# Pythagoras

## A Counter-intuitive PYTHAGOREAN SURPRISE

Imagine a tightly stretched rope from one end of the field to be 100m; so the length of the rope is 100m. Now replace this rope by one that is slightly longer, say by 20 cm. There is some slack in the rope, so we should be able to lift the midpoint of the rope to some height. Imagine pinching the rope at its midpoint and raising the rope till it is taut. Question: To what height can you raise the midpoint?

I n the March 2017 issue, Khushboo Awasthi had described an investigation of the familiar Square Root Spiral, which had taken her along unexpected paths filled with mathematical discoveries. At the end of the article, she posed some questions for the reader to investigate and we did just that! We share our bonanza of findings with you, and as usual, the tasks are arranged from Low Floor to High Ceiling. This time, we include some investigations with the free dynamic geometry software GeoGebra; regular constructions with compass and ruler will do the job just as well!

## Tonic Ambiguity

It has been found (right from Pythagorean times) that the frequency of the tonic and the frequencies of the rest of the tones and semi-tones form a simple ratio. A particular musical tone always has the same frequency ratio relationship with the tonic. The western solfege syllables corresponding to Sa, Ri, Ga, Ma, Pa, Dha, Ni are Do, Re, Mi, Fa, So, La, Ti respectively.

## A strange case of banned numbers

They say the pen is mightier than the sword, and authorities have often agreed. From outlawed religious tracts and revolutionary manifestos to censored and burned books, we know the potential power of words to overturn the social order. But as strange as it may seem, some numbers have also been considered dangerous enough to ban. Alessandra King details the history behind illegal numbers.

## 3 4 5 strikes again

These are excellent GeoGebra exercises for students helping them to develop and practise skills of visualisation, logical sequencing, making connections and recalling theory. Read on.

## Spiral of Square Roots

Take the perennial favourite Pythagoras theorem to the classroom and pose these investigative questions. Spiral of square roots perks up the proceedings!

## A Potpourri of Problems ~ CoMac

We present once again a miscellaneous collection of nice problems, followed by their solutions. We state the problems first so you have a chance to try them out on your own.

## A Baby One Quarter the Size of its Parents

The Pythagorean theorem tells you how beautifully the baby circle is quarter the size of its parents! Check it for yourself.

## Keeping Things in Proportion

‘Low Floor High Ceiling’ activities are simple age-appropriate tasks which can be attempted by all the students in the classroom. The complexity of the tasks builds up as the activity proceeds so that each student is pushed to his or her maximum as they attempt their work. There is enough work for all, but as the level gets higher, fewer students are able to complete the tasks.

## 3, 4, 5 And other memorable triples – Part I

What’s interesting about the triple of consecutive integers 3, 4, 5? Almost anyone knows the answer to that: we have the beautiful relation 32 + 42 = 52, and therefore, as a consequence of the converse of Pythagoras’ theorem, a triangle with sides 3, 4, 5 is right-angled.

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