© 2016 Scott Adamson

2016 Excellence in Mathematics Contest

Team Project

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 2 2C l w 2A r

Area of a rectangle Perimeter of a rectangle Area of a circle

2C r 1

2A bh 2 1

2 1

y ym

x x

Circumference of a circle Area of a triangle Slope

12 inches = 1 foot 5280 feet = 1 mile 3 feet = 1 yard

16 ounces = 1 pound 2.54 centimeters ≈ 1 inch

cos

sintan

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.

2V r h V lwh

34

3V r

Volume of cylinder Volume of rectangular prism Volume of a sphere

1000 grams = 1 kilogram y = ax + b a2 + b2 = c2

Lateral SA = hr 2 a

acbbx

2

42

21

3V r h

Lateral surface area of cylinder Quadratic Formula 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…

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.

24

3V ab

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

Part 4 – Prolate Ellipsoid

Images retrieved 12/23/2015 from http://blog.recursiveprocess.com/2014/05/30/mms-mega/

Mega M&M

1 unit = 1 mm

Part 4 – continues…