March Math Madness:
The Mathematics of the NCAA Basketball Tournament*
Mark Colgan
Taylor University
Upland, IN 46989
mrcolgan@tayloru.edu
A. What are some of your overall goals when teaching mathematics?
Please list some of them as we begin:
1.
3.
2.
4.
Beyond Crossroads (AMATYC 2006) states, “To build problem-solving skills, faculty need to engage
students actively in the learning process,” and “Mathematics should be presented in the context of
realistic, understandable, applied problems that help students develop an appreciation of the nature,
history, and usefulness of the discipline.”
B. Basics of the NCAA basketball tournament
The annual NCAA basketball tournament (both men’s and women’s) features a 64-team bracket divided
into 4 regions of 16 teams each. There are approximately 30 teams that get in the tournament by winning
their post-season conference tournament and the other teams are selected as the best at-large teams. The
teams are each seeded from 1-16 in each region. How the teams are seeded and which teams get in the
tournament are always subjects for debate.
C. RPI (Rating Percentage Index)
“The RPI (Rating Percentage Index) is a measure of strength of schedule and how a team does against
that schedule. It is used by the NCAA as one of their factors in deciding which teams to invite to the
NCAA tournament and where to seed them. The basic formula is 25% team winning percentage (WP),
50% opponents' average winning percentage (OWP), and 25% opponents' opponents' average winning
percentage (OOWP).” www.collegerpi.com
The basic formula is:
RPI = .25 (WP) + .50 (OWP) + .25 (OOWP)
A team's winning percentage (WP) is calculated by first adjusting the win total and loss total:
Win Total = 1.4 (road wins) + (neutral wins) + 0.6 (home wins)
Loss Total = 0.6 (road losses) + (neutral losses) + 1.4 (home loss)
D. RPI Examples
As of Sunday evening, March 13, 2005 (Selection Sunday), the Duke University men’s basketball team
had a record of 25-5 with 5 road wins, 6 neutral site wins, 14 home wins, 1 home loss and 4 road losses.
Calculate the RPI for Duke, given Duke's OWP=.6113, and Duke's OOWP=.5585.
As of December 12, 2005, Kansas St. was 6-1 as follows:
4-4
Georgia So.
W
4-3
New Mexico W
3-2
Steph. F. Aust. W
3-2
5-2
1-7
7-1
Cal St. Fuller.
Washington St.
Longwood
Colorado St.
W
L
W
W
E. Comparing Data Sets
Explore the RPI and ESPN rankings on Selection Sunday with the placement of the teams into the
bracket. Which teams received higher than expected seeds and which received lower than expected seeds?
* More details can be found at "March Math Madness: The Mathematics of the NCAA Basketball Tournament" by
Mark Colgan (Mathematics Teaching in the Middle School, 11 [March 2006]: 334-342).
Shooting Hoops
Work in groups of four to collect data for the following table. You will need a wastebasket, a ping-pong
ball, a yardstick, and three distance markers (pens or pencils work well). Each person takes 5 shots at the
basket. For each shot, you may choose to take a 1-pointer (from 5 feet), a 2-pointer (from 10 feet), or a 3pointer (from 14 feet) in any combination. Record the result of each shot in your table. When finished
shooting, record your totals on the overhead or chalkboard with the rest of the class.
Name
1-Ptr
Made
1-Ptr
Missed
2-Ptr
Made
2-Ptr
Missed
3-Ptr
Made
3-Ptr
Missed
Total
Points
Bracket Math
While you wait for all the data to be recorded, work in your groups to solve the problems below. In
addition to receiving points for making baskets, you will earn points for correctly answering the
problems. A small prize will be given for the group that scores the most combined points from shooting
the baskets and solving the problems.
A. (1-pointer) If there are 64 teams, how many games are played in
the entire tournament?
(2-pointer) If they played a double-elimination tournament, how
many games would be played?
(3-pointer) If they played a round-robin tournament (each team
played every other team), how many games would be played?
B. (1-pointer) How many games must a team win in order to win the championship?
(2-pointer) How many games would a team have to win if there were 256 teams in the tournament?
(3-pointer) If your team is ranked 10th with an RPI of 12, how many games would you expect to win?
C. (1-pointer) How many of the 64 teams lose their last game?
(2-pointer) How many teams are eliminated after 3 rounds of the tournament?
(3-pointer) What percentage of the 64 teams are left after 4 rounds?
D. (1-pt) If every team has the same probability of winning the championship, what is the probability
your team will win?
(2-pt) If the Big Ten gets 7 teams in the tournament, what is the probability that a Big Ten team will win?
(3-pt) If the probability IU will win each game is 1/2, what is the probability IU will win the championship?
