Activities: The Conic Sections in Taxicab Geometry: Some Investigations for High School Students

1998 ◽  
Vol 91 (4) ◽  
pp. 304-341 ◽  
Author(s):  
Fernand J. Prevost
Keyword(s):  
High School ◽  
High School Students ◽  
Euclidean Distance ◽  
Euclidean Geometry ◽  
Pythagorean Theorem ◽  
Conic Sections ◽  
School Students ◽  
Taxicab Geometry

The urban world in which many of us live does not lend itself to the metric of Euclidean geometry. Assuming that the avenues are perpendicular to the streets in a city, the distance from “fifth and fifty-first” to “seventh and thirty-fourth” is not the familiar Euclidean distance found by applying the Pythagorean theorem. The distance must instead be measured in blocks from fifth to seventh avenues and then from fifty-first to thirty-fourth streets. This taxicab metric, one of several me tries used in mathematics (Eisenberg and Khabbaz 1992), is practical for many applications and helps students pursue interesting investigations while deepening their understanding of familiar topics.

Non-Euclidean Geometry Lesson Promotes Mathematical Reasoning

2021 ◽  
Vol 114 (11) ◽  
pp. 869-877
Author(s):  
Derek A. Williams ◽  
Kelly Fulton ◽  
Travis Silver ◽  
Alec Nehring
Keyword(s):  
High School ◽  
High School Students ◽  
Euclidean Geometry ◽  
Mathematical Reasoning ◽  
School Students ◽  
Non Euclidean Geometry ◽  
Taxicab Geometry

A two-day lesson on taxicab geometry introduces high school students to a unit on proof.

Sharing Teaching Ideas: Bringing Pythagoras to Life

1995 ◽  
Vol 88 (9) ◽  
pp. 744-747
Author(s):  
Donna Ericksen ◽  
John Stasiuk ◽  
Martha Frank
Keyword(s):  
High School ◽  
High School Students ◽  
Classroom Teacher ◽  
Pythagorean Theorem ◽  
School Students ◽  
Geometric Figures ◽  
Experience Teaching ◽  
High School Classroom ◽  
The University ◽  
Teaching Ideas

The Curriculum and Evaluation Standards for School Mathematics (NCTM 1989) states that “[o]ne of the most important properties in geometry, the Pythagorean theorem, is introduced in the middle grades” (p. 113). Although the Standards document assigns much prominence to the Pythagorean theorem, our experience teaching at the university level has revealed that students know the theorem by name and can recite a2 + b2 = c2 but that they often cannot handle even simple computations using the formula. Students' experience with the Pythagorean theorem in high school needs to be broadened by their continually using the standard formula as well as applying the formula to geometric figures and special right triangles-in particular, the 30°-60°-90° and the 45°-45°-90° right triangles. The following game was developed to afford high school students more opportunity for practicing the formula in an engaging way. This game was created by the second author of the article, a high school classroom teacher, while he was a student in a class taught by another of the authors.

Con-fusing Pairs: An Intriguing Investigation of LCMs

2006 ◽  
Vol 100 (2) ◽  
pp. 140-144
Author(s):  
Curtis D. Bennett ◽  
Mary J. DeYoung ◽  
James J. Rutledge ◽  
Elaine Young
Keyword(s):  
High School ◽  
High School Students ◽  
Driving Force ◽  
Mathematical Knowledge ◽  
Conic Sections ◽  
School Students ◽  
New Knowledge ◽  
Young Students

As today's high school students and undergraduates contemplate the conic sections known to the Greeks or learn the calculus dating back to Newton, they rarely stop to realize that mathematics is continually being discovered and invented. A great deal of current mathematical knowledge has been established since these young students were born. What is the driving force behind this new knowledge? The simple answer is curiosity.

Tessellating the Sphere with Regular Polygons

2004 ◽  
Vol 97 (3) ◽  
pp. 165-167
Author(s):  
Hortensia Soto-Johnson ◽  
Dawn Bechthold
Keyword(s):  
High School ◽  
High School Students ◽  
Euclidean Geometry ◽  
Spherical Geometry ◽  
Regular Polygons ◽  
School Students ◽  
Disney World ◽  
Spaceship Earth

Spherical geometry can be used to entice students into a deeper understanding of Euclidean geometry. Determining which regular spherical polygons tessellate the sphere is another motivating topic that is accessible to high school students. The most recognizable tessellations of the sphere are found on balls, such as soccer balls, volleyballs, and golf balls. Even Spaceship Earth at Epcot Center in Disney World involves tessellation.

