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Make three equally spaced notches along the four sides of a square. Choose one of these three notches on each side. Connect them up to form a simple quadrilateral.

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For example, here we have chosen the middle notch on each side. What is the area of this diamond-shaped quadrilateral compared to the initial square?

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What happens if you chose a different set of points as the corners of your quadrilateral? Calculate the area. You win if the area is exactly half of that of the big square.

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Ask yourself a question:

How many ways can you win and lose?


What chance would I win if I randomly selected the notches on each side?

Half Area Quadrilateral

(Dan Finkel, 2013)

Students practice calculating area of right triangles and making hypotheses. I saw Dan present this puzzle at Nueva school and was thrilled with the beautiful discovery that was made by the students. The results can be generalized for squares with more notches, but this is the perfect size to begin.

Visit Dan’s web site, Math 4 Love.

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 😉 For a free poster of MathPickle's ideas on elementary math education go here.

Gordon Hamilton

(MMath, PhD)