TEACHER'S GUIDE
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Math Hoops Game created by Tim Scheidt Editorial Director James Fina Writers Rich Crowe, Tim Scheidt Creative Director Aileen Hengeveld Additional Design & Production K-Hwa Park All player/team statistics and other references are true as of the end of the 2011–12 season. The NBA and individual NBA member team identifications reproduced on this product are trademarks and copyrighted designs, and/or other forms of intellectual property, that are the exclusive property of NBA Properties, Inc. and the respective NBA member teams and may not be used, in whole or in part, without the prior written consent of NBA Properties, Inc. © 2012 NBA Properties, Inc. All rights reserved. Copyright © 2012 by Math Hoops, Inc. All rights reserved. Printed in the U.S.
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Table of Contents Introduction Making the Most of Your NBA Math Hoops Classroom Kit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v Creating a Math Hoops Roadmap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
Part I: Game Rules UNIT 1
Basic Game Rules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
UNIT 2
Advanced Game Rules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
UNIT 3
Keeping Score. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
UNIT 4
The Shot Clock. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
Part II: Creating a Math Hoops League UNIT 5
Practice Games (Pre-Season) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
UNIT 6
Organizing the League. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
UNIT 7
Keeping Statistics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
UNIT 8
Mid-Season Review/All-Star Game. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
UNIT 9
League Playoffs and Championships. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
Part III: Strategies and Math Explorations for Successful UNIT 10
Understanding Player Cards. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
UNIT 11
Analyzing the Game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
UNIT 12
Creating Teams and a Season Schedule. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
UNIT 13
Winning Strategies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
UNIT 14
Improving Your Team. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
UNIT 15
Why Do Coaches Need Statistics?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
Appendix Season Schedule Template. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 Game Summary Stat Sheet. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 Player Season Totals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 Mid-Season Assessment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 All-Star Nomination Form. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 Team Analysis Chart. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 Player Card Template. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 NBA Math Hoops: Program Roadmap. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
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introduction Making the Most of Your NBA Math Hoops Classroom Kit Congratulations! If you are reading this copy, the assumption is that you’ve found a way to have in your possession, a comprehensive NBA Math Hoops classroom kit. If you do, you’ll quickly notice that there is a LOT of stuff in the kit. Let me tell you, every bit of “stuff” has a designated purpose. The classroom kit has been carefully and creatively designed for use in school classrooms and out-of-school programs with groups of up to 32 students. The target audience is middle school students although upper elementary and younger high school students have a great deal to gain from the Math Hoops experience as well. Regardless of the instructional level of the students you’re instructing, you’ll want to be sure and make the most of what’s included in the kit. Let’s take a quick loo at some of the key components:
Teacher’s Guide This comprehensive Teacher’s Guide contains the instructional and organizational support you’ll need to run a successful NBA Math Hoops program. Before diving into the program, you’ll want to carefully read through the section that follows, titled Creating a Math Hoops Roadmap. Mapping out a detailed plan or roadmap specific to your group of students and learning environment is an important preliminary step that will put you and your students on the path to success. You’ll note that the Teacher’s Guide is divided into three parts: Part I: Basic and Advanced Game Rules Part II: Creating a Math Hoops League Part III: Strategies and Math Explorations for Successful Coaches
We suggest that you take the time to become familiar with each part before jumping in with students.
MAKING THE MOST OF YOUR NBA MATH HOOPS CLASSROOM KIT • v
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Part I: Basic and Advanced Game Rules is all about the board game. Here you’ll find specific
instructions, rules, relevant examples, and gaming tips. If your time with students is limited and your primary focus is to give them experience with playing the game, you’ll find everything you need in this section. Note that the Teacher’s Guide contains the main rules for both the Basic and Advanced Game and guidance on how to handle the rules in your classroom. (A more detailed explanation of each rule can be found in Part I of the Coach’s Manual.) On some pages, you’ll also see callouts in the margins with the labels Classroom Connections or Math Connections. Classroom Connections will direct you to relevant pages in Part II that will provide information on how to treat some of the game rules in the context of a classroom or school “league”. Math Connections will direct you to pages in Part III that can be used to delve into the mathematical or strategic thinking embedded in different aspects of the game. provides guidance in how to organize class or after-school time to create a total classroom experience. If you’re able to incorporate Math Hoops into your schedule as a recurring event (e.g., one or more sessions per week for 4–6 weeks), this section will give you options and tips on managing the logistics of running a Math Hoops league in your classroom. Part II covers techniques for introducing the game to your students, setting up a Pre-Season, organizing a league (or helping students organize their own league) and developing and running a season schedule of games. You’ll also learn how to guide students through a Mid-Season Review, organize a mid-season All-Star Game, and wrap up your league with a Playoff Series and Championship Game. The Math Connections callouts here will direct you to Part III explorations you can use in conjunction with the development of a full classroom experience. Part II: Creating a Math Hoops League
Part III: Strategies and Math Explorations for Successful Coaches contains information that lies at the heart of the Math Hoops classroom experience: instruction and guidance in how to use NBA Math Hoops to reach educational objectives. This section represents the bulk of the Teacher’s Guide. Here is where you’ll find detailed support for engaging students in meaningful discussion and exploration directly tied into the game experience. You can use specific explorations to highlight areas of math or critical thinking that are part of your current curriculum or you can use them to augment the curriculum. The writing of these explorations has been informed by the new Common Core State Standards with a concentrated focus on integration of the eight Mathematical Practices.
Coach’s Manual (32) The Coach’s Manual is a consumable guide/workbook/rule book created for student use. It follows the same 3-part organization as the Teacher’s Guide. In fact, you will find
vi • INTRODUCTION
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that the two books align exactly in terms of unit division and content. Part I contains the comprehensive set of rules for playing the Basic Game and Advanced Game. Because the text is intended for students to read and refer back to, the rules are described in more precise detail than in the Teacher’s Guide, so you will want to be familiar with the information given in the Coach’s Manual. Part II provides students with the basic outline of how to set up their own teams and a Math Hoops league. There is enough information in these pages to allow them to develop their own informal versions of a pre-season, draft, regular season, and tournament, but you should expect to step in to give explicit structure and details here. Neither Part I nor Part II requires student written participation in the manual itself. Part III has specifically been designed with interactivity in mind and can serve as a running portfolio of student work. These pages provide concentrated space for students to respond to contextual questions and tasks, in addition to some special features and extensions, oral and written, which provide some further challenges. All the explorations found in this section connect directly to a feature, rule, or strategy in the game, so the motivations for mathematical investigation come from interest in or desire to become better coaches or win more games. The Appendix at the back of the manual includes blank forms and charts needed to complete a number of the explorations.
Player Cards (8 sets of 32 cards) Each game comes complete with a set of 32 specially designed Player Cards featuring current NBA and WNBA players. Each position is represented by 6 or 7 cards with a designated background color. Data found on the backs of the cards and reflected in the graphical displays are representative of the 2011–12 NBA season and the 2011 WNBA season. Creating teams with these attractive, colorful cards gives students the opportunity to make a quasi-personal connection with popular, professional basketball players. Imagine how good it feels for a middle school student to be able to say, “Yeah, Chris Paul plays for MY team!” Game Board (8) The double-sided game board is the centerpiece of the Math Hoops game. One side is used to play the Basic version of the game and the other side is used in the Advanced version of the game. The Basic Game side (marked with a red border) has twenty possible shot locations on each end of the board. Here you’ll note that each color/player has an equal number (4) of possible shot locations. On the Advanced Game side (marked with a blue border), there are thirty possible shot locations on each end. Here you’ll note that the number of shot locations for each color/player varies. The Advanced Game also includes “passing lanes:” dotted lines connecting two different shot locations. During the game, a player who passes the ball to a teammate who then makes a shot gets credit for an assist.
MAKING THE MOST OF YOUR NBA MATH HOOPS CLASSROOM KIT • vii
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Shot Planner (16) The Shot Planner is essentially the coach’s clipboard. Here students will find functional chart templates for both the Basic and Advanced versions of the game. The roll of the dice and corresponding calculations are recorded using dry-erase pens and can be easily cleared when the planner is full. Shot Clock (4) This component will garner as much if not more excitement than any other included in the kit. The clock starts, stops, and is reset by depressing the top of the basketball. The toggle switch on the side allows for a setting of 24-seconds or 35-seconds. But it’s the realistic game-like buzzer that will get your students bustling with enthusiasm. In your role as league commissioner, you’ll want to establish parameters that warrant use of the shot clock in the simulated games. All students should works towards the target of bringing the shot clock, at some level, into their game-play experience. Scoreboard (8) If you’re looking for a colorful, quick-start tool that will allow students to jump right into the Math Hoops game, well, here it is! The front of the Scoreboard gives a brief set of guidelines—the fundamental rules needed to play either the Basic or Advanced Game. Of course, the main function of the Scoreboard is provided by the number wheels, which allow you to keep a running score of the game. This is extremely important— even more so when participating in a close, competitive game! As you can see, you’ve inherited a goldmine of instructional materials designed to excite, engage, and motivate your students to achieve at the highest level possible. Make the most of each and every kit component and you and your students will be on your way to one of the most comprehensive, educationally-integrated teaching/learning experiences you’ll ever encounter. Well, what are you waiting for?!
viii • INTRODUCTION
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Creating a Math Hoops Roadmap Greetings! If you’re reading these words, you’ve decided to take the plunge and engage your students in the classroom experience of NBA Math Hoops. What exactly does this mean? Well, for one, you might want to start having students refer to you as Ms., Mrs., or Mr. “Commissioner”! By assuming the role of Math Hoops Commissioner, you will be taking the lead in structuring an experience that may include: pre-season games, forming a league and drafting teams, creating a season schedule, playing regular season games, keeping statistics (team and individual player), hosting an All-Star Game, and the pinnacle—a League Tournament Series! In addition, you may want to have student “coaches” delve into some of the meaningful math emerging naturally from game play. How much you do and how far you go is entirely up to you! Throughout this Teacher’s Guide, you will find detailed instructions and helpful tips to guide you through your personal NBA Math Hoops journey. While we expect this information will prove to be an invaluable resource, there are some basic decisions you will first need to make to get the most out of your experience: what parts of the program you want to focus on, how to most effectively structure your sessions, and how in-depth to go. The first place to start is by looking at your specific situation. By considering a few questions, you will be better prepared to begin crafting your personal NBA Math Hoops roadmap. QUESTIONS TO CONSIDER:
• Will this program be implemented during the regular school day or as some form of after-school or club program? What are the minimum/maximum numbers of students that will participate in the program?
• How much time will be allocated to the NBA Math Hoops experience on a daily/ weekly basis? What is the total number of sessions/hours scheduled for this program? What will be the duration of implementation?
• What is the grade level(s) of the students you’ll be working with? Are they performing below, at, or above grade level expectations?
• What level of focus needs to be on the development of math skills and understanding? If students are actively engaged in game play, is that enough or do you want/need to do more?
• If it’s important to develop deeper understanding of math concepts students will experience strictly with game play, what specific content focus should be the priority?
• How do you envision the conclusion of your program? Will there be a tournament? A classroom celebration? Will parents and other guests be invited?
