Yellow Band: by Sea, Week 5

Welcome back everyone! A short post this afternoon, just thinking about the great math work we did this week, and thought I’d share a bit.

A few weeks ago, while we were working on the constellation project, we started studying the Babylonian/Sumerian number and counting system. The ancient Babylonians did not use base 10–the system that we use, and is also the foundation of the metric measurement system. Rather, they counted in base 60. But, hit the brakes. We didn’t start there. We started by looking closely at how we use our hands to count. And we started to have some really neat discoveries!

Emilio records in his journal how he would count up to a few different numbers, some big, some small.

When we got everyone’s hands drawn on the board, we noticed some really neat things. Some people started counting on their thumbs, some on their pointers. Some held their hands palm up, others palm down. And, to count to big numbers, like 47, some people counted up by 5 and others by 10.

From there, we learned how the Babylonians counted on their hands. They did a neat thing: they used their right and left hands differently. On their right hand, they used their thumb as a pointer, and counted out each joint on each finger. This allowed them to count up to 12 on one hand. On their left hand, they kept track of the dozens that they counted on their right. In this way, they could count up to 60 on just their two hands. These first few explorations really focused on our hands, our most concrete way to count, most literal connection to the abstract concept of number.

Samira shows how she is practicing counting on her hands like a Babylonian!

Sakira helps Emilio record in his journal how to count up to a few different numbers like a Babylonian.

Then, we started to talk about base 10 and base 60. We watched a couple of videos about mathematical archaeology, which pointed out to us a few fundamentals of our number system, and contrasted them to the Babylonian base 60 system. Both systems work from left to right, and as we move up an order of magnitude, we add a numeral to the left. In our system, we use a 0-9 pattern, and when we get to 9 in a place, we add one to the place value to the left in order move from 9 to 0. Each place value represents the numeral in that place multiplied by a power of 10. For example, for first place is 10^0, or 1. So, a numeral in the ‘ones place’ is equal to that numeral x 10^0. When you want to move from 9 up to the next order of magnitude, you add one to the ‘tens place,’ or 1 x 10^1 = 10, and the 9 in the ones place turns back to a 0. Well, the Babylonians basically did the same thing, except replace all the 10s with 60s. Or, as Emilio so helpfully put it for us, in base 10 your ‘silent alarm’ goes off at 10, but if you’re Babylonian, your ‘silent alarm’ goes off at 60.

All of this with 7 and 8 year olds! And they really stepped up to the plate! We learned the symbols the Babylonians used (really just 2 different symbols), then started working each morning to practice writing numbers in base 60, which got really interesting when we wanted to write big numbers.

Reyahn works on adding symbols to our ‘glossary’ of Babylonian numbers, to help folks work on translating some different numbers.

Reyahn shows how to carefully organize your symbols. The Babylonians didn’t string out their symbols, the never wrote more than three in a row. Instead, they started to stack the symbols.

Solin shares her strategy for writing a particular number in Babylonian. Solin organizes herself into the first two place values in base 60:  x 60     x 1 . This helped her see that, though she wrote the same symbol twice, one is worth 60, and other 1, so the whole number she wrote is 61.

Sakira shows how she figured out how to write 104 in Babylonian. All these numbers with only 2 symbols! You can see in this picture how Solin organizes herself into the first two place values in base 60:  x 60     x 1 . This helped her see that, though she wrote the same symbol twice, one is worth 60, and other 1, so the whole number she wrote is 61.

Once we were comfortable with the system and the symbols, we could really hit the gas. We worked mostly in just the first to places, which you can see in the picture above–which will take you all the way up to 3,599! This exploration has been so rich with number sense and operations–addition, multiplication and division–and our understanding of our own base 10 system has really gelled. By taking this step outside of our comfort zone, and essentially learning a different counting language, we noticed some really important things about our own number system that will inform the way we work with numbers forever. Woah!