Transcript 48x36 poster template - University of Cincinnati

Triangle Transformations:
A
th
10
Grade Geometry Lesson on Triangle Properties
Carol
Rebecca
2
Richmond
Department of Civil and Environmental Engineering, University of Cincinnati, Cincinnati OH; 2 Hughes Center, Cincinnati, OH
Abstract
Activity
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.
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
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).
Conclusions
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)
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
Pictures
Student Improvement
% score (classwork) or %
improvement
1
1
Clinton ,
1200
1000
800
600
Classwork score
400
% Improvement
200
0
-200 0
5
10
15
Student ID
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;
State Standards
2004
• Andres, Richard J. and Bernstein, Joyce; Preparing for the OGT in Mathematics; Amsco
School Publications, Inc.; 2004
• Tessellations from http://www.tessellations.org. They have a great database of hundreds
of Tessellations by subject.
• http://www.mathcats.com also has some great arts and crafts activities with geometric
concepts.
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.
• 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,
http://www.eng.uc.edu/STEP/activities/descriptions/brent_spence_bridge.htm
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