CREATING A MATH HOOPS ROADMAP • ix
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Your Personal NBA Math Hoops Roadmap As you answer the questions above, keep in mind that your original plans will most likely change over time and evolve in a way that best meets the needs of your students. Understanding and expecting this up front should help alleviate any concerns you may have once fully immersed in the NBA Math Hoops experience. Maintaining flexibility and an openness to “adapting as needed” as you progress down your path will serve you and your students well. This is a good time to read through the unit openers in Parts II and III. You’ll note that along with each summary introduction, each activity displays a rough timeframe. While the time allocated for each section will vary by site and facilitator, this should provide an adequate framework for initial planning purposes. Following is a working example of a 5th/6th grade teacher opting to create an NBA Math Hoops classroom experience via an after-school club. This particular teacher has decided to run the club for 8 weeks, 3 sessions (one hour per session) per week for a group of 24 students. Given the group of students, she has decided that focusing exclusively on the Basic Game would be the most beneficial. Please note that this is the teacher’s initial roadmap and was drafted with full understanding that adjustments will be made along the way.
x • INTRODUCTION
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Okay, now it’s your turn. In the Appendix at the back of this book, you will find a blank template that you can use to sketch out your own Program Roadmap. [See page 142.] You’ll note that it contains spaces for up to 32 sessions to allow for greater flexibility. Of course, you should modify it to fit your situation. As an additional aid, the next few pages contain examples of roadmaps covering a variety of scenarios. These are not intended to be prescriptive; they are meant solely to give you ideas for your own roadmap. After you finish mapping out a plan, take a close look at what you put together. Remember that this is an initial draft. Your goal for now should be to prepare a plan that’s detailed enough to enable you to see an overall structure to the experience you want to create and flexible enough to allow you to drop some activities or even add some! Keep in mind that the first time through any program usually takes longer than anticipated so better to be conservative with what you hope to accomplish during each session and add on accordingly. Also, pencil is much easier to erase than pen! Once you’re satisfied with your plan, it’s time to start moving the ball up and down the court. Pull out your Classroom Kit and call in your students. Your Math Hoops season is about to begin!!
Program Roadmap, p.142
NBA Math Hoops: Comprehensive Roadmap 32 SESSIONS—Eight weeks @ 4 sessions per week WEEK 1
Basketball 101 Introduce Basic Game
Play Sample Game (whole class)
Play Sample Game (small groups)
Form League (2 Divisions) Draft Teams
Create/Adopt Schedule Prep for Regular Season (RS)
RS Game 1 (Basic) (end of game statistics)
RS Game 2 (Basic) (end of game statistics)
Math Exploration #1
WEEK 2
WEEK 3
RS Game 3 (Basic) (end of game statistics)
RS Game 4 (Basic) (end of game statistics)
Introduce Advanced Game Play Sample Game (WG)
Play Sample Game (small groups)
Math Exploration #2
RS Game 5 (Advanced) (end of game statistics)
RS Game 6 (Advanced) (end of game statistics)
Math Exploration #3
WEEK 5
Mid-Season Team Stats League Leader Stats
Mid-Season Draft Mid-Season Trades
All Star Game Preparation (reps from all teams)
RS Game 7 (Advanced) (end of game statistics)
WEEK 6
All Star Game (reps from all teams)
RS Game 8 (Advanced) (end of game statistics)
Math Exploration # 4
RS Game 9 (Advanced) (end of game statistics)
WEEK 7
RS Game 10 (Advanced) (end of game statistics)
Math Exploration # 5
RS Game 11 (Advanced) (end of game statistics)
RS Game 12 (Advanced) (end of game statistics)
WEEK 8
End of Season Stats League Leader Stats
1st Round Playoffs (Division Leaders—1st & 2nd)
Math Exploration # 6
Championship Game (Division Champions)
WEEK 4
CREATING A MATH HOOPS ROADMAP • xi
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NBA Math Hoops: Comprehensive Roadmap 32 SESSIONS—Twelve weeks @ 2-3 sessions per week WEEK 1
Basketball 101 Introduce Basic Game
Play Sample Game (whole class)
WEEK 2
Play Sample Game (small groups)
Form League (2 Divisions) Draft Teams
Create/Adopt Schedule Prep for Regular Season (RS)
WEEK 3
RS Game 1 (Basic) (end of game statistics)
RS Game 2 (Basic) (end of game statistics)
Math Exploration #1
WEEK 4
RS Game 3 (Basic) (end of game statistics)
RS Game 4 (Basic) (end of game statistics)
WEEK 5
Introduce Advanced Game Play Sample Game (WG)
Play Sample Game (small groups)
Math Exploration #2
WEEK 6
RS Game 5 (Advanced) (end of game statistics)
RS Game 6 (Advanced) (end of game statistics)
Math Exploration #3
WEEK 7
Mid-Season Team Stats League Leader Stats
Mid-Season Draft Mid-Season Trades
All Star Game Preparation (reps from all teams)
WEEK 8
RS Game 7 (Advanced) (end of game statistics)
All Star Game (reps from all teams)
RS Game 8 (Advanced) (end of game statistics)
Math Exploration # 4
RS Game 9 (Advanced) (end of game statistics)
RS Game 10 (Advanced) (end of game statistics)
Math Exploration # 5
RS Game 11 (Advanced) (end of game statistics)
WEEK 11
RS Game 12 (Advanced) (end of game statistics)
End of Season Stats League Leader Stats
WEEK 12
1st Round Playoffs (Division Leaders – 1st & 2nd)
Math Exploration # 6
WEEK 9
WEEK 10
Championship Game (Division Champions)
xii • INTRODUCTION
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NBA Math Hoops: Intermediate Roadmap A 24 SESSIONS—Six weeks @ 4 sessions per week WEEK 1
Basketball 101 Introduce Basic Game
Play Sample Game (whole class)
Play Sample Game (small groups)
Form League (2 Divisions) Draft Teams
WEEK 2
Create/Adopt Schedule Prep for Regular Season (RS)
RS Game 1 (Basic) (end of game statistics)
RS Game 2 (Basic) (end of game statistics)
Math Exploration #1
WEEK 3
RS Game 3 (Basic) (end of game statistics)
Introduce Advanced Game Play Sample Game (WG)
Play Sample Game (small groups)
RS Game 4 (Advanced) (end of game statistics)
Math Exploration #2
RS Game 5 (Advanced) (end of game statistics)
All Star Game Preparation (reps from all teams)
RS Game 6 (Advanced) (end of game statistics)
WEEK 5
All Star Game (reps from all teams)
RS Game 7 (Advanced) (end of game statistics)
Math Exploration #3
RS Game 8 (Advanced) (end of game statistics)
WEEK 6
End of Season Stats League Leader Stats
1st Round Playoffs (Division Leaders – 1st & 2nd)
Math Exploration # 4
Championship Game (Division Champions)
WEEK 4
NBA Math Hoops: Intermediate Roadmap B 24 SESSIONS—Nine weeks @ 2–3 sessions per week WEEK 1
Basketball 101 Introduce Basic Game
Play Sample Game (whole class)
WEEK 2
Play Sample Game (small groups)
Form League (2 Divisions) Draft Teams
Create/Adopt Schedule Prep for Regular Season (RS)
WEEK 3
RS Game 1 (Basic) (end of game statistics)
RS Game 2 (Basic) (end of game statistics)
Math Exploration #1
WEEK 4
RS Game 3 (Basic) (end of game statistics)
Introduce Advanced Game Play Sample Game (WG)
WEEK 5
Play Sample Game (small groups)
RS Game 4 (Advanced) (end of game statistics)
Math Exploration #2
WEEK 6
RS Game 5 (Advanced) (end of game statistics)
All Star Game Preparation (reps from all teams)
RS Game 6 (Advanced) (end of game statistics)
WEEK 7
All Star Game (reps from all teams)
RS Game 7 (Advanced) (end of game statistics)
Math Exploration #3
WEEK 8
RS Game 8 (Advanced) (end of game statistics)
End of Season Stats League Leader Stats
WEEK 9
1st Round Playoffs (Division Leaders – 1st & 2nd)
Math Exploration # 4
Championship Game (Division Champions)
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NBA Math Hoops: Compact Roadmap A 16 SESSIONS—Four weeks @ 4 sessions per week WEEK 1
Basketball 101 Introduce Basic Game
Play Sample Game (whole class)
Play Sample Game (small groups)
Form League (2 Divisions) Draft Teams
WEEK 2
Create/Adopt Schedule Prep for Regular Season (RS)
RS Game 1 (Basic) (end of game statistics)
RS Game 2 (Basic) (end of game statistics)
RS Game 3 (Basic) (end of game statistics)
WEEK 3
Introduce Advanced Game Play Sample Game (WG)
Play Sample Game (small groups)
RS Game 4 (Advanced) (end of game statistics)
RS Game 5 (Advanced) (end of game statistics)
WEEK 4
RS Game 6 (Advanced) (end of game statistics)
RS Game 7 (Advanced) (end of game statistics)
RS Game 8 (Advanced) (end of game statistics)
Championship Game (Division Champions)
NBA Math Hoops: Compact Roadmap B 16 SESSIONS—Eight weeks @ 2 sessions per week WEEK 1
Basketball 101 Introduce Basic Game
Play Sample Game (whole class)
WEEK 2
Play Sample Game (small groups)
Form League (2 Divisions) Draft Teams
WEEK 3
Create/Adopt Schedule Prep for Regular Season (RS)
RS Game 1 (Basic) (end of game statistics)
WEEK 4
RS Game 2 (Basic) (end of game statistics)
RS Game 3 (Basic) (end of game statistics)
WEEK 5
Introduce Advanced Game Play Sample Game (WG)
Play Sample Game (small groups)
WEEK 6
RS Game 4 (Advanced) (end of game statistics)
RS Game 5 (Advanced) (end of game statistics)
WEEK 7
RS Game 6 (Advanced) (end of game statistics)
RS Game 7 (Advanced) (end of game statistics)
WEEK 8
RS Game 8 (Advanced) (end of game statistics)
Championship Game (Division Champions)
xiv • INTRODUCTION
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PA RT
I
Game Rules
t the core of the NBA Math Hoops experience are the simulated basketball games that students play using the enclosed board game materials. In this section (Part I), you will find an explanation of the rules and step-by-step processes for playing both the Basic and Advanced versions of the game.
A
Out-of-school providers looking to quickly engage students in high-interest game play will find this the ideal place to begin. Simulated games can then be modified to best meet allotted times and individual schedules. Take note that Part I is strictly about setting up for and engaging in game play. Teachers and out-of-school providers looking to provide students with a more comprehensive classroom experience—including organization and implementation of an exhibition season, regular season, and possibly a playoff and/or tournament structure—will want to reference Part II: Creating a Math Hoops League. Those desiring to progress further and have students delve into some of the rich mathematical exercises and investigations that emerge from the game and classroom experience should turn to Part III: Strategies and Math Explorations for Successful Math Coaches. For help in planning how much and what parts of NBA Math Hoops are right for you, turn to Creating a Math Hoops Roadmap on page vii.
In this section Unit 1: Basic Game Rules Unit 2: Advanced Game Rules Unit 3: Keeping Score Unit 4: The Shot Clock
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UNIT
1
BASIC GAME RULES M
ath Hoops is played in the same way that basketball is played: opposing teams take shots to score points, and, at the end of a game, the team with more points wins. This section explains how to play Math Hoops. The following rules can be used in both the Basic Game and Advanced Game. Additional rules pertaining solely to the Advanced Game are found in Unit 2: Advanced Game Rules.
Creating a Classroom Experience If you elect to incorporate playing Math Hoops into a larger classroom experience, it is highly recommended that you first introduce your students to the Basic Game regardless of their academic level. Once they are comfortable with basic game play, you can have them turn to the more complex rules for the Advanced Game and begin playing League games.
Use the following units in Part II to help create a classroom experience: UNIT 5: PRACTICE GAMES (PRE-SEASON) UNIT 6: ORGANIZING THE LEAGUE
Developing Math Skills Whether or not you elect to create a complete classroom experience, you will find numerous opportunities to investigate math concepts that are inherent in the Basic Game. The explorations found in Part III can be introduced while students are learning the rules and starting to play the Basic Game. The following explorations should flow naturally from the situations students will face at this time. GAME RULES
RELATED EXPLORATIONS
MATH CONTENT
1.1 Setting Up for Play
Unit 10 Understanding Player Cards Unit 11 Analyzing the Game 12.1 How Do You Make a Player Draft That's Fair?
