# 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.

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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.

# Algebra: Teaching Concepts

When we teach algebra, most teachers find that getting across the manipulation of expressions is far easier than teaching the big ideas that underlie algebra. Lately I’ve run across sites that help middle school students grasp those concepts. I’d like to share them with you in this post and ask for your ideas in return.

First, two excellent ideas on helping students walk the bridge from arithmetic to algebra:

Building Bridges In this lesson, students move from arithmetical to algebraic thinking by exploring problems that are not limited to single-solution responses. These are common, not complex, problems that are developed through questioning to a higher level. Within real-world contexts, students organize values into tables and graphs, then note the patterns, and finally express them symbolically.

Difference of Squares uses a series of related arithmetic experiences to prompt students to generalize into more abstract ideas. In particular, students explore arithmetic statements leading to a result that is the factoring pattern for the difference of two squares. Very well done.

Equivalence is one of those underlying concepts that make algebraic reasoning possible. Everything Balances Out in the End offers a unit in which students use online pan balances to study key aspects of equivalence. The three lessons focus on balancing shapes to study equality, then balancing algebraic statements in order to explore simplifying expressions, order of operations, and determining if algebraic expressions are equal.

The next lesson, Equations of Attack, is a game but developed to uncover the algebra beneath the strategies. The two players each plot points on a coordinate grid to represent their ships and points along the y-axis to represent cannons. Slopes are chosen randomly (from a deck of prepared cards) to determine the line and its equation of attack. Students use their algebraic skills to sink their opponent’s ships and win the game. Afterwards, the algebraic approach to the game is investigated.

Walk the Plank is also a game. You need to place one end of a wooden board on a bathroom scale and the other end on a textbook. Students can “walk the plank” and record the weight measurement as their distance from the scale changes. This investigation leads to a real world occurrence of negative slope.

A final teaching idea develops students’ understanding of algebraic symbols: Extending to Symbols.  As students begin to use symbolic representations, they use variables as unknowns. To help their concept of symbolic representation to grow, they need to explore questions such as: What is an identity? and When are two symbolic representations equal? This activity engages students in work with an online algebraic balance.

Each of these lessons comes from NCTM’s Illuminations site, a rich source for K-12 teaching.

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# Writing Math

As in other subjects, writing forces us to order our thoughts, make them clear to others. In mathematics, writing, as difficult as it is, helps students organize their understandings of concepts and set out for themselves their reasoning about a problem and its solution.  As stated in the Principles and Standards for School Mathematics, “students who have opportunities, encouragement, and support for writing, reading, and listening in mathematics classes reap dual benefits: they communicate to learn mathematics, and they learn to communicate mathematically” (National Council of Teachers of Mathematics 2000).

Writing comes easily to few of us, and that includes our students. These resources offer practical advice learned from the experiences of teachers in the classroom.  If you would, share with your colleagues your ideas on using writing as a teaching/learning tool by commenting on this blog post.

A Case for Using Reading and Writing in a Mathematics Classroom
Speaking from her own experiences as a math teacher, Sarah Kasten tells how — and why — she introduced reading and writing in her classroom. She shares how she directed her classes to do 5-minute, impromptu writing assignments, explain their problem-solving process, or even explain a new concept and create their own example problems.

Writing in Mathematics
A brief teacher-to-teacher article on getting started with writing in math class — moving from think-pair-share to a less-known model: think-write-pair-share.  A set of helpful links to other teachers’ experiences is given.

Math and Communication
You’ll find solid tips on encouraging and supporting math talk in this brief piece by well-known math teacher Kay Toliver.

Adapting Literacy Strategies to Improve Student Performance on Constructed-Response Items
This article discusses ways of adapting various reading strategies to help students improve their answers to extended-response questions on the mathematics portion of high-stakes tests. A practical article directed to teachers.

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# Fractions: Multiplying and Dividing

Is there anything more difficult to explain at 5th and 6th grade level than the rules for multiplying and dividing fractions? These resources offer support in explaining the concepts that underlie the rules. Visual, interactive models are provided where possible. You will also find opportunities for your students to practice their skills in this area of arithmetic. If you have other approaches to teaching this topic, please share! Just use the comment box below!

Multiplication of Fractions
Visualize and practice multiplying fractions using an area representation. With the “Show Me” option selected, the virtual manipulative is used to graphically demonstrate, explore, and practice multiplying fractions. A rectangular grid, representing a whole, shows the areas of two fractions to be multiplied, one fraction in red on the left and another in blue at the bottom. The area of the overlapping region, shown in purple, represents the product of their multiplication. The “Test Me” option provides problems to be solved using the same graphical representation.

Multiplying Fractions
This tutorial site offers instruction as well as practice in multiplication of fractions. The fractions are modeled with either circles or lines (rectangular areas). The visual display matched with the numerical makes an effective demonstration.

Divide and Conquer
This lesson is based on the idea that middle school students can better understand the procedure for dividing fractions if they analyze division through a sequence of problems. Students start with division of whole numbers, followed by division of a whole number by a unit fraction, division of a whole number by a non-unit fraction, and finally division of a fraction by a fraction. Activity sheets and guiding questions are included.

Dividing Fractions
In this activity, students divide fractions using area models. They can adjust the numerators and denominators of the divisor and dividend and see how the area model and calculation change. Full access to ExploreLearning is available through an annual subscription, but you can apply for a month’s free access in order to test out the applets.

Fraction Bars
This applet offers a classroom-adaptable idea of how to explain division of fractions. Adjustable colored bars are used to illustrate arithmetic operations with fractions on the number line. The initial seeding shows a division problem, dividing 7/5 by ½.

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# Testing! Beyond One-Step Math Problems

Middle school students often do well on straight calculations but feel lost when faced with more complex problems. And many tests these days require critical thinking and ask for an extended response. As in all test preparation, students need practice, especially on problems requiring more than one-step computation. Here are sites that offer test items you might use in class reviews, challenges, quizzes, etc.  Please let us know of other treasure troves of practice problems or how you prepare your students for testing.

Searching for Solutions
Within this WebQuest is a set of individual lessons on several problem-solving techniques, such as finding patterns, making a table, working backward, and solving a simpler problem. Each strategy is explained simply, with students in mind, then activities requiring that strategy are presented. An excellent guide from an e-learning specialist on ways to attack math problems!

Balanced Assessment
A set of more than 300 assessment tasks, indexed for grades K-12. Each incorporates a story problem format and includes hands-on activities. Rubrics provided.

Figure This! Math Challenges for FamiliesCreated for students in grades 6 to 8, the site offers math challenges that focus on everyday life, such as how fast your heart beats, what shape container holds the most popcorn, and how much of you shows in a small wall mirror.

Word Problems for Kids
A wide range of carefully selected problems! Organized by grade level from 5 through 12, each problem links to a helpful hint and to the answer; the more difficult problems offer complete solutions.

Problems with a Point
Here you can search for word problems by topic, lesson time, required mathematical background, and problem-solving strategy. Take the time to do the short guided tours of the site, and then look at favorite problems selected by teachers — a good set of problems at the middle school level.

Fermi Questions
Fermi questions emphasize estimation, numerical reasoning, communicating in mathematics, and questioning skills. Students often believe that word problems have one exact answer and that the answer is derived in a unique manner. Fermi questions encourage multiple approaches, emphasize process rather than the answer, and promote non-traditional problem solving strategies.

NAEP Questions
Over 2000 questions archived.  Online tools allow you to search the collection by content area, grade level, and difficulty. The site also shows what students at each achievement level are likely to know and how NAEP questions are scored.

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