PATTERNS, PROBABILITY, AND STATISTICS

SPORTS

Knight Foundation Summer Institute

Jennifer Fisher, Bryn Mawr College

Introduction:

This lesson focuses on patterns, probability and statistics in a sports setting. It would be beneficial for the students to have this lesson in an interdisciplinary unit where other subjects were also teaching units in relation to sports. Students do not realize how much of their world is ruled by mathematics or relates in some way to mathematics. By relating mathematics to their favorite sports, or by disguising mathematics as everyday sports problems, hopefully students will take an interest in the topic at hand and be able to apply the concepts from a sports scene to other parts of life. Sports are also a way to encourage the curiosity of students. Many of them will know terms such as batting average, standings, GAA, etc. but are unable to calculate these results. They can soon realize that these problems are simple mathematics which they could learn in school and further their understanding of the terms that they hear in sports.

There are three parts to this lesson. As a warm up each day, the teacher could review the vocabulary from that section and quickly go over the concepts from the previous day. Then the teacher can begin the new activity, teaching the concepts where appropriate.

Objectives:

  1. To have the students understand the meaning of mean, median, mode, range and mph
  2. To have the students be able to figure out the above concept given the appropriate information
  3. To help the students find their own batting average
  4. To have the students be able to apply this concept of probability to other aspects of mathematics and their lives
  5. To have the students be able to predict the next number in a series of numbers
  6. To have the students be able to recognize patterns
  7. To have the students be able to create pattern problems for other students
  8. To get the students moving and active
  9. To show the students that math can be applied to areas that they are interested in

Vocabulary:

#1

Central Tendency

Mean

Mode

Median

Range

MPH

#2

Probability

Batting Average

Fractional Batting Average

Decimal Batting Average

Accurate

#3

Pattern

Order

Increasing

Decreasing

CLASS ACTIVITIES

Exercise #1 Running Average

Materials:

Procedure:

  1. Hand out a sheet with all the students names and a space to mark their times
  2. Take the students to an area of known distance
  3. Have one of the students volunteer to go to the end of the running area and time the students as they finish
  4. Have each student in the class run the distance and record their time
  5. If there is still time left, each student can run twice
  6. Put all of the times on the board in the classroom or on an overhead projector or wipe off board
  7. Let the students copy down all of the times for each student

    8. Note: if the students were able to run twice, have the students find out the average of their two times and use this as their input to the class

  8. Help the class find the mean and give them a chance to do the calculation The mean is another word for average and the average is how the class does as a whole.
  9. Help the class find the mode and give the students a chance to do the calculations The mode is the number that appears most often in a set of data. To find this, the students
  10. Help the class find the median and give the students a chance to calculate this for their set of data The median is the number in the middle of a set of data. To find this value, the students
  11. Help the class figure out the range of the running times. The range is an area that includes all of the data in a given set. It is usually represented by the lowest value followed by a dash and then the highest value.
  12. Help the students to figure out their miles per hour, using the below equation if needed

    13. Time in seconds X ( 1 minute / 60 seconds) X ( I hour / 60 minutes) = Time in hours

    Distance in meters X (1 mile/ 1609 meters) = Distance in miles

    Distance in miles / Time in hours = MPH

  13. Have each student make a chart with their information and the class information
  14. Graph the results of each student and check if the mean, mode, median, and range that were calculated are correct by analyzing the data
  15. Compare the results; How is the mean different from the median? mode? Are the numbers similar? very different?
  16. Lead a discussion as to why these might be similar or different

Assessments:

A vocabulary test could help to demonstrate the students' understanding of the concepts of mean, mode, median, and range as could an evaluation of the students' final charts. Each student could also be asked to explain one of these concepts using an example from their favorite sport. If the students are given a set of data from another class, they should last be able to find the mean, mode, median, and range of this new set of information.

Extension:

Have the students make a chart of everyone's birthday month. Convert the names of the months into numbers; January = l, February = 2, ..., December = 12; the students must find the mean, mode, median, and range for the class birthdays.

