2016 Excellence in Mathematics Contest
Team Project
School Name:
Group Members:
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© 2016 Scott Adamson
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
A   r2
Area of a circle
C  2l  2w
Perimeter of a rectangle
y2  y1
x2  x1
Slope
m
Circumference of a circle
1
A  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
tan  
1 kilogram ≈ 2.2 pounds
1 ton = 2000 pounds
1 mile = 1609 meters
1 gallon ≈ 3.8 liters
1 sq. yd. = 9 sq. ft
1 cu. ft. of water ≈ 7.48 gallons
C  2 r
V   r 2h
Volume of cylinder
V  lwh
Volume of rectangular prism
1000 grams = 1 kilogram
Lateral SA = 2  r  h
Lateral surface area of cylinder
y = ax + b
 b  b 2  4ac
2a
Quadratic Formula
x
sin 
cos 
1 gigabyte = 1000 megabytes
1 square mile = 640 acres
1 ml = 1 cu. cm.
4
V   r3
3
Volume of a sphere
a2 + b2 = c2
1
V   r 2h
3
Volume of a Cone
TEAM PROJECT
2016 Excellence in Mathematics Contest
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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
This Team Project is an exploration related to
Mega M&M candy. Before you start the
mathematical investigation, please read the
following history of the original M&M candy.
The Wartime Origins of the M&M
JUNE 2, 2014 By Laura Schumm
It may not surprise you to learn that many
amazing discoveries and inventions are spawned from war, but did you know the hugely popular M&M candies
beloved by kids and adults of all ages around the world are one such innovation?
After clashing with his father—the creator of the Milky Way bar—for a few years at Mars Inc., Forrest Mars Sr.
moved to England, where in 1932 he began manufacturing the Mars bar for troops in the United Kingdom. It was
during the Spanish Civil War that Mars purportedly encountered soldiers eating small chocolate beads encased in
a hard sugar shell as part of their rations. In an age when sales of chocolate typically dropped off during summer
months due to the lack of air conditioning, Forrest was thrilled by the prospect of developing a product that would
be able to resist melting in high temperatures. He returned to the United States and, shortly thereafter,
approached Bruce Murrie, the son of Hershey executive William Murrie, to join him in his new business venture.
Anticipating a shortage of chocolate and sugar as World War II raged on in Europe, Mars sought a partnership
that would ensure a steady supply of resources to produce his new candy. In return, Murrie was given a 20 percent
stake in the M&M product, which was named to represent ‘Mars’ and ‘Murrie.’
In March of 1941, Mars was granted a patent for his manufacturing process and production began in Newark, New
Jersey. Originally sold in cardboard tubes, M&M’s were covered with a brown, red, orange, yellow, green or violet
coating. After the United States entered the war, the candies were exclusively sold to the military, enabling the
heat-resistant and easy-to-transport chocolate to be included in American soldiers’ rations. By the time the war
was over and GIs returned home, they were hooked.
Shortly after wartime quotas ended and the candies were made available to the general public, Forrest Mars
bought out Murrie’s shares in the company and took sole ownership of the M&M brand. The familiar brown bag
package that remains in use today was introduced in 1948. In 1950, the candies were imprinted with a black “m”
(which changed to white in 1954) and customers were encouraged to “Look for the M on every piece” to ensure
they were getting the real thing. Peanut M&M’s made their debut in 1954, along with the cartoon characters Mr.
Plain and Mr. Peanut, and by 1956 M&M’s had become the No. 1 candy in the United States.
In 1964, Forrest merged his various businesses (which by then included pet food and rice, among other products)
with his father’s company, Mars Inc., and soon began to phase out external chocolate suppliers like Hershey’s.
Upon request by the crew aboard NASA’s first space shuttle, Columbia, M&M’s were the first candy to rocket into
space in 1981. Three years later, they were advertised as the Official Snack of the 1984 Olympics in Los Angeles.
Today, the crowd-pleasing and satisfying candies continue to sweeten a soldier’s day as a welcome part of their
individual Meal, Ready to Eat (MRE) field ration.
http://www.history.com/news/hungry-history/the-wartime-origins-of-the-mm
Part 2 – Mega M&Ms Claim “3 times the chocolate”
For Part 2 of this Team Project, you are to investigate the claim that the Mega M&Ms have “3 times the chocolate”.
Your goal is to creatively, completely, computationally, and mathematically explore this claim and communicate your
results. The judges will be looking for the team that demonstrates sound methods for testing this claim that clearly
communicates their results and that demonstrates a wide variety of ways in which the claim could be true. Use this and
the next 3 pages (as needed) to record your results. You may want to work on scratch paper initially and then work to
create a well communicated response which you will record on these pages.
Part 2 – continues…
Part 2 – continues…
Part 2 – continues…
Part 3 – More Investigations
Suppose someone used the following method to investigate the claim that Mega M&Ms have “3 times the chocolate”:
Pour regular M&Ms into a cylindrical container to a height of 3 inches. Then pour Mega M&Ms into a similar
cylindrical container to a height of 3 inches (see pictures).
Create a mathematical argument that investigates the “3 times the chocolate” claim using this method. Clearly indicate
if the claim is accurate or not. If the claim is not accurate, clearly indicate “how many times the chocolate” does this
method support.
You may use the next page to show your work. Write a couple of sentences explaining any limitations observed in
investigating the Mega M&M claim using this method.
radius of container: 4.2 cm
number of M&Ms: 41
radius of container: 4.2 cm
number of M&Ms: 11
Part 3 – continues…
The idea is for students to find the volume of a cylinder with height 3 cm in each case and then find the volume of an
individual M&M.
The volume of the cylinder is V    4.2   3  166.25 cm3 .
2
Therefore, the volume of a regular M&M is
166.25 cm3
 4.1 cubic centimeters per candy piece .
41 pieces
166.25 cm3
The volume of a Mega M&M is
 15.1 cubic centimeters per candy piece .
11 pieces
Using this method, Mega M&M’s are
15.1
 3.68 times as large as regular M&M’s.
4.1
Students are also asked to write a little bit about limitations to this method. Possible ideas:
 Is the height really 3 cm? How thick is the bottom of the container? Note that the “students” measured from
the end of the ruler rather than from true zero.
 Is there really all the M&Ms contained as possible or could another be squeezed into the space? Or are there
too many?
Part 4 – Prolate Ellipsoid
Technically speaking, an M&M candy is known as a prolate ellipsoid or less technically speaking, it is a “squished”
sphere. The volume of a prolate ellipsoid is found as follows.
4
V   ab 2
3
To help you to make sense of this volume formula, consider the following diagram.
Use the following images of a regular M&M and a Mega M&M and the volume formula presented above to explore
the “3 times the chocolate” theme. Note that the images attempt to focus only on the amount of chocolate rather than
also including the candy shell.
Regular M&M
1 unit = 1 mm
4
V   ab 2
3
4
2
    6.5    3.5 
3
 333.5 mm3  0.3335 cm3
Part 4 – Prolate Ellipsoid
Mega M&M
1 unit = 1 mm
4
V   ab 2
3
4
2
   10.3   4.5 
3
 873.7 mm3  0.8737 cm3
Images retrieved 12/23/2015 from http://blog.recursiveprocess.com/2014/05/30/mms-mega/
Part 4 – continues…
873.7 mm3
The volume of the Mega M&M is about
 2.62 times as large as the volume of a regular M&M.
333.5 mm3
Note that this method excludes the candy shell so that the volumes should be slightly less than the volumes
found in Parts 2 and 3.