A General Intersection Formula for Subspaces of N-Dimensions

1975 ◽  
Vol 68 (2) ◽  
pp. 153
Author(s):  
J. Taylor Hollist
Keyword(s):  
High School ◽  
High School Students ◽  
Euclidean Geometry ◽  
Algebraic Model ◽  
Analytic Model ◽  
School Students ◽  
Four Dimensions ◽  
Intersection Formula

There are many approaches one could take in making leaps from the plane and space into four dimensions and then into n-space. One approach would be to extend the algebraic model with which most high school students are familiar. In this article, I want to look at an interesting formula concerning intersections by starting with the familiar analytic model for Euclidean geometry.

Using Puzzles to Teach the Pythagorean Theorem

1989 ◽  
Vol 82 (5) ◽  
pp. 336-339 ◽  
Author(s):  
James E. Beamer
Keyword(s):  
High School ◽  
High School Students ◽  
Careful Analysis ◽  
Pythagorean Theorem ◽  
Alternative Proof ◽  
School Students ◽  
High School Geometry ◽  
School Geometry ◽  
Starting Point ◽  
Senior High School Students

One of the aims of a mathematics prog- gram is to familiarize the students with the Pythagorean theorem. The result, stated algebraically, is c2 = a2 + b2. Stated geometrically, the Pythagorean theorem refers to squares drawn on the three sides of a right triangle. The theorem states that the square drawn on the longest side has exactly the same area as that of the other two squares combined. Is it possible systematically to dissect the two smaller squares into pieces that will cover the larger square? The answer to this question is the focus of this article, which offers sugges-tions about how the Pythagorean theorem can be introduced to students in the middle school years. Enrichment challenges in the form of proofs suitable for high school geometry students are also included. Finally, three proofs of the Pythagorean theorem based on careful analysis of the puzzles are discussed. Senior high school students can be asked to prove that the pieces actually fit and to use this tessellation as a starting point to provide an alternative proof of the theorem.

Articulatory Intervention for the Visually Impaired

1979 ◽  
Vol 10 (3) ◽  
pp. 139-144
Author(s):  
Cheri L. Florance ◽  
Judith O’Keefe
Keyword(s):  
High School ◽  
High School Students ◽  
Behavior Change ◽  
Visually Impaired ◽  
School Students ◽  
Handicapped Children ◽  
Paired Stimuli ◽  
Visually Handicapped ◽  
Parent Program

A modification of the Paired-Stimuli Parent Program (Florance, 1977) was adapted for the treatment of articulatory errors of visually handicapped children. Blind high school students served as clinical aides. A discussion of treatment methodology, and the results of administrating the program to 32 children, including a two-year follow-up evaluation to measure permanence of behavior change, is presented.

A Triarchic Analysis of an Aptitude-Treatment Interaction

1999 ◽  
Vol 15 (1) ◽  
pp. 3-13 ◽  
Author(s):  
Robert J. Sternberg ◽  
Elena L. Grigorenko ◽  
Michel Ferrari ◽  
Pamela Clinkenbeard
Keyword(s):  
High School ◽  
High School Students ◽  
Introductory Psychology ◽  
College Level ◽  
School Students ◽  
Treatment Interaction ◽  
Aptitude Treatment Interaction ◽  
Psychology Course

Summary: This article describes a triarchic analysis of an aptitude-treatment interaction in a college-level introductory-psychology course given to selected high-school students. Of the 326 total participants, 199 were selected to be high in analytical, creative, or practical abilities, or in all three abilities, or in none of the three abilities. The selected students were placed in a course that either well matched or did not match their pattern of analytical, creative, and practical abilities. All students were assessed for memory, analytical, creative, and practical achievement. The data showed an aptitude-treatment interaction between students' varied ability patterns and the match or mismatch of these abilities to the different instructional groups.

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