© 2014 Scott Adamson

2014 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 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.

2V r h V lwh 34

3V r

Volume of cylinder Volume of rectangular prism Volume of a sphere

Lateral SA = hr 2 a

acbbx

2

42

cos

sintan

Lateral surface area of cylinder Quadratic Formula

TEAM PROJECT

2014 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 – The Winter Olympics

The Winter Olympic Games were first held in 1924 in

Chamonix, France. Except for the 1940 and 1944

games, the Winter Games have been held every four

years from 1924 – 1992 in the same year as the

Summer Olympic Games. In 1986, the International

Olympic Committee decided to place the Summer and

Winter Games on separate four-year cycles so that an

Olympic Games would be held every two years.

Therefore, the next Winter Games after 1992 were held

in 1994 in Lillehammer, Norway. The Winter Games

continue to be held every four years and will be held in

2014 in Sochi, Russia.

The following excerpt, from

www.olympic.org/sochi-2014-winter-olympics,

provides background information related to the host city for the 2014 Winter Games.

The 2014 Olympic Winter Games will be the first time that the Russian Federation will have hosted the Winter Games; the Soviet Union hosted the 1980 Summer Games in Moscow. The host city Sochi has a population of 400,000 people and is situated in Krasnodar, which is the third largest region in Russia.

The Games will be organised in two clusters: a coastal cluster for ice events in Sochi, and a mountain cluster located in the Krasnaya Polyana Mountains. This will make it one of the most compact Games

ever, with around 30 minutes travel time from the coastal to mountain cluster. The Sochi Olympic Park will be built along the Black Sea coast in the Imeretinskaya Valley, where all the ice venues such as the Bolshoi Ice Palace, the Maly Ice Palace, the Olympic Oval, the Sochi Olympic Skating Centre, the Olympic Curling Centre, the Central Stadium, the Main Olympic Village and the International Broadcast Centre and Main Press Centre, will be built anew for the 2014 Games. The Park will ensure a very compact concept with an average distance of 6km between the Olympic Village and the other coastal venues.

The mountain cluster in Krasnaya Polyana will be home to all the skiing and sliding sports. The mountain concept is again a very compact one with only an average distance of 4km between the mountain sub-village and the venues. There will also be a sub-media centre in the mountain cluster.

Each part of this Team Project will focus on some aspect of the 2014 Olympic Games. Have fun!

Part 2 – Figure Skating

Each participant in the women’s figure skating event is observed by nine

judges. Each judge awards points for different elements of the skater’s

program. Ultimately, each judge gives a final score for each performance.

However, only 5 of the judge’s scores will count in the final tabulations.

Two judge’s scores are randomly eliminated.

The highest score is eliminated.

The lowest score is eliminated.

This leaves 5 scores from 5 of the judges. These scores are averaged.

Consider the following scores from the 2010 Winter Games held in

Vancouver, British Columbia, Canada. They represent the scores of each of

the 9 judges for one aspect of the free skating program for the gold and

silver medalists for the event.

Kim Yu Na

(Korea)

8.75 9.00* 9.00 9.50 8.75 9.00* 9.25 9.25 9.00

Mao Asada

(Japan)

8.50* 8.75 8.25 8.75 9.25 7.75 8.50 8.25* 8.00

1. Suppose that the scores with the asterisks (*) were the scores that were randomly eliminated. Find the average

score for Kim Yu Na.

9.00 8.75 9.25 9.25 9.00 45.259.05

5 5

2. Suppose that the scores with the asterisks (*) were the scores that were randomly eliminated. Find the average

score for Mao Asada.

8.75 8.25 8.75 8.50 8.00 42.258.45

5 5

3. Write a detailed, mathematical explanation of what the average score for Kim Yu Na means. To be clear, make

sure you explain what the average score means and NOT how it was computed.

The average score of 9.05 takes the total points (45.25) and equally distributes the points among the 5 remaining

judges so that it is as if each judge gave the same score of 9.05. Hypothetically, if each judge awarded a score of

9.05, the skater would have earned the same total points, 42.25, as actually were awarded.

Part 3 – Speed Skating

Another Winter Olympic event that was made

most popular by Apollo Ono (to the far left in the

picture) of the United States is speed skating.

There are several different speed skating events

held on both a short track and a long track. The

short track oval has a circumference of 111.2

meters. The long track oval has a circumference

of 400 meters. At the short track, races are held at

five different distances, 500 meters, 1000 meters,

1500 meters, 5000 meters, and 10,000 meters.

The table shows the Olympic records in each of

the different short track distances.

Distance Olympic Record Holder Stopwatch Time* Year and Place

500 meters Casey FitzRandolph ( ) 34.42 2002 Salt Lake City

1000 meters Gerard van Velde ( ) 1:07.18 2002 Salt Lake City

1500 meters Derek Parra ( ) 1:43.95 2002 Salt Lake City

5000 meters Sven Kramer ( ) 6:14.60 2010 Vancouver

10000 meters Lee Seung-hoon ( ) 12:58.55 2010 Vancouver

*Stopwatch time is of form MINUTES:SECOND – for example 6:14.60 means 6 minutes, 14.60 seconds

1. By finding the average speed of each in miles per hour, rank the skaters in order from greatest to least average

speed. Fill in your rankings in the table below. Round each speed to the hundredths place.

500 m 3600 seconds 0.000621371 miles32.49 MPH

34.42 seconds 1 hour 1 meter

1000 m 3600 seconds 0.000621371 miles33.30 MPH

67.18 seconds 1 hour 1 meter

1500 m 3600 seconds 0.000621371 miles32.28 MPH

103.95 seconds 1 hour 1 meter

5000 m 3600 seconds 0.000621371 miles29.86 MPH

374.60 seconds 1 hour 1 meter

10000 m 3600 seconds 0.000621371 miles28.73 MPH

778.55 seconds 1 hour 1 meter

Rank Name of Skater Speed (MPH)

Greatest (1) Gerard van Velde (1000 m) 33.30

(2) Casey FitzRandolph (500 m) 32.49

(3) Derek Parra (1500 m) 32.28

(4) Sven Kramer (5000 m) 29.86

Least (5) Lee Seung-hoon (10000 m) 28.73

Part 3 Continues…

2. Suppose Casey FitzRandolph skated at his average speed throughout the entire race. Create a graph of Casey’s

distance (in meters) from the starting line as a function of time (in seconds) for the duration of the race. Write a

brief explanation of why you made the graph the way you made it. Be as precise as possible.

Assuming a constant speed of 33.30 MPH (14.53 meters per second), the graph of distance from start (in

meters) versus elapsed time (in seconds) is linear. If his speed was constant, he would cover the same amount of

distance in any particular given interval of time during any part of the race.

3. If possible, write a function formula for the graph you created in #2 above. Explain how you were able to create

this function formula. If it is not possible, explain why.

The function formula is 14.53D t where D represents the distance from start (in meters) and t represents the

elapsed time in the race. At an average speed of 14.53 meters per second, we multiply this by the time elapsed to

find the total distance traveled.

4. Suppose Casey FitzRandolph was able to compete in the 10,000 meter race and was able to maintain his average

speed found in #1. What would his finish time have been? Would he have set a record? Comment on whether or

not Casey would be able to complete the 10,000 meters in this way.

10000 14.53

10000

14.53

688.23 seconds

t

t

t

At an average speed of 14.53 meters per second, Casey would have completed the 10,000 meter race in 688.23

seconds or 11:28.23. This would be an Olympic record. It is unlikely that Casey would be able to keep this pace

for the entire 10,000 meter race.

Dis

tance

fro

m S

tart

(m

eter

s)

Elapsed Time (seconds)

Part 3 Continues…

5. The following graph shows the progression of average world record speeds for the different race lengths. Clearly

describe the details of the most dramatic increase in the record speed that can be observed in the graph. When

did it occur? Which race and gender? How do you know?

www.vorbridge.com

The women’s 5,000 meter race experienced the greatest increase in average world record speed in

approximately 1985. We see that the average speed had been holding steady for about 20 or 25 years at

approximately 21 MPH and then increased to about 24 MPH around 1985.

6. The website www.nbclearn.com/olympics shares a video

where two different 5000 meter racing scenarios are

presented. In one case (Case I), it is proposed that a racer

starts out fast but then tires out and slows down as he

finishes the race. In the other case (Case II), it is proposed

that the racer starts out slower but then finishes the race

relatively quickly. The following graph is used to represent

these situations. Which graph, A or B, corresponds with

which scenario, I or II? Explain how you know.

Case I – B

Case II – A

A good response will clearly describe the relationship

between distance and time in justifying their conclusion. A mediocre response will only point to the “steepness”

of the graph with no attempt to describe the covarying nature of the quantities.

7. According to the graph, which skater, A or B, won the race?

According to the graph, the race was a tie. Each skater traveled the required 5000 meters in the same amount of

time thus making the race a tie.

Part 4 – Gold Medal Count

Since 1976, the United States and Germany have been among the top countries that have earned medals (gold,

silver, and bronze) in the Winter Olympic Games. The following table shows the total number of medals earned by

each country since the 1976 games.

