S h a n n o n R. O v e r b a y
a n d M a r y J e an B r o d
Magic with
Mayan Math
T
he ancient Mayan system of numera-
tion is quite sophisticated, yet the basics are
beautifully simple. It is an appealing system
to teach to students of almost any age. Mayan numerals are composed of only three symbols,
but with the power of place value and a symbol for
zero, large numbers can be concisely written. With
their interest in astronomy and the passage of time,
the Maya (located in what is now Central America
and Southern Mexico) developed accurate calendars
using this numeration system. Mayan merchants often used cacao beans in making computations to keep
track of their business transactions (Closs 1986).
The Mayan numerical symbols are a dot • for 1, a
bar
for 5, and a shell
for 0.
The numerals • (1) through
(19) are writ-
ten in the first place-value position. Since the Maya
used a base-twenty system, the number 20
is written with a dot in the vertical second place-value position and a shell (zero) as a placeholder in the
first place-value position. Thus,
represents 7, whereas
Shannon Overbay, [email protected], is an assistant
professor of mathematics at Gonzaga University, Spokane,
WA 99258. She also teaches a mathematics enrichment
class for seventh and eighth graders at St. Aloysius School.
Mary Jean Brod, [email protected], is retired from
the department of mathematical sciences at the University
of Montana, Missoula, MT 59812. Her experience includes
teaching mathematics in summer camps for Native American middle school children.
340
MATHEM ATICS TEACHIN G IN THE M IDDL E SCHOOL
Copyright © 2007 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved.
This material may not be copied or distributed electronically or in any other format without written permission from NCTM.
PHOTOGRAPH BY SHANNON OVERBAY; ALL RIGHTS RESERVED
represents 45 (2 × 20 + 5 × 1) in this vigesimal, basetwenty, system (Morley 1983).
Activities
the Worksheets titLed ACTIVITY 1 throuGh
Activity 4 merge two fascinating mathematical topics:
the Mayan system of numeration and magic squares.
students tend to love solving puzzles and are easily motivated by these activities. in particular, magic
squares have been enjoyed by a variety of cultures
throughout the ages (anderson 2001). While using
their problem-solving skills, students are introduced
to the numeration system of the ancient culture of the
Maya. these activities have been very popular with
native american middle school students in sMiLe
(science and Mathematics interactive Learning experience) summer camps. the exercises work well
when done individually or in a group setting.
Activity 1: Mayan Numerals
Activity 1 acquaints students with the basic notation of Mayan numerals. students can work in small
groups to help each other become familiar with the
numeration system. one way to start is by counting using Mayan numerals. (Fig. 1 is a student example of counting up to 50.) this can be completed
with pencil and paper or manipulatives. shells (real
shells or uncooked macaroni) can be used for the
0 placeholder; flat toothpicks or craft sticks can be
used for 5; anything small and round (such as candy
or pennies) can be used for 1. first, students should
Fig. 1 Student example of counting to 50 in Mayan. [Errors were made in the
symbols for 38 and 49.—Ed.]
voL. 12, no. 6 . februar y 2007
341
learn to exchange 5 dots for a bar and to count up
through 19. Then students can be introduced to the
use of the Mayan symbol for 0 when writing 20. This
reinforces the concept of place value and the importance of a symbol for 0 used in our conventional
Hindu-Arabic decimal system (NCTM 2000).
After learning to count with Mayan numerals,
students practice converting Mayan numerals to
standard decimal notation. If work is done in groups,
each member can take turns writing a Mayan numeral while the rest of the group converts it to standard notation. Next, students translate numbers
from standard notation to Mayan numerals. At this
stage, it is recommended that the numbers used are
less than 360, since the third place-value position
is a 360 (rather than 400) in Mayan solar calendar
mathematics. Working with larger numbers can be
done later as an extension exercise.
Activity 2: Working with Mayan Numerals
Activity 2 focuses on the operation of addition using
Mayan numerals, which prepares students for working
with magic squares. A possible supplemental activity
would be to create addition tables with Mayan numerals. (See fig. 2.) Students may need to be reminded
that no more than 4 dots or 3 bars are to be used in a
level. A possible group activity would be to write down
the ages of all members of a group and then find the
sum of the ages using Mayan numerals. Students can
also make up their own addition problems.
With a little practice, each student should be able
to complete the addition in problem 1. Finding the
missing addends not only prepares them for completing the magic squares but gives them a good
background for algebra, since they must find the
unknown quantity in each equation (NCTM 2000).
Fig. 2 Student addition table for Mayan numerals
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MATHEM ATICS TEACHIN G IN THE M IDDL E SCHOOL
Problem 2 gives some historical information on the
Mayan solar calendar and provides students the opportunity to think about why a system based on 5
and 20 would be a natural system.
Activity 3: Mayan Magic Squares (3 × 3)
Activity 3 introduces the properties of a magic
square by verifying the sums in all rows, columns,
and diagonals. This is a continuation of the addition
from the previous activity. After completing problem 1, students should understand the definition of
a magic square.
In problem 2, students actually begin to complete
the magic squares. This should help them develop
logical thinking skills. Problem 3 is an extension
of the traditional 3 × 3 magic square where Mayan
numbers for 3 through 11 are used. (See fig. 3 for
student work.)
Activity 4: Mayan Magic Squares (4 × 4)
Activity 4 provides more challenging problems. If
completed in a group setting, students can think out
loud and share problem-solving strategies with one
another. They may also verify their work to determine if all the conditions for a magic square are met.
Problem 3 causes students to think about the effect
on each sum in the magic square when each entry
has been increased by 2.
