, students focus on the prerequisite topics of reading a map and identifying basic transla-tions, reflections, and rotations in the two-dimensional coordinate plane. Then, in activity 2, they apply these concepts to follow a series of clues

carryn Bellomo, carryn.bellomo@unlv.edu, is an assistant professor at the University of Nevada Las Vegas, Las Vegas, NV 89154. She specializes in mathematical biology and mathematics education.

a treasure hunt:Reflecting, Translating, and Rotating Points on a Coordinate Map

Edited by denisse thompson, Thompson@tempest.coedu.usf.edu, Mathematics Depart-ment, University of South Florida, Tampa, FL 33620, and gwendolyn Webb-Johnson, gjohnson@coedu.usf.edu, Secondary Education, University of South Florida. This department is designed to provide activities appropriate for students in grades 5–9. The material may be reproduced by classroom teachers for use in their classes. Readers who have developed successful classroom activities are encouraged to submit manuscripts in a format similar to this “Mathematical Explorations.” Of particular interest are activities focusing on the Council’s Content and Process Standards and Curriculum Focal Points. Send submissions by accessing mtms.msubmit.net.

Carryn Bellomo

to find treasure on the map. The two activities can be completed in a single class period. Teachers who have used the activities have found them to be easy to implement while promoting student interest and excitement.

nctM’s standardsThe activities and discussions in this lesson address the following Standards from Principles and Standards for School Mathematics (NCTM 2000, p. 232):

• Specify locations and describe spa-tial relationships using coordinate geometry and other representa-tional systems

• Apply transformations and use symmetry to analyze mathematical situations

• Describe sizes, positions, and ori-entations of shapes under informal transformations such as flips, turns, slides, and scaling

Copyright © 2007 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved.This material may not be copied or distributed electronically or in any other format without written permission from NCTM.

Vol. 13, No. 5, December 2007/January 2008 ● MatheMatics teaching in the Middle school 317

Preparation for the ActivityTeachers who want to develop their own clues should—

• find a map of interest for the local community or school and enlarge it to fit on an 8 1/2 × 11 inch sheet of paper;

• photocopy the map onto a coordi-nate grid; and

• develop a set of reflections, trans-lations, and rotations to direct students to points of interest.

sUMMarYThese activities offer a fun and chal-lenging approach to viewing geometric reflections, rotations, and translations. By using a map to locate sites, stu-dents have a means to self-check their work. As a summative assessment and extension, students can be challenged to make their own instructions to find a special location on a map.

solUtionsReading a Map1. Paradise School2. P15 (Teachers might want to dis-

cuss how a building can be close to several points.)

3. a. R11 b. TAC4. O5, RESc5. P8 is made with the horizontal

line Line 8 and vertical line Line P. Rotating 90 degrees with P8 as the center is equivalent to moving up 3 and right 3, arriving at P11.

6. P15

Following the Clues(See the solutions in the table above.)

reFerenceNational Council of Teachers of Math-

ematics (NCTM). Principles and Stan-dards for School Mathematics. Reston, VA: NCTM, 2000. ●

Movement Finish Point

Building name

Translate 3 units left and 1 unit down. I7 TMC

Reflect through Line 10. I13 MPEc

Translate 1 unit left and 2 units up. H15 LAC

Rotate 90 degrees clockwise centered at H10. M10 LIB

Translate 3 units right. P10 WRI

Rotate 90 degrees counterclockwise centered at P12. R12 HFA

Translate 2 units left and 1 unit down. P11 BSL

Rotate 90 degrees clockwise centered at M11, then translate 1 unit left.

L8 CBCb

from the December 2007/January 2008 issue of

Name ______________________________

reading a MapFor this activity, you will need the map of the University of Nevada Las Vegas campus that has been placed on a grid.

how to read the mapThe points and lines on the map are referenced using the following guidelines:

• All vertical and horizontal lines on the grid are referenced by the row or column name. Vertical lines are identified by the column heading, and horizontal lines are identified by the row heading. For instance, the Maryland Parkway can be identified most easily as Line T.

• All points on the grid are referenced using the point of intersection between two lines. For instance, the Thomas and Mack Center (TMC) is located at the intersection of Line I and Line 7; thus, it is refer-enced as I7. Because this building is so large, it could also be referenced as J7.

Finding map locationsUse the map to answer each of the following questions:

1. Identify the building located at point J16. _________________________________________________

2. Identify the most appropriate coordinates for the Parking Garage. ______________________________

3. a. Begin at point N11 and translate 4 units right. What are the coordinates? ______________________

b. What building is located at the translated coordinates in 3a? ________________________________

4. Begin at point M8 and translate 2 units right and 3 units down. Identify the coordinates after this trans-lation. What building is located at these coordinates? ________________________________________

5. Begin at point M8 and rotate 90 degrees clockwise centered at P8. What are the coordinates? ________

6. Reflect the point H15 through Line L and identify the coordinates of the reflection. _______________

7. Write three clues of your own that use translations, reflections, or rotations. Have a classmate follow your clues.

Clue 1: _____________________________________________________________________________

___________________________________________________________________________________

Clue 2: _____________________________________________________________________________

___________________________________________________________________________________

Clue 3: _____________________________________________________________________________

___________________________________________________________________________________

activity sheet 1

from the December 2007/January 2008 issue of

Name ______________________________

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2

1A B C D E F G H I J K L M N O P Q R S T U

from the December 2007/January 2008 issue of

Name ______________________________

activity sheet 2

Following the cluesStart at the point L8 (CBCb building). Find the next point on the map by following the clues in the table below to track down the treasure. Record each new point or ordered pair in the table, along with the build-ing name. In what building is the treasure found?

Movement Finish Point

Building name

Translate 3 units left and 1 unit down.

Reflect through Line 10.

Translate 1 unit left and 2 units up.

Rotate 90 degrees clockwise centered at H10.

Translate 3 units right.

Rotate 90 degrees counterclockwise centered at P12.

Translate 2 units left and 1 unit down.

Rotate 90 degrees clockwise centered at M11, then translate 1 unit left.