Project Template

This project will use baseball as a central theme to develop a high school curriculum for a mathematics and a biology course. Students will work in small ‘ownership groups’ to investigate focused questions that might face the owners of a professional baseball team, questions chosen to cover the full range of standards-based algebra and biology topics. These investigations involve significant mathematical and scientific content, yet provide a clear relevance to engage students. We believe that the richness of this subject, its engaging topic, and a spirit of healthy competition will inspire students to a deep mastery of the subject knowledge. This approach will invite students to make connections between math, science, and everyday life, and to change their attitudes towards mathematics and science.

Driving Question

How do you use mathematics in designing, owning, and managing a professional baseball team?

Overall Goals of the Project

In designing, managing and maintaining a professional baseball team, students will investigate topics such as statistics and probability in game management, biological, chemical and ethical issues stemming from performance enhancing drugs, the construction of professional ballparks, and modeling the motions of balls using computer based technologies. Students will work with professionals in fields such as athletics, medicine, and architecture to create a professional baseball organization that includes a player roster, ballpark design, a medical staff, and the necessary business plan to ensure a profitable return.

Project Objectives

· Analyze data in histograms, scatterplots and stem-leaf plots with statistical software.

· Apply various methods (such as mean, median, and mode, variance and standard deviation) for characterizing data, and recognize what method is appropriate in a given circumstance

· Fit a line to a linear data set; characterize such a data set by an initial value and rate of change; and recognize the quality and appropriateness of a linear fit to a given data set.

· Relate linear equations to constant speed motion and constant rate-of-change phenomena.

· Calculate the surface area and volume of composite shapes (such as a baseball field or stadium), and recognize how those quantities change as the dimensions of the shape are altered.

· Use computer software to explore the effects of changing dimensions on surface area

· Use simulation software to tune a real-world design (their ballpark) to fit certain constraints.

· Discuss the properties of proteins, carbohydrates, and fats and how they function in the body

· Discuss the properties of energy consumption and how they apply to metabolism

· Analyze the fad diets that exist today and why people use them and how they work

· Discuss hormones in both men and women and how they creates differences among the two sexes

· Describe the size, shape, and structure of different macromolecules and how they apply to the human body.

· Discuss the different types of energy the body uses and which is most useful in the long and short term

· Analyze the structure of nerves and how neurotransmitters how they apply to muscle function

Rationale

It is a sad reality that today, more than ever, generations of students in the United States are falling behind the international accomplishments of their peers in mathematics and the sciences. Numerous educational studies indicate that, among other factors, a lack of motivation from students and their perceived notions that mathematics and sciences are not beneficial to the real world contribute to such low interest and abilities in these fields.

To address this national concern, we have designed a project that is rich in mathematical and scientific content, is considered a worthwhile interest of study to an overwhelming majority of the population, and most importantly, provides a bridge clearly tying mathematical and scientific principles to real-world scenarios. In designing, managing and maintaining a professional baseball team, students will investigate topics such as statistics and probability in game management, biological, chemical and ethical issues stemming from performance enhancing drugs, the construction of professional ballparks, and modeling the motions of balls using computer based technologies. Students will work with professionals in fields such as athletics, medicine, and architecture to create a professional baseball organization that includes a player roster, ballpark design, a medical staff, and the necessary business plan to ensure a profitable return.

This is a project that will unite students from all ability levels, race, religion, and socioeconomic status together to work for a common goal. What better opportunity will students have to realize the usefulness and necessity for mathematics and science in our world than to see it in use with a sport such as baseball, which is recognizable and loved throughout the world? We believe that this project will engage students, provide them with a playground in which to explore the beauty of mathematics and science in our everyday life, and perhaps plant a seed from which the next generation of great mathematicians and scientists shall grow.

Background

We’ll assume that readers of this document are passingly familiar with the basic play of baseball – if not, watching a game with a knowledgeable fan is sufficient introduction to baseball, and superior to any explanation we could give in this short space. (For one such attempt see http://www.pbs.org/kenburns/baseball/beginners/.) For our course, students will play an actual game of baseball (during class time or in conjunction with the Phys Ed department) at the start of the project.

