Activity III: Batting Averages and More (Grades 7-10)
Standards
Standard 2: Patterns, Functions, and Algebra
Standard 5: Data Analysis, Statistics, and Probability
Standard 6: Problem Solving
Standard 7: Reasoning and Proof
Standard 8: Communication,
Standard 9: Connections
Standard 10: Representations
* These standards have been adopted and are based on the information from
Principals and Standards for School Mathematics: Discussion Draft, October l998,
National Council of Teachers of Mathematics.
Objectives
 set up and solve a system of equations
 apply their understanding of percentages and their calculation
 set up a spreadsheet and do calculations using columns
 reason from data
Activity3
A. The following data is for eleven of the greatest hitters in the baseball
hall of fame.
B.
At
Batting
Runs Batted Home
Player
Games
Hits
Bats
Average (BA) In (RBI)
Runs
Hank Aaron 3298
12364 3771 0.305
2297
755
Rod Carew 2469
9315
3053 0.328
1015
92
Ty Cobb
11429 4191 0.367
1961
118
Lou Gehrig 2164
8001
2721 0.34
1990
493
Rogers
Hornsby
2259
8173
2930 0.358
1584
301
Reggie
Jackson
2820
9864
2584 0.262
1702
563
3034
Mickey
Mantle
2401
8102
2415 0.298
1509
536
Willie Mays 2992
10881 3283 0.302
1903
660
Stan Musial 3026
10972 3630 0.331
1951
475
Babe Ruth
2503
8399
2873 0.342
2211
714
Ted
Williams
2292
7706
2654 0.344
1839
521
1. Place this data in a spreadsheet using either a computer or in a graphing
calculator. Determine the percentage of home runs per games, the
percentage of home runs per at bat, and the home runs per hit.
2. Rank these eleven players in descending order of the ratio of home runs per
game.
3. Rank these eleven players in descending order of the ratio of home runs per
at bat.
4. Rank these eleven players in descending order of the ratio of home runs per
hit.
5. Based on all of the data, make an argument that Babe Ruth is the best hitter
in baseball history.
6. Based on all of the data, make an argument that one of these eleven is the
number two greatest hitter in baseball history.
7. Make a statistically based argument that Henry Aaron is the greatest home
run hitter of all time.
B. The morning edition of the daily newspaper reported that Mark McGwire had a
batting average of .350. After the Saturday night game in which McGuire had 3 hits
in 5 times at bat, his new batting average was reported to be .352. Determine the
number of at bats and hits McGwire had prior to the Saturday night game.
C. At different times in the season, when Mark McGuire had 3 hits for 5 times at
bat, his batting average jumped from .350 to.366, from .350 to .358, and from
.350 to .355. Explain why the increase is not the same amount each time.
Activity 3: Batting Averages and More
Activity 3
A. The following data is for the eleven of the greatest hitters in the baseball hall of fame.
Player Games
Hank Aaron
Rod Carew
Ty Cobb
Lou Gehrig
Rogers
Hornsby
Reggie
Jackson
Mickey
Mantle
Willie Mays
Stan Musial
Babe Ruth
Ted Williams
At Bats
Hits
3771
3053
4191
2721
2930
Batting Average
(BA)
0.305
0.328
0.367
0.34
0.358
Runs Batted In
(RBI)
2297
1015
1961
1990
1584
Home
Runs
755
92
118
493
301
3298
2469
3034
2164
2259
12364
9315
11429
8001
8173
2820
9864
2584
0.262
1702
563
2401
8102
2415
0.298
1509
536
2992
3026
2503
2292
10881
10972
8399
7706
3283
3630
2873
2654
0.302
0.331
0.342
0.344
1903
1951
2211
1839
660
475
714
521
1. Place this data in a spreadsheet using either a computer or in a graphing calculator.
Determine the percentage of home runs per games, the percentage of home runs per at
bat, and the home runs per hit.
2. Rank these eleven players in descending order of the ratio of home runs per game.
3. Rank these eleven players in descending order of the ratio of home runs per at bat.
4. Rank these eleven players in descending order of the ratio of home runs per hit.
5. Based on all of the data, make an argument that Babe Ruth is the best hitter in baseball
history.
6. Based on all of the data, make an argument that one of these eleven is the number two
greatest hitter in baseball history.
7. Make a statistically based argument that Henry Aaron is the greatest home run hitter of
all time.
B. The morning edition of the daily newspaper reported that Mark McGuire had a batting
average of .350. After the Saturday night game in which McGuire had 3 hits in 5 times at
bat, his new batting average was reported to be .352. Determine the number of at bats
and hits McGuire had prior to the Saturday night game.
