Connecting with Mathematics

What is Mathematical Investigation?
Dissection and Area
Linearity and Proportional Reasoning
Pythagoras and Cousins
Pascal's Revenge: Combinatorial Algebra
Games and Number Theory

Pythagoras and cousins

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The Pythagoras and cousins module, as you might guess, delves into the Pythagorean Theorem (the sum of the squares of the lengths of the legs of a right triangle is equal to the square of the length of its hypotenuse) and related topics. Some of the topics include various proofs and generalizations of the Theorem - did you know that the book The Pythagorean Proposition contains 366 proofs of the Theorem? - and properties of Pythagorean triangles (right triangle with integer-valued legs and hypotenuse). We'll also derive an algorithm for producing every Pythagorean triple (an ordered triple of positive integers, (a,b,c), so that a2 + b2 = c2) and use this algorithm to solve a variety of problems. Some of the questions we'll ask:

  • What happens in triangles that don't contain a right angle?
  • Does the Pythagorean Theorem generalize to higher dimensions?
  • Is it true that every Pythagorean triple contains a multiple of 3? of 4? of 5? of . . . ?
  • Which 17th century mathematician is one of Pythagoras' cousins?
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