March Mathness

There are more than nine quintillion (9 x 1018) ways to fill out a 64-team March Madness bracket — and almost 150 quintillion permutations for the 68 college basketball teams in this year’s men’s tournament of the National Collegiate Athletic Association (NCAA).

The Princeton University Press March Mathness blog includes interviews of sports rankings experts, coaches, and mathematicians. Their predictions take the power of mathematical methods of rating and ranking, and bring them to bear on the NCAA hoops tournaments. The blog will also provide updates on the group’s collective performance, and the best method for picking the winner.

Blog posts, which date back to March, 2011, have described how math is used during tournaments, as detailed in Princeton University Press books such as Mathletics: How Gamblers, Managers, and Sports Enthusiasts Use Mathematics in Baseball, Basketball, and Football by Wayne Winston and Amy Langville and Carl Meyer’s Who’s #1? [Thanks to the Math Forum for putting this information in their weekly newsletter!]

There are all sorts of ways people fill out their brackets. Google has filled out a bracket based on search volume http://www.google.com/insidesearch/collegebasketball.html. Check back often to see how they’re doing.

We’ve blogged about the integration of math and sports in the past, too – check them out at http://msms.ehe.osu.edu/category/sports/.


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We want and need your ideas, suggestions, and observations. What would you like to know more about? What questions have your students asked? We invite you to share with us and other readers by posting your comments. Please check back often for our newest posts or download the RSS feed for this blog. Let us know what you think and tell us how we can serve you better. We appreciate your feedback on all of our Middle School Portal 2 publications. You can also email us at msp@msteacher.org.

What Can Batting Averages Tell Us?

It’s the bottom of the 9th, 2 outs, bases loaded in the 7th game of the World Series. On the mound is the opposing team’s left-handed pitcher trying to close out the game. As the Head Coach you have a decision to make: let your left-handed 9th batter hitting .270 for the season go up and take his hacks, or pinch hit with your young, recently called-up rookie batting .350?

The first question we need to answer before making a decision is: What do the batting average numbers mean?

Batting averages are a simple decimal that approximates the number of hits per at-bat, or more simply the probability that a batter reached first base on a hit during his previous at-bats. The equation used to calculate batting average is simple: # Hits/# At-Bats.

A batting average is written in decimal form using 3 digits after the decimal point. Avid baseball readers read these as large numbers, so .400 would be read as “four-hundred” and .283 would be read as “two eighty-three.” Each individual thousandth is called a “point,” so .400 would be considered 117 points higher than .283.

But not all batting averages can be read equally. Two players can have the same batting average, take .300 for example, and have very different statistics. Player 1 could have 3 hits in 10 at-bats while player 2 may have 120 hits in 400 at-bats.

So which is a more accurate description of a player’s ability? Let’s take a look at what happens to the players after their next at-bat.

If they were to both get a hit in the next at-bat, their averages would indicate that Player 1 is much more likely to get a hit, yet if they both made an out the numbers would swing heavily in favor of Player 2.

The key to this discrepancy lies in the number of total at-bats. With more at-bats, the denominator for the fraction becomes larger and is less affected by adding 0 or 1 to the numerator. Referring to the chart, the next at-bat for Player 1 will either increase his average by 64 points or decrease it by 27. Player 2 will see either a 2 point increase or a 1 point decrease. So batting averages are less affected with larger numbers of at-bats, and can more accurately describe a hitter’s tendency over a period of time.

Now, looking back to the original question, I will add more context to the problem. In an average 162-game season a player might amass about 450 at-bats, and back-ups could see 100 at-bats. Rookies and recent call-ups (players invited to the major-league team from the minor leagues) will usually be on the team for the final 50 games of the season.

Knowing this information and having seen the chart from above, does this change your original decision for what to do? Why or why not? There is no definitive correct answer to this question, but I do ask that you use numbers to support your reasoning. Please post your decisions in the comments.


We Want Your Feedback
We want and need your ideas, suggestions, and observations. What would you like to know more about? What questions have your students asked? We invite you to share with us and other readers by posting your comments. Please check back often for our newest posts or download the RSS feed for this blog. Let us know what you think and tell us how we can serve you better. We appreciate your feedback on all of our Middle School Portal 2 publications. You can also email us at msp@msteacher.org. Post updated 11/27/2011.


This post was originally posted in the Everyday Explanations – Answers to Questions Posed on a Middle School Bus Ride by Sean Mittleman. We have his permission to re-post in the MSP2 blog.

Happy Belated Pi Day!

I’m sure lots of folks celebrated Pi Day on 3/14 – this post is for those folks that sorta forgot! Anyway – these great ideas, lessons, and activities can be used throughout the year. Please share your favorite Pi lessons in the comment area.

Ways to Think About Pi – from the Ohio Resource Center for Math, Science, and Reading – includes facts about Pi and lessons and activities

Going Around In Circles – from the Middle School Portal 2 (MSP2) project

These next three resources come from Wired Science and include background information and lesson ideas:

How Do You Determine Pi Without a Circle?

Activities for Pi Day

Pi: How Many Digits Do You Need?

