Geometry Visualization Tasks

This is a fun fractal pattern I came across yesterday. Questions to ask students:

* How many squares are in stage n? (Can you write both a recursive and an explicit formula?)

* How big is the area of each square in stage n? (Can you write both a recursive and an explicit formula?)

* How big is the total shaded area in stage n?

* How big is the total perimeter in stage n?

* Are these arithmetic or geometric sequence patterns, or neither? Justify your answer.

* At what stage will the total shaded area be less than 1/1000 of the original area?

* Can you design your own square-based fractal and repeat the steps above?



And this is another fun/short geometry problem I came across today.

Find the area of the shaded region, if each circle has radius r and A, B, C are the centers of the three circles.

(And for extra fun, they can also find the areas of the other sections of the Venn D.)

Both of these are attributable to IB resources. The latter came from Mathematics For The International Student: Mathematics SL and the former is a modified version of a task from the MYP Taskbank.

PS. I keep missing my 6:04pm bus, because at around 5:30pm daily I start playing with random math problems. sigh. This is no good, because the bus only comes once every 20 minutes, and even after I get on the bus, it still takes me another 60 minutes to get home. I need an iPhone so that I can get an app called iProcrastinateAboutGoingHome!! ...This also means that I am at work for 11 hours almost daily (not including my one-hour-each-way commute). Work-life balance during year 1 is always a toughie for me, since I have naturally obsessive working tendencies. :(

Addendum March 9, 2012: I was playing around with circular fractals yesterday. If you keep replacing circles with two smaller inscribed circles (whose radii are 1/2 of the original circle), then the total perimeter of the shape stays the same and its total area decreases at each stage by a factor of 2. Neat eh? And not very intuitive! If you repeat this indefinitely, the final "shape" approaches two slightly squiggly (basically straight) lines, and together they still have the same length as the original circumference!