A set of plastic circle fractions got me thinking. How many ways can a circle be constructed if your smallest piece is 1/6?
If your smallest piece is 1/6, there are six ways to complete the circle. Here we see the front and back view of one proposed solution. Is it correct?

1/6 + 1/6  + 1/3 + 1/6 + 1/12 + 1/12 = 1

The sum is correct, but 1/12 is less than 1/6 so this is not a solution.

Find all six solutions if you start with 1/6.

Here is the most obvious solution…
Students should think carefully how to organize their search for new results.

There is only one left… 

Wait – did I make a mistake! Yes! Lhianna Bodiford from Math Explorers in Philadelphia, PA, USA contacted me in March 2026 to say that there are more solutions that I’ve missed! She’s right! Think for a moment and then go to the next slide.

Lhianna says I forgot solutions like 1/4 + 1/4 + 1/6 + 1/6 + 1/6 and 1/4 + 1/4 + 1/3 + 1/6. Of course these can be be rearranged in many ways. I’ll leave your classroom to fix my error. 

Clearly this was too difficult a puzzle for me to start on. Let me go back and try to think of simpler stuff…

 

There is only one way of creating a circle where the smallest fraction is 1/1.

How many ways can you complete the circle if the smallest fraction is 1/2?

Just one.

How many ways can you complete the circle if the smallest fraction is 1/3?

Just one again.

How many ways can you complete the circle if the smallest fraction is 1/4?

There are two. This one and…

 

How many ways can you complete the circle if the smallest fraction is 1/5?

 

Just one.

We have already seen the six ways to complete the circle if the smallest fraction is 1/6. What about 1/7?

 

Do your students see a pattern? Do a lot more of these questions have just one solution?

How many ways are there to complete the circle if the smallest slice is 1/8? There are 20-30 so this is not an easy exercise. In the next slides I’ll take you rapidly through them.

Hey – did you catch that – I’ve made a mistake. There are only 28 solutions. You can go back and figure out which two of my images are mirror images of one another!
Hey – did you catch that – I’ve made a mistake. There are only 28 solutions. You can go back and figure out which two of my images are mirror images of one another!
Hey – did you catch that – I’ve made a mistake. There are only 28 solutions. You can go back and figure out which two of my images are mirror images of one another!
Hey – did you catch that – I’ve made a mistake. There are only 28 solutions. You can go back and figure out which two of my images are mirror images of one another!
Hey – did you catch that – I’ve made a mistake. There are only 28 solutions. You can go back and figure out which two of my images are mirror images of one another!
Hey – did you catch that – I’ve made a mistake. There are only 28 solutions. You can go back and figure out which two of my images are mirror images of one another!
Hey – did you catch that – I’ve made a mistake. There are only 28 solutions. You can go back and figure out which two of my images are mirror images of one another!
Okay – so there are only 28 of these solutions – and I made an error. That’s okay. I’m not even going to go back and correct it 😉

How many ways are there to complete the circle if 1/9 is the smallest fraction?  Do you think less or more than 1/8?

Things get completely out of control for pleasant hand-calculations now. The number of solutions for 1/10 is larger than 100, but I’m too lazy or maybe I just have better things to do with my time than to calculate it 😉
I might as well just show you one example…

Eleven and all future prime numbers have a unique answer.

If anybody writes a program to determine the number of solutions for 10, I’d love to hear about it. I’ll give you credit 😉

Plastic Fraction Circles

(MathPickle, 2015)

This puzzle holds the record for the longest time a big, glorious mistake has survived on MathPickle… 11 years! It is good pedagogy to keep and celebrate your mistakes, so I’ve left it in the slide show. Mistakes are so much more interesting than sterilized perfection. 😉

A big thank you to Lhianna Bodiford of Math Explorers in Philadelphia, Pennsylvania, USA, for the discovery. 😉

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?

(http://www.corestandards.org/Math/Practice/)

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)