A Geometric Puzzles Unit

The Geometric Puzzles home page contains many links, and it is organized reasonably well. Still, someone who is not familiar with all these puzzles may need help in turning them into a curriculum unit. This is what I hope to do here. I will try to not duplicate the overview of some puzzles from this article, or the philosophical underpinnings I shared in this one.)

I call this a unit, but I do not expect that it would be taught as one continuous stream of lessons. It could be done that way in a math club or summer program, but that would be not an option in most classes. It is a unit in the sense that it is organized in a reasonable sequence pedagogically and mathematically. This would allow a teacher to schedule the various parts of a puzzle strand in a way that makes sense for their classes — whether their students tackle geometric puzzles routinely, weekly, or just occasionally.

A feature of geometric puzzles is that the same puzzles work with an enormous range of grade levels, although the curricular connections for a given activity are most relevant to a more specific age group. One consequence of this state of affairs is that no activities in the unit are absolutely required. You can pick and choose based on your preferences and available materials. Another consequence is that in some cases students are more capable as puzzle solvers than their teacher. That is not a bad thing!

Anyway, without further ado, here is a suggested geometric puzzles unit. Note that almost all the links below are to materials that are freely downloadable for any non-commercial use.

Tangrams are a good place to start. They involve few pieces, are relatively inexpensive, and they lend themselves to be used in conjunction with basic geometric vocabulary.

On occasion, I've had students who had previously memorized the solution to the basic tangram puzzle (the seven-piece square), but that is not the focus of the work I do with tangrams. In fact, that solution comes up with the first activity.

• Tearing tangrams — an activity you can do prior to handing out plastic tangram sets.
• In Geometry Labs, Section 2 is all about tangrams, with activities involving measurement, vocabulary, symmetry, and convexity.

You can use my Virtual Tangrams if you need to project solutions on a screen, or create images for handouts.

Tip: Hand out the tangrams so that neighbors have different-colored sets. This makes it more difficult for pieces to migrate to a different bag. It also makes it more obvious if a student "cheats" by using a piece from another set. (This suggestion also applies to pentominoes and supertangrams.)

Polyominoes are the shapes made by joining squares edge-to-edge. They can readily be explored on grid paper. Perhaps most convenient for polyomino exploration is this centimeter grid paper. Or you can use Virtual Grid Paper.

In Geometry Labs, Section 4 is all about polyominoes and some generalizations to other shapes. It is largely based on  Polyomino Lessons, pp. 1-17, where you can find much the same material in a somewhat different presentation.

Even if you don't do many of these activities, I would encourage you to at least have students do "Finding the Polyominoes" from the start of one of the above, in preparation for work with pentominoes. (With younger students, that initial activity could be done using interlocking cubes, or square tiles.)

Consider making copies of the "Polyomino Names" sheet from one or the other book, as it may come in handy in later activities.

Pentominoes are the shapes made by joining five squares edge-to-edge. They lend themselves to many, many puzzles. Plastic pentominoes are available from many math education publishers, but if you have access to a laser cutter, you should make your own. (See my Pentominoes home page for the needed file and many pentomino links.)

Here are some introductory activities:

• Pentomino holes: challenge students to arrange the pentominoes so as to have a maximum number of holes in the resulting figure. You will need to set some parameters as to what constitutes a legitimate solution. The photo shows an impressive 15-hole solution found by an eighth grader, but it is not necessary to aim for any particular number. If your class record is 12, that's perfectly fine.

• Three-Pentomino Puzzles. The worksheet includes the pentomino names, and can be used to have students familiarize themselves with the names. Have them tell you which ones they used for each figure (for example LTY), and keep track of the solutions on the board.
• Congruent pairs: make a two-pentomino figure, and cover it with another two pentominoes, thus creating a two-layer arrangement. Try for three layers!
• Pentomino rectangles: what are the dimensions of rectangles that can be covered exactly with pentominoes? This can be an open exploration, or it can be structured using pp. 14-23 of Working with Pentominoes (available from Didax). 

This only scratches the surface! You will find hundreds of additional pentomino puzzles in these three (free) books, and dozens of pentomino lessons in Working with Pentominoes. 

One of the most visited pages on my website is Virtual Pentominoes. It is especially useful if you can't use physical pentominoes, as a way to project solutions onto a screen, and as a tool to create images to copy-paste into documents.

Tiling (or tessellation) is the covering of a surface by using  the same shape (or shapes) repeatedly, with no overlaps and leaving no gaps. You can read more about how to use tiling to introduce geometric concepts in this blog post and its sequels. You can find links to many activities on my Tiling home page, and in Chapter 7 of Geometry Labs.

For this geometric puzzles unit, see pp. 18-30 of Polyomino Lessons where you will find a wide range of tiling challenges to be carried out on grid paper  or Virtual Grid Paper.

Supertangrams are the shapes made by joining four isosceles right triangles edge-to-edge. They offer a way to build on and extend activities previously done with tangrams or pentominoes in a more challenging environment. See SuperTangram Labs #1-3.

Plastic supertangrams are only available from me ($3/set + shipping/handling. Find more information on my Supertangrams home page.)

All the puzzle types we've discussed in this unit lend themselves to the exploration of similar figures, scaling factor, and ratio of areas. You can do these activities as a culmination of the unit, or tackle them along the way:

• Tangram Similarity: Geometry Labs, 10.6
• Tiling scaled polyominoes: Polyomino Lessons, pp. 31-33
• Pentomino Blowups: Pentomino Labs #4, and/or Working with Pentominoes, pp. 52-58.
• Scaling SuperTangrams: SuperTangram Labs #4

Finally, an extension: Rep-Tiles. (My Geometry Labs template is helpful for this activity, and for tiling in general.)

The above is a reasonably meaningful sequence of lessons and activities involving geometric puzzles. They can be spread across the year, or even in a collaborative math department over more than one year, especially since some of the content spirals and appears in different guises at various points. It is most appropriate to middle school and high school.

As you figure out how to implement this unit, remember that almost any part of it can be omitted. And on the other hand, almost any part of it can be tackled in greater depth using the additional activities and books I link to.

An alternative or complementary approach is the one I used back in the day when I taught grades 4-5: a weekly math lab during which students could choose what geometric puzzles they wanted to work on. I had two or three laminated copies of each puzzle, organized in various folders. To get a sense of that setup, look at the checklists at the front of my tangram, pentomino, and superTangram books: those are keyed to the relevant folders. Each student had their own copy of all the checklists. Once they had solved a puzzle, they would ask me to confirm they did it correctly and checked it off their list. This allowed kids to choose their own challenge level, and undermined any unhealthy racing and competition. (In addition to geometric puzzles, the folders included pattern block activities, symmetry puzzles —like the ones on pp. 1-11 of this packet— and any hands-on mathematical activities I found that kids could do on their own.)