The math of a circular sukkah

When I first studied Sukkot 8a in my studies at Midreshet Lindenbaum, I was warned it was a tough Gemara to crack. Without Artscroll to illuminate it, I struggled through it the first time. The mishna in Sukkah has lots of geometry, but most of it deals with Sukkot that are rectangular prisms. This Gemara asks, what if your Sukkah is circular? What will be its minimum size?

Many years later, I wound up with a new curriculum in my 6th grade class, who needed substantial enrichment. This curriculum happened to teach circumference in the third week of school - usually right before Sukkot. I went back to the Gemara in Sukkot 8a and asked, "Can I take this apart so that talented 6th graders can grasp its intricacies?"

The short answer is "somewhat." I designed an investigation that focuses on circumference, pi and unit conversions, while only delicately touching on the aspects of the Gemara that require an understanding of the Pythagorean Theorem. My worksheet begins where the Gemara does - with Rabbi Yochanan's assertion that a circular Sukkah needs to have enough space in its circumference to seat 24 people. (You can find a blank worksheet here and a partially completed student copy here.)

This lesson is really a two-day lesson, unless you have at least an hour for math. The first day is really active. I never have 24 students in my class, Baruch Hashem, so I usually either have the students make a circle of 24 chairs, or have 12 students sit in a circle and tell them to double all the numbers. The goal here is to get the kids to realize just how big a 24-person minimum is. Then we measure the outer circumference and the inner circumference of the 12 students seated in a circle. (You need a bunch of soft measuring tapes - we got ours as promotional items from a health care provider once, but you can also ask colleagues or students to bring them in.) You're better off doing this with enough measuring tapes to go around the circle - about 4-6 depending on their length.

After the students record the inner and outer circumference and then scale it up for 24 people, (with a small class you can use eight students and triple it,) we then continue in the Gemara, which explains that R. Yochanan is following Rebbi, and requires a Sukkah to be 4x4 amot. Then it introduces the idea that one person takes up 1 amah of space, and that for every 3 units in the circumference, there is 1 unit in the diameter. This is essentially approximating pi as 3. (If you're eager for fractions and decimals, don't worry - they're coming in just a few lines.) I don't recommend measuring a student to see if they take up 1 amah of space - that seems too likely to lead to bullying based on size. Instead, I always talk to the kids about how, as kids, they are smaller than adults, so we can expect our estimate to be lower than it would be if we were using adults, as Rabbi Yochanan was.

This activity is rife with opportunities for error, which is a great topic of conversation with the kids. (You can record them on the worksheet where it says "problem #2d.") For the inner circumference, even if you have each student hold the tape measure to his or her chest, you will find you have more of a polygon than a circle. (Around the outside there is less of an issue.) As I mentioned previously, middle schoolers are smaller than adults. What are your other sources of measurement error? Are the ends of the measuring tapes matching up properly? Is your measuring tape slanted, making it appear to be longer than it is? You cannot eliminate these sources of error, so be certain to point them out. It is important that students realize measurement is not simple or tidy.

Before we try to figure out the connection between a circle with a circumference of 24 amot and a square that is 4x4 amot, we convert the measurements to inches so the students can grasp them. I have the students use both the standard that an amah is 18 inches and the standard that an amah is 24 inches to convert. That way, we can compare our personal circle to the Rabbi Yochanan's. In most cases, your number will be on the low side. (Last year, for example, we got 430.4 inches, which is just below the range of 432-576 inches. After we have the conversation about measurement error, we then move on to the square and the circle.

To truly understand how this works, we need the Pythagorean Theorem. If we inscribe a square in a circle (which is what we are doing when we say the circular sukkah needs to enclose the minimum square sukkah), the diagonal of the square is also the diameter of the circle. To find the length of the diagonal of the square, we imagine that it is really two isosceles right triangles. Then we can figure out that the minimum side length of the square is 4 amot, so the diagonal is 4*root2. In fact, the Gemara says this, using 1 and 2/5 as its approximation of root 2. (This is actually quite close.) Then it "scales up" to figure out that 4*1.4 is 5.6 amot.  Essentially, we have concluded that the diagonal of the minimum sukkah is 5.6, so the diameter of the circle should be at least 5.6 amot. Before I return to the obvious Gemara issues, I note here that this part of the Gemara, the most difficult, is really well suited for older students - 9th or 10th graders who have studied Geometry could probably work independently through most of this, given a sheet written at their level. I've purposely over-explained some things to my 6th graders, but with a little less scaffolding, this activity could be beneficial to older students.

Now for the obvious Gemara issues. We just concluded that the minimum diameter should be 5.6 amot. Rabbi Yochanan wants to put a square with a diagonal of this length inside a circle which we concluded above actually has a minimum diameter of 8 amot (using the Gemara's pi.) This is a large discrepancy - an extra 2.4 amot bigger than the minimum diameter needed to encircle the minimum square. Rav Assi saves the day by clarifiying that 24 amot is around the outside, not counting the people inside the sukkah. Since the average person is 1 amah deep, he says (I leave this assertion unverified), we are dealing with a diameter of 6 amot - we cut off 1 amah on each end for the people who are standing there.

Rav Assi says we don't count the space that person A and person B on each end take up, leaving just a 6 amah diameter.

Now, if d is 6 amot, then c is 18 amot using the Gemara's calculations. The Gemara, familiar with fractions, notes that 5.6 * 3 = 16.8 and that even 18 amot is rounded up. They conclude that a lack of precision led Rabbi Yochanan to be stringent. I love this conclusion because at this age, I am often urging my students to be more precise. "Just about" and "almost" and "sort of" are the enemies of precision. (You'll note that this is a very different lesson than the one we did on pomegranates - math can have dialectics, too.) We said at the beginning of the lesson that is is incredibly difficult to be precise when measuring, so it's great to come back later to the importance of precision - at least when calculating. Taking pride in attention to detail is a value that is central to both Torah and Madda.

How many seeds in a pomegranate?

When I told a fellow Judaic studies teacher I was planning to have my students count the number of seeds in a pomegranate, he said something like, "Don't do that! You'll ruin everyone's childhood!"

If you've been to Jewish kindergarten, you know the number 613. If you look at the pomegranates below, surely you doubt that each contains 613 seeds. I know I do.

This colleague's statement was tremendously helpful because it helped me ask myself a few key questions:

1) Why does Judaism like to assign numbers to rituals and historical events? (Apropos of Yom Kippur - the 100 Shofar blasts and Sisera's mother, for example.)

2) What are we supposed to do with these numbers? How literally are they to be taken? In the Sisera's mother example, I could make a compelling argument that if someone is weeping and mourning, it would be incredibly difficult if not downright impossible to separate and count all her cries.

3) What underlying messages about numbers and about intellectual values are we supposed to glean from the way the Torah deals with numbers?

Let's turn to our main example, the pomegranate. The short answer to the three above questions are as follows:

1) The pomegranate's seeds are said to number 613 so they can symbolize the Torah. (We take our counting of 613 mitzvot VERY seriously. That number is by no means an estimate.) We associate the pomegranate with the number 613 to show on Rosh Hashana a bridge between the physical and the spiritual world. The Torah, an intellectual concept, is reflected in the physical world, which God created and is a source of great religious potential. (Essentially, Torah and Madda are forever intertwined - and that's a good thing.)

2) I don't imagine anyone expects that every pomegranate has exactly 613 seeds. I confess, I was not sure when I embarked on this project for the first time whether the number was anywhere close. My first estimate of the number of seeds was between 100 and 200.

3) This question is by far the most central to our lesson. The fact that the number 613 is so important in Judaism is worth a look. 613 is not a likely choice. If 100 is a paradigmatic round number, 613 is its exact opposite (although not in mathematical terms!) being prime, and a special prime at that.

I'm going to argue to my students that the example of the pomegranate teaches us about the importance of precision, estimation, and attention to detail, among other things. The first two might seem to be mutually exclusive, but they are both essential components of mathematical study, and I'm going to to try to convince you (and my students) that they are important to Judaism, too.

We'll be doing this activity today, on the first day of school, in addition to the "meet the teacher" basics. Here's our lesson plan, based on my trial run at home with an almost 5-year-old:

1) Buy a pomegranate and a plastic tablecloth. Make sure you have 1 good knife on hand but keep it away from the kids.

2) Introduce the topic, and ask every kid to record an estimate of the number of seeds in the pomegranate. Slips of paper work for this, or if you use clickers or other mobile devices in your classroom, this is an ideal way to collect the data quickly. For more information about using estimation in your classroom, I recommend Andrew Stadel's Estimation180 website. What you can't allow the kids to do is make a wishy-washy estimate like I did above. 100-200 is really not one estimate - it's at least two. Tell the kids to pick one and stick with it. I decided not to share my estimate with my students - I didn't want them thinking I was pushing them in a particular direction.

3) A good estimate has multiple parts. Mr. Stadel likes to ask students, "What's too low?," "What's too high?," "What is your estimate?" and "What is your reasoning?" This all takes time, but is worth a quick discussion. If you're not using clickers, a think-pair-share is also a good venue for every kid to get a chance to speak.

4) Cut up the pomegranate. scoring the sides and breaking them apart by hand so you don't cut the seeds and get an even larger mess. With gentle hands, you won't have too much pomegranate juice to deal with. Lay out its seeds in orderly rows. Make sure the kids wash their hands first, and try not to let them drop seeds all over the floor. I'll talk about measurement error more in my Sukkot post (coming on 9/24), but in the meanwhile, we'll attempt to be precise by:
a) not dropping seeds
and
b) putting the seeds in clear rows of 10.
Counting to 200 or more would be difficult to do precisely, so groups of 10 allow us to just count rows.
Since this is the first day of school, some students will be setting up their binders, so the seed-counting will be done by those who finish quickly. The more kids, the faster it will happen.
I did the whole activity in 33 minutes, essentially alone because the 5-year-old opted out. For this to work in my classroom, it needs to take less than 20 minutes, so the more students helping, the faster it will go. I hope.

As a technical note, popping the seeds out is easy and fun, but putting them in rows is tricky on a plastic surface. Today's 6th grade session will hopefully be on a tablecloth that is paper on top and plastic on the bottom, to make the seeds easier to work with. You'll also note that the seeds don't photograph well on a shiny background. (If you're reading this post more than 36 hours after I post it, I'll probably have replaced the photos here with ones that don't look terrible.)

In conclusion, we note our final number and maybe even graph it on a number line or a histogram with everyone's estimates. Then we make the following points:

i) Judaism cares about numbers, quantities and math. Studying math can be a spiritual pursuit because God created, and generally works through, the natural world.

ii) Precision matters. The rabbis care very much that there are 613 laws. They will double up laws and leave some out to arrive at 613. In our math studies, we apply this by taking care in our work and learning how to precisely answer challenging questions.

iii) Estimation is a useful skill.

The first pomegranate contained 592 seeds. My largest estimate was 3 times too small. Wowza! Estimation is a skill and if you don't practice it, you won't develop it. I always tell my students - especially the strong ones - you should know before calculating whether your answer will be closer to 40 or 400. In this case, I flunked that test. (I could have gone back and edited my original estimate but honestly, being that far off really drives home my point.)

Without multiple trials, I can't calculate variance, standard deviation or perform any other statistical analysis to predict whether other pomegranates will be the same. I encourage you to share your pomegranate counts with me using the form linked below if you do this activity so that I can build a data set over time.

I can tell you that 592 seeds in the pomegranate means that it was missing less than 4% from the traditional estimate. That's incredibly close. Not only is my childhood not ruined, I am impressed yet again with the mathematical insight of our Sages - a theme which will feature prominently on this blog.

Shana tova! Add your pomegranate data here.

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Update: My 6th graders did the activity in about 20 minutes. I scored the pomegranate and divided into sections and they did the rest. They counted 610 seeds. It was a nice way to start the year.
You'll note that they grouped their tens randomly instead of in grids. Most heartening was the way almost everyone was able to participate.