Transcript 48x36 poster template - University of Cincinnati

Tantalizing Triangles:
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
Concepts of triangle properties and classifications are reinforced through use of a
manipulative called a geoboard. Students gain experience with using Cartesian
coordinates, classifying triangles, solving for missing angles, measuring angles,
defining angle bisectors and congruent and similar figures, and more.
The lesson begins with photographs of local structures where triangle geometry is
foundational (airport runways, bridge trusses) and with discussion from the STEP
fellow about personal experiences involving use of geometry in actual structural
design projects. This demonstrates that geometry (like all the STEM disciplines) is
useful and relevant in everyday life in this community.
Students’ prior theoretical knowledge of triangle classifications and properties is
refreshed through questioning and demonstration. Then they extend it to a physical
understand by constructing and “playing with” various specified types of triangles
through a series of worksheet exercises.
Future lessons in other classes will deal directly with my current research on
drinking water protection. But already, the STEP program has shown me how to
translate my knowledge of the applications of math and science into experiences that
help the students understand why math and science are useful and interesting.
Goals
Students should understand triangle, side, and angle concepts; lines in coordinate
system; congruence, and similarity. They should gain experience constructing
triangles, calculating and measuring angles, and classifying triangles.
Objectives
Students will correctly construct triangles on a Geoboard based on
Cartesian coordinates of the vertices, classify the triangles, solve missing
angle problems, and create congruent and similar triangles.
Activity
Conclusions
The lesson begins with the students identifying photos of familiar structures comprised of
triangles, particularly local bridges. They proceed to an activity in groups of two or three
using diagrams of portions of trusses from one of the local bridges. One student verbally
describes the section to the other, who attempts to draw it without seeing the diagram.
They switch roles and try with another section of the bridge truss. The challenging
process and occasional comic results point out the need for more precise ways to
describe triangles, and the value of learning how to construct and use triangles. Results
of the pre-test are also discussed to identify areas where their knowledge can be
improved.
The teacher then demonstrates the GeoBoard, a peg-board device where rubber bands are
used to construct geometric shapes. The teacher reviews the types of triangles
(reinforcing key vocabulary), use of the Cartesian coordinate system, angle and side
relationships, congruence and similarity.
(With these classes, I also mentioned
transformations and line slope formulas , to tie into a future lesson.)
The students learn cooperatively in their groups by completing a worksheet. Triangles
are constructed on the GeoBoard. Results are shown on dot paper and the worksheet.
Students are encouraged to take turns within the groups between using the GeoBoard and
drawing the figures on the dot paper. The teacher circulates working with each group to
ensure that the content is grasped. The worksheet includes extra credit activities for
students who complete the worksheet. Extra credit is also given for solving tantagram
puzzles -- creating a specified outline by placing a set of geometric shapes in the correct
orientation. The next day the post-test is administered.
Scores rose an average of 14% from the pre-test to the post test. Roughly half of the
students showed no change. Roughly one third saw increases up to 200%. However, nearly
20% of the students actually showed decreased performance on the post-test.
Pictures
Change in Student's Performance
16
20
15
8
Number of Students 10
5
1
2
1
2
1
# students
0
-100
-66.7
-50
-33.3
0
100
200
Change in Performance
A slight positive correlation was observed between classwork grades and test performance.
Correlating Classwork vs. Test Scores
Test Performance
Change
1
1
Clinton ,
performance
change
300
200
Linear
(performance
change)
100
0
-100 0
50
100
150
y = 1.2316x - 74.173
R2 = 0.0627
-200
Classwork Grades
References
State Standards
Number, Number Sense and Operations Standard: G
G. Estimate, compute and solve problems involving real numbers, including
ratio, proportion and percent, and explain solutions.
Measurement Standard: D, E
D. Use proportional reasoning and apply indirect measurement techniques,
including right triangle trigonometry and properties of similar triangles, to
solve problems involving measurements.
E. Estimate and compute various attributes, including length, angle measure, to
a specified level of precision.
Geometry and Spatial Sense Standard : A, B, C, D E, F, I
A. Formally define geometric figures.
B. Describe and apply the properties of similar and congruent figures; and justify
conjectures involving similarity and congruence.
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.
I. Use right triangle trigonometric relationships to determine lengths and angle
measurements.
Patterns, Functions and Algebra Standard: CC. Translate information from one representation (words, table, graph or
equation) to another representation of a relation or function.
• Bass, Laurie E. and Johnson, Art; Prentice Hall Mathematics Geometry,
Pearson Education, Inc.; Saddle River, New Jersey; 2004
• Andres, Richard J. and Bernstein, Joyce; Preparing for the OGT in
Mathematics; Amsco School Publications, Inc.; 2004
• 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;
School Supply Company; Oak Lawn, IL; 1971
Ideal
• 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;
2004
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