Fig. 3 Student work on Mayan magic squares (3 × 3)
Discussion
These activities work well in a summer
camp setting or can be used in the classroom during
the school year. The material is new to most students,
so groups of varying ages and diverse mathematics
backgrounds can work together, starting with the
basics. The beginning activities can be done with
manipulatives. Younger, less mathematically experienced students can count in Mayan, do simple addition problems (learning to exchange 5 dots for 1 bar),
and complete some of the beginning magic squares.
Many students will even find it easier mentally to
check the sums of a 3 × 3 magic square in Mayan
than in conventional Hindu-Arabic notation.
Those who want more challenging problems will
move on quickly to the more difficult magic squares
and work with numbers greater than 19. While having fun with the puzzle-solving aspect of completing
the magic squares, students can master a different
numeration system. They can see the importance of
using 0 as a placeholder and appreciate the genius
of the Maya who devised the system long ago.
Extensions
Depending on the ages and mathematics
levels of the students, natural extensions of the ac-
tivities presented may be explored. For example,
must the 5 always be in the middle of a 3 × 3 magic
square? Why? How does one create a 4 × 4 magic
square? Is there a 4 × 4 magic square that is different from those found in the activity sheets? If we add
the same number to every entry of a magic square,
is the result also a magic square? If so, how do the
sums change? Are there 5 × 5 magic squares? How
are Mayan numerals 360 or greater written? What
goes in the third place-value level? Was there more
than one Mayan calendar? There are many possibilities, including integrating Mayan mathematics with
a study of the culture of the Maya people.
References
Anderson, Dawn L. “Magic Squares: Discovering Their
History and Their Magic.” Mathematics Teaching in
the Middle School 6 (April 2001): 466–71.
Closs, Michael P. Native American Mathematics. Austin,
TX: University of Texas Press, 1986.
Morley, Sylvanus G., and George W. Brainerd. The Ancient Maya. 4th ed. Stanford, CA: Stanford University
Press, 1983.
National Council of Teachers of Mathematics (NCTM).
Principles and Standards for School Mathematics. Reston, VA: NCTM, 2000. l
VOL . 1 2 , NO. 6 . Fe b r u a r y 2 0 0 7
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Activity 1: Mayan Numerals
Name _______________________________
The Maya used a vigesimal (base twenty) system with three symbols. One bar
represents
5 units, 1 dot • represents 1 unit, and 1 shell
represents 0. The table below gives the numbers
1−29 in both Mayan and standard Hindu-Arabic notation.
To convert from Mayan (base twenty) to standard (base ten) notation, the lowest level represents the
1’s position, which in the Mayan system includes numbers 0−19, where each dot represents a 1 and
each bar represents a 5. The second level represents the 20s position. At this level each dot represents
a 20 and each bar represents five 20s, or 100.
For example, the Mayan numeral
converts to 129 in standard notation, since we have six 20s
(1 bar and 1 dot) and nine 1’s (1 bar and 4 dots) for a total of 6 × 20 + 9 = 129.
1.Convert the following Mayan numerals into standard notation:
.
a...
b.
c.
To convert a number such as 256 from standard notation to Mayan, first divide by 20. Since 256 ÷
20 = 12 with a remainder of 16, we place a 12 in the 20s position and a 16 in the bottom position.
In Mayan, the number 256 is written as
2.Convert the following numbers into Mayan numerals:
a. 26
b. 40
c. 57
From the February 2007 issue of
d. 342
Activity 2: Working with Mayan Numerals
Name ______________________________
1.Fill in the boxes with the proper Mayan numerals.
a.
b.
c.
d.
e.
f.
2.The Mayan solar calendar year (called a Vague year because it only approximates the 365-day
calendar) is different from the yearly calendar that we now use. Answer the following questions
to learn more about the Mayan calendar.
a. The answer to addition problem 1(d) is the number of full months (uinals) in a Vague year.
Convert the answer to part 1(d) from a Mayan numeral to a standard number.
b. The answer to addition problem 1(f) is the number of days (kins) in a full Mayan month. Convert the answer to part 1(f) from a Mayan numeral to a standard number.
c. The first part of the year consisting of the full months is called a tun. Determine the number of
days in a tun by finding the product of the answers from part 2(a) and part 2(b).
d. The remaining few days at the end of each year form a short month called Uayeb. Assuming
that there are 365 days in a year, how many days are in this short month? (Hint: Subtract the
answer for part 2(c) from 365.)
3. Why do you think that 5 and 20 were chosen as key numbers in Mayan numerals?
From the February 2007 issue of
Activity 3: Mayan Magic Squares (3 x 3)
Name _______________________________
A magic square is an arrangement of numbers in a square so that every row, column, and diagonal has
the same sum and each number is used exactly once.
1. This square uses each of the Mayan numerals 1 through 9 exactly once.
a. What is the sum of each row of this square?
b. What is the sum of each column of this square?
c. What is the sum of each diagonal of this square?
d. Is this square a magic square? Explain your answer.
2. Complete the following magic squares using the Mayan numerals 1 through 9.
3. Complete the following magic squares using the Mayan numerals 3 through 11.
From the February 2007 issue of
Activity 4: Mayan Magic Squares (4 x 4)
Name _______________________________
Recall that a magic square is an arrangement of numbers in a square so that every row, column, and
diagonal has the same sum and each number is used exactly once.
1.Complete the following magic squares using the Mayan numerals 1 through 16.
2. What is the sum of each row, column, and diagonal of these magic squares?
3.Use the magic squares from part 1 to create new magic squares with the Mayan numerals 3
through 18. (Hint: Add the Mayan numeral • • to each entry in the squares from part 1.)
a. What is the sum of each row,
column, and diagonal of these
two new magic squares?
b. How is this sum related to the
answer from part 2? Why?
From the February 2007 issue of