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I need to begin with a few notes. My daughter attends a private school. Why does this matter? For one thing, the class size is small. This 1st grade class has 17 children, all of whom recognize and are comfortable around me.
Also, the school’s teaching style is not at all what I experienced in public school in Queens, NY (back in my day…). The teachers provide a unique learning program for each student matching the student’s individual learning style.
Putting those things together, we have a group of students who are not afraid to try new things, even with someone they’re not used to learning from. The following lesson plan happened to work in my daughter’s class, but I don’t know if it would work in a larger class, for example. Anyway, let’s get started.
I introduced myself as a mathematician. My daughter’s class is full of girls. Of the 17 students, only 5 are boys. So it was important to me to tell the girls that I am not just someone who likes math. I am Fr. Dr. Mimi. I spent more years in school after high school than before it. I spent 15 years studying mathematics! And now that I studied so much more math than most people in the world, it’s my responsibility to teach math to anybody who wants to learn about math from me.
And today, I want to talk to you all about math because today is a very special math day. Today is Pi Day! This letter on my T-shirt is a symbol that represents a very special number. You all spent a lot of time this year learning about counting numbers. This is a special number that isn’t a counting number.
This letter [latex]\pi[/latex] is actually a letter from the Greek alphabet. You all know what an alphabet is, right? A, B, C, D, … Some languages have different letters in their alphabet. In German there’s an ö and ü and ä. In Spanish there are letters that are different, like ñ. And in Greek there are also a bunch of letters like A.
What letter do you think “A” is? This is the letter alpha. This is the capital letter alpha. Here’s the lowercase letter alpha: [latex]\alpha[/latex]. Here’s capital and lowercase beta: B [latex]\beta[/latex]. And the next letter in the Greek alphabet is gamma: [latex]\Gamma \; \gamma[/latex]. And this is the letter pi: [latex]\pi[/latex].
The Greeks also have symbols for numbers. This is the symbol for one: I. And this is the symbol for two: II. Here’s the symbol for five: V. It actually doesn’t matter what symbol we use for numbers, as long as we all agree to use it. Numbers aren’t real. You can see or touch numbers the way you can see or touch a cat. Zero is a number that represents nothingness. And three is a number that represents how many things you have, but not the actual thing. Numbers are just ideas. Some numbers can be used for counting. But not all numbers are for counting. Pi is not a counting number. Pi is a very special number that has something to do with circles. And we’ll talk more about that later.
So far you might think that math is all about numbers. But that’s only because you just started learning math. When you start to read, you start with the alphabet, but eventually you start to form words using those letters. And then you combine words together using special rules to make sentences. And you combine sentences together to write stories.
Numbers are just like the alphabet. So 1 and 5 and 200 are all letters. Right now you’re learning about plus and minus. Well, 1+5=6 is a sentence in mathematics. And I’m going to show you today that you can tell stories in mathematics without using any numbers at all!
I am a very special kind of mathematician. I am a geometer. I study geometry. I don’t study numbers. I study shapes. So I’m going to talk to you all today about one of my favorite shapes. The circle.
Before we begin we have to start with a special word we use in mathematics: distance. Distance is used when talking about something far away. Distance is expressing exactly *how* far away that something far away is. Sometimes you might hear someone talk about short distances using a different word. Length. How long something is.
Now take a look at these two lines. Can someone tell me whether the length of these two lines are the same or different? What’s your guess?
A guess is also called a hypothesis. And that’s great place to start. Now I want you to show me that your guess is correct. How can you show me whether these two lines are the same or different?
I gave the kids a chance to play around drawing other lines on the board and drawing more lines with the giant triangle I used to draw the lines. Then someone suggested marking the giant triangle.
Great idea! You want to compare one line with the triangle and then compare the other line with the same triangle? I’ve actually brought some tools with me that you might be able to use to do exactly that.
