Mayan Numbers: Base 20 Number System vs. Base 10
The Maya number system is very different from the system you use daily—the Maya used only three symbols to represent all numbers! They used a dot to represent 1, a line to represent 5, and a shell to
represent 0. The Maya wrote their numbers vertically and used zero as a placeholder. Many believe that the Maya were the first people to use a symbol for zero. The Maya used a place value system based on 20s, not 10s like the number system we use today.
So the place values were multiples of 20s: 1s, 20s (20x1), 400s (20x20), 8,000s (20x400), and so on. Numbers 1 through 4 were written using a row of dots. The number 5 was written as a horizontal line. Numbers 6 through 19 were written using a combination of lines and dots, or 5s and 1s.
For example:
6 was written as one line with one dot above it: (5+1)
10 was written using two lines: (5+5)
19 was written as three stacked lines with a row of four dots on top of them: (5+5+5+1+1+1+1)
For numbers greater than 19, the symbols were arranged vertically in place values, with the greatest value on top. Each place value was 20 times greater than the one that came before it. Look at the examples below.
For example:
- 303 was written as three lines in the 20s place, and
three dots in the 1s place: ((5+5+5) x 20) + (1+1+1)
- 420 was written with one dot in the 400s place, one
dot in the 20s place, and one shell in the 1s place:
(1x400) + (1x20) + (0x1)
- 4,008 was written as two lines in the 400s place, one
shell in the 20s place, and three dots above a line in
the 1s place: (10x400) + (0x20) + (8x1)
The first two pdfs entitled Mayan Numbers 100s Chart gives the Mayan numbers from 0-100. The first challenge for all students was for them to fill in a hundreds chart with missing numbers in Mayan Base 20. For grades 2-5, there were two pages of challenges using four levels or place values (1s, 20s, 400s, and 8000s). Finally, there were make your own worksheets allowing the children to convert any Mayan Base 20 Number to Base 10.
The pdf entitled Mayan Chichen Itza History-Calendar.pdf is a brief history of the Mayan people, pictures of their major city, their number system, and their two calendars.
This was the challenge the first week in January and week two for Monday and Thursday Mathletes.
The following lesson was 2018 week two for Tuesday, Wednesday and Friday classes. The Monday and Thursday Mathlete classes may want to try these challenges as well.
Week 2 of 2018, Mathletes learned how to covert Base 10 numbers Mayan Base 20 numbers. This process is much more challenging involving the following process:
If the number is 19 or less, it only requires the bottom level of 1s.
If the number is 399 or less, it requires the bottom level of 1s and the second level of 20s.
If the number is 7,999 or less, it requires the bottom level of 1s, the second level of 20s, and the third level of 400s.
If the number is 160,000 or less, it requires the bottom level of 1s, the second level of 20s, the third level of 400s, and the fourth level of 8,000s.
Start with the top level required and count off up to 19 of those values. For example, if your number is 824, count one 400 and a second 400 for a total of 800 (so two dots in the 400 level), subtract 800 from 824 and look at the difference, 24. You will need one 20 and 4 ones (so one dot on the 20s level and 4 dots at the 1s level).
Using repeated addition or multiplication is required to find the maximum number at a certain level, then use subtraction to find out what is left for the next level.
I chose numbers that have specific significance, and on the answer key, I identified that significance.
K-1st Grade:
5,10,15,20,25,30,35,40 are the first 8 multiples of 5.
2,4,8,16,32,64,128,256 are the first 8 powers of 2.
100,200,300,400,500,600,700,800 are the first 8 multiples of 100.
2-3rd Grade:
1,4,9,16,25,36,49,64,81,100,121,144,169,196,225, and 256 are the first 16 square numbers.
100,200,300,400,500,600,700,800 are the first 8 multiples of 100.
45, 90,180, 360, 720, 1080, and 1440 are angle measurements of a circle (1/8 rotation, 1/4 rotation, 1/2 rotation, 1 rotation, 2 rotations, 3 rotations, and 4 rotations.
365 days in a year, 5280 feet in a mile, 314 for pi 3.14, 1618 for the Golden Ratio 1.618, 666 is the number of the beast, 911 emergency call, 555 phone number used in movies, 888 luckiest number in many cultures.
4-5th Grade:
In addition to all of the numbers above, the super challenge page included:
8675309 Jenny was a 1982 song my Tommy Tutone that caused hundreds of people to give up their telephone numbers since thousands of people called the number asking for Jenny.
92,960,000 is the average number of miles from the Earth to the Sun.
238,900 is the average number of miles from the Earth to the Moon.
7,917 is the average diameter of Earth.
24,901 is the average circumference of the Earth.
1,728 is the number of cubic inches in one cubic foot: 12 cubed, 144 x 12 or a super gross.
27,878,400 is the number of square feet in one square mile: 5,280 x 5,280.
1,000,000,000 or one billion is the average number of atoms that if lined up side by side would span one linear centimeter.
For some of the super challenge numbers above, Mathletes will have to go beyond the fourth level at 8000s. The next level is 160,000s, then 3,200,000s, then 64,000,000s.
| Attachment | Size |
|---|---|
| Mayan_Chichen_Itza_HistoryCalendar.pdf | 1.4 MB |
| Mayan_Numbers_100_chart_K-1.pdf | 666.22 KB |
| Mayan_Numbers_100chart_4_levels.pdf | 1.55 MB |
| Base_10_Convert_to_Mayan_Base_20_K-1.pdf | 880.82 KB |
| Base_10_convert_to_Mayan_Base_20_Grades_2-3.pdf | 1.46 MB |
| Base_10_Convert_to_Mayan_Base_20_Grades_4-5.pdf | 1.57 MB |