• Understanding Fractions, Decimals, and Percents • Operating with Whole Numbers and Decimals • Using Proportional Reasoning • Understanding and Using Angle Measures and Circle Graphs • Using Number Patterns to Solve Problems
1.2 Playing the Game
12.2 Which Players Should You Choose for Your Team? 13.1 Who Gets the Ball? 13.2 Go for Two Points or Three Points? 13.5 Foul Play: When Is the Best Time to Foul?
• Comparing Decimals and Percents • Using Probability and Expected Value • Analyzing and Interpreting Data
2 • PART I: GAME RULES
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1.1 Setting Up for Play Your Classroom Kit includes enough materials for 8 complete Math Hoops games. Group students so that there are at least 2 coaches for each team. In a group of 32 students, you can have up to 8 games of Math Hoops in play at the same time. Each game consists of the following items: • 1 Game Board
• 2 Game Scoresheets
• 32 Player Cards
• 8 Foul Cards
• 2 Shot Planners (dry-erase
• 1 Game Token
boards with pens)
• 1 Scoreboard
• 2 spinners • 2 ten-sided dice
In addition, the Classroom Kit contains 4 Shot Clocks. Using the Shot Clock is optional and you may want to wait until students are comfortable with the game before introducing it.
Before the Game Starts Two teams play Math Hoops, with at least two coaches on each team. Each team needs five Player Cards, one of each different color. The process of selecting which Player Cards will constitute a team will be part of the fun for many students. Make sure you allot sufficient time for students to review the Player Cards before deciding on their players. Before play begins, a coach from each team writes its players’ names on its Game Scoresheet, making sure to use the color of the Player Card to write each player’s name in the correct space. Once scoresheets are filled out, have teams exchange them. During the game, coaches will record statistics for their opponents. Having coaches complete Game Scoresheets for the opposing team keeps both teams actively involved in each play of the game.
Timing the Game Math Hoops is played in two halves, just like professional basketball. The length of a game can be dictated by your classroom constraints, but the suggested time is 15 minutes per half for the Basic Game and 20 minutes per half for the Advanced Game. You will need to use a timer or clock to let students know when to start and end each half.
CLASSROOM CONNECTION A more formal process for drafting players is explained in Part II. 6.1: CREATING TEAMS AND LEAGUE DIVISIONS PP. 37–38
MATH CONNECTION For math explorations related to drafting a team and determining which players to select, take a look at the first two explorations in UNIT 12: CREATING TEAMS AND A SEASON SCHEDULE PP. 84–92
The Game Board The Game Board has two sides—one for the Basic Game (identified by the red border) and one for the Advanced Game (identified by the blue border). Both sides of the Game Board show a picture of a basketball court with even numbers
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on one end of the court and odd numbers on the other end. Teams start the first half on one end of the court and then switch to the other end for the second half. The purpose of this switch is to simulate what occurs in a real basketball game and to have students deal with the different scenarios that arise from odd and even numbers in the game.
1.2 Playing the Game CLASSROOM CONNECTION Tips and guidelines for how to introduce the game to your students can be found in 5.1 INTRODUCING THE BASIC GAME PP. 30–33
The first play of the game is like a tip-off (the initial jump ball) in a real basketball game and you can refer to it as such. To see who wins the tip-off, have a coach from each team roll the two dice and find their sum. The team with the greater sum gets possession of the ball and takes the first turn. If the higher sum is an odd number, they start on the Odd end of the court and if it’s an even number, they start on the Even end. (If the sums are equal, have both teams roll again.)
First Turn To get students into the game quickly, no math is required for the first turn of the game. Whichever team won the tip-off should move the game token to the number equal to their winning sum and take the shot. No fouls are allowed on this first shot. Roll and Record After the first shot, teams will alternate turns by rolling the dice and using the Basic Game Shot Planner to determine which number on their end of the court they can place the ball on to take a shot. One coach rolls the dice and a second coach records the rolls, larger number first, in the boxes on the Shot Planner labeled ROLL.
4 • PART I: GAME RULES
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To the right of the two recorded rolls on the Basic Game Shot Planner, there are four shaded cells with the operation symbols: +, –, ×, and ÷. To determine the number that belongs in each cell, coaches perform the appropriate operation with the two numbers they rolled, operating from left to right. If one of the results is a number on their end of the court, they get to take a shot. To use a division result, they must get a whole number quotient. EXAMPLE A team’s two rolls are 5 and 2. Performing the four operations yields: 5 + 2 = 7 5 − 2 = 3 5 × 2 = 10 5 ÷ 2 = 2.5 The picture below shows what the Even team should record on its Shot Planner. ROLL
5 2
+
7 3 10 –
×
MATH CONNECTION
÷
Notice that there is no number written in the space where division is recorded. When the quotient isn’t a whole number, teams put a slash to indicate that this square on the Shot Planner isn’t being used.
Division Rule Options
To give students some insight into and practice with the Shot Planner, use the first half of the exploration in 11.1 INVESTIGATING THE SHOT PLANNER PP. 75–77
CLASSROOM TIP
For the Basic Game, the intent is to first focus on pure whole number operations. The first three operations make this easy. With division, whole number quotients surface less than half of the time so when they don’t, the diagonal line or “slash” is used. Once students get comfortable with this process, you may want to introduce the division rule for the Advanced Game. Here students are allowed to round up or down, depending on which works to their advantage. What they will quickly discover is that unless the roll of the dice includes a 0 or 1, the quotient will always be a 1, 2, 3, 4, or 5.
STEALS
While a team does the arithmetic on the Shot Planner, their opponents check their work. If the team records any incorrect result, an opposing team coach can declare “Stolen ball!” At that point, the opposing team must prove their accusation. If they’re correct, they gain possession of the ball. They move the token onto any circle on their end of the court that is the same color as the player who had the ball at the time the Steal was declared. The team that lost the ball can call a foul on that player. Otherwise, their opponents take a 2-pt. or 3-pt. shot
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depending on where they placed the game token. The Shot Planner is not used when taking a shot after a Steal.
Dealing with Steals
CLASSROOM TIP
Because steals are a part of the game and can work to a team’s advantage, students should be encouraged to keep a careful eye on their opponent’s calculations. One thing they may want to do is have one co-coach watch what their opponents are recording on their Shot Planner while the other co-coach uses her or his own Shot Planner to check the calculations. This way when the opponent goes to move the ball to the selected shooting location, they can be asked to show the specific calculation and compare results. As Commissioner, be prepared to moderate should any computation disputes arise!
After the team that steals the ball selects a player, a Steal is recorded for that player on the Game Scoresheet. If a team challenges a math calculation that turns out to be correct, they do not get credited with a steal. Instead, the team on offense is automatically awarded 2 points or 3 points depending on where they had placed the ball. TURNOVERS MATH CONNECTION An investigation into odd and even numbers can be found in 11.2 WOULD YOU RATHER BE ODD OR EVEN? PP. 77–80
If on their turn a team can’t get the ball to a number on their end of the court, a turnover is called and their turn ends. Their opponents pick up the dice and start their turn. Turnovers aren’t recorded on the Game Scoresheets. Turnovers will occur in the Basic Game only. They most likely will not occur often but they will come up. Curious students can be prompted to investigate the likelihood of occurrence and whether the likelihood is greater with Even or Odd numbers. DOUBLE ZEROS (FAST BREAK)
A special situation occurs when a team rolls two zeroes. Every operation results in zero, which is not a number on the Game Board. Normally, this would be considered a turnover, but instead of penalizing a team, who won’t even get a chance to use the Shot Planner, this becomes a favorable circumstance called a Fast Break. In basketball, a Fast Break is a play that moves the ball down the court quickly as soon as a team gets possession of it. A team that gets a Fast Break is usually able to take a shot (or a slam-dunk) before the opposition has had a chance to regroup. In Math Hoops, a team that gets a Fast Break can put the game token on any number on their end of the court and take a shot and their opponents can’t foul that player.
6 • PART I: GAME RULES
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Pick a Shooter After filling out the Shot Planner, a team chooses any applicable number and moves the game token to that circle on the Game Board. The color of the numbered circle indicates which player on the team will take the shot. If none of the Shot Planner results match a number on their end of the court, it’s a turnover and the other team begins their turn. EXAMPLE 1 A team is playing on the Even end of the court: ROLL
7 3
10 4 21
+
–
×
÷
MATH CONNECTION Deciding which shooter to pick when there are multiple options is a skill that will improve if coaches field the same team over many games. An exploration to help students think critically about their choices can be found in 13.1 WHO GETS THE BALL? PP. 94–96
This team has a choice of having the player corresponding to #10 or the player corresponding to #4 take a shot. A shot from #10 would be a 2-point shot from a blue Forward. A shot from #4 would be a 2-point shot from a red Guard. If the coaches choose #10, they would declare “10 to shoot,” and place the game token on that circle. Then they would place the transparent spinner on the circle graph of their blue card and spin.
EXAMPLE 2 A team is playing on the Odd end of the court: ROLL
8 4
12 4 32 2
+
–
×
÷
Because all of the calculations yield an Even number, this team has no options for moving the ball to their end of the court. This is a turnover and the dice are passed to the Even team to begin their turn.
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FOULS MATH CONNECTION A discussion on how to help students strategize when and how to use fouls, along with an explanation of the math behind some strategies, can be found in 13.5 FOUL PLAY: WHEN IS THE BEST TIME TO FOUL? PP. 103–107
After a team decides on a shooter and moves the game token to that circle on the Game Board, their opponents have the option of fouling the player. Make sure students know that fouls must be called BEFORE the offense takes a shot. This will keep the defense actively engaged since they’ll need to be ready to call a foul quickly after the offense moves the game token. It will also prevent the offense from spending too much time on their turn, which will step up the pace of the game. This will create more of the feel of a real basketball game and it will be good practice for working up to using the Shot Clock. Because fouls add complexity to the game, you may want to have students play a number of games without fouls and add those rules when you see that they are comfortable with the regular sequence of steps in a turn. Here are the nitty-gritty rules regarding fouls: • To call a foul, the defense needs to lay down one of the Foul Cards before the offense spins the spinner.
• Each team is allowed 5 fouls per half. • When a foul is called, the foul is recorded on the defense’s Game Scoresheet. The defensive player with the corresponding color to the shooter chosen by the offense is credited with the foul.
• No player can have more than 3 fouls per game so they should be used strategically. ODD AND EVEN ENDS OF THE GAME BOARD
Each end of the Basic Game Board has 20 numbered circles on it—6 circles for 3-point shots and 14 circles for 2-point shots. The Even end has almost every even number you can get with operations from two 10-sided dice. (It doesn’t have 54 or 72). The Odd end has all the odd numbers you can get with operations from two 10-sided dice, but there are fewer combinations that result in odd numbers than in even numbers. Instead, there are three odd numbers that are repeated: 15, 35, and 63 can be found in two different places on the Game Board. A team with any of those results has an option to take either a 2-point shot or a 3-point shot.
Odd and Even Inequities
CLASSROOM TIP
As noted above, the Odd end has all of the odd numbers you can get with operations using two decahedra dice but the Even end does not. This is because viable operations yield twenty-two possible even numbers and only seventeen possible odd numbers. To make things semi-equitable and add another interesting twist, three of the Odd numbers are duplicated so that the number is found in both a 2-pt. and 3-pt. shooting location. Another fun side-exploration some of your more inquisitive students may want to investigate!
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Take the Shot 2-POINT SHOTS
A team attempts a 2-point shot whenever they place the ball on a circle inside the 3-point line. (Those circles are labeled “2 POINTS”.) To take the shot, a coach places the transparent spinner over the circle graph on the appropriate Player Card and spins it forcefully. You may want to require that the spin make a minimum of two revolutions to be considered a legitimate spin. If the point of the spinner lands in one of the orange zones, the shooter makes the shot. The scorekeeper on the other team records this by circling a 2 in the Field Goal column for the shooter. If the point of the spinner lands in one of the gray zones, the shooter misses the shot. The other team records this by drawing a diagonal slash through a 2 in the Field Goal column for the shooter. If the spinner arrow lands on a line, the offense should spin again.
