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The 14 Different Types of Convex Pentagons that Tile the Plane
Many thanks to Branko Grunbaum for assistance with this page.

Doris Schattschneider's interactive tiling page is here. Bob Jenkin's half-bath has a beautiful tiling based on the Hirschhorn Medalion. This is one pentagon in the Chaos Tiles. Bob Jenkins has made a page about non-convex pentagons. Marjorie Rice, who discovered many of these, has a page here. A bit more is at MathWorld and Wikipedia. An interactive demonstration of pentagon tilings is available. Some of these might be interesting to study in the context of the Clean Tile Problem, a gambling game with interesting odds and probabilities that probably won't be seen at an online casino anytime soon.


Type 1 : D + E = 180	Type 2 : C + E = 180, a = d

Type 3 : A = C = D = 120, a = b, d = c + e	Type 4 : A = C = 90, a = b, c = d

Type 5 : C = 2A = 120, a = b, c = d	Type 6 : C + E = 180, A = 2C, a = b = e, c = d

Type 7 : 2B + C = 360, 2D + A = 360, a = b = c = d	Type 8 : 2A + B = 360, 2D + C = 360, a = b = c = d

Type 9 : 2E + B = 360, 2D + C = 360, a = b = c = d	Type 10 : E = 90, A + D = 180, 2B - D = 180, 2C + D = 360, a = e = b + d

Type 11 : A = 90, C + E = 180, 2B + C = 360, d = e = 2a + c	Type 12 : A = 90, C + E = 180, 2B + C = 360, 2a = c + e = d

Type 13 : A = C = 90, 2B = 2E = 360 - D, c = d, 2c = e	Type 14 : D = 90, 2E + A = 360, C + A = 180, B + D + E = 360, 2e = 2c = a

Types 1-5 were found by K. Reinhardt in 1918.
Types 6-8 were found by R. B. Kershner in 1968.
Type 10 was found R. James in 1975.
Types 9, 11-13 were found by M. Rice in 1976-1977.
Type 14 was found by R Stein in 1985.

Most math teachers know that the best way for students to improve at mathematics, is for them to regularly practice solving mathematical problems. The challenge that teachers of course face is how to make practicing the same activity again and again, always seem new, interesting and engaging to students.

One technique that many teachers use, is to integrate educational games and puzzles into their lessons - provided the games are carefully chosen, students can learn while having fun. A preferred game amongst educators is bingo, for younger students a simple game can help students calculate simple maths more advance students can discuss the odds and probability of the game.

The Convex Pentagons require a great understanding of mathematics. A student who is educated properly and is open minded to new ideas will surely be able to wrap his mind around the convex pentagons.

Sources:
The Penguin Dictionary of Curious and Interesting Geometry, David Wells, 1991.
Mathematical Recreations: A Collection in Honor of Martin Gardner, David Klarner, 1981.
Tilings and Patterns, Branko Grunbaum and G. C. Shephard