 

 

Exercise #2 Batting Average

Materials:

Procedure:

  1. Hand out the recording sheets
  2. Line the students up, the second student should be several meters away from the first one
  3. Have the students who can throw the ball under control pitch five balls to each student
  4. The next person in line will record which pitches were hit and which were not on the sheet
  5. On the board or an overhead projector, have all of the students record their results
  6. Let the students copy all of this information onto their sheets
  7. Explain to the students that the number of hits over the total number of tries is their fractional batting average
  8. Have each student find the fractional batting average of every other students in the class
  9. Explain to the students that this can be changed into a decimal and demonstrate how that is done The students must divide the numerator by the denominator
  10. Allow the students to find the decimal batting average of every member of the class
  11. Teach the students to find the percent of pitches that they were able to hit
  1. This is the decimal multiplied by 100%
  1. Allow time for the students to find the percentage of hits of everyone in the class
  2. Graph the results of the entire class
  3. Discuss the differences in the fractions, decimals and percentages and the connections between them

Assessments:

The students should be graded on their understanding of the vocabulary words and on their understanding of applying these concepts to the project. They could be given a set of data from another class and asked to repeat the calculations they have learned on this new set of information. The students may also be asked to look in a newspaper or on the internet and find the batting averages of a baseball team or a few players. The teacher could explain how to convert these decimals into fractions. (Move the decimal place over and place it over a multiple of ten that corresponds with the number of places of the decimal, e.g.; .354 = 354/1000).

Extension:

Have the students repeat the activity and see how many keep the same batting average. If they were to use all ten tries now what would their new batting average be? Lead a discussion as to if using ten tries would give a more accurate result about their hitting ability. Explain that as more information is taken into account, the results become more accurate. Another project would be to look at the standings in the Newspaper and figure out what all of the categories mean.

 

 

Exercise #3 Patterns in Sports

Materials:

Procedure:

  1. Put a simple problem on the board e.g.; 2,4,6,8,_ what comes next? and give them some time to think about the problem and let a students write the answer on the board
  2. Let the students work in groups or individually to complete the ditto sheet
  3. Offer help while they try and do the problems
  4. Have them each create their own pattern problem
  5. Compile a new sheet with all of the students' problems
  6. Give the students time to do these new problems and find the patterns

Assessments:

The students should have a full understanding of the vocabulary and the concept of finding patterns. They should be graded on the creation of their own pattern problem because this will be the best indicator if they completely understand the concept. As a review, the students could solve a pattern problem a day that will be put on the board. The first one to solve the problem could get to put one up the following day or get extra credit or a prize.

Extension:

The students could find patterns in systems that are not only numbers, for example in pictures, names, colors etc. The students would have to find a pattern in school or in their own lives and report it back to the class.

Philadelphia Mathematics Content Standards:

Mathematics Content Standard 1- Number Systems: Arithmetic, Relationships and Theory

As part one of this standard requires, graphing information to see relationships can be found throughout this lesson. The second part requires students to "develop and use relations for whole numbers, fractions, decimals..." which is completed while evaluating the baseball bat activity. Also in this part of the activity, the student must convert fractions to decimals to percents and back again which satisfies part seven. This same issue is dealt with in part nine where the students must "investigate relationships among fractions, decimals and percents."

Mathematics Content Standard 4- Patterns, Algebra, and Fractions

Part five states that students must be able to "detect patterns and functions from statistical data and predict future outcomes" and this is completed by the worksheet where the students must predict the next number once a pattern is established.

Mathematics Content Standard 5- Using Data, Statistics, and Probability

Part two says that students must "demonstrate and use measures of central tendency such as mean, median, and mode and measures of variability such as range and quartiles." The next one states students need to "hypothesize conclusions and make convincing arguments based on data analysis" which the students will do as part of the running exercise. Finally part six states that students will "compare experimental results with mathematical expectations" which is done when the students repeat the baseball activity and must see if their averages stay the same.

Cross References:

The students will be using scientific techniques while running the experiments and recording the information. The students will be using reading and writing skills while doing and creating the pattern problems. They will also be doing some research to find the batting averages and other information of major league baseball stats.