**Three Exponential Explorations**

##### (MathPickle, Polignac, Armstrong)

Give your students practice working with bases and exponents while in pursuit of the answers to bigger questions. In this video you will explore three such… including the infamous Polignac Conjecture:

Polignac Conjecture – Sometimes mathematicians make strong conjectures that can be disproven easily by tenacious middle school students. This is true of the humiliating Polignac Conjecture.

Here are two extensions of the 291/292 puzzle for math geeks.

1) Add the additional constraint that no two bases are equal. Is there a largest integer that has no solutions? Not sure–there may be an infinite number of them.

2) Add the additional constraint that no two bases are equal and no base equals an exponent. Is there a largest integer that has no solutions?

**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 Know the tools.

Students master the tools at their fingertips - whether it's a pencil or an online app.

##### 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 😉 If you are using MathPickle with adults, please mention the website. If you are from a private school, please ask parents or administrators to donate.