We had so much fun verifying that the difference in the lengths of string was really the same no matter how big or small your original circle seemed to be. One student even did the activity in reverse. He noticed that drawing a new circle with a radius that was 1/2″ longer added 3″ to the string. When he did a circle with a radius that was 1″ longer, it added 6″ to the amount of string needed to go around the new circle (as compared to the original circle). So he guessed if he had a 9″ difference that the radius would increase by 1.5″ compared to the original circle. Then he tested his hypothesis! I was so excited by his thinking process! It was a bit challenging to use string and get accurate measurements.
Another neat question that arose, when we showed that the difference in circumference is always the same amount no matter the original circle size, was WHY!? Two of my students were really annoyed and wanted to know “Why! Mrs. Nash. How is that possible!” After racking my brain for a moment we settled on the analogy of making rice. It’s a proportion, and that’s where Pi really started to click (even for me!) Just like making rice is always 1 part rice : 2 parts water; circles are always increasing by 2π.
So if I need 9 cups of rice I measure 3:6; if I need 24 cups of rice I measure 8:16; but no matter how BIG the amount of rice I need, the proportion will be the same. That’s when they got it—that’s when I got it! No matter how big my starting circle (even if it’s the earth), the proportion I add is connected to 2π . It’s a ratio!
If you’d like to try the sparks too, watch for our next Inspired by Calculus class.