Year 1976 Innsbruck

1980 Lake

Placid

1984 Sarajevo

1988 Calgary

1992 Albertville

1994 Lillehammer

1998 Nagano

2002 Salt Lake

City

2006 Turin

2010 Vancouver

USA 19 12 8 6 11 13 13 34 25 37

Germany* 10 23 24 25 26 24 29 36 29 30

*East Germany until 1992

1. Create a five-number summary for the total medal count for the USA.

Minimum Lower

Quartile Median

Upper

Quartile Maximum

6 9.5 or 11 13 29.5 or 25 37

NOTE: Q1 and Q3 answers can vary depending on method.

2. Create a five-number summary for the total medal count for the Germany.

Minimum Lower

Quartile Median

Upper

Quartile Maximum

10 23.5 or 24 25.5 29.5 or 29 36

NOTE: Q1 and Q3 answers can vary depending on method.

3. On the graph below, create a box-and-whisker plot for both the data showing the medal counts for the USA and

Germany.

Part 4 Continues…

4. Write a summary statement (several sentences) that compares and contrasts what the box-and-whisker plots for

the United States and Germany show regarding the medal counts in the Winter Olympic Games in the given

time period.

Look for some of the following important features:

The relative “shortness” of the box for Germany shows a consistent number of high medal counts

compared to the relative “longness” of the box for the US which shows that the number of medals

earned is more spread out.

The median medal count for Germany is much higher than for the United States. In fact, the median

medal count for Germany is nearly the same as the upper quartile medal count for the US.

The lower quartile for the US is much more compact than the lower quartile for Germany. This speaks to

the relative spread of the data in this lower quartile.

The data in the upper quartiles are nearly equally spread when comparing the two nations.

Part 5 – The Skeleton Event

Skeleton is an event at the Winter Olympic Games

where participants ride a sled, head first, down the

bobsled track. They are allowed a running start but

then must jump on the sled, belly down, and navigate

the course. The course in Sochi has a length of 1,814

meters and participants will experience a vertical drop

of 131.9 meters. The course is designed to have 19

curves that must successfully be navigated by the

participants. The goal is to finish the course in the

fastest time. There

has been some

controversy

surrounding the design of the course in Sochi. It is required that the local

event team submits the downhill grade to Olympic officials to ensure safety.

It was reported that the downhill grade of seven sights have been submitted

but Olympic officials have rejected them all due to high downhill grade on the

track.

While the downhill grade varies at different places along the course, your task

is to use the given quantities and report an overall downhill grade for the course. In the space below, clearly

communicate the mathematical thinking involved in your response. You may use words, mathematical symbols,

computations, and/or pictures as needed to best communicate your strategy.

Answers may vary. Here is a strategy…

Suppose we straighten out the track (hypothetically). Using the given quantities, we can create the following

picture (not to scale). Use the Pythagorean Theorem to compute the length of the horizontal side.

2 2 2

2 2 2

2

131.9 1814

1814 131.9

3273198.39

1809.198 meters

L

L

L

L

Therefore, the grade is: 131.9

0.07291809.198

or about 7.3%

1814 meters

131.9 meters

1809.19 meters

Part 6 – Ice Hockey

An understanding of fundamental geometry may be helpful in

the sport of ice hockey. Particularly, hockey players can benefit

by understanding the idea of angle measure. You will need to

use the protractor that has been provided to respond to the items

in this part of the project.

First, study the images and descriptions below. Then answer the

questions.

The following images show different locations of the puck that

the offensive team is trying to shoot into the opposing team’s

net. The goaltender would do his or her best to prevent the puck

from going into the net. Consider the angle of attack (in yellow)

which is the angle formed where the puck is the vertex of the

angle and the sides of the angle extend to the corners of the goalie’s crease (in blue). We will call the angle that the

puck makes with the center of the back of the crease and the horizontal red line the observed angle. In Figure 1,

the observed angle is 90o. In Figure 2, the observed angle is 45

o.

Figure 1: The puck is located directly in front of the net. Figure 2: The puck is located at a 45

o angle as shown.

1. Think about and discuss with your team the mathematical definition of “angle.” As clearly and precisely as

possible, write a definition of angle. That is, if you were to describe the idea of “angle” to someone who does

not know what one is, what would you say? Include in your description both what an angle is and how an angle

is measured.

Answers will vary. Here is the general idea that we are looking for…an angle is an object and we can measure a

particular attribute of that object.

An angle is the union of two rays that have a common endpoint. The measure of the angle’s “openness” can be

described as being the fraction of a circle’s circumference that the angle cuts out (assuming that the vertex of the

angle is located at the center of the circle).

