Figure 1. How can we place this next square? Can this vertex pattern possibly work on the whole plane?


and squares. Tat subquestion narrows the scope, al- lowing for more detailed involvement with needs (2) and (3).


Notation. For the purposes of record-keeping and com- munication, it may be useful to introduce a standard- ized vertex notation at some point, if teachers don’t come up with their own. A vertex at which, cyclically, we see a “triangle, square, triangle, triangle, square” can be described as a 3.4.3.3.4 vertex.


Question 2: For each vertex pattern that works, can you continue the tiling forever? Of all the possible tilings around a vertex, we focus on three that highlight different intricacies of mathemati- cal justification.


Figure 2. Is this a semiregular tiling? What is the vertex pattern?


Example 1: 3.3.4.3.4 Tis rich example will likely arise early in the explo- ration, because equilateral triangles and squares are familiar shapes, with familiar angle measures. It’s well timed if it does arise early, because its complexity motivates participants to approach later vertex patterns with care. It naturally leads to two important math- ematical questions about the general scenario at hand: 1) How do I continue the pattern beyond a single vertex? 2) How do I know the pattern will continue forever? When participants work to continue this pattern,


they will quickly find themselves having to make choices about whether to place a square or triangle along a given edge. It can be very productive (so don’t try to stop it!) for some groups of participants to make


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MTCircular · Summer/Autumn 2016 · American Institute of Mathematics


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