The Thinking of Students: My Application of the Pythagorean Theorem

One of my seventh-grade-algebra students, Paul Pollack, shared a discovery he had made. We had finished a unit of study on the Pythagorean theorem, including the Pythagorean triples. Paul noticed that in several triples, the hypotenuse was equal to one of the legs plus 1. For example, 3-4-5, 5-12-13, and 7-24-25 triples have two sides whose values are consecutive integers. Paul was intrigued and developed a pattern to derive triples wherein the hypotenuse and one leg differ by 1, or, for that matter, by any desired quantity. Note that these “triples” often take a little finagling to result in integers (e.g., in example 2). He found thallhe lengths of the three sides of which two are consecutive integers would fit the pattern

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A kaleidoscope of solutions for a Diophantine system

The Mathematical Gazette ◽

10.1017/s0025557200007130 ◽

2010 ◽

Vol 94 (529) ◽

pp. 42-50

Author(s):

Juan Pla

Keyword(s):

Analogous Problem ◽

Pell Equation ◽

Pythagorean Triples ◽

Diophantine System ◽

Consecutive Integers ◽

Image Position

A classical exercise in recreational mathematics is to find Pythagorean triples such that the legs are consecutive integers. It is equivalent to solve the Pell equation with k = 2. In this case it provides all the solutions (see [1] for details). But to obtain all the solutions of a Diophantine system in one stroke is rather exceptional. Actually this note will show that the analogous problem of finding four integers A, B, C and D such that

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Delving Deeper: Visualizing Pythagorean Triples and Beyond

Mathematics Teacher ◽

10.5951/mt.104.5.0393 ◽

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.

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An experimental approach to the Pythagorean theorem

The Arithmetic Teacher ◽

10.5951/at.17.2.0129 ◽

1970 ◽

Vol 17 (2) ◽

pp. 129-132

Author(s):

Aaron L. Buchman

Keyword(s):

Elementary School ◽

Experimental Approach ◽

Pythagorean Theorem ◽

The Other ◽

Elementary Grades ◽

Two Sides ◽

Modern Mathematics

Modern mathematics programs for the elementary school stress the involvement of a greater amount of geometry. Among the geometric relations that can acquire meaning in the elementary grades is the Pythagorean relation: In a right triangle, the area of the square on the hypotenuse is equal to the sum of the areas of the squares on the other two sides.

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Characterization of defects in tabular silver halide grains

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100178367 ◽

1990 ◽

Vol 48 (4) ◽

pp. 1046-1047

Author(s):

C. Goessens ◽

D. Schryvers ◽

J. Van Landuyt ◽

A. Verbeeck ◽

R. De Keyzer

Keyword(s):

Radiation Damage ◽

Silver Halide ◽

Chemical Characteristics ◽

X Ray ◽

Free Region ◽

Central Defect ◽

Crystallographic Defects ◽

Physical And Chemical ◽

Two Sides

Silver halide grains (AgX, X=Cl,Br,I) are commonly recognized as important entities in photographic applications. Depending on the preparation specifications one can grow cubic, octahedral, tabular a.o. morphologies, each with its own physical and chemical characteristics. In the present study crystallographic defects introduced by the mixing of 5-20% iodide in a growing AgBr tabular grain are investigated. X-ray diffractometry reveals the existence of a homogeneous Ag(Br1-xIx) region, expected to be formed around the AgBr kernel. In fig. 1 a two-beam BF image, taken at T≈100 K to diminish radiation damage, of a triangular tabular grain is presented, clearly showing defect contrast fringes along four of the six directions; the remaining two sides show similar contrast under relevant diffraction conditions. The width of the central defect free region corresponds with the pure AgBr kernel grown before the mixing with I. The thickness of a given grain lies between 0.15 and 0.3 μm: as indicated in fig. 2 triangular (resp. hexagonal) grains exhibit an uneven (resp. even) number of twin interfaces (i.e., between + and - twin variants) parallel with the (111) surfaces. The thickness of the grains and the existence of the twin variants was confirmed from CTEM images of perpendicular cuts.

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The Eleventh Circuit Holds that Agreements in which Pharmaceutical Companies Pay Generic Companies Not to Compete May be Valid

The Journal of Law Medicine & Ethics ◽

10.1017/s107311050000749x ◽

2004 ◽

Vol 32 (1) ◽

pp. 181-184

Author(s):

Amy Garrigues

Keyword(s):

Pharmaceutical Companies ◽

Court Decisions ◽

Patent Holder ◽

Trial Court ◽

Court Of Appeals ◽

The Us ◽

Antitrust Laws ◽

Per Se ◽

Appeals Court ◽

Two Sides

On September 15, 2003, the US. Court of Appeals for the Eleventh Circuit held that agreements between pharmaceutical and generic companies not to compete are not per se unlawful if these agreements do not expand the existing exclusionary right of a patent. The Valley DrugCo.v.Geneva Pharmaceuticals decision emphasizes that the nature of a patent gives the patent holder exclusive rights, and if an agreement merely confirms that exclusivity, then it is not per se unlawful. With this holding, the appeals court reversed the decision of the trial court, which held that agreements under which competitors are paid to stay out of the market are per se violations of the antitrust laws. An examination of the Valley Drugtrial and appeals court decisions sheds light on the two sides of an emerging legal debate concerning the validity of pay-not-to-compete agreements, and more broadly, on the appropriate balance between the seemingly competing interests of patent and antitrust laws.

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The Two Sides of Temporal Orienting

Experimental Psychology (formerly Zeitschrift für Experimentelle Psychologie) ◽

10.1027/1618-3169/a000018 ◽

2010 ◽

Vol 57 (2) ◽

pp. 142-148 ◽

Cited By ~ 26

Author(s):

Ángel Correa ◽

Paola Cappucci ◽

Anna C. Nobre ◽

Juan Lupiáñez

Keyword(s):

Executive Control ◽

Stroop Effect ◽

Traffic Situation ◽

Target Onset ◽

Stimulus Features ◽

Two Sides ◽

Spatial Stroop Effect ◽

Probable Interval ◽

Spatial Stroop ◽

Temporal Orienting

Would it be helpful to inform a driver about when a conflicting traffic situation is going to occur? We tested whether temporal orienting of attention could enhance executive control to select among conflicting stimuli and responses. Temporal orienting was induced by presenting explicit cues predicting the most probable interval for target onset, which could be short (400 ms) or long (1,300 ms). Executive control was measured both by flanker and Simon tasks involving conflict between incompatible responses and by the spatial Stroop task involving conflict between perceptual stimulus features. The results showed that temporal orienting facilitated the resolution of perceptual conflict by reducing the spatial Stroop effect, whereas it interfered with the resolution of response conflict by increasing flanker and Simon effects. Such opposite effects suggest that temporal orienting of attention modulates executive control through dissociable mechanisms, depending on whether the competition between conflicting representations is located at perceptual or response levels.

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