1. Triangle Transformations:
A 10th Grade Geometry Lesson on Triangle Properties
Carol Clinton1, Rebecca Richmond2
1 Department of Civil and Environmental Engineering, University of Cincinnati, Cincinnati OH; 2 Hughes Center, Cincinnati, OH
Abstract
Activity
Conclusions
Who knew that math could be art and art could be math? The lesson begins with examples of tesselation art, made by repeating and translating common shapes.
Engineers’ use of these concepts is demonstrated by examination of Cincinnati bridge trusses. Students identify translations, rotations and reflections in the structures. The need for engineers to be able to quickly and accurately compute angles and side lengths.
Instruction:
Define translation, reflection over x-axis, reflection over y-axis, rotations (clockwise, counterclockwise).
Show class (with help from volunteers) how to do the questions from the pre-test using a geoboard and drawing on the classroom board. This demonstrates all the operations that they will be doing for the worksheet.
Distribute and explain the worksheet. Allow work as individuals or groups of 2.
If lesson goes long assign completion as homework. Discuss results next day.
If lesson goes short, demonstrate with class assistance how to find/construct:
the perpendicular bisector of a line segment
a line perpendicular to a given line from a point not on the line
a line perpendicular to a given line from a point on the line
a line parallel to a given line through a given point not on the line
circumscribe a circle about a triangle
inscribe a circle in a triangle
Average improvement in class scores from the pre test to the post test was 102%.
For those students who took both the pre and post test, the average improvement was 158%. (show on graph)
There was no correlation between scores on classwork and improvement in test scores. (shown on graph)
Students will gain physical understanding of triangle transformations (translations, reflections, rotations) and relate the physical realities to mathematical concepts including coordinated, slopes, line equations.
Applications to art and engineering bridge truss design are discussed. Cincinnati bridge trusses are used as examples to show how engineers design structures, and to demonstrate why being able to recognize transformed shapes and calculate the angles and side lengths, and describe the location of the triangles in space (coordinate system).
Goals
Students should understand triangle transformations (translations, reflections, rotations) and relate the physical realities to mathematical concepts including coordinated, slopes, line equations.
Objectives
Pictures
Students will
perform transformations (translations, reflections and rotations) of triangles
compute coordinates of transformed shapes, lengths of line segments given the coordinates of vertices, and slopes and equations of lines that comprise the shapes.
References
Bass, Laurie E. and Johnson, Art; Prentice Hall Mathematics Geometry, Pearson Education, Inc.; Saddle River, New Jersey;
Andres, Richard J. and Bernstein, Joyce; Preparing for the OGT in Mathematics; Amsco School Publications, Inc.; 2004
Tessellations from They have a great database of hundreds of Tessellations by subject.
also has some great arts and crafts activities with geometric concepts.
Lund, Charles; Dot Paper Geometry With or Without a Geoboard; Cuisenaire Co. of America, Inc.; New Rochelle, NY; 1990
Cech, Joseph P. and Tate, Joseph B.; Geo-board Activity Sheets; Ideal School Supply Company; Oak Lawn, IL; 1971
Dimmerling, Amy; Vice Bowling, Bethany; and Massie, Emma; Brent Spence Bridge STEP lesson;University of Cincinnati,
State Standards
Number, Number Sense and Operations Standard: G
Geometry and Spatial Sense Standard –
A. Formally define geometric figures.
C. Recognize and apply angle relationships in situations involving intersecting lines, perpendicular lines and parallel lines.
D. Use coordinate geometry to represent and examine the properties of geometric figures.
E. Draw and construct representations of two- dimensional geometric objects using a variety of tools, such as straightedge, compass and technology.
F. Represent and model transformations in a coordinate plane and describe the results.
Patterns, Functions and Algebra Standard -
C. Translate information from one representation (words, table, graph or equation) to another representation of a relation or function.
Acknowledgments
Project STEP is funded through NSF Grant # DGE058532
Appreciation is particularly given to the following for their assistance in development and implementation of this lesson:
Ms. Rebecca Richmond – Hughes High School
Dr. Richard Miller – University of Cincinnati
Andrea Burrows – University of Cincinnati