28July2015 – It is like 90 degrees in here!

I saw this idea in a post by a math department on Google+ and knew that Luke and I had to do this, it looked like such fun. At first glance, it seems like a fun way to concretely make different angles and find out what 100 degrees means in the real world, but it developed into a math history lesson and an exploration of constructive geometry.

I put down the strip of yellow tape against the edge of the door to represent zero, and then asked Luke “how do we make the other angles?”. “How do we make 90 degrees?”. He held up his index finger and thumb for a right angle, but when I asked him if he would have wanted our house built with “finger” angles, he said “no”. Smart boy realizes importance of precision in house building! But back to the original question…how would the ancients have made precise angles? Luke knows about protractors and builder’s squares, but we continued on with thinking how the Ancient Greeks would have done this (we have read some Living Math books about Pythagoras and other Greek mathematicians). We were a little stumped…and then Daddy showed up and said “Triangles, of course!”. (He really should be the one homeschooling these kids, but we need to eat sometime).

And so we were off.

We were onto something with the triangles, and when I asked Luke if there is a triangle that we can make where we can absolutely know the measurement of the angles, he mentioned a triangle with equal sides and something about the Greeks using a rope with knots. We got out some string, and yes, you can make knots so that you have three equal pieces of string  – you don’t cut the string, the knots tell you where the beginning and ends of the three equal pieces are. This is important when you make the triangle because when you pull the string taut, keeping it all one pieces allows you make a rather precise equilateral triangle. And voila, we have three 60 degree angles without anything more than a piece of string!

But now what do we do? Daddy then took one side of the triangle,  folded it in half, and the connected the three knots once again. Luke realized that we now had one 90 degree angle, one 30 degree angle and an original 60 degree angle. It is important to just barely touch the other string to the halved string when re-connecting the triangle, it makes for the most precise right angle. And then we discovered you could halve that side once again, and now you end up with a 15, 90 and 75 degree angles. It was cool using ancient practical geometry techniques to play with angles.

Ah, but one angle seems out of reach, what about 45 degree angles? It turns out that you can’t use the halving trick for those, but what you can do is keep two sides the same, but make the third piece of string longer (essentially stretching out one of the 60 degree angles, and narrowing the other two angles). As long as two of the sides are identical (easy to do with the knots), you will have two 45 degree angles and one 90 degree angle.

Once we figured out how the Greeks would have determined the angles, Luke decided to use a protractor to make the angles with the tape. It was just a little unwieldy using the string, although I am sure a practiced Greek geometer would have found it easy. One more crucial lesson that he figured out on his own after making repeated mistakes: you need to have the protractor on the origin, or your angles will be wrong. Once he figured that out, it was smooth sailing and the end of our math adventure for the day. But we are keeping the tape down for as long as we can, it brightens the whole foyer 🙂

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