Mathematics

In this short note, we present a proof of the generalised Pythagoras theorem. We use the ‘ordinary’ Pythagoras theorem for the proof.

Theorem. In any triangle ABC, we have:

AC2 + BC2 > AB2 ⇐⇒

The topic of ‘proof by induction’ is now a standard part of the syllabus of mathematics at the 11-12 level. Most students consider it a ‘scoring topic’ – they generally master the mechanics of proof by induction quickly, as the proofs follow a standard trajectory and are easy to mimic.

In this short note, we present a proof of the generalised Pythagoras theorem. We use the ‘ordinary’ Pythagoras theorem for the proof.

Theorem. In any triangle ABC, we have:

AC2 + BC2 > AB2 <--> C < 90 ,

AC2 + BC2 < AB2  <--> C > 90◦ .

Children are taught many kinds of measures in the primary grades, like length, weight, volume, money, time and finally area and perimeter (which is nothing but measurement of length). While for teaching length, weight and volume, efforts are made to give students an exposure to informal/ non-standard units of measurement (and thus some implicit understanding of what we may use to measure something), the other measurement ideas start with the standard units, conversions and formulae.

Problem-solving, which has always been an important part of learning mathematics, received considerable attention after it was recognised as a route for promoting mathematical thinking in the Position Paper on Teaching of Mathematics (NCERT, 2005). For problem-solving to flourish in its true spirit, two ingredients are essential: adequate knowledge and skill to solve the problems and acumen to generate good meaningful problems