Happy Irrational Number day!
As many of you know, today is Pi day - but have you ever thought about Pi being a ratio? One of the best lessons in my classroom is one where the learners discover Pi! In one class, I had a student measure the circumference and diameter of his truck tire and compared that ratio with the circumference and diameter of his chap stick. The expression on his face is one that I will never forget when he discovered that their ratio was about the same. I live for teaching moments like this! Yes, it is easier to tell students that Pi is approximately 3.14 but when learners can create those ah-ha moments the learning becomes apart of them. Here is a link to a lesson similar to mine: http://illuminations.nctm.org/lesson.aspx?id=1849
I hope you have several ah-ha moments in your classrooms!
Brooke Istas
Comments
I LOVE having students discover pi!
I pass around tape measures and circular objects of varying sizes, everything from a platter to a poker chips. In groups they measure everything you can about a circle with a tape measure- around and across. We talk about the vocabulary:definition of a circle, diameter, circumference, perimeter,etc... As they measure we record their data on the board in a chart: circle number, diameter and circumference. I tell them, it doesn't matter if they choose inches or cm, as long as their label is consistent. After we have good variety of information, we step back and look at the data, and look for a pattern. There is always someone who notices that the circumference is approximately three times as big as the diameter. We talk about that our measurements aren't the most accurate, because of our tools, but the more accurate you get, the closer you get to pi. Not only is this a GREAT way to discover pi, it is also a great way to learn that formulas aren't "invented", but rather they are a pattern/relationship that was discovered and then recorded.
Today I want to try extending it. So after we "predict the circumference is about 3 times as big", taking circles we haven't measured, measure the diameter, and predict the circumference, and then again starting with the circumference and predict/estimate the diameter.
I'm so excited to do this in both of my classes today!
Hope you all have a fun pi day!!!
If you were introduce the lesson as an Algebra one - many learners would run for the hills or simply say they can't do algebra! I like how you are asking them to look at the data and find a pattern. Additionally, you are asking them to estimate! I think estimation is often not something we teach because we assume learners have this ability. I spoke to a learner once about estimation and she said that estimation is only used in the real-world and not in math. This was an eye-opening moment for me about the importance not to assume.
Thank you for sharing! And Happy Pi Day!
Brooke
Greetings all!!
Brooke, I am so glad you started this thread!
I have done something similar to this lesson as well, and combining it with vocabulary like Rebecca S suggested as well. I had students take pictures of various circles in their homes, and they emailed these to me ahead of class. I printed out 8.5" x 11" posters of the pictures and gave several to each group of students working together. We talked about the vocabulary first: what makes a circle, a diameter, etc., and then we began looking at the ratio of circumference to diameter. I had them predict what they thought would happen to a ratio of a smaller circle and a larger circle. We went through a lot of measuring (we also discussed whether it mattered what units were used - and modeled our answer). There was a great deal of really good discussion during the predicting process, and like your student Brooke, mine were amazed to see all circles had the same ratio between circumference and diameter.
Libby
That's the question I was asking myself today as I thought about the lesson that you're all talking about. I hadn't thought about doing this lesson until after I got to school today and the stuff I usually use for the lesson was home in a nice little kit. Determined to do it anyway, I found a paper plate, a coffee can, and the lid to the plastic container that we keep sugar in. Now for measuring.
I had rulers to measure the diameter but what to do about measuring the circumference of the circle? I headed to my office to look around for something akin to string, but before I got there I realized I had a lanyard hanging around my neck. That'll work!
It was a great lesson on Pi and in teaching on a shoestring, or in this case, on a lanyard:)
Patricia, I have been there - having to find items that are circular when my learners forgot to bring their items to class for the lesson. It is amazing how if we just open our #matheyes we can begin to see that circles are everywhere. Thanks for sharing!
Happy Pi Day!
Brooke
I love that you have learners email you with pictures from their homes!! I know my learners often forget to bring their items to class, even though, I have reminded them 24 hours in advance (which is why we used a chap stick and a truck tire). I love those ah-ha moments! Thanks for sharing!
Happy Pi Day!
Brooke
Probably everyone has heard that 22/7 is a reasonably good approximation to pi. But how might that appear while students are measuring circles?
Well, suppose they measure a cycle wheel and find that the diameter D is 28 inches, and then they measure the circumference C and find that as close as they can tell it seems to be 88 inches. Then C/D = 88/28 = 22/7 = 3+1/7.
Here is another way it might occur without using any standard measuring units. Pull a non-stretchable piece of ribbon tightly around the outside of something round and cut it so its length is equal to the circumference C. Now take one end of this ribbon and stretch it across the center of the circular object and cut off a length D that is the diameter, so that the remainder has length C -- D. Now use that shorter length D to mark and cut off another piece of ribbon of length D, and then repeat this. You now have 3 pieces of length D and the remaining much shorter piece of length C -- 3*D. Now see how many times you can lay off this shorter piece, one after the other along one of the diameter pieces. You will see that you can do that 7 times (if you are very careful), with seemingly nothing left over -- or maybe a remaining bit less than 1/100 of D in length.
Thus 7*(C -- 3*D) is just a little less than D, that is, C -- 3*D is almost (1/7)*D. i.e. C/D is approximately 3+1/7.
Ladnor, Thank you for reminding us that 22/7 is another reasonably good approximation to pi. I know that learners get used to knowing that pi is about 3.14 because the majority of our materials reference it. BUT, I would be curious how many other instructors have curriculum that uses 22/7? I know our learners have an aversion to fractions but many this could be another layer that we can add in, hmmmm? Thoughts?
Brooke