This only takes about 5 minutes of class time, maybe 10. But it went well for me.

Here’s the question as it was written in the book (Discovering Geometry by M. Serra):

“Make a set of sketches to show how two quadrilaterals can intersect in exactly one point, two points, three points, four points, five points, six points, seven points, and eight points.”

The answers are all practically there already–it’s just a formality for the students to draw them.

But, ENOboard space being limited, I decided to shorten it up and wrote this on the board:

“Make a set of sketches to explore if 2 quadrilaterals can intersect in exactly 1 point, 2 points, 3 points, and so on. How many different points can they intersect in?”

Then, when I listened in, I heard some groups arguing about whether you could have ODD numbers of intersections. One student said “put a kite inside a square, that’s 3 points.” Surprisingly, one group found a way to make 16 intersections. It turns out that by shortening the question, I opened the question up, and the kids surprised themselves and me!

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michael SerraThanks for alerting me to the exercise. Although it wasn’t intended to be a performance task but simply an exercise in visual thinking, your opening it up changed the nature of the exercise and made it much more interesting. I just tried out the open version with a class of 5th graders this morning and we did indeed find all 16 possible cases. At first of course everyone was sketching convex quadrilaterals but once someone shared that they drew a “dented” quadrilateral the race was on. Well worth the time in 5th grade.

joeltpattersonPost authorHey, Michael! I just want to THANK YOU for all the work you did putting together Discovering Geometry & your Patty Paper Geometry book. I’ve been using both in my 10th grade Geometry class, and the questions in those books do a lot to grow my students’ creativity & visual thinking. Most of your approach is very open-ended, and I think I stumbled into one of the few limited questions in the book. Also, I’m going to be doing your Donut Polygons soon with them, so that should be good! When I finished up my Bachelor’s degree in Physics and started teaching high school in 1995, I had no idea the kind of rich questions I would explore and deep thinking I would be doing about math with my students.

michael SerraJoe, Thanks for the kind words. I was just searching for that exercise and couldn’t find it. Which edition was it in and what page?

joeltpattersonPost author2nd edition. somewhere in chapter 2, I think. The chapter where the terms like quadrilateral, parallelogram, etc get defined. Not the inductive reasoning chapter.

Joel Pattersonpage 129 #52, Chapter 2 Review Questions. 2nd edition