When the Spinner Lands on a Line
CLASSROOM TIP
Although it won't happen often, “liners” will occur. It’s best to anticipate this happening and establish proper protocol before starting a game. If the offense claims the arrow is on the line, the defense should confirm it. If there is disagreement, as Commissioner, you can step in and make the call.
The Game Scoresheet below shows a player who, in one game, missed one 2-point shot and then made three 2-point shots in a row. MATH CONNECTION An investigation into strategies students can use when deciding whether it would be better to take a 2-pt or 3-pt. shot can be found in 13.2 GO FOR TWO POINTS OR THREE POINTS? PP.97–99
3-POINT SHOTS
On each end of the Game Board, six circles lie outside the 3-point lines. Each of these circles is marked “3 POINTS.” To attempt a 3-point shot, use the same spinner as for a 2-point shot, but record success only if the spinner point lands inside one of the zones marked by black crosshatch. These zones always lie within orange zones. When a 3-point attempt is successful, the opponent records this by circling a 3 in the Field Goal column for the shooter. When a 3-point attempt is unsuccessful, the opponent records this by drawing a diagonal slash through a 3 in the Field Goal column for the shooter. UNIT 1: BASIC GAME RULES • 9
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FREE THROWS
When a team has decided which player will shoot for their turn, the opposing team has the option to foul that player. They do this by playing one of their foul cards BEFORE the shooting team spins the spinner. A player who is fouled takes free throw shots instead of 2-pt. and 3-pt. field goals. To take a free throw shot, a coach rolls both dice. On the right side of the shooter’s Player Card, there is a free throw grid. The top row of the grid lists numbers for the red die and the left-hand column lists numbers for the black die. To see if a free throw shot is successful, look at the square that corresponds to the dice. An orange square means the shot was made. A gray square means the shot was missed. EXAMPLE A team rolls a black 3 and red 5 for their shooter. The picture below shows how to read the results. Since the free throw grid shows a grey square, the shooter missed the shot. 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9
Grey = Miss
Rules for how many free throws a player gets are different in each half of the game: IN THE FIRST HALF: A
player fouled while attempting a 2-point field goal is put in a “One and One” situation. The player attempts one foul shot. If unsuccessful, possession of the ball passes to the other team. If successful, the player attempts a second foul shot. After the second shot, possession of the ball passes to the other team. IN THE SECOND HALF: A player fouled while attempting a 2-point field goal is awarded two foul shots. After the second shot, possession of the ball passes to the other team.
A player fouled while attempting a 3-point field goal is awarded three foul shots. After the third shot, possession of the ball passes to the other team.
AT ANY POINT IN THE GAME:
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Foul shots are recorded on the Game Scoresheet under the Free Throw column. A successful free throw is recorded by circling the number 1. A missed shot is recorded by writing a slash across a number 1.
Optional Rules The Advanced Game contains a number of different rules that make it more challenging, the most obvious being the use of a different Game Board! If your students are performing at an advanced level, you’ll probably want to introduce them to the Advanced Game quickly so they can experience the richer array of strategies that are involved. If your students are struggling, let them play the Basic Game until they gain proficiency with the basic rules. You can then start acquainting them with some additional rules that can add complexity to the game a little at a time. This can add variation to the Basic Game as well as provide an easier transition to the Advanced Game. DEFENSIVE REBOUNDS
Students will see two columns on their Game Scoresheet that aren’t used in the Basic Game: Rebounds and Assists. Recording Assists will make sense only when they begin using the Advanced Game Board, which includes passing lanes. You can, however, introduce them to Rebounds much earlier, especially if you’re looking to use Math Hoops to develop skills involving data collection and statistical reasoning. The Advanced Game rules explain two different types of rebounds: Offensive Rebounds and Defensive Rebounds. For the Basic Game, you’ll want to focus on Defensive Rebounds. A Defensive Rebound is awarded to a player who gets the ball after an opponent misses a field goal or the final free throw of their possession. Defensive Rebounds are tracked on the Game Scoresheet by circling an R in the Rebounds column for the player.
Why Include Defensive Rebounds
CLASSROOM TIP
Students may wonder what purpose tracking defensive rebounds serves. Although this data doesn’t give a complete picture, it can provide some indication of how an opponent’s player is performing. For example, a blue Forward with 5 defensive rebounds in the first half of the game tells that team’s coaches that the blue Forward for their opponent has missed 5 shots in the first half of the game. Riding this wave, they may want to elect not to foul this player in the second half as she/he appears to be having an “off” night from the floor!
ADVANCED DIVISION RULES
Another concept from the Advanced Game that you can introduce to the Basic Game is how to deal with division. In the Basic Game, because you can’t use
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division results that aren’t whole numbers, students will often be choosing their shooter from one of the other operation results. In the Advanced Game, students will be asked to round quotients up or down, which will allow them to almost always use a division result. If your students are comfortable with all the aspects of the Basic Game, try putting this rule into effect. Allowing your students to round their quotient either up or down is a good way to start building their estimating skills. As they use this rule more, they will begin to see that they don’t need to figure out exact quotients as it is more important to determine what numbers the quotient is between. As a consequence, students will have a division option much more frequently and there will be far fewer turnovers.
1.3 End of Half / End of Game Beginning the Second Half At the end of the first half, teams switch from one end of the court to the other. So the team playing the Odd end now plays the Even end and vice versa. The team that did NOT start the first half starts the second half of the game. On the first turn, a team rolls the dice and fills in the Shot Planner, but they can then choose a player whose color corresponds to ANY number in their results. Fouls can’t be called on this first turn. The game comes to a halt when time expires for either half. If the team last in possession of the ball has not yet rolled its dice, the half ends immediately. If the team last in possession has rolled its dice, it is allowed to complete its play. Should the ball be stolen (as explained above) after time expires for the half, play does not continue and the half ends immediately. No statistic is then recorded for steals. When the second half ends, the team with the higher score is the winner. Have teams exchange scoresheets to verify each other’s totals. Any discrepancies uncovered are to be mediated judiciously by the League Commissioner of Math Hoops, i.e., you. MATH CONNECTION If you want to engage students in thinking about strategies related to the lightning round, turn to 13.6 LIGHTNING ROUND STRATEGIES PP. 107–108
The Lightning Round Should the game end in a tie, overtime is played in a lightning round. Starting with the team that ended the game as Even, teams take turns having three different players on their team attempt a 2-point or 3-point field goal. No fouls are allowed in overtime and teams can select any players to shoot, as long as three different players are used. At the conclusion of the lightning round, the team with the higher score is the winner. Should the game remain tied at the end of the lightning round, it is declared a tie. When a game is finished, teams exchange Game Scoresheets and coaches complete their own sheets as a final check. This includes adding up point totals for each
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player and filling in the final scores. An example of a final Game Scoresheet is shown below.
UNIT 1: BASIC GAME RULES • 13
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UNIT
2
Advanced Game Rules T he Advanced Game of Math Hoops uses the same Player Cards as the Basic Game, but the game board looks a little different. As shown in the picture on the following page, the board for the Advanced Game has 30 circles on each end of the court and dotted line segments connecting many pairs of circles. Students also use a more complex Shot Planner to work out the more complicated math operations required by the Advanced Game. On the following pages, you will find the rules that are different from or additional to the ones in the Basic Game.
Creating a Classroom Experience If you elect to incorporate playing the Advanced Game into a larger classroom experience, you should introduce the Advanced Game Rules only after your students feel comfortable with the Basic Game. The Advanced Game has more complex and challenging features and can best be understood if students are building on the knowledge and comfort they have with the Basic Game.
Use the following units in Part II to help create a classroom experience: UNIT 5: PRACTICE GAMES (PRE-SEASON) UNIT 6: ORGANIZING THE LEAGUE
Developing Math Skills Whether or not you elect to create a larger classroom experience, you will find numerous opportunities to investigate the math concepts that are inherent in the Advanced Game. The following explorations from Part III should flow naturally from the situations students will face in the Advanced Game. GAME RULES
RELATED EXPLORATIONS
MATH CONTENT
2.1 Playing the Game
13.1 Who Gets the Ball? • Comparing Decimals and Percents 13.2 Go for Two Points or Three Points? • Using Probability and Expected Value 13.5 Foul Play: When Is the Best Time to Foul?
2.2 Assists and Rebounds
13.3 Should You Pass or Should You Shoot?
• Using Probability and Expected Value
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2.1 Playing the Game Flip the Basic Game Board over to play the Advanced Game. The immediate differences students will notice are that the Advanced Game Board has more circles on it (30 on each end of the court) and there are "passing lanes" connecting many of them. During the game, students will use the same Game Scoresheets as in the Basic Game but will now keep track of Rebounds and Assists in addition to the other statistics.
Beginning the Game The game starts in the same way as the Basic Game does: each team rolls the dice and the higher total goes first, placing the game token on the sum they rolled. Whether the sum is even or odd determines which end of the court the teams take to begin the first half. As in the Basic Game, the first shot is taken without the need to do any other math and without the opposition being allowed to foul. Unlike the Basic Game, the shooter’s number is important for the next turn, so it should be recorded in the Ball On space of the defense’s Shot Planner.
Continuing Play After the first turn, the ball is given to the other team. They roll the two dice and then BOTH TEAMS complete the first (gray) row of their Advanced Game Shot Planners. The steps are the same as explained for the Basic Game. Coaches should mark a forward slash (/) in the appropriate gray cell if one of the operations results in a number that isn’t found on the Game Board. (This will occur when a team rolls a zero and when multiplication yields a product greater than 60.)
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When the top row is completed, there is a second set of calculations coaches need to do before they can determine a shooter: • In the space labeled “Ball On,” EACH team records the number on which the ball was last placed.
• Each team then completes the 4 × 4 array next to the “Ball On” number. Each column in the array is completed by using the “Ball On” number and the number in the gray cell at the top of that column to perform the four operations: +, −, ×, and ÷.
You’ll notice that instructions have been given to always operate with numbers from greatest to least. Sometimes the lesser number appears in the gray shaded top row where coaches record the results of the operations from the roll of the dice. Other times, the lesser number is the “Ball On” number. EXAMPLE Suppose that the Even team begins the game with the ball on the number 6 and the Odd team rolls 9 and 3. Here is how the two teams would fill out their Shot Planners. Numbers greater than 60 can be marked with a slash because they do not appear on the Game Board.
ROLL
9 3
BALL ON
6
12 6 27 3 + + + + 18 12 33 9 – – – – 6 0 21 3 × × × × 36 18 ÷ ÷ ÷ ÷ 2 1 5 2
+
–
×
÷
So the Odd team can choose any of these numbers: 1, 3, 5, 9, 21, or 33. Note that 27 is NOT an option. The numbers in the top row are used only to get the results below them.
Introducing the Advanced Game Shot Planner
CLASSROOM TIP
Given that the Shot Planner for the Advanced Game is more involved, you may find it helpful to unfold this process two operations at a time. For example, once students complete the top row of operations using the roll of the dice, they can work with those answers and the “Ball On” number to complete the rows for addition and subtraction. Shot selections can initially come from these rows as they play the game. Once they’re comfortable with completing and working from these two rows, bring in the multiplication and division rows. This will provide students with even more options to consider. Because multiplication and division play a special role in the Advanced Game, coaches might eventually find they want to work from the bottom up in order to complete those first. [See OFFENSIVE REBOUNDS on p. 20.]