You can also do a search in the Middle School Portal 2 Digital Library – you’ll get back resources that right on track for middle school math students.

We Want Your Feedback
We want and need your ideas, suggestions, and observations. What would you like to know more about? What questions have your students asked? We invite you to share with us and other readers by posting your comments. Please check back often for our newest posts or download the RSS feed for this blog. Let us know what you think and tell us how we can serve you better. We appreciate your feedback on all of our Middle School Portal 2 publications. You can also email us at msp@msteacher.org. Post updated 4/05/2012.

Math and Baseball

The baseball season is just warming up and the playoffs are around the corner. Why not bring “America’s Pastime” into the math classroom? The following problems challenge students to exercise some of the skills they learn in the middle school curriculum.

Baseball Fantasy
These two activities from PBS Mathline have pairs of students act as co-managers of a baseball team. Each pair receives a pack of baseball cards. They compute and analyze the key statistical data of the given players, make decisions on who they want to keep and who they want to trade, arrange their lineups, and play simulated games. Students use the key statistical data to construct individual player spinners, determine the line up, and play a simulated baseball game. The ultimate goal is to be the manager of the winning team of the Fantasy Baseball World Series.

What Is Round, Hard and Sold for $3 Million?
This activity challenges students to determine which is worth more today: Babe Ruth’s 1927 home-run record-breaking ball or Mark McGwire’s 70th home-run ball that sold in 1999 for $3 million. Compound interest is the main topic.

Who’s On First Today?
In this activity, students use hits and at-bat statistics to determine which of two baseball players has a better batting average.

Fun with Baseball Stats
In this lesson plan, students use baseball cards to convert statistics to decimals, fractions and percentages. Then, they use their statistics in playing a game. Activity sheets can be downloaded.

Come to the Middle School Portal 2: Math and Science Pathways online network to discuss this and many other topics and connect with colleagues!


We Want Your Feedback
We want and need your ideas, suggestions, and observations. What would you like to know more about? What questions have your students asked? We invite you to share with us and other readers by posting your comments. Please check back often for our newest posts or download the RSS feed for this blog. Let us know what you think and tell us how we can serve you better. We appreciate your feedback on all of our Middle School Portal 2 publications. You can also email us at msp@msteacher.org. Post updated 11/29/2011.

Taking Advantage of Technology

The computer can be a distraction and a frustration, but it can also be a teaching tool. Usually, you hear that you should be using technology in your teaching, but no one gives an example of a site that works for middle school curriculum. Here are a few online resources that actually show the potential of the Internet as a teaching strategy.


The MegaPenny Project

This site shows arrangements of large quantities of U.S. pennies. It begins with only 16 pennies, which measure one inch when stacked and one foot when laid in a row. The visuals build to a thousand pennies and in progressive steps to a million and even a quintillion pennies! All pages have tables at the bottom listing the value of the pennies on the page, size of the pile, weight, and area (if laid flat). The site can be used to launch lessons on large numbers, volume versus area, or multiplication by a factor of 10.

Cynthia Lanius’ Fractal Unit
In this unit developed for middle school students, the lessons begin with a discussion of why we study fractals and then provide step-by-step explanations of how to make fractals, first by hand and then using Java applets—an excellent strategy! But the unit goes further; it actually explains the properties of fractals in terms that make sense to students and teachers alike.

The Pythagorean Theorem
[This site is temporarily unavailable – we are going to leave this link in place and continue to check back in case it revives – 6/26/2010]
This site invites learners to discover for themselves “an important relationship between the three sides of a right triangle.” Five interactive, visual exercises require students to delve deeper into the mystery; each exercise is a hint that motivates and entices. The tutorial ends with information on Pythagoras and problems that rely on the theorem for their solutions.  

Fraction Sorter
A visual support to understanding the magnitude of fractions!  Using this applet, the student represents two to four fractions by dividing and shading areas of squares or circles and then ordering the fractions from smallest to largest on a number line. The applet even checks if a fraction is correctly modeled and keeps score. From Project Interactivate Activities.

Algebra Balance Scales — Negatives
This virtual balance scale offers students an experimental way to learn about solving linear equations involving negative numbers. The applet presents an equation for the student to illustrate by balancing the scale using blue blocks for positives and red balloons for negatives. The student then solves the equation while a record of the steps taken, written in algebraic terms, is shown on the screen. The exercise reinforces the idea that what is done to one side of an equation must be done to the other side to maintain balance. From the National Library of Virtual Manipulatives.

Geometric Solids
This tool allows learners to investigate various geometric solids and their properties. They can manipulate and color each shape to explore the number of faces, edges, and vertices, and to answer the following question: For any polyhedron, what is the relationship between the number of faces, vertices, and edges?  From Illuminations, National Council of Teachers of Mathematics Vision for School Mathematics.

We Want Your Feedback
We want and need your ideas, suggestions, and observations. What would you like to know more about? What questions have your students asked? We invite you to share with us and other readers by posting your comments. Please check back often for our newest posts or download the RSS feed for this blog. Let us know what you think and tell us how we can serve you better. We appreciate your feedback on all of our Middle School Portal 2 publications. You can also email us at msp@msteacher.org. Post updated 4/05/2012.