What we discovered was that circumference was about 3 times larger than the diameter. Or the diameter was about circumference divided by 3.
So today we examined that relationship a little more by making these foldables. We did the inside first. A copy of the cut-outs is available here: [link]
- Students had to color and then cut out the circles and the strip. The circles' circumferences and the strip had to be the same color; and the diameters had to all be the same color, but one different from the circumference.
- Students then drew a "starting line" on the left of their page. They glued the strip down touching the line. I explained that this line was the same length as a circle's circumference "unrolled" into a straight line.
- Then we glued down the circles to see how many diameters could fit along that circumference. We drew an ending line where the diameters stopped and could tell the circumference was longer. We added a second line to show where the circumference stopped and looked at that distance in-between them. I asked students to estimate about how much of a diameter that extra bit was.
- I told students that mathematicians have been looking at the extra little bit for thousands of years because it turns out, it's not such a simple number to get. I shared this YouTube video about the history of pi [link]. I prefer to use ViewPure to show less clutter on the screen (it strips out all the adds and the sometimes questionable video links that follow a YouTube video).
- Then we go back and add the information on the above the circles, that pi is a symbole used to represent the circumference divided by diameter (the idea of how many diameters fit on a circumference); and that two estimates of pi (mentioned in the video) are 3.14 and 22/7. Any time we use those numbers (3.14 or 22/7), we are estimating the number of diameters on a cirucmference... just like we did when we estimated that there were only three. But 3.14 and 22/7 are better estimates -- they are not precise measurements.
- We next add that + ~ 0.14 to the diagram and we go to the cover next to learn a few ways that people write the pi symbol (I have learned that students need to be taught some method of writing pi, or they end up with REALLY sloppy approximations). So they can choose any of the representations on the cover to write (and maybe there are other variations, but these are pretty common).