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TIME STRATEGY
As explained later in Unit 4: The Shot Clock (p. 24–25), you have the option of limiting the time a team has to shoot the ball. With that in mind, the prospect of performing 16 extra arithmetic procedures may seem daunting at first to players. The pedagogical intention is that players will realize on their own various strategies to “speed up their game.” As the Advanced Game begins, the most obvious strategy to suggest is that teams divide the arithmetic among the coaches whenever they get the ball. Other strategies will develop over time (some of which are discussed in Part III).
Advanced Division Rules ROUNDING QUOTIENTS
In the Basic Game, a division result that isn’t a whole number is discarded and division is not an option for the team’s shooter. In the Advanced Game, the division rules are a little different: If division produces a quotient that is not a whole number, a team can round that quotient down or up to the either of the two nearest whole numbers. This is true both for the top row of calculations on the Shot Planner and for the 4 × 4 array of calculations below the top row. As a consequence, teams will almost always have division results that are usable. EXAMPLE Suppose a team rolls a 9 and 4 in the Advanced Game. The quotient of 9 ÷ 4 is not a whole number. It’s 2 R1 or 2.25. So, in the division cell in the top row of the Shot Planner, a team can write either 2 OR 3.
Whichever way a team decides to round the quotient in the top row will make a difference in the final results on the Shot Planner since that number is then used with the Ball On number. Students may think that they should always round that first result to match which end of the court they’re on—i.e., the Even team would round to an even number. This reasoning is appropriate when using division with the Ball On number, but for the top calculation, this is not necessarily true. You may want to have students try rounding both ways for a few games to see how the decision they make in the top row affects their options.
MATH CONNECTION To help students explore how the new rules for division might affect games, turn to 11.3 HOW WILL THE ADVANCED DIVISION RULES AFFECT YOUR STRATEGY? PP. 81–83
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division by zero
CLASSROOM TIP
When a team rolls the same number on both dice, they will get 0 when subtraction is performed. In the Basic Game, this simply means that the team has no play for that number. In the Advanced Game, the entry of 0 must be used to perform further arithmetic with the Ball On number. Addition, subtraction, and multiplication should pose no problem, but some students may get frazzled when it comes to division. For example, suppose a team has to combine 0 with 12 using division. A student might think that 12 ÷ 0 = 0. The simple truth is that 0 ÷ 12 = 0 but 12 ÷ 0 cannot be performed! A simple way to convince students of this is as follows: You know that any division fact has a related multiplication fact. For example, 12 ÷ 3 = 4 only because 3 × 4 = 12 and 182 ÷ 13 = 14 only because 13 × 14 = 182. What about 12 ÷ 0 = 0? If this were true, it must also be true that 12 = 0 × 0. But that statement is ridiculous. And so 12 ÷ 0 = 0 is also ridiculous. If any student is still troubled following the above explanation, ask the student to imagine dividing 12 by progressively smaller numbers that get closer and closer to 0. For example, divide 12 by 0.1, 0.01, 0.001, 0.0001, and so on. As the divisor gets closer to 0, the quotient gets greater and greater: 120; 1,200; 12,000; 120,000, and there is no end in sight! This idea of dividing by a number that approaches but never reaches 0 is treated formally in calculus.
Beginning the Second Half As in the Basic Game, teams switch sides at halftime, with Odd becoming Even and Even becoming Odd. The team that did not start the game will start the second half. For the first turn, a coach rolls the dice and fills out the top row only of the Shot Planner. There is no Ball On number yet, so the team uses the results from the top row to select a shooter. For this turn only, the team can use any of the results, whether odd or even, to determine placement of the game token. A player can’t be fouled on the first turn. The Lightning Round The end-of-game rules are the same as those for the Basic Game, with one difference: teams must keep track of rebounds during the lightning round of the Advanced Game.
2.2 Assists and Rebounds Two new concepts are introduced in the Advanced Game. Because the Advanced Game Board contains passing lanes, teams now have the option of placing the ball on a number and then passing it to a player connected by a passing lane.
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Also, teams will now keep track of Defensive and Offensive Rebounds. With Offensive Rebounds, it now becomes possible for a player to miss a shot and get a second chance to shoot.
Assists If you look at the Advanced Game Board, you will see that most (but not all) of the numbers have a dotted line connecting them to another number. The dotted line means that a team, after using the Shot Planner to get the numbers for potential shooters, can choose to have one of these players take the shot or can pass the ball to the number connected by the dotted line and have THAT player shoot instead. EXAMPLE Go back to the Shot Planner results shown on page 16. The options for the Odd team were 1, 3, 5, 9, 21, and 33. Suppose the Odd team chooses 21 to get the ball. They would move the game token to the number 21. They can then declare “21 to shoot” or they can say “21 passes the ball to 19 to shoot” and then move the game token to 19.
MATH CONNECTION You can have students investigate strategies around passing the ball by turning to 13.3 SHOULD YOU PASS OR SHOULD YOU SHOOT? PP. 99–100
If the player receiving a pass scores a field goal, the player that made the pass is awarded an assist. This is recorded in the Assists column of the Game Scoresheet. (Assists are not awarded if the player misses the field goal.) For example, if 21 passes the ball to 19 and 19 scores a 2-point field goal, 19 gets credit for the points and 21 gets credit for an assist on the Game Scoresheet. As always, it is the responsibility of the opposing team to record these statistics. When a pass is made, the defense has the option of fouling the player making the pass or the player receiving the pass. In the example above, the Even team could foul either 21 before the pass is made or 19 following the pass. Assists are not credited when free throws are scored.
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Rebounds In basketball, rebounds play a vital role to a team’s success. Math Hoops incorporates both Defensive and Offensive Rebounds into the Advanced Game. Offensive Rebounds are a little more complex so you may wish to wait until students are comfortable with the other aspects of the Advanced Game before introducing offensive rebounds into game play. DEFENSIVE REBOUNDS
When a field goal or the final free throw of a team’s possession is unsuccessful, the player on the opposing team that first gets the ball is credited with a Defensive Rebound. This is recorded by circling an R in the Rebounds column of the Game Scoresheet for the player. OFFENSIVE REBOUNDS
Students faced with the additional calculations and choices available in the Advanced Game will find that it takes longer to decide which player should take a shot than in the Basic Game. The urge to move the game along may mean that initially coaches will focus less on strategy than on simply getting the ball to a player who can take a shot. (This will be especially true if you involve the Shot Clock.) As they gain more facility, they will try harder to steer the ball to their best shooters. At that point, the Offensive Rebound rule will become a very attractive option. In basketball, an offensive rebound is given when a player on offense misses a shot and another offensive player recovers the ball. This gives the offense another chance to make a play and score. To simulate offensive rebounds in Math Hoops, teams on offense are given a chance to take a second shot under the following circumstances: • After working out Shot Planner options, a team places the ball on a number that is the result of multiplication or division.
• The player corresponding to that number can’t pass the ball but must be the one to take the shot.
• The opposing team does not foul that player.
If these criteria are met, and the player on offense misses the shot, the team earns the right to try for an Offensive Rebound. (If the player makes the shot, the ball is turned over to the defense as usual.) To determine if an Offensive Rebound is made, a coach from each team rolls a die after the shot is missed. (If you are using the Shot Clock, you can reset it for the next turn. You don’t use the Shot Clock for an Offensive Rebound itself.)
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• If the offense rolls a number GREATER THAN than the defense, they are awarded an Offensive Rebound and get to take another shot. The offense can have the same player take a second shot, or if that player has a passing lane connecting to another player, they can pass the ball first. The defense can choose to foul either the player who makes a pass or the player who takes a shot.
• If the offense rolls a number LESS THAN OR EQUAL TO the defense, there is no Offensive Rebound. The ball is turned over to the defense who starts their turn as usual.
When a team gets an offensive rebound, the original shooter is credited with the rebound by drawing a square around an R in the Rebounds column of the Game Scoresheet. Only one offensive rebound is allowed on any turn. If a team gets an offensive rebound, but misses the shot, there is no possibility for a second offensive rebound on that turn. The ball is simply turned over to the defense.
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UNIT
3
Keeping Score Creating a Classroom Experience The Game Scoresheets will allow your students to keep track of how well their players are performing in a number of different categories throughout the season. Keeping track of player and team performance and win/ loss records will help bring the idea of a Math Hoops league to life. In addition, the two Classroom Posters included in the Classroom Kit (Team Standings/League Leaders and Math Hoops Tournament posters) will allow you to build excitement throughout the season leading up to a playoff series and championship event.
Use the following units in Part II to help create a classroom experience: UNIT 7: KEEPING STATISTICS UNIT 8: MID-SEASON REVIEW / ALL-STAR GAME UNIT 9: LEAGUE PLAYOFFS AND CHAMPIONSHIPS
Developing Math Skills Keeping score can involve much more than just finding the winner of any particular game. Have your students use their Game Scoresheets to analyze how well their team is doing, whether they should make adjustments to their coaching strategies, and which players, if any, they might want to replace. The Student Manual Appendix contains some templates that will be particularly useful for team analysis. In addition, you can use the following math explorations to help students gain a better understanding of statistics and probability. GAME RULES
RELATED EXPLORATIONS
MATH CONTENT
Game Scoresheets
Unit 15: Why Do Coaches Need Statistics?
• Comparing and Ordering Decimals and Percents • Collecting, Analyzing, and Displaying Data
3.2
3.1 Using the Scoreboard Each Math Hoops game comes with a Scoreboard, which teams will use to keep track of the score as games are played. The Scoreboard tracks the total number of points only. To keep track of all other statistics, the Game Scoresheets are used. You may wish to have students use the Scoreboard alone when they’re first starting out.
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Under the Scoreboard wheels, there is a summary of the basic rules for the Basic and Advanced Games. As League Commissioner, you will likely be asked to give a ruling on all sorts of specific circumstances that arise. The brief summary will hopefully give students a resource they can turn to for the most common questions that come up.
3.2 Game Scoresheets Before a game begins, each team prepares its Game Scoresheet by filling in its players’ names. The scoresheets are then exchanged between teams, since teams keep the game stats for their opponents. Along with carefully watching each spin, this keeps all coaches actively engaged throughout each game. The Game Scoresheet has eight columns: • Player
• Fouls
• Field Goals
• Rebounds (Advanced Game only)
• Free Throws
• Assists (Advanced Game only)
• Steals
• Total Points
A coach keeps records as the game is played by: • Circling the appropriate number when a field goal (2, 3) or free throw (1) is made.
• Placing a slash through the appropriate number when a field goal or free throw is missed.
• Circling the appropriate letter or number when a Steal, Foul, Defensive Rebound, or Assist occurs.
• Drawing a square around an R in the Rebounds column when an Offensive Rebound occurs.
At the conclusion of a game, teams give back the scoresheets to their opponents. Coaches then complete the Total Points column for each player on their own team. To do this, simply add together the circled 1s, 2s and 3s for each player. After calculating each player’s total points, a coach adds the five numbers together to get the total score. These numbers should match the score shown on the Scoreboard at the end of the game. If there is any discrepancy, the Game Scoresheet should be considered the official record. If there is a discrepancy that affects the outcome of the game, you will need to step in as arbitrator.
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UNIT
4
The Shot Clock Creating a Classroom Experience It is not necessary to use the Shot Clock to create a Math Hoops league and a regular season schedule. However, the introduction of a time limit will speed up the pace of the games and add to the similarity to real basketball games.