I brought with me a bunch of rolls of colorful string. (Hemp string about 10+m each in 6 colors. Was on clearance at Target for $2.50. Cut them to be about 4m per student.) I used the string and measured out the length of one line, remembered it, and then brought it over to the other line to compare. I handed out a string to each student.
So these two lines are the same length! Is everybody here convinced? Does anybody still think that these two lines are different?
Next I pulled out some colored paper. (I bought 3 giant poster-sized paper, also at Target, at about $0.69 each. They were not soft like construction paper, but rather stiff. I cut them to be about 4″x18″.) I handed one out to each student.
Now do you think these two (pink) edges of this paper are the same or different. Show me!
Now tell me if these two edges are the same or different. Show me! And see if you can find a way to show me without using the string. (Solution: Just bend the paper over to put the two edges together.)
Now we’re going to do something much harder. I got the paper and got some tape and taped the two short edges of the paper together to form a cylinder. I showed the two circles that form on the boundary of the cylinder. And I asked are these two circles the same length or different? What does it mean to ask about the length of something that’s curved?
To measure the circumference of the circles on the cylinder, I showed them that we could just use more tape and go around the circle using the string again. Because the string bends just like the paper bends. Once we get back to where we started taping the string down, we could probably cut the string. Then reuse that string to compare the circumference of the other circle.
But actually, I told them, we already knew that the two circles are the same. How?? Just open up the cylinder again and we get back the original rectangle! We can prove that the circles are the same “length” by reusing an older proof about the length of two edges of a rectangle. We never changed how long the edges are, so the lengths should still be the same even if it’s curved!
At this point, I think it would have been a good idea to spend a bit of time letting the kids play with this idea of the “length” of a circle. I feel now that I didn’t give them enough time here. I think it would have been better to let kids come up with their own ideas. For example, a diameter would be a perfectly reasonable attempt to measure the length of a circle. It’s unfair to assume that a circumference is the only length of a circle. And it would have been a great way to segue back to the conversation about [latex]\pi[/latex].
But I got ahead of myself and way too excited about the next thing I wanted to show them: a Möbius strip.
Once the cylinder was back to rectangle again, I showed them that we can just introduce a half twist to obtain what is called a Möbius strip. (I had to do the actual construction myself for every student.) I then asked them whether the circles on the edges of the Möbius strip are the same or different.
A bunch of kids immediately said they must be the same for the same reason as before. I think they had had enough by this point and few had the patience or attention span to complete this final task. But about 2 or 3 kids were still intrigued. They tried the trick of using tape to go around the edge and realized there’s only one circle. This seemed to surprise them, but not amaze them, which is the reaction I was hoping for.
At the very end, I spent about two or three minutes demonstrating to them what [latex]\pi[/latex] is. I got a string and measured the diameter of a circle and cut it. Then I cut another string for the circumference of a circle. And then show them that I need to get a little bit more than 3 of the short strings to get the long string. I explained that it doesn’t matter how big or small the circle is. You’ll always need a little bit more than 3 of the short ones to get the long one. And that number keeps showing up so many times in nature that we decided to give that number a special symbol [latex]\pi[/latex].
I’m pretty sure nobody got the message about [latex]\pi[/latex]. But my goal to begin with was to show the students that math does not have to be about numbers. Hopefully that message got through.
This lesson ended up taking just over an hour. I planned only to spend about 40 mins because I felt that’s the absolute maximum that I would be able to keep their attention. I think that I lost a bunch of students way before 40 mins. About a third of the kids were wearing the cylinders on their heads at around 20 mins. But I was so encouraged by the handful of students who were coming back to me with proper proofs that I just kept going.
This was my first time teaching such a large group of such young kids. It was a very educational experience for me. I have ever more respect for my daughter’s teacher, Fr. Huhnke, who handles these kids expertly on a daily basis.
This lesson plan was just one and the first iteration of what one could try to do with a classroom full of first graders. I hope it will be useful to some of you. I welcome any feedback or suggestions you may have in the comments.