**Full Integer Collatz**

##### (MathPickle, 2012)

Students can play with variations of the famous 1937 Collatz conjecture. Here we look at two. Students can also come up with their own Collatz-like conjectures. Brian Kehrig, a 15 year old student asked about the behaviour of the following Collatz-like idea:

Start with a positive integer. If it is a multiple of 3… then multiply it by five and add one.

Otherwise… divide by three and round to the closest integer.

Logic and mathematics are nothing but specialized linguistic structures.

**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. MathPickle recommends pairing up students for all its puzzles.

##### 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. MathPickle encourages students not only to use the precise terms of others, but to 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.