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My favorite pi Day activity might not fit with a geometry class as much as it does my eighth grade math class. It is an NCTM illumination’s activity where they investigate pi as slope. We measure different circular items and plot the ratio of their radius to their circumference- radius on the X circumference on the Y. The ratio should equal about pi 3.14. It helps them understand The meaning of pi.
I'm a big fan of getting outside, so I take my students around the school searching for circles to measure and calculate pi for. They usually latch on to our lunch tables and the basketball courts, for example. It gets fun when they find circles bigger than their measuring tapes and they have to get creative to problem-solve for that. They usually get results slightly (or greatly) off from pi and we discuss possible reasons why.
I love having my students bring in "circular" objects from home. They then use yarn to measure the circumference and radius of their project. They can the relate the circumference to pi.