Use the following units in Part II to help create a classroom experience: UNIT 5: PRACTICE GAMES (PRE-SEASON) UNIT 6: ORGANIZING THE LEAGUE
Developing Math Skills The addition of the Shot Clock to either the Basic or Advanced Game will raise game play to more challenging levels. In the Advanced Game, students will find it difficult to complete all the squares in the Shot Planner. This is to be expected and you can help coaches realize that making decisions on which operations to complete is part of the game. Understanding number patterns will help them target just those operations that will give them an odd or even result or will give them a result that isn’t on the Game Board (zero or any number greater than 60). Coaches may also realize that analyzing the statistics on their Player Cards beforehand will help them save time during the game when deciding the best player to get the ball in different situations. GAME RULES
RELATED EXPLORATIONS
MATH CONTENT
Unit 4: The Shot Clock
13.4: Can You Beat the Shot Clock
• Using Properties of Integers to Solve Problems
Math Hoops comes with a shot clock that can be used in either the Basic Game or the Advanced Game. A toggle switch on the shot clock can be set to 24 seconds (professional level) or 35 seconds (collegiate level). As soon as the team with possession of the ball rolls its two dice, its opponent starts the shot clock clicking. (As commissioner, you can set the alternative rule that the shot clock starts ticking once a team records its two rolls, or, in the Advanced Game, once a team records the top row of its Shot Planner.) The team with the ball must then perform its arithmetic and indicate which of its players is getting the ball before the shot clock’s buzzer sounds. If
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the buzzer sounds before the team selects a number that it has written on its Shot Planner, possession of the ball immediately passes to its opponent. No statistics are recorded in this situation.
When to Use the Shot Clock
CLASSROOM TIP
The shot clock unquestionably adds even more excitement to an already action-packed game! We suggest you use this as an incentive for students to work towards greater fluency and proficiency with their operations—regardless of level. It’s intentional that the Classroom Kit includes only 4 shot clocks with 8 complete games as the shot clock should not be used at all times. Some games should focus on speed and accuracy—others more on employing deliberate strategy. Balance is the key operative and making sure that all students have the opportunity to build this feature into their Math Hoops experience.
When Math Hoops is first played, the shot clock can be excluded as players learn the ins and outs of the game. But, once students have a few games under their belts, its inclusion is highly recommended, as it adds a dynamic and pressure-packed dimension to the game.
How the 24-Second Clock Found Its Way to the NBA In 1950, George Mikan played center for the Minneapolis Lakers and was a dominant figure, averaging 28 points per game in the National Basketball Association. The Lakers also played a zone defense that other teams had trouble scoring against. When the Fort Wayne Pistons came to Minneapolis on November 22, it had been nearly a year since a visiting team had defeated the Lakers at home. Fort Wayne’s coach, Murray Mendenhall, decided to try a new strategy. Instead of pushing forward on offense to score, he instructed his team to stall. Instead of shooting, they would pass, hoping to force the Lakers to abandon their zone defense. The strategy worked. At a time when a typical NBA final score was 82–77, Fort Wayne defeated Minneapolis 19–18 that night.
While the strategy worked for Fort Wayne and Coach Mendenhall, NBA officials were not thrilled: fans come to games to see scoring. The league decided to incorporate a new 24-second rule into the game. Once in possession of the ball, a team must take a shot within 24 seconds or surrender possession. The number 24 had been proposed by Danny Briscoe, the owner of the Syracuse Nationals. He had figured that two teams normally average about 120 field goal attempts per game. Dividing the length of a game, 48 minutes, by 120 yielded exactly 24 seconds! The 24-second rule was instituted for the 1954 season and remains in place to this day.
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PA RT
II
Creating a Math Hoops League
rganizing a Math Hoops League is, in essence, creating a total classroom experience and an awesome one at that! With a formal “season” of games and optional Playoffs, you’ll find students completely embracing the role of a Math Hoops coach and all that involves. Your role, as League Commissioner, is to chart the course and make sure students see the “Big Picture” right from the get-go.
O
So what do you need to consider as you begin to plan your league? The following table lists criteria you can use to construct the basic framework of the class experience.
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DURATION OF THE LEAGUE
Determine how much time you want to spend overall on this experience. Six weeks? Nine weeks? Longer?
NUMBER OF STUDENT MEETINGS
How often will your students meet in their league? Twice a week? Three times? More? How much time will they have each time they meet?
DEPTH AND COMPLEXITY OF THE GAME
What level of focus and attention do you want there to be on the game itself? You may want to concentrate on the Basic Game solely, or you may plan to increase the complexity of the game in stages, gradually introducing students to more advanced rules. If so, you can have students progress to the next level as a group or allow for movement from one level to the next as students demonstrate proficiency.
LEVEL OF STUDENT INVOLVEMENT
How directed do you want this experience to be? You can set up all the league parameters in advance or you can allow for some shared decision-making. There are many areas where you can have students help determine how the league will work: deciding on the number and organization of teams, working out a season schedule, deciding whether the Shot Clock will be used. Just be sure you factor in the time needed to have students help with the planning process.
MATH ACTIVITIES
What role will the Math & Strategy Explorations (found in Part III) play in the total classroom experience? Which ones fit best with your curriculum? Make sure you build in time for the ones you feel are most important for your students to do.
LENGTH OF THE SEASON
How long will your pre-season and regular season be? Before starting the regular season, you’ll want to be sure your students are comfortable with the level of games they’ll be expected to play once the scores “count”.
CULMINATING EVENT
Will your league culminate with playoffs and ultimately, a Championship Game? How about an All-Star Game—will that be part of your total classroom experience? These events generate a lot of excitement and drive motivation throughout a season.
PARENT AND/OR STAFF PARTICIPATION
What role will others play in your league? Other staff? Parents? How might you involve those interested in participating to enhance the total experience?
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The more you think through in advance, the more smoothly your league will run. Some advice to get you started—SMALL STEPS. Start with small, manageable steps and build from there. Success breeds success AND motivation to do more! If your students see the league as something you’re building together, even if you’ve established the initial parameters, the buy-in will be much greater. Use the information in this section as your guide but don’t be afraid to veer off the beaten path if you feel doing so would be of benefit to your students.
IN THIS SECTION Unit 5: Practice Games (Pre-Season) Unit 6: Organizing the League Unit 7: Keeping Statistics Unit 8: Mid-Season Review/All-Star Game Unit 9: League Playoffs and Championships
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UNIT
5
Practice Games (Pre-Season) T
he playing of pre-season games in the NBA serves a variety of purposes. Coaches use this opportunity to assess player strengths and weaknesses, develop team cohesiveness, and craft strategic plans for the upcoming regular season. Successful coaches make the most of this opportunity. As Commissioner of your about-to-be-formed league, your job is to provide your student coaches with the understandings and tools they’ll need to be successful, as well as set the tone and expectations for the upcoming season. The introduction of game play and providing the opportunity to compete in simulated head-to-head competition prior to forming a league and creating teams are critical steps in the progression of the NBA Math Hoops classroom experience.
Scheduling This Unit The table below outlines the collection of classroom activities found in this unit and a rough time estimate for each activity. Note that the estimated times are based on 45-60-minute sessions and will vary by academic level of students as well as familiarity and comfort with the content. Also listed are suggestions for math explorations from Part III that you can incorporate into the overall classroom experience. Feel free to pick and choose activities you feel will best serve your students based on interest and need. These explorations have been designed to further students’ mathematical understanding within the context of strategic game play. If you choose to incorporate any of the Part III explorations, be sure to make room in your schedule for the instruction related to them. CLASSROOM ACTIVITY
RELATED EXPLORATIONS
MATH CONTENT
5.1 Introducing the Basic Game
Unit 10: Understanding Player Cards
• Relating Fractions, Decimals, and Percents • Comparing and Ordering Percents • Operating with Whole Numbers and Decimals • Understanding and Using Ratio and Proportional Reasoning
EST. TIME:
1 session
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CLASSROOM ACTIVITY
RELATED EXPLORATIONS
MATH CONTENT
2–3 sessions
11.1 Investigating the Shot Planner 11.2 Would You Rather Be Odd or Even?
•Operating with Whole Numbers • Investigating Commutative Properties of Arithmetic Operations • Analyzing Properties of Integers • Exploring Number Patterns to Solve Problems
5.4 Introducing the Shot Clock (Optional)
13.4 Can You Beat the Shot Clock?
• Understanding and Using Properties of Integers
5.3
Playing Sample Games
EST. TIME:
EST. TIME:
1 session
Introducing the Advanced Game (Optional) 5.5
EST. TIME:
1–2 sessions
11.3 How Will the Advanced • Operating with Whole Numbers Division Rules Affect Your • Using Estimation and Rounding to Strategy? Solve Problems
Before You Begin The more familiar you are with the game and game play, the smoother your introduction will be to your students. This is a good time to refer back to Part I and review the particulars for playing the Basic Game and Keeping Score. It’s important that students are well grounded in specifics for playing the Basic Game as the Advanced Game builds on that knowledge. The best way for you to get fully grounded is to both read the directions and actively engage in game play—either with a colleague or small group of students. Personal, hands-on experience will go a long way towards you facilitating a smooth-running Math Hoops experience for your students.
5.1 Introducing the Basic Game Setting Tone and Expectations Prior to formally introducing students to game play, it will be in your best interest to set both the desired tone and coaching expectations. A well-run league will result from everyone working together to make sure classroom and league guidelines are clear and followed; time together is focused and well spent; and competition is fair and cordial. All will benefit from a highly collaborative atmosphere that promotes new learning, fosters creativity, and encourages positive and meaningful communication. Collectively establishing and posting a clear set of 5–7 “league guidelines” will go a long way towards building a strong collaborative learning environment. Establishing the Rules Students will want to start playing games right away, and you probably will too, but you may want to delay game play until you’ve had a chance to review the basic rules. Consider the level of sophistication of your students to determine if you want 30 • PART II: CREATING A MATH HOOPS LEAGUE
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to pare down the rules for even the Basic Game at the beginning. For example, you may decide to skip using the Game Scoresheet at the beginning and just use the Scoreboard to keep score. Other options would be to hold off on fouls and/or steals until students are confident with the basic mechanics. On the other hand, if your students are very self-directed, you can have them first read through the rules in the Coach’s Manual for themselves and then gather them together to ensure that everyone is operating with the same level of understanding. (All the rules for the Basic and Advanced Game are found in the Coach’s Manual and a concise summary is provided on the Scoreboard for easy reference during game play.) The most efficient way to get students acclimated to the rules is to actively engage them in game play. Introducing the Basic Game in a whole group environment has proven to be a good avenue for ensuring all students are acquiring the same understandings. Once you’re confident that students have the basic understandings, they will be ready to pair off and participate in small group competition.
Modeling the Games as a Class Divide the class into two groups. Set up a Game Board (Basic side up) and Scoreboard. Place a transparent spinner on each side and a pair of decahedra dice and game token at center court. A Shot Planner and dry-erase pen should be placed on each side. Blank scoresheets can be provided to each student or if enlargement services are available, print larger scoresheets and display one for each team that everyone can see.
MATH CONNECTION There are several explorations in Part III that you can use to draw out the mathematical underpinnings of the game. Depending on the level of your students and the time available, you may find it helpful to review any of the explorations in UNIT 10: UNDERSTANDING PLAYER CARDS PP. 63–73 UNIT 11: ANALYZING THE GAME PP. 74–83
Randomly select five Player Cards for each team—one per color/position and place them along the side of each board so that students will have easy access to them. It is not important to conduct individual player analysis at this time. The purpose of this exercise is to expose students to game play. Analysis of individual player statistics will preclude the drafting of teams prior to the regular season. The Basic Game is played in two 15-minute halves. For the purpose of this initial introduction, you will want to be flexible with the time. Again, what’s important here is that students are exposed to and become comfortable with the particulars of the game before engaging in small group competition. Walk through the game setup before starting game play. Then select two students for each side and have each side roll the dice to determine who goes first. Have the winning side take the even or odd end of the court depending on what the winning sum was. Step through how to use the Shot Planner, emphasizing that both sides should do this because if the opposing team catches an error they will get an advantage. Move the game token and model using the transparent spinner on the Player Card circle graph. Rotate students in on each side, two at a time. Once each side has been allowed the opportunity for 3–4 rolls of the dice, a new group of students can assume the role of team coaches. This is an ideal time to field student questions or for you to ask UNIT 5: PRACTICE GAMES (PRE-SEASON) • 31
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questions to gauge understanding. Designate an ending point to the first half and conduct a quick review of game play and scoring processes. Let teams know they will switch to the opposite ends of the Game Board for the second half of the game. Again rotate pairs of students and as they get comfortable with the process, encourage acceleration of the pace of game play. Upon completion of whole group game play, engage students in discussion about the experience. This can be done through informal discussion and/or via a collection of questions. Sample questions can include: • What do you like about the game? What do you dislike? Why? • Does the game seem fair? Why or why not? • What impact does/will the speed of game play have on the game? • Do you think the choice of players has an impact on the outcome of the game? Why or why not? • How will this whole group experience help you in small group play?
