What makes a good problem?


Watch MTC leaders Chris Bolognese, Fawn Nguyen, Paul Zeitz, and Joshua Zucker discuss this question in this Educator Innovator webinar:


www.educatorinnovator.org/webinars/ what-makes-a-good-problem


Figure 3. What goes wrong when we try to continue this vertex pattern? Is there a way to fix it? Why or why not?


the wrong choices, leading to a tiling dead end that is not easily rectifiable without serious backtrack (see Figure 1). Even once participants have found the right choices to make in laying down the tiles, an example of a path that didn’t work will hopefully motivate the need to truly convince ourselves that the pattern can continue forever.


Example 2: 3.3.6.6 Tis example is tricky because, starting with this vertex pattern, you can tile the plane. However, the most natural tiling that arises conflicts with the definition of semiregular tiling (see Figure 2). Tis is an opportunity to revisit the definition (and the importance of defini- tions in general), and to justify why the pattern cannot be continued in a fashion that fits our definition.


Example 3: 5.5.10 Although it is clear almost immediately that you won’t be able to continue this vertex pattern (see Figure 3), articulating exactly why this is the case is a great exer- cise in mathematical argumentation.


Extensions Changing the constraints in our definition allows for many additional questions. For example: What if we don’t require our shapes to be regular? What if we don’t require our vertex patterns to be uniform? What if we delete our “edge-to-edge” constraint? Tis session could be part of a larger series of ques-


tions about tilings. See, for example, Rodin’s article on Escher-like tilings in the Summer/Autumn 2014 MTCircular. ⊆


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