C. At different times in the season, when Mark McGuire had 3 hits for 5 times at bat, his
batting average jumped from .350 to.366, from .350 to .358, and from .350 to .355.
Explain why the increase is not the same amount each time.
Activity 3: Batting Averages and More
Solutions
A.
1.
Player
Games
At
Bats
Hits
Home
Runs
HR/
Games
HR/
At Bats
HR/
Hits
Hank Aaron
3298
12364
3771
0.305
2297
755
0.229
0.061
0.200
Rod Carew
2469
9315
3053
0.328
1015
92
0.037
0.010
0.030
Ty Cobb
3034
11429
4191
0.367
1961
118
0.039
0.010
0.028
Lou Gehrig
2164
8001
2721
0.340
1990
493
0.228
0.062
0.181
Rogers
Hornsby
2259
8173
2930
0.358
1584
301
0.133
0.037
0.103
Reggie
Jackson
2820
9864
2584
0.262
1702
563
0.200
0.057
0.218
Mickey Mantle
2401
8102
2415
0.298
1509
536
0.223
0.066
0.222
Willie Mays
2992
10881
3283
0.302
1903
660
0.221
0.061
0.201
Stan Musial
3026
10972
3630
0.331
1951
475
0.157
0.043
0.131
Babe Ruth
2503
8399
2873
0.342
2211
714
0.285
0.085
0.249
Ted Williams
2292
7706
2654
0.344
1839
521
0.227
0.068
0.196
2.
Player
Babe Ruth
Hank Aaron
Lou Gehrig
Ted Williams
Mickey Mantel
Willie Mays
Reggie Jackson
Stan Musial
Rogers Hornsby
Ty Cobb
Rod Carew
3.
HR/Game
0.285
0.229
0.228
0.227
0.223
0.221
0.2
0.157
0.133
0.039
0.037
Batting
Runs Batted
Average (BA)
In (RBI)
Player
Babe Ruth
Ted Williams
Mickey Mantel
Lou Gehrig
Willie Mays
Hank Aaron
Reggie Jackson
Stan Musial
Rogers Hornsby
Rod Carew
Ty Cobb
HR/At bat
0.085
0.068
0.066
0.062
0.061
0.061
0.057
0.043
0.037
0.01
0.01
4.
Player
Babe Ruth
Mickey Mantel
Reggie Jackson
Willie Mays
Hank Aaron
Ted Williams
Lou Gehrig
Stan Musial
Rogers Hornsnby
Rod Carew
Ty Cobb
HR/Hit
0.249
0.222
0.218
0.201
0.2
0.196
0.181
0.131
0.103
0.03
0.028
5. Student answers will vary.
Sample: Babe Ruth has the highest ratio in all three categories computed.
6. Student answers will vary.
Sample: This will be a very personal or statistical argument and difficult to provide an example.
7. Student answers will vary.
Sample: Henry Aaron has the most home runs in a career and ranks second in home runs per game. He ranks
sixth in home runs per time at bat and fifth in home runs per hit. No other player, outside of Babe Ruth, ranks
that consistently high.
B. Solution:
Let H = the number of hits prior to Saturday night’s game.
Let AB = the number of times at bat prior to Saturday night’s game.
H/AB = 0.350 and (H + 3)/(AB + 5) = 0.352
H = 0.350AB and H + 3 = 0.352(AB + 5)
H + 3 = 0.352AB + 1.760
H = 0.352AB + 1.760 – 3
H = 0.352AB – 1.24
By substitution: 0.350AB = 0.352AB – 1.24
1.24 = 0.002AB
1.240.002 = AB
620 = AB
0.35(620) = H
217 = H
Prior to that Saturday night’s game Mark McGuire had 217 hits and 620 times at bat.
C. Solution:
Percentage, which is what batting averages measure, is a relative measurement. The measure is relative to the
base, which represents 100%. For example, we know that 50% is always half, however, 50% of 100 items is 50
while 50% of 40 items is only 20. Increasing the 50 items to 53 and the base number to 105 produces a
percentage of 50.5%, while increasing 20 to 23 and 40 to 45 produces a percentage of 51.1%.
Since the base (number of times at bat) is constantly changing, the increase in percentage relative to the increase
of 3 hits and 5 times at bat will also change. Simply, the total number of hits and at bats is never the same.
Hence, the ratio of hits to times at bat will have different values because the numerator and denominator of each
ratio are different numbers.