A worksheet that guides students through a coloring exercise that leads to an infinite sum that turns out to be finite.

1. An infinite sum?
1. Divide the square above inhalf bydrawinga diagonal.Colorinhalf the square.If the entire square hasarea1,
what'sthe areaof the triangle youhave coloredin?
2. Nowdivide the uncoloredtriangle intotwocongruenttriangles(thatis,twotriangleswiththe same shape and
size) andcolorin one of those triangles.Whatisthe area of the small triangle?Whatisthe total area of the two
trianglesyouhave coloredin?

2. 3. Continue inthisway,dividingthe uncoloredtriangleinto
twocongruenttrianglesandcoloringinone of those two.
Complete the chartonthe right – Challenge:canyoufigure
out the numbersforstages20 and 100?
4. Imagine thatyou couldfinishingdoingthisshadinginfinitelymanytimes(thatis,imagine thatall of the
infinitely-manystepsof the coloringare done).Canyouwrite downasumthat givesyouthe area youhave
coloredinafterthissuper-humanfeat?Canyouaddup thissum?Is the total areayou coloredafterthissuper-
humanfeatfinite orinfinite?
Stage Area of smallest
triangle
Total shaded area
1 ½ ½
2 ¼
3
4
5
6
20
100