Delving Deeper: Visualizing Pythagorean Triples and Beyond

2010 ◽  
Vol 104 (5) ◽  
pp. 393-398
Author(s):  
David A. Booze
Keyword(s):  
High School ◽  
Algebraic Representation ◽  
Visual Method ◽  
High School Algebra ◽  
Algebra Students ◽  
Pythagorean Triples ◽  
Considerable Success ◽  
Algebraic Properties ◽  
Integer Solutions

Exploring the algebraic properties that form the framework for analytical generation of Pythagorean triples and quadruples is a challenging topic for high school algebra students. I have had considerable success in motivating my students to explore these properties by using a visual method. Students can use this visual method to find Pythagorean triples and quadruples easily and can capably and colorfully supply as many integer solutions as they desire to these well-known equations. Here I will present a visual method for producing integer solutions to the equations a2 + b2 = c2 and a2 + b2 + c2 = d2 and then develop an algebraic representation of this method.

scholarly journals Studies of Diagnosis and Remediation with High School Algebra Students

1989 ◽  
Vol 13 (4) ◽  
pp. 551-568 ◽  
Author(s):  
D. Sleeman ◽  
A.E. Kelly ◽  
R. Martinak ◽  
R.D. Ward ◽  
J.L. Moore
Keyword(s):  
High School ◽  
High School Algebra ◽  
Algebra Students

Studies of Diagnosis and Remediation with High School Algebra Students

1988 ◽  
Author(s):  
R. Martinak ◽  
Anthony E. Kelly ◽  
D. Sleeman ◽  
J. Moore ◽  
R. Ward
Keyword(s):  
High School ◽  
High School Algebra ◽  
Algebra Students

scholarly journals UPAYA MENINGKATKAN AKTIVITAS DAN HASIL BELAJAR SISWA DENGAN MEDIA AUDIO VISUAL PADA MATA PELAJARAN SISTEM REM KELAS XI SMK ISLAM SALAK BROJO PEKALONGAN TAHUN AJARAN 2014/2015

TAMAN VOKASI ◽  
2016 ◽  
Vol 4 (2) ◽  
pp. 206
Author(s):  
Muh Fredy Hermawan ◽  
Tarto Sentono
Keyword(s):  
High School ◽  
Student Learning ◽  
Learning Outcomes ◽  
Research Result ◽  
Student Learning Outcomes ◽  
Brake System ◽  
Vocational High School ◽  
Vehicle Engineering ◽  
Visual Method ◽  
Average Value

The purpose of this research to imporove the activity and student learning outcomes of class XI Islam Salak Brojo Pekalongan Vocational High School by audio-visual menthod on the brake system subjects.This education is clasroom action researsh that been conducted in Islam Salak Brojo Pekalongan.Vocational High School in the academic year of 2014/2015. The subject in this research is a class XI student of Light Vehicle Engineering Islam Salak Brojo Pekalongan Vocational High School that amounted to 26 student. The data collection techniques used observation and tests are divided into pre-test and post test. The data were anilyzed using descriptive statistics.The research result showed that the application of audio-visual method on the brake system subjects can improve activity and student learning outcomes class XI Islam Salak Brojo Pekalongan Vocational High Shool. The result can be seen in the successful development of the I cycle, the average value of learning activities amounted to 53,84 increased in the II cicle to 56,34 then on the III cycle increased to 59,42. On the learning outcomes of I cycle on average reached 5,11 increased in the II cycle to 6,51 then on the III cycle increased to 18,32. Therefore can be concluded that audio-visual method can increase the activity and student learning outcomes on the  break system subjects of class XI student of Islam Salak Brojo Pekalongan Vocational High School.

scholarly journals Quadratic ideals, indefinite quadratic forms and some specific diophantine equations

2018 ◽  
Vol 36 (3) ◽  
pp. 173-192
Author(s):  
Ahmet Tekcan ◽  
Seyma Kutlu
Keyword(s):  
Diophantine Equation ◽  
Quadratic Forms ◽  
Diophantine Equations ◽  
Algebraic Properties ◽  
Integer Solutions ◽  
Indefinite Quadratic

Let $k\geq 1$ be an integer and let $P=k+2,Q=k$ and $D=k^{2}+4$. In this paper, we derived some algebraic properties of quadratic ideals $I_{\gamma}$ and indefinite quadratic forms $F_{\gamma }$ for quadratic irrationals $\gamma$, and then we determine the set of all integer solutions of the Diophantine equation $F_{\gamma }^{\pm k}(x,y)=\pm Q$.

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