You will need to determine whether one round of whole group play will suffice or if another whole group experience would be beneficial before moving into small group play.
5.2 Classroom Set-Up While there are several ways to organize a Math Hoops game, the diagram on the next page presents an arrangement we strongly suggest for good reason. You’ll note that Co-Coaches are placed next to one another with one responsible for working with the Player Cards/spinner and Shot Planner and the other keeping score for the opposing team and managing the foul cards. Sitting side-by-side allows for maximum collaboration—a key attribute of successful coaches. Coaches working with the Player Cards can easily access the Scoreboard and update the running score when shots are made. If you use a shot clock, the coach in charge of their opponent’s Game Scoresheet can also take the responsibility of starting and monitoring the shot clock for the opposing team. By having the coaches working with the Player Cards sit directly across from one another, each can easily monitor the other to make sure calculations and placement of the ball are done correctly. Likewise, with coaches keeping score for the opposing team, they will want to monitor closely to make sure scoring is done correctly. Using this type of game configuration will help to ensure that all coaches are fully engaged throughout the entire game.
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As co-coaches become comfortable working with each other, they will most likely tailor the division of labor to fit their coaching style. What’s important is that they’re clear on the assortment of tasks associated with being on offense and those associated with being on defense. TEAM B COACH 1
TEAM B SHOT PLANNER
TEAM B COACH 2
TEAM A GAME SCORESHEET
SCOREBOARD
TEAM B PLAYER CARDS
TEAM B FOUL CARDS
SHOT CLOCK
GAMEBOARD
TEAM A SHOT PLANNER
TEAM A PLAYER CARDS TEAM A COACH 1
TEAM A COACH 2
TASKS ON OFFENSE
• roll decahedra dice • work out calculations on Shot Planner • review Player Cards and select shooter • move game token (ball) to desired spot • place the spinner on the proper Player Card and spin (take the shot) OR roll decahedra dice and reference 10 × 10 free throw grid (when fouled)
• update score on scoreboard (when shot is made)
• verify that opponents are correctly updating the Game Scoresheet
TEAM A FOUL CARDS
TEAM B GAME SCORESHEET
TASKS ON DEFENSE
• watch roll of dice (carefully) by the opposing team
• start shot clock • verify opponent’s Shot Planner calculations
• verify that opponents accurately place the game token (ball)
• decide whether or not to foul • verify accuracy of opponent’s shot
• record player data on scoresheet for opponent
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5.3 Playing Sample Games Once you feel comfortable that students have a good understanding of the Basic Game, pair them off for some 2-on-2 competition. This continues to be practice so it’s not important to determine official Co-Coaches at this time. Each group of 4 students should have access to a complete game. Player Cards can either be randomly distributed by you or quickly selected by each pair of students. Again, avoid getting into player analysis at this time as these are only practice games. Establish the parameters for game play and turn students loose. Your role at this time is to monitor games and address any questions and/or issues that may arise. By making sure students are comfortable and confident with game play, you increase the likelihood that the transition to forming a league and beginning regular season game play will be a smooth and uneventful one.
MATH CONNECTION For an extended discussion on how to incorporate the shot clock into games, turn to 13.4 CAN YOU BEAT THE SHOT CLOCK? PP. 100–103
5.4 Introducing the Shot Clock In Part I you learned about the purpose and effective use of the shot clock. The shot clock is intended to allow students to compete at various levels—when ready to do so. Depending on your personal situation, you will need to decide whether or not to introduce this feature to students at this time or phase it in during regular season game play.
5.5 Introducing the Advanced Game While we don’t advocate introducing students to the Advanced Game at this time, you will need to make that determination based on your personal teaching situation. The Advanced Game builds on a firm understanding of the Basic Game while building in new features and complexities. Making sure that students are ultra comfortable with the Basic Game—almost to the point of boredom—serves as a good indicator that it’s time to move on to the Advanced Game. Don’t feel the need to rush the process; mastery of the Basic Game (including effective use of the shot clock) can take some time. Please note that formal introduction of the Advanced Game is factored into the Sample Roadmaps and should be introduced when you feel your students are ready for the challenge. If you do elect to introduce students to the Advanced Game in the Pre-Season, make sure they’re clear on the level of game they’ll be playing once the regular season begins. If they’ll be jumping right in to the Advanced level, lots of experience and practice at this level will be most beneficial.
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THINKING AHEAD Once students have a firm grasp of the Basic Game, it’s time to begin thinking about forming a league and creating teams. Following are some questions you can pose to students as they begin to think about league play. These are questions students can ponder on their own or you can use as small or large group discussion prompts. • Knowing what you do about how the Basic Game is played, how important is the role of Co-Coach? What attributes will you be looking for in your CoCoach? • What do you want your team name (and logo!) to be? • How important do you think player selection is to fielding a competitive team? What will you be looking for when you have the opportunity to draft players? • What do you feel you need to work on to be as competitive as possible during Regular Season game play? What’s your plan for improving these skills?
While there are no “correct” or even standard responses to the questions listed above, you should begin to see how thoughtfully students are deliberating over them. For example, students will ideally recognize that they would be best served by pairing up with a Co-Coach who will work hard, stay focused, and embrace the idea of shared responsibility rather than one who would opt to sit back and let their partner do all the work. Reaching agreement on a team name and logo is a great initial indicator of how well coaches will work together. From here, it gets much more involved as they talk about player selection and game skills/strategies they feel they’ll need to work on to be as competitive as possible. The discussion itself, whether conducted in small groups or with the whole class, will serve as a means for you to gauge the current level of student interest. Don’t be discouraged by those who appear somewhat disinterested at this point. It’s amazing what a little bit of exciting league competition will do!
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UNIT
6
Organizing a League Scheduling This Unit The table below outlines the collection of classroom activities found in this unit and some estimated times you can expect to spend on each activity. The Estimated Time is based on 45–60 minute sessions and can vary based on the amount of student participation and length of classroom experience that you want. Also shown are suggestions for math explorations from Part III that can be used in conjunction with the classroom activities. Approximate times for math explorations can be found in the Part III Unit openers. CLASSROOM ACTIVITY
RELATED EXPLORATIONS
MATH CONTENT
6.1 Creating Teams and League Divisions
12.1 How Do You Make a Player Draft That’s Fair? 12.2 Which Players Should You Choose for Your Team?
• Exploring the Concept of Fairness • Devising Math-Based Rules to Create Fair Game Play • Comparing Decimals and Percents • Using Number Sense to Evaluate Data
12.3 How Can You Design a Creative Season Schedule?
• Analyzing Data to Create Schedules • Understanding and Using Combinations
UNIT 13: Winning Strategies
• Using Probability and Expected Value • Using Properties of Integers to Solve Problems
EST. TIME:
1–2 sessions
Creating a Season Schedule 6.2
EST. TIME:
1 session
6.3 Conducting the Regular Season EST. TIME:
6–10 sessions
Before You Begin Turn back to the table of criteria in the introduction to Part II on p. 27. Now is a great time to re-visit them. You may even want to add some others to the mix. Advance planning will serve you well and once you determine the level of involvement you want students to have in the creation of the league, you’ll be off and running!
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6.1 Creating Teams and League Divisions Each Math Hoops game contains 32 Player Cards in five color-coded categories: • 7 gold Guards
• 6 purple Forwards
• 6 red Guards
• 6 green Centers
• 7 blue Forwards
Each team must pick at least one player from each of the five categories above. Students will begin with a team of five players, although you may decide to allow additional players. How to group students is left to your discretion as you will know the best process to follow for your specific group. Preferably all teams will consist of two coaches but that may not be possible depending on the number of students in your class. Three coaches on a team will work as long as students work cooperatively to clearly define roles and divvy up responsibilities. Due to the number and variety of tasks taking place in the course of a game, having one student coach a team is not advised. Whenever possible, try to create an even number of teams. This will prevent having one team be idle while others are competing in games. Each team’s coaches must agree upon a team name, subject to your approval. Team names will be used for recordkeeping throughout the league’s season of play. As done in the National Basketball Association, break the league into two halves, with an equal number of teams in each half. You can name one half the East Division and the other half the West Division or use some other terms favored by you or your students. (The NBA uses the word “Conference” instead of “Division.”) This division of teams will come into play during the All-Star Game (see Unit 8) and the Tournament Series (see Unit 9).
Initial Player Draft One of the explorations in Part III (12.1) involves students creating a draft process they feel is fair. If you’re pressed for time and feel it’s best to direct the process yourself, here’s one possible method you can use:
MATH CONNECTION To find information on conducting a student exploration around the player draft, turn to 12.1: HOW DO YOU MAKE A PLAYER DRAFT THAT’S FAIR? PP. 85–86
• Group teams into pairs and present a set of 32 Player Cards to each. Each pair of teams will conduct a separate player draft.
• To start, have a coach from each team roll the two decahedra dice. The team with the higher sum will draft first.
• Each pair of teams should place all 32 Player Cards on a table, data side up and sorted into five rows by color. Coaches can now begin choosing players by following these steps:
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MATH CONNECTION The step of selecting players to make up one’s team is the first point in the game where students can think strategically. For details on how to engage students in this type of strategic thinking, turn to 12.2: WHICH PLAYERS SHOULD YOU CHOOSE FOR YOUR TEAM? PP. 86–90
1. Have each team select a Player Card, starting with the team that rolled the higher sum. Teams can select a Player Card of any color. 2. For the second draft round, switch the drafting order and have the team that rolled the lower sum pick first. Each team should select a Player Card of any of the four colors that it has not yet chosen. 3. Teams should select an additional card each in the third and fourth rounds, alternating who selects first. Each round, a team should only select a Player Card of one of the colors that it has not yet chosen. 4. For the fifth round, have each team roll the two decahedra dice one more time. Each team should then select a final Player Card, starting with the team that rolled the higher sum. The final Player Card must be from whatever color that team needs to complete a starting lineup.
The process above describes a draft in which coaches select 5 players only for their teams—one per color/position. If you’re building a Math Hoops league, we recommend allowing coaches to work with teams of 8 players: the initial draft of five plus three substitute or “bench” players. (The Coach’s Manual makes reference to this as an option.) This adds a new and interesting twist—one that will capture student interest and could potentially help a team turn their season around. You may want to wait to have students draft these additional players until they have a few games under their belt. If you’d prefer, however, to have students draft all eight players in one comprehensive draft, Rounds 5 through 8 can replicate the process used for Rounds 1 through 4 above. Before actually starting the draft, ask students if they feel the process is fair and why they feel it is or isn’t. You might also ask them whether or not they feel there’s any advantage or disadvantage to having the first pick in the draft. Once each team has its five players, the draft is finished and teams are ready to begin pre-season play. The opportunity to draft additional players to replace low performers can be introduced later in the season.
6.2 Creating a Season Schedule Before the NBA Math Hoops season begins, it is important to lay out a full schedule of regular season games. In this way, you can guarantee relative fairness in the way games are distributed among the teams. If you’ve created an NBA Math Hoops Roadmap, refer back to it to determine the scheduling parameters for your particular program. You can read about how to make a Math Hoops Roadmap on pp. vii–xii of this Guide.
