
06512812
j
2016a.00587
Griffiths, Martin
Squeezing bubbles into corners.
Math. Teach. (Reston) 107, No. 6, 473477 (2014).
2014
National Council of Teachers of Mathematics (NCTM), Reston, VA
EN
G40
I30
filling a square with circles
total area
geometric series
plane geometry
solid geometry
filling a cube with spheres
total volume
generalization
multidimensional spaces
hypercubes
hyperspheres
student activities
discovery learning
http://www.nctm.org/Publications/mathematicsteacher/2014/Vol107/Issue6/DelvingDeeper_SqueezingBubblesintoCorners/
From the text: I always seek activities that might stretch my students yet would be accessible to them; that might require logical thought yet would contain counterintuitive elements; that might provide the opportunity to venture into new mathematical realms yet would have a simple starting point. This article and the activity that inspired it did indeed arise by way of a relatively straightforward problem that I proposed to one of my classes. I asked students to draw a square $S$ of side length 2 units containing a circle $C_1$ of radius 1 unit. The challenge was to find the radius $r$ of the largest circle $C_2$ that could fit into one of the ``corners" between $S$ and $C$. This problem can be solved in several ways. One approach is to imagine an infinite series of everdecreasing circles. Later we deal with the extended problem of finding the total area of the infinite number of circles that cover $S$. Moving into three dimensions gives rise to the ``bubbles" referred to in the article's title. And this problem generalizes beautifully into $n$ dimensions.