At the start of each play, the pitcher throws the ball towards home plate to fool or overpower the batter. The actual pitch depends on the delivery of the pitcher. A fastball usually travels at 85 to 100 mph (38 to 45 m/s) and “breaks” (deviates from parabolic straight down) or “rises.” (A rising fastball obviously descends from the pitchers hand to the plate, but the spin of the ball causes it to cross the plate above where a parabolic path would predict). A good changeup is thrown with exactly the same physical motion (setup, windup, arm angle, delivery) as a fastball but travels at 60 to 80 mph (27 to 36 m/s). The slower speed not only causes a later arrival time but different vertical movement –a batter expecting a fastball will tend to swing early and above the ball. There are several breaking ball pitches (chiefly the curveball and slider) that use the rotation of the ball to cause significant movement either sideways or downwards. They travel at only 75 to 90 mph (33 to 40 m/s) but they can “move” as much as 14 inches (36 cm) from their apparent path. This means, for instance, that a left-handed pitcher can throw a slider that will initially look like it is heading straight for a left-handed batter, yet deflect by about a foot and actually cross the strike zone. When 6’10” tall Randy Johnson does this at 80+ mph, even professional batters often dive for cover. One last pitch of note is the knuckleball, which is thrown softly (60-75 mph, 27-36 m/s) but with as little spin as possible. The baseball’s stitches make it aerodynamically unstable, and the ball will dance to the plate according to the vicissitudes of the intervening air. Neither the pitcher, the batter, or the catcher has any idea where is will go – catcher Bob Uecker once remarked that the best way to catch a knuckleball “is to wait until it stops rolling and then pick it up.” All of these aspects in baseball remarkably deal with two science concepts, both math and biology. The biological makeup of an athlete can drastically change whether or not he/she is able to throw certain pitches at different speeds, or hit these pitches depending on how fast their body reacts. This deals directly on an internal level all the way down to a cellular level. While, on a mathematical basis, we can calculate how fast an athlete must react or angle his/her arm to throw or hit a ball. This dualism that exists in this sport allows for a in scope view from two different perspectives both intone with one another.

An excellent reference for this and other physical aspects of the game is The Physics of Baseball by Robert Adair. (Adair is a professor of physics at Yale and serves as the official physicist of Major League Baseball). Here are other important characteristic speeds, times and distances of baseball:

Well-hit batted ball	100 to 160 mph (45 to 72 m/s), with significant deflection due to spin
Crisply thrown fielded ball	Similar to fastball – 85 to 100 mph (38 to 45 m/s).
Fast outfielder, on grass	30 feet/s (9 m/s)
Fast baserunner, on dirt	33 feet/s (10 m/s)
Fielder’s reaction time plus transfer (glove to throwing hand) time	About 1 second, less for a good fielder
Delivery time for a pitcher (with runners on base, after committing to delivery)	About 0.8 seconds
Variation in baserunners’ reaction times	About 0.2 seconds
Lead (starting distance) of a very fast baserunner	About 12 feet (4 m)
Distance between 1^st and 2^nd base	90 feet (27.4 m)
Distance from pitching rubber to home	60 feet 6 inches (18.4 m)
Distance between home plate and 2^nd base	127.3 feet (38.8 m)

With these as reference – and estimation or experiment – there are an abundance of opportunities to answer baseball questions that rely on linear functions and distance-rate-speed equations.

Baseball has a rich historical background, and its own arsenal of statistics to analyze the game. With a few primitive statistics,

At-Bat	AB	Walk	BB	Run	R
Plate Appearance	PA	Hit	H	Run Batted In	RBI
Innings Pitched	IP	Single	1B	Sacrifice Fly	SF
Strikeout	K	Double	2B	Stolen Base	SB
Hit by Pitch	HBP	Triple	3B	Caught Stealing	CS
Pitches	#P	Home Run	HR	Ground into Double Play	GIDP

a variety of useful synthetic statistics help characterize the strength and balance of a player’s offensive and defensive contributions (full glossary here):

Batting Average – roughly, what fraction of balls put in play result in hits?	AVG	H / AB
On-Base Percentage – in what fraction of plate appearances does the batter reach base?	OBP	(H + BB + HBP) / PA
Total Bases – bases resulting from hits.	TB	1B + 22B + 33B + 4*HR
Slugging Percentage – what is the expectation for total bases for each at-bat?	SLG	TB / AB
On-Base plus Slugging – a rough-and-ready but surprisingly effective summary of offensive output.	OPS	OBP + SLG
Secondary Average – “a way to look at a player's extra bases gained, independent of Batting Average”	SecA	(TB - H + BB + SB - CS) / AB
Walk-to-Strikeout and Walk-to-Plate Appearance ratios – how disciplined is this hitter?	BB/K BB/PA	BB / K BB / PA
Earned Run Average – what is a pitcher’s run expectation for a complete game of 9 innings?	ERA	(ER * 9) / IP
ERA+ – The “Z-Score” for ERA (normalized to the mean and standard deviation for the league).	ERA+	(ERA – lgERA) / σ(ERA)

Besides the regulations of baseball and fan demands, major league ballparks are subject to constraints of finances, politics, and accessibility. Obviously both the baseball rulebook and the accessibility guidelines are too dense for high school students (or this casual reader, for that matter), though there’s no harm in letting the students see where they come from. For our lessons we have selected a few important constraints from these documents, simplified them, and translated them into human language.

