id: 06512812
dt: j
an: 2016a.00587
au: Griffiths, Martin
ti: Squeezing bubbles into corners.
so: Math. Teach. (Reston) 107, No. 6, 473-477 (2014).
py: 2014
pu: National Council of Teachers of Mathematics (NCTM), Reston, VA
la: EN
cc: G40 I30
ut: 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
ci:
li: http://www.nctm.org/Publications/mathematics-teacher/2014/Vol107/Issue6/Delving-Deeper_-Squeezing-Bubbles-into-Corners/
ab: 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 ever-decreasing 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.
rv: