Pumpkin Numbers and Math Circle Names (Eye of Horus 6)
OCTOBER 22, 2013
This week, we finalized our Math Circle names, explored the number theory behind some interesting numbers, did some mathematical thinking, discussed some important ideas, and then presented our own Pumpkin Numbers.
NUMBER THEORY and MATHEMATICAL THINKING
“Can you explain Narcissistic Numbers?” asked R, S, K, and E at the beginning of class.
“I know what they are,” said V. “They’re numbers that equal each other.” No one was sure exactly how they “equal each other,” so I passed out and explained a handout showing how they work.* This may have been the first time I have ever given an explanatory handout in math circle. It was helpful, though, since comprehension of this concept had been so ephemeral. Since this was the final session of our circle, the kids really wanted to take the concept home.
The last line of the handout said “Why isn’t 351 a Narcissistic Number?”
“That was my question last week,” said E excitedly. I apologized for using his work without giving credit, and promised to insert the proper credit before using the handout again. The kids were amazed to hear that I really needed to do this.
Some of the kids were less interested in the handout and were playing with a hoberman sphere that I had set out. I directed everyone’s attention to this object. I explained that as the Keeper of the Squares, I had been endowed with the superpower of expanding the square without actually moving my body. I then opened it as slowly as possible, claiming that my hands were not moving.** I intended this activity as a straightforward mindfulness activity – to focus everyone’s attention toward a common goal. But some interesting math came out of it. The kids debated whether the sphere was made out of straight lines or curves.
We left that question open as we turned our attention to an Eye of Horus drawing on the board. We compared that to R’s own very impressive drawings of falcon eyes, then formed some conjectures about what the numbers on the Horus Eye (1/2, ¼, 1/8…) mean.
I gave each child a square piece of paper. “Fold your paper in half,” I instructed.
“Which type of half?” asked V, as he folded his diagonally. As usual in math, a clarifying question was needed for understanding. I meant the other type – perpendicular folds. We worked together to repeatedly fold each piece in half, and label one side with its numerical value (1/2, ¼, 1/8, etc.). Before each fold, the kids predicted what the fraction would be. V, E, and others predicted that the third fraction would be 1/6, and were surprised to unfold their papers and see that the third fold produced 1/8. At this point we started listing the numbers on the board too. Soon everyone realized that we were generating the same numbers that were on the Eye of Horus. D predicted even smaller numbers in the sequence (i.e. 1/128), leading the group to wonder if the Eye was missing some parts. We unfolded the paper, labeled up to 1/64 and wondered as a group about the missing section where the smaller numbers would be. If we had more weeks in this math circle, we would follow this thread into the infinite nature of harmonic series of fractions. Yes, most of these kids are eight or nine, but one of the great beauties of mathematics is that ideas can be expressed and made accessible in so many different ways
I told them some info I had read about the Eye of Horus being used for measurement. We wondered how that could be if the fractions didn’t total up to a whole. (Not everyone quite understood the math of this, but the folded paper helped.) Then I read a quote from a website that at one point says that the numbers do add up to one. “Well,” said D, in a sing-song voice, “you can’t believe everything you read on the internet!”
I went on the say that until recently (within the lifetime of some of our participants), many scholars believed that these “Horus Eye Numbers” were actually a form of numeral writing. I showed a copy of the Rhind Mathematical Papyrus, the study of which gave rise to this theory. Then explained, to the kids’ enjoyment, that Jim Ritter debunked that theory in 2003 with his paper “Closing the Eye of Horus: the Rise and Fall of Horus Eye Fractions.”***
PUMPKIN NUMBERS and MATH CIRCLE NAMES
Since today was 2 days before Halloween, I asked the kids to invent Pumpkin Numbers. This was an individual, versus collaborative, task – extremely unusual for this math circle. I gave each student a small orange paper pumpkin, a pencil, and a few minutes. Then I asked each person to present. We worked on some new skills that a mathematician must have: the ability to not only explain your idea in words, but to do so while facing an audience and holding your diagram toward your audience instead of yourself. What follows is a list of Pumpkin Numbers, paraphrases of students’ explanations, and the kids’ Math Circle names – the culminating achievement of our last six weeks.
S (Master of Spinning): 377. It’s round (3), and the 7s look like they would be on top, where the stem is.
D (math circle name still TBD – my suggestion of Questioner of Terms still stands): 38. It really looks like a pumpkin with its curves and bumps.
R (The Keeper of Hawks): “a way to organize numbers.” One number goes in the stem (i.e. 2), and then each of its multiples (i.e. 2×1, 2×2, 2×3,…) gets listed in a section between the vertical creases from left to right. I asked her whether this was a Pumpkin Function instead of a Pumpkin Number, but she, the inventor and therefore owner, declared that no, it’s an organizational system as opposed to a function.
Y (The Prince of Lightning): 0150. Actually 0151 is just an example of a Pumpkin Number. K’s Number is a rule whereby the first 2 digits had to be 0 then 1. Kids quizzed him on other examples, and one student pointed out some numbers that could be both Bodyguard Numbers (K’s earlier invention, as you may recall) and Pumpkin Numbers. I asked whether these numbers had a useful purpose. “They’re just for fun,” answered K. (I gave homage again here to Vi Hart and the field of Recreational Mathematics.)
E (The Keeper of Science): ½. Again, just an example. For E, a Pumpkin Number is a fraction with a maximum number of 3 digits in the numerator and 3 digits in the denominator, as “pumpkins are only so big.” I asked a lot of clarifying questions of E based upon the examples he gave. This discussion led to my talking about the theory that Egyptians of that time period only used unit fractions.
O (King of Earth): 1,111. You couldn’t tell by looking that the number drawn onto O’s paper was 1,111. In a merging of math and art, the number was distorted, with displaced crooked digits. “It reminds me of a rotten pumpkin. The digits are all rotted into each other.”
Z (The Caretaker of All Animals; The Remember): a pumpkin that “carves itself by subtracting.” You start with 500. Subtract any number from it. Then you can carve a shape, any shape, into your pumpkin. Then subtract another number from your prior result. This allows you to carve another shape. And so on. No numbers need be shown on the pumpkin. The math actions prompts a physical action.
M (The Secrets of the Mythical Creatures; Athena): any number with the first digit 1 and the final digit 0, a minimum of 3 digits. Zero, after all, looks like a ghost, and one looks like the sides of a triangle, like a jack-o-lantern eye.
L (Queen of the Black World; The Namer): 6 7 12. L’s Pumpkin Number is a list of three numbers that are “numbers supposed to be lucky.” To add in a design element, the edge of her pumpkin was a mirror image of the standard clock. This “number” prompted a group discussion on the differences in our thought processes; some Pumpkin Numbers were unique, while others were rules that could generate multiple numbers.
J (The Keeper of Awesomeness): “The Lucky Jack-O-Lantern.” The Lucky Jack-O-Lantern is “a way to explain the sides of the pumpkin. The first digit is the number of stems, the second the number of bumps on top, the third the number of bumps on the side, and the fourth and final digit the number of bumps on the bottom. 1,223, for instance, describes a pumpkin with only one stem. What really floored me about this particular Pumpkin Number is that it was almost exactly the same as the one I had created in my own mind, and J, as many of you know, is very closely related to me. In fact, as I thought of the Pumpkin Number idea earlier in the day, I think I assumed that everyone’s would somehow use numbers to count a jack-o-lantern’s physical features. Yet this is the only one that did that. Is there some kind of similarity in mathematical thinking among family members? If so, is it nature or is it nurture? What do you think? For now though, to quote Iris DeMent, I’ll just “let the mystery be.”
V (King of Sports): 0, 8, 10, 6, etc. V’s Pumpkin Numbers must contain a digit that physically resembles the round shape of a pumpkin. That digit must have a closed circle, so 3 won’t do. The number must have in it a 0, 6, 8, or 9. A good example of an acceptable Pumpkin Number under this rule is 10.
While I may have remained cool and detached in my classroom and writing affect, I’ve been jumping for joy inside at the amount of math coupled with the amount of joy that came out of these past six weeks. Those of you who’ve been reading these reports all along know that I’ve struggled with helping kids really own the mathematics. In fact, the topic of this course in general may have been too arithmetic-heavy for this age group and/or range. The good news is that these kids struggled both emotionally and intellectually, questioned, and persevered to culminate with their own mathematical inventions. They are recreational mathematicians. I just went back and italicized many of the math ideas that came up just in the Pumpkin Numbers activity, and counted at least twenty.
It’s also inspiring to notice that the Pumpkin Numbers fell into very distinct categories:
- those defined by digits physically resembling the shape of a pumpkin
- those defined by functionality
- those with only one numerical value
- those with multiple possibilities for value
- those representing a system
- those representing a number or a rule
I owe special thanks this week to R’s mom M for her excellent photography. (So many photos are posted on the Talking Stick website.) Thanks to all of you for sending your children to become part of this collaborative mathematical inquiry. I hope to see them again in Math Circle.
*Email me if you’d like a copy of this handout. Not every child took one home.
**I learned this activity from Wynne Kinder at the 2011 Mindfulness in Education Network Conference.
***See footnote 16 in the Wikipedia article on the Eye of Horus