2010 Excellence in Mathematics Contest
Team Project
The Babylonian tablet Plimpton 322
School Name:
Group Members:
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Reference Sheet
Formulas and Facts
You may need to use some of the following formulas and facts in working through this project. You may not need
to use every formula or each fact.
A bh
Area of a rectangle
C 2l 2w
Perimeter of a rectangle
A
r2
Area of a circle
y2 y1
x2 x1
Slope
Circumference of a circle
1
bh
2
Area of a triangle
12 inches = 1 foot
5280 feet = 1 mile
3 feet = 1 yard
16 ounces = 1 pound
2.54 centimeters = 1 inch
100¢ = $1
1 kilogram = 2.2 pounds
1 ton = 2000 pounds
1 gigabyte = 1000 megabytes
1 mile = 1609 meters
1 gallon = 3.8 liters
1 square mile = 640 acres
1 sq. yd. = 9 sq. ft
1 cu. ft. of water = 7.48 gallons
1 ml = 1 cu. cm.
C
V
A
2 r
r 2h
V
Volume of cylinder
Lateral SA = 2
lwh
Volume of rectangular prism
r h
Lateral surface area of cylinder
b 2 4ac
2a
Quadratic Formula
x
b
m
4 3
r
3
Volume of a sphere
V
tan
sin
cos
2
TEAM PROJECT
2010 Excellence in Mathematics Contest
____________________________________________________________________________________________
The Team Project is a group activity in which the students are presented an open ended, problem situation
relating to a specific theme. The team members are to solve the problems and write a narrative about the theme
which answers all the mathematical questions posed. Teams are graded on accuracy of mathematical content,
clarity of explanations, and creativity in their narrative.
Part 1 – Introduction
The clay tablet, Plimpton 322, is so named because it has catalog number 322 in the G. A. Plimpton Collection at
Columbia University. It is said to have been “written” in about 1800 BC by the Babylonian‟s (current day Iraq)
and was discovered in Iraq in the early 19th century. Experts have struggled to determine the exact meaning of the
numbers written on the tablet, but it is currently thought that the numbers represent relationships among the sides
of triangles.
An interesting feature of the tablet, which will be part of this project, is that the Babylonians expressed numbers
using the sexagesimal (base-60) number systems. That is, rather than using the numerals 0 through 9 (base-10) to
express all numbers, the Babylonians used 1 through 59 as shown below.
When numbers larger than 59 were needed, the Babylonians wrote multiple symbol blocks of this form using place
value notation. This is much like what we do with our base-10 number system. When we want to express a number
larger than 9, we use the place value notation shown. The number 43,569 is expressed as follows.
4
__________
ten thousands
3
__________
thousands
5
__________
hundreds
6
__________
tens
9
__________
ones
Different number systems are used often in different applications. For example, the binary number system, which
uses only the numerals 0 and 1, is used internally in all computers. Computers also use the hexadecimal number
system (base-16) which uses the numerals 0-9 and the letters A-F.
Exploring different number bases may not only help you if you are working in some particular application (like
computers or electronics), but can also help you make sense of the number system with which you are most
familiar – the base-10 number system.
In this activity, you will explore different number systems and demonstrate your understanding of these number
systems by
writing numbers using a particular number system
performing operations using a particular number system
converting from one number system to another
3
Part 2 – Base-5 Numbers
Suppose that the following symbols are used as numerals to represent the number of items in the designated oval.


For example, if someone had this many dollar bills, they would say “I have $ .”
Express each of the following amounts of money using the base-5 symbols as shown above.
1.
2.
3.
(Hint: If we saw $41, we would know that this does not mean $4 + $1 = $5 – this would mean 4 tens and 1 one or
forty-one dollars.)
4.
5.
6.
7.
PART 2 ANSWERS
1.
$
2.
$
3.
$ 
4.
$ 
5.
$
6.
$
7.
$

5
Part 3 – Converting from Base 5 to Base 10
For each of the following, express the base-5 numeral using traditional base-10 symbols. For example, the number
would be expressed in base-10 using the numeral „3.‟ Write your base-10 number in the box given.
1.

