Composite Critters
(Inspired by Solomon Golomb, 1981)
In my twenty years of designing puzzles for the classroom, this is the most important because it reliably engages K-2 students in rich mathematical ideas that were traditionally introduced many years later in their education. It is not original. Solomon Golomb highlighted the potential of pentominoes in 1981, but his ideas did not get adopted by elementary school curricula. Please let us change that.
Composite Critters should be in all schools that can afford the pentominoes. Better still, this should be an outreach activity coordinated by a school district. Not all schools should acquire this mass of plastic.
As of 2025, pentominoes can be economically bought in 6-colour sets. This is too many colours. Much better, coordinate with another school or district and take Red-Yellow-Blue or Purple-Orange-Green. Ask students which pentominoes can tile the classroom floor so that no identical colours share an edge.
Here are the puzzle sheets:
Level 1 (This is a great place for everyone to start. The pentominoes are given.)
Level 2 (The pentominoes are given. The critters are a bit bigger.)
Level 3 (The correct pentomino and some incorrect ones are given.)
Level 4 (Some of the shapes are infinitely long and must be tiled in a regular, repeating pattern. A large number of correct and incorrect pentominoes are given. The word “prime” is raised as an option, meaning that the critter is not composite and cannot be tiled with the same pentomino. However, all of these critters are composite. They can be covered if the correct pentomino is chosen.)
Level 5 (Only the top kindergarten or grade 1 students will get to this level in 45 minutes. Some of these critters are prime. They cannot be covered using the same pentomino.
Level 6 (We do not have a nice manipulative to help solve these puzzles, so they are not as good as the previous levels. Composite critters may not have any pentomino factors, but may have a factor of a different size of polyomino.)
Teaching is an experimental science. Don’t expect to be a great teacher your first year standing in front of a class.
Standards for Mathematical Practice
MathPickle puzzle and game designs engage a wide spectrum of student abilities while targeting the following Standards for Mathematical Practice:
MP1 Toughen up!
Students develop grit and resiliency in the face of nasty, thorny problems. It is the most sought after skill for our students.
MP2 Think abstractly!
Students take problems and reformat them mathematically. This is helpful because mathematics lets them use powerful operations like addition.
MP3 Work together!
Students discuss their strategies to collaboratively solve a problem and identify missteps in a failed solution. Try pairing up elementary students and getting older students to work in threes.
MP4 Model reality!
Students create a model that mimics the real world. Discoveries made by manipulating the model often hint at something in the real world.
MP5 Use the right tools!
Students should use the right tools: 0-99 wall charts, graph paper, mathigon.org. etc.
MP6 Be precise!
Students learn to communicate using precise terminology. Students should not only use the precise terms of others but invent and rigorously define their own terms.
MP7 Be observant!
Students learn to identify patterns. This is one of the things that the human brain does very well. We sometimes even identify patterns that don't really exist! 😉
MP8 Be lazy!?!
Students learn to seek for shortcuts. Why would you want to add the numbers one through a hundred if you can find an easier way to do it?
Please use MathPickle in your classrooms. If you have improvements to make, please contact me. I'll give you credit and kudos 😉 For a free poster of MathPickle's ideas on elementary math education go here.
Gordon Hamilton
(MMath, PhD)
