Informal Derivation of Pi

7.GM.5 Understand the formulas for area and circumference of a circle and use them to solve real-world and other mathematical problems; give an informal derivation of the relationship between circumference and area of a circle.

I gave the students hula hoops and some coasters, and next year I might save some container lids or something to create a few more things for them to measure and work with. Then i gave them strings and measuring tapes (there were measuring tapes in some of the manipulative kits I found at the end of last year!) and had them figure out the relationship between the circumference and diameter. They all came pretty close to pi, which was cool.

I need to review the instruction sheets I gave them to make sure the language is as clear as possible, or else create an instruction video.

I liked this activity and the students seemed to as well, but it’s frequently really challenging for my students to reverse the direction of the problem. They have the idea that circumference/diameter = pi, and some of them even got that pi is like a unit rate, or a very specific variable, but they can’t seem to remember the process that circumference = diameter * pi. I need to think of more ways to extend this activity to help with that.


Tacit Knowledge and the I-We-You Trap

Long story short: I became a mathematics teacher like my forefathers before me.


Basically, I went and took the Middle School Mathematics certification test. And because this is how my life works, that’s what I got hired to teach. But I have an admission to make – I like teaching math, but I really struggle with it.

Most of my mathematics knowledge is tacit knowledge. I don’t think about problem solving, particularly not at the pre-algebra level that my students are learning at. I just solve the problem. My parents described the situation as having a lot of power, but not much finesse. So my challenge is turning that tacit knowledge into something that I can share with the kids.

There’s also a huge problem in the culture of education right now, which you probably know about if you are reading this. Your measure of success, career stability, and financial value is a test score. The test is written and graded by a gigantic corporate entity that apparently has no experience whatsoever with children in general and seventh graders in specific.

There’s a standard approach to mathematics instructions that has been used in the classroom for over a century -I’ve read about it being called I-We-You, or sometimes it’s called Gradual Release of Responsibility. Basically the teacher explains the concept, then we solve some problems as a class together, and then the student is expected to solve the problems. Despite it being the prevalent practice, it’s not actually the best practice. Kids and adults hate and despise math and are missing out on the best part of it because they are so caught up in algorithms and computation and I-We-You.

The practice that’s considered most effective is You-You All – We. In this sequence, a student takes a problem and tries to solve it. Then they work with a small group to discuss their solutions, and then the teacher leads a whole group discussion of solutions. If there are textbooks written this way, I don’t have one. (Actually I have no textbook whatsoever).

The two problems with this approach are as follows: 1. This approach takes time, which, when I consider the number of standards I am supposed to teach them and that they may be  tested on, I am not sure I have. 2. If kids are used to I-We-You they seem to be incredibly resistant to trying any other approach. Frequently attempts at doing so end in tears, even for students who do well in math. Sometimes particularly for students who do well in math.

Or maybe I’m just doing it wrong. It’s not like there’s anybody who can tell me if I am; ¬†it seems like these are pretty uncharted waters.




This is a game I learned years ago at Girl Scout camp (I was not cool enough for band camp). I dug it out last spring to entertain some middle-school girls, who thought it was hilarious.

Players stand in a circle with their hands held palms up on either side of them. Depending on which way you want to clap, one hand should be over the hand of the player next to you and one hand should be underneath. So everybody is right-hand over and left-hand under, or the reverse. As you chant the song, players clap their top hand onto the top hand of the person next to them, to the beat.

Down by the banks of the hanky panky,
There’s a bullfrog yelling hanky panky,
You go fee fi fo fum,
Move to the rhythm of the bullfrog.

The clap moves around the circle and the player who’s hand is clapped on the word “frog” is out. Play continues with the clap started by the next player.

When there are only two players remaining, players use both hands, clapping down on their opponent, and then turning their hands over to be clapped in turn.

It is usually acceptable for the person who is about to be clapped on “frog” to pull their hand out of the way so it cannot be slapped. 

I think there are different versions of the chant. If you remember one, post it!