Tantalizing Triangles: A 10 th Grade Geometry Lesson on Triangle Properties Carol Clinton 1, Rebecca...
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Tantalizing Triangles: A 10 th Grade Geometry Lesson on Triangle Properties Carol Clinton 1 , Rebecca Richmond 2 1 Department of Civil and Environmental Engineering, University of Cincinnati, Cincinnati OH; 2 Hughes Center, Cincinnati, OH Abstract Activity Conclusions Goals Pictures References Objectives State Standards 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 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. 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. 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. 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: C- 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. A slight positive correlation was observed between classwork grades and 1 1 2 1 16 8 2 0 5 10 15 20 Num berofStudents -100 -66.7 -50 -33.3 0 100 200 Change in Perform ance C hange in Student's Perform ance # students C orrelating C lassw ork vs.TestScores y= 1.2316x-74.173 R 2 = 0.0627 -200 -100 0 100 200 300 0 50 100 150 Classw ork Grades TestPerformance C hange perform ance change Linear (perform ance change) • 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; 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_spenc e_bridge.htm ; 2004
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Transcript of Tantalizing Triangles: A 10 th Grade Geometry Lesson on Triangle Properties Carol Clinton 1, Rebecca...
Tantalizing Triangles: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
Goals
Pictures
References
Objectives
State Standards
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
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.
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.
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.
Number, Number Sense and Operations Standard: GG. Estimate, compute and solve problems involving real numbers, including
ratio, proportion and percent, and explain solutions.
Measurement Standard: D, ED. 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, IA. 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: C- C. Translate information from one representation (words, table, graph or
equation) to another representation of a relation or function.
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.
A slight positive correlation was observed between classwork grades and test performance.
1 1 2 1
16
8
2
0
5
10
15
20
Number of Students
-100 -66.7 -50 -33.3 0 100 200
Change in Performance
Change in Student's Performance
# students
Correlating Classwork vs. Test Scores
y = 1.2316x - 74.173
R2 = 0.0627-200
-100
0
100
200
300
0 50 100 150
Classwork Grades
Te
st
Pe
rfo
rma
nc
e
Ch
an
ge
performancechange
Linear(performancechange)
• 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; 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; 2004