45o

90o

Part 6 Continues…

2. Measure the angle of attack in Figure 1 using a protractor. Record that angle measure here.

15

3. Measure the angle of attack in Figure 2 using a protractor. Record that angle measure here.

10

4. Let the measure of the angle of attack be represented with the variable A. Let the measure of the observed angle

be represented by the variable O. Create a rough graph of the measure of angle A as a function of the measure of

angle O.

5. Write a description of the graph you created in #4. That is, what information does it provide relative to the

situation? Be as clear and precise as possible.

Answers will vary. As we judge, look for sophistication both in the description and in the graph provided. For

example, does the response precisely indicate a maximum? Is the graph made up (incorrectly) of linear

segments or is it curved? Either way, does the response provide an explanation using rate of change ideas? For

example, a good response would recognize that the angle of attack only changed by about 4 degrees for the first

increase of 45 degrees in the observed angle. In the last 45 degrees (taking the puck to the red line), the angle

would change from about 11 degrees to 0 degrees…a much more dramatic change. Thus, the graph should be

concave down. A good response might indicate that hockey players might take as a strategy the idea of working

to keep the puck within the “first” 45 degrees on either side of the front of the net. Moving the puck within this

range does not change the angle of attack as much as it does in the “lower” 45 degrees on either side of the net.

Mea

sure

of

Angle

A

Measure of Angle O

Part 7 – The Economic Impact of the Olympics in Sochi

Consider the following excerpt from www.economist.com.

Sochi has already set one record. At an estimated cost of $50 billion, these will be the most expensive games in history. When Russia placed its bid in 2007 it proposed to spend $12 billion, already more than any other country. Within a year the budget had been replaced by a seven-year plan to develop Sochi as a mountain resort. Most of the money is coming from the public purse or from state-owned banks.

Allison Stewart, of the SAID Business School at Oxford, says that Olympics tend to have cost overruns of about 180% on average. For Sochi the overrun is now 500%%. But Russia made clear that money was not an issue, says Ms Stewart. She also notes that relations between the government and construction companies appear closer in Sochi than in other games. Large construction projects often have a side-effect of corruption. But in Russia corruption is not a side-effect: it is a product almost as important as the sporting event itself.

1. Note the blocked out portion of the article. It reports the cost “overrun is now _____%.” Based on the

information in the article, what number should go in the blank? Show and/or explain the mathematical work

needed to answer this question.

383.17 317%

12

2. The unit of currency in Sochi is the Russian Ruble. Using Google, the following information was discovered

(retrieved 12/28/13). Use this information to determine the cost, in Rubles, of hosting the Winter Olympic

Games assuming that the estimated $50 billion amount is correct.

$50,000,000,0001,612,903,226,000

0.031 Russian Rubles.

Part 7 Continues…

3. According to sochimagazine.com, the average monthly

salary in Sochi is 32,100 rubles. The population of Sochi

is 343,334 (google.com). Considering that not all of the

population is working (e.g. kids, elderly, retired), we will

assume that 40% of the population is working (similar to

the US). To gain perspective on the cost of hosting the

Olympics, determine how many month’s salary would the

workers in Sochi hypothetically have to donate to the

Olympic cause in order to cover the $50 billion cost?

Explain your results.

0.40 343,334 137,334 workers in Sochi

1,612,903,226,000 Russian Rubles50,246,207 monthly salaries

32,100 rubles per person per month

50,246,207 365.87

137,334

A total of about 50,246,207monthly salaries or about 366 monthly salaries from each worker in Sochi would

have to be donated in order to cover the cost of hosting the Olympic Games.

4. According to the Bureau of Labor Statistics, the 128.2 million workers in the United States as of May 2011 hold

jobs ranging from fast-food workers requiring no education and earning barely above minimum wage to CEOs

with MBAs who earned more than nine times that amount. The population of the US is about 313.9 million

people. The population of Salt Lake City, UT is 189,314 people. The average monthly salary in Salt Lake City is

$4,333. How many monthly salaries from the people in Salt Lake City would hypothetically have to be donated

to the Olympic cause in order to cover the $50 billion cost? Assume that the percentage of workers in Salt Lake

City is the same as for the US. Explain your results.

$50,000,000,00011,539,349 monthly salaries

$4,333 per person per month

The percentage of workers in the US (and therefore Salt Lake City) is 128,200,000

0.41 41%313,900,000

0.41 189,314 77,619 workers in Salt Lake City

11,539,349148.67

77,619

A total of about 11,539,349monthly salaries or about 149 monthly salaries from each worker in Salt Lake City

would have to be donated in order to cover the cost of hosting the Olympic Games.