Standards Addressed

- The student represents relationships among quantities using concrete models, tables, graphs, diagrams, verbal descriptions, equations, and inequalities.

- In solving problems, the student collects and organizes data, makes and interprets scatterplots, and models, predicts, and makes decisions and critical judgments.

- The student investigates, describes, and predicts the effects of changes in a on the graph of y = ax².

- The student investigates, describes, and predicts the effects of changes in c on the graph of y = x² + c.

- For problem situations, the student analyzes graphs of quadratic functions and draws conclusions.

(a) General requirements. Students shall be awarded one credit for successful completion of this course. Prerequisites: none. This course is recommended for students in Grades 9, 10, or 11.

(1) In Biology, students conduct field and laboratory investigations, use scientific methods during investigations, and make informed decisions using critical-thinking and scientific problem-solving. Students in Biology study a variety of topics that include: structures and functions of cells and viruses; growth and development of organisms; cells, tissues, and organs; nucleic acids and genetics; biological evolution; taxonomy; metabolism and energy transfers in living organisms; living systems; homeostasis; ecosystems; and plants and the environment.

(2) Science is a way of learning about the natural world. Students should know how science has built a vast body of changing and increasing knowledge described by physical, mathematical, and conceptual models, and also should know that science may not answer all questions.

(3) A system is a collection of cycles, structures, and processes that interact. Students should understand a whole in terms of its components and how these components relate to each other and to the whole. All systems have basic properties that can be described in terms of space, time, energy, and matter. Change and constancy occur in systems and can be observed and measured as patterns. These patterns help to predict what will happen next and can change over time.

(4) Investigations are used to learn about the natural world. Students should understand that certain types of questions can be answered by investigations, and that methods, models, and conclusions built from these investigations change as new observations are made. Models of objects and events are tools for understanding the natural world and can show how systems work. They have limitations and based on new discoveries are constantly being modified to more closely reflect the natural world.

3) Scientific processes. The student uses critical thinking and scientific problem solving to make informed decisions. The student is expected to:

(A) analyze, review, and critique scientific explanations, including hypotheses and theories, as to their strengths and weaknesses using scientific evidence and information;

(B) evaluate promotional claims that relate to biological issues such as product labeling and advertisements;

(E) evaluate models according to their adequacy in representing biological objects or events; and

B) investigate and identify cellular processes including homeostasis, permeability, energy production, transportation of molecules, disposal of wastes, function of cellular parts, and synthesis of new molecules; (F) research and describe the history of biology and contributions of scientists.

(5) Science concepts. The student knows how an organism grows and how specialized cells, tissues, and organs develop. The student is expected to:

(A) compare cells from different parts of plants and animals including roots, stems, leaves, epithelia, muscles, and bones to show specialization of structure and function;

- Understand and perform transformations such as arithmetically combining, composing, and inverting commonly used functions, using technology to perform such operations on more-complicated symbolic expressions

- Represent and analyze mathematical situations and structures using algebraic symbols

- write equivalent forms of equations, inequalities, and systems of equations and solve them with fluency—mentally or with paper and pencil in simple cases and using technology in all cases

- Use mathematical models to represent and understand quantitative relationships

- identify essential quantitative relationships in a situation and determine the class or classes of functions that might model the relationships

- -Use visualization, spatial reasoning, and geometric modeling to solve problems

- -use geometric models to gain insights into, and answer questions in, other areas of mathematics

- -use geometric ideas to solve problems in, and gain insights into, other disciplines and other areas of interest such as art and architecture

- -understand and use formulas for the area, surface area, and volume of geometric figures

- -Formulate questions that can be addressed with data and collect, organize, and display relevant data to answer them

- -understand histograms, parallel box plots, and scatterplots and use them to display data

- -compute basic statistics and understand the distinction between a statistic and a parameter

- -understand how basic statistical techniques are used to monitor process characteristics in the workplace

- -communicate their mathematical thinking coherently and clearly to peers, teachers, and others

- -understand how mathematical ideas interconnect and build on one another to produce a coherent whole