E. (1-pt) What percentage of 1-pointers were made by our class?
(2-pt) What percentage of overall shots were made by our class?
(3-pt) What strategy appears to be the best: shooting all 1-pointers, or all 2-pointers, or all 3-pointers?
ANSWERS:
A. (1-pointer) If there are 64 teams, how many games are played in the entire tournament?
(Answer: 63, since each team but one loses)
(2-pointer) If they played a double-elimination tournament, how many games would be
played? (Answer: 126 or 127, since 63 teams lose twice and the winner may lose once)
(3-pointer) If they played a round-robin tournament (each team played every other team),
how many games would be played? (Answer: 2016= 64x632)
B. (1-pointer) How many games must a team win in order to win the championship? (Answer:
6)
(2-pointer) How many games would a team have to win if there were 256 teams in the
tournament? (Answer: 8)
(3-pointer) If your team is ranked 10th in the coaches poll and 12th in the media poll, how
many games would you expect your team to win? (Answer: 2, since you would be seeded 3rd
in your region and likely lose to the 2nd seed in the third round)
C. (1-pointer) How many of the 64 teams lose their last game? (Answer: 63)
(2-pointer) How many teams are eliminated after 3 rounds of the tournament? (Answer: 56,
with 8 left)
(3-pointer) What percentage of the 64 teams are left after 4 rounds? (Answer: 6.25%, which
is 4 of the 64)
D. (1-pointer) If every team has the same probability of winning the championship, what is the
probability your team will win? (Answer: 1/64)
(2-pointer) If the Big Ten gets 7 teams into the tournament, what is the probability that a Big
Ten team will win? (Answer: 7/64)
(3-pointer) If the probability your team will win each game is 1/2, what is the probability
your team will win the championship? (Answer: (1/2)6= 1/64)
E. (1-pointer) What percentage of 1-pointers were made by our class? (Answer: divide the
number made by the number attempted)
(2-pointer) What percentage of overall shots were made by the class? (Answer: divide the
number made by the number attempted)
(3-pointer) What strategy appears to be the best? For example, should a student shoot all 1pointers, or all 2-pointers, or all 3-pointers? (Answer: see explanation below.)
DISCUSSING STRATEGY
When all the data have been recorded on the board, discuss the answers to the Bracket Questions,
including what shooting strategy appears to be the best. For example, should a student shoot all 1pointers, or all 2-pointers, or all 3-pointers? (Answer: Most students should make about 90% of the 1pointers, about 60% of the 2-pointers, and about 25% of the 3-pointers. Individual classes will vary.
However, if they shoot five 1-pointers, they would expect to make 0.90 x 5 = 4.5 shots x 1 point each =
4.5 points. If they shoot five 2-pointers, they would expect to make 0.60 x 5 = 3 shots x 2 points each = 6
points. If they shoot five 3-pointers, they would expect to make 0.25 x 5=1.25 shots x 3 points each =
3.75 points. So, the average student would score the highest taking all 2-point shots.)
Class NCAA Math Contest
Another fun class activity (either with or without the above activities) is to hold a class contest to pick
winners in the NCAA basketball tournament. Each student simply picks the winner of each of the four
regions, the champion, and the number of wins (0-6) they think the most local team will have (as a tiebreaker). I provide them a copy of the bracket (and paste the following Contest Rules inside the bracket),
so those who claim they know nothing about basketball can always pick from the top seeds. Points are
awarded as follows: 5 points for picking each correct Final Four team, 10 points for picking the
champion, and 3 points for your team losing in the first round (which makes for interesting strategy and
makes you feel better when your favorite team gets beat in the first round). This contest may serve as a
catalyst for the previous activities.
NCAA MATH CONTEST RULES
1. Pick your teams as follows and fill in the blanks below:
A. Pick one Final Four team from each of the 4 regions.
B. Pick the national champion from these 4 teams.
C. For the tie-breakers, choose the number of games IU will win
and the number of games Michigan State will win (0 to 6).
2. Points will be awarded as follows:
A. Five points will be given for each correct Final Four team.
B. Ten points will be given for the correct national champion.
C. Three points will be given for a team losing in round one.
3. The person in each class with the most points wins.
East ____________
Midwest ________
South ________
West _________
Champion _____________
IU Wins __ MSU Wins __
CONCLUSION
These activities help create a classroom environment where mathematics is seen as interesting
and a fun part of everyday life. Students are engaged in problem solving in an area of their own interest,
and by analyzing data they can make predictions about future results. The NCAA basketball tournament
offers a creative context for bringing excitement to the mathematics classroom and gives us a new way to
look at March Madness.