1.
4
2. 16
2.
3.
4.
3. 53


4.
214
5. 1876
5.

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Part 4 – Operations with Base 5 Numbers
Find the sum. Explain your thinking and process. Keep in mind that the written explanation of your thought
processes will be scored as well as your numerical solution.
+


Explanation:
7
Part 4 continued…
Find the difference. Explain your thinking and process. Keep in mind that the written explanation of your thought
processes will be scored as well as your numerical solution.

Explanation:
8
Part 5 – The Binary Number System
You have been focusing on different number systems throughout this activity. Hopefully, you are seeing that the
main idea is the same whether working in base-10, base-5, or base-2. And that idea is…place value! In this part of
the project, you will apply the place value idea to base-2 – binary numbers. The only 2 symbols needed in base-2
are 0 and 1. Binary numbers are used extensively in computer hardware applications (switches can be ON (1) or
OFF (0)).
“Why is the binary number system used in computers?”
To answer this question, decode the message below. Read the code like a book where each binary number
corresponds to a letter, number, or punctuation as shown in the decoder. That is, read the numbers across the rows
not down.
000100
000101
001111
000011
010111
010010
001111
010110
011001
010100
000001
001110
001111
000101
000101
000100
010111
010010
000011
010011
001101
001001
010100
001000
010011
000001
001110
010010
010100
001001
010011
000101
001111
010111
010000
000101
001000
000101
000001
010100
000101
001111
000101
000111
010100
010100
001101
001111
010101
010111
001001
010111
000011
001001
010011
000101
000011
001001
001000
001111
010000
010010
010100
000101
001110
001111
001111
001111
000001
010011
010100
010100
000101
000110
010101
001011
000101
010110
000111
010010
001101
001110
001110
101001
001001
001011
010011
001000
010100
100101
010010
000101
001001
001100
000010
001111
000100
010110
001110
000101
001111
000101
000011
010011
010010
001110
000100
001001
000110
011010
Decoder
1 = A 2 = B 3 = C 4 = D … 26 = Z
0 = 27 1 = 28 2 = 29 3 = 30…9 = 36
. = 37 , = 38 „ = 39 ? = 40 ! = 41
Write your answer here:
Detective Digit knows the secret of how computers work. Computers view
everything in the world as a combination of ones and zeroes!
Part 6 – The Babylonians and Base-60 Numbers
You did it! You made it to Part 6. Here you will go back to the Plimpton 322 tablet and try to figure out what part
of it is saying.
Suppose that you were looking at part of the tablet and saw the following sequences of symbols. Each row is
intended to represent a particular relationship between the given three numbers. A larger version of the Babylonian
numbers is provided on the next page for your convenience. Your task is to figure out what each set of three
numbers represents in our base-10 system. Once you have done that, your next task is to try to determine what the
relationship is between these three numbers. Write your idea in the box below.
3
4
5
5
12
13
7
24
25
9
40
41
12
35
____________, ____________, ____________
,
,
,
____________, ____________, ____________
,
,
____________, ____________, ____________
,
,
____________, ____________, ____________
,
,
,
37
____________, ____________, ____________
Write your response here.
Pythagorean Triples
10
Part 7 – Tiebreaker
In past years, there have often been more than one team project that has been done extremely well and the judges
have had a difficult time deciding on a winner. To help eliminate that possibility, teams will respond to the
following scenario, if time permits. Again, as in the first part of the group project, teams will be graded on
accuracy of mathematical content, clarity of explanations, and creativity in their narrative.
Scenario: In the base-10 system, we have the following place values, for example.
Ten thousands
Thousands
Hundreds
Tens
Ones
One-Tenths
One-Hundredths
Suppose that you were creating the base-3 number system. Instead of the labels shown above, what labels would
you have? Fully explain why the labels would be as you say. Fully unpack the meaning of each of these labels. Use
pictures, drawings, and verbal explanations as necessary to make your explanations clear. You can use the rest of
this page and the next to create your explanation
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