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CLASSROOM TIP
Whose Task is this Anyway?
Depending on the amount of time allocated for this program, you may elect to craft a regular season schedule yourself or engage students in designing the schedule. There are definite advantages to both options. One saves time and allows you full control to manipulate the schedule however you see fit. The other takes additional time yet promotes student buy-in and responsibility and lends itself to a meaningful mathematical investigation. The choice is yours!
Student Schedulers If you embrace the philosophy of having students contribute as much as possible to the creation of the Math Hoops league, involving them in creating the season schedule is a necessity. One option is to work through this task with the entire class, soliciting input as you go along. Another possibility is to have two teams (four students) work together to create a schedule to be presented to the class. Schedules can be reviewed and critiqued and a class vote can be conducted to select one schedule that will be used for the league’s regular season. League Commissioner as Scheduler One method for creating a simple schedule is to divide your teams into “Divisions” of four teams each. For example, let’s assume that you have 8 teams and decide to break them into the Eastern Division (Teams A, B, C and D) and Western Division (Teams E, F, G and H). Working from the Rookie or Professional roadmap, you would look to create a schedule that consists of 8 regular season games. You could have each team play 2 games each (one Home, one Away) against the remaining three teams in their division and either assign or allow students to self-select 2 crossdivision games. Here is one way to set up such a schedule that allows students to self-select their cross-division opponents:
MATH CONNECTION If your inclination is to involve your students in the development of the season schedule, you can find a student exploration on this topic in 12.3: HOW CAN YOU DESIGN A CREATIVE SEASON SCHEDULE? PP. 90–92
EASTERN DIVISION (TEAMS A, B, C AND D)
WESTERN DIVISION (TEAMS E, F, G AND H)
GAME
1
AB
CD
GAME
1
EF
GAME
2
AC
BD
GAME
2
EG
GH FH
GAME
3
DA
CB
GAME
3
HE
GF
GAME
4
Cross-Division Game
GAME
4
Cross-Division Game
GAME
5
BA
DC
GAME
5
FE
GAME
6
CA
DB
GAME
6
GE
HG HF
GAME
7
AD
BC
GAME
7
EH
FG
GAME
8
Cross-Division Game
GAME
8
Cross-Division Game
Another method would be to keep everyone together in one league and create an 8-game schedule where each team plays each other team once and one team twice.
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The schedule should also allow for an equal number of Home and Away games. Here is an example of that type of schedule: SINGLE-DIVISION LEAGUE (8 TEAMS) AB
CD
EF
GAME 2
AC
BD
EG
FH
GAME 3
DA
CB
HE
GF
GAME 4
HA
BG
FC
DE
GAME 5
AE
HB
CG
FD
GAME 6
AF
BE
HC
GD
GAME 7
GA
FB
EC
DH
GAME 8
EA
DG
CF
BH
GAME 1
GH
Notice that the first seven games of the schedule above has each team playing every other team exactly once. The eighth game has them facing an opponent for the second time. Students may see this schedule as fair or unfair depending on what team their opponent is for Game 8! Please note that these are intended to serve as viable examples and can be adapted in the event you have more teams or expanded should you use a schedule that allows for 12 regular season games. These examples should help you gain some insights into scheduling your own league. As you move forward, keep these two principles in mind: • Have the teams play each other in their division or league the same number of times.
• Have each team play the same number of home games. • If you can’t achieve the two goals above, try to get as close as possible. Never favor one team in the scheduling, for example, by giving them two more home games than other teams.
6.3 Conducting the Regular Season The regular season schedule either you or your students have created should be posted for easy public viewing. On designated game days, students can refer to the schedule, move to the appropriate “arena” and prepare for game play. You’ve probably seen a great deal of student excitement already but the regular season is where the real fun begins! Think about it—your students have been introduced to the game, decided on a co-coach, broken off into divisions, drafted their team of NBA and WNBA players, and (possibly) created a league schedule. Wow! Of course they’re going to be anxious to start the regular season! How you run your league will be entirely up to you. The environment you create will go a long way towards generating and sustaining student interest and enthusiasm. There are a number of decisions you’ll need to make along the way. The following 40 • PART II: CREATING A MATH HOOPS LEAGUE
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list contains details you can consider in advance to make classroom management of the regular season go much more smoothly. • Set up distinct basketball “arenas” in the classroom where games will be played. You may want to have a specific classroom arrangement that you create on game days. If you plan to have teams play several games, think ahead of time how you will manage the movement of teams from game to game, or arena to arena.
• Figure out where games will be stored and who will be responsible for getting them out and putting them away. Establish a procedure to use so that students are “game ready” as quickly as possible upon entering the classroom.
• Do you want all teams to play games at the same time? An alternative would be to stagger the games so that some teams are playing and others are working on Math & Strategy Explorations.
MATH CONNECTION
• Rather than have all teams play at the same level, you may want to allow the level of play to vary depending on demonstrated proficiency.
The regular season is a great time to engage students in discussion and investigations about game strategy as they will immediately see the applications of them. Specifically, you might want to take periodic breaks in the regular schedule to focus attention on any of the explorations in UNIT 13: WINNING STRATEGIES PP. 93–108
• The Shot Clock injects an extra jolt of excitement to the game, but you won’t want to introduce it until students feel comfortable enough playing without it that it won’t frustrate them. Present the Shot Clock as a goal they can look forward to. You might even want to describe this incentive as graduating to college level of play (with a 35-second clock) and eventually pro level of play (with a 24–second clock).
• Expect real-world situations to sometimes disrupt your season schedule. Think about how you will handle absences. Will teams be allowed to make up missed games? Will you allow teams in the league to end with an unequal number of games played? (If so, prepare students for how this will affect win/loss records.)
• Questions and disputes will undoubtedly arise during the course of the games. Think about what protocols you can put in place to help things run as smoothly as possible.
• To make sure the season schedule is followed accurately, consider assigning the job to one or more students. You can also have students be in charge of keeping track of team standings and league leaders, as well as assuming responsibility for recording scores at the end of games.
• If space allows, set up a designated NBA Math Hoops bulletin board for the posters included in the classroom kit as well as samples of student work.
While the logistics of running a league in your classroom may at first seem sizable, you’ll find that as you make clear-cut, definitive decisions, your Math Hoops league will come into much clearer focus. Involving students in the decision-making process creates a feeling of empowerment and ownership and once they’re locked in, you will be well on your way to a very successful Math Hoops season!
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UNIT
7
Keeping Statistics D
uring each Math Hoops game, coaches keep track of player data on their Game Scoresheets. But there are a lot more statistics that can be tracked and analyzed over the course of a regular season. Unit 15 (Why Do Coaches Need Statistics?) provides discussion on how to guide students to use the statistics they keep to their advantage. Those explorations will help students see how statistics are useful in making team decisions. You may want to involve students in all the record-keeping that’s offered in Math Hoops—or you may just want to focus on a couple.
Scheduling This Unit This unit provides an informational overview of the various statistics and record-keeping you and your students can explore in Math Hoops. The table below outlines the collection of statistics forms found in this unit with suggestions for math explorations from Part III that can be incorporated into the activities. CLASSROOM ACTIVITY
7.3
Season Totals
7.4 Player Analysis Charts
RELATED EXPLORATIONS
MATH CONTENT
15.2 Per-Game Statistics
Finding Rates Computing Averages of Data Sets Collecting, Analyzing, and Displaying Data
15.2 Per-Game Statistics 15.3 Team Analysis
Finding rates Computing Averages of Data Sets Collecting, Analyzing, and Displaying Data Comparing and Operating with Whole Numbers and Decimals
7.5
Team Standings
15.1 Team Standings
Comparing and Ordering Decimals and Percents Collecting, Analyzing, and Displaying Data
7.6
League Leaders
15.4 League Leaders
Collecting and Organizing Data in Tables
Before You Begin Some of the statistic sheets described below can be found in the Coach’s Manual and some are given only in this Teacher’s Guide. Be sure you’re aware of which ones you will need to provide for students. Note that to track their players’ season totals, students will require multiple copies of the Player Season Totals forms—one for each player on their team—so you may wish to make enough copies of this form ahead of time.
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7.1 Game Scoresheets A pad of Game Scoresheets is included in each NBA Math Hoops Classroom Kit. These can be kept in your possession and distributed at the beginning of every game. Once a game is completed and scores recorded, students can place their completed scoresheet in a designated Math Hoops folder. Students should be encouraged to review completed scoresheets from time to time in order to gauge player performance and adjust game-play strategies accordingly. In addition, students will need access to player data recorded on the scoresheets when completing the mid-season review.
7.2 Game Summary Stat Sheet A Game Summary Stat Sheet template can be found in the Coach’s Manual Appendix as well as on p. 136. Game Summary Stat Sheets allow students to get a quick picture of their players’ performance in a specific game. Even though coaches will already have Game Scoresheets, the Game Scoresheet is more useful as a tool to keep a tally of players’ statistics AS the game is being played. For students to be able to analyze the game data, it’s far easier to read the Game Summary Stat Sheets. These are to be completed by coaches at the conclusion of every game and can be kept in the same folder as the Game Scoresheets. You may want to spot check completed sheets from time to time to be sure students are keeping current with this important task. To complete these sheets, coaches convert each row of the original Game Scoresheet into a numerical record, as shown in the example below. GAME SCORESHEET 2-Pt Player/Position 1. Kevin
Love
3-Pt
Free Throws
FGM
FGA
FGM
FGA
FTM
FTA
Total Points
S
F
R
A
2
3
2
4
1
2
11
1
1
2
0
After a team enters its players’ data, it adds the numbers in each column to find the Team Totals for the game.
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7.3 Season Totals MATH CONNECTION Explorations related to the statistics discussed here can be found in UNIT 15: WHY DO COACHES NEED STATISTICS? PP. 121–133
The Game Scoresheet and the Game Summary Stat Sheet provide information for a single game. To keep track of season data for their players, coaches should use Player Season Totals forms. The player data can be easily pulled from the Game Summary Stat Sheets. The most efficient way of keeping track of Player Season Totals is to fill them out at the end of a game day. The Player Season Totals forms are important because students will need to have access to information on the them in order to conduct a more thorough analysis of player performance. This data will be used to calculate shooting percentages and per-game statistics to be recorded on the Team Analysis Chart. The more students can collect, organize, and thoughtfully analyze player and team data, the more proficient coaches they will become. Here’s a typical Player Season Totals sheet as it might look after three games. Notice that the season totals are filled in at the bottom of the page. A coach should record the season totals in pencil, since they will be adjusted after each game.
PLAYER SEASON TOTALS 2-Pt
Date of Game
Opponent
10/11/11
SUNS
2
10/12/11
PACERS
10/14/11
LAKERS
Free Throws
3-Pt
FGM FGA
Total Points
S
F
R
A
2
12
1
1
2
0
0
0
8
0
2
3
1
1
2
3
7
1
0
2
2
6
3
5
26
2
3
7
3
FGM
FGA
FTM FTA
3
2
4
1
4
5
0
1
1
4
1
7
12
3
SEASON TOTALS TOTAL GAMES PLAYED
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7.4 Team Analysis Charts At the conclusion of the regular season, information recorded on the Player Season Totals sheet can be utilized to conduct a reflective, in-depth performance analysis of the team. On page 86 of the Coach’s Manual and on page 140 at the back of this Teacher’s Guide, you will find a Team Analysis Chart. Unlike the Mid-Season Assessment form where students first transferred player data and then performed calculations, this form is a bit different. Students will again use the data from the Player Season Totals sheet, but this time around, they will perform various calculations they’ve learned around shooting percentages and per-game statistics to complete this chart. Completing the Team Analysis Chart prior to teams entering playoff or tournament play is highly recommended. Here they’ll be able to identify definitive strengths and weaknesses that should help them to clarify and/or refine their game-play strategies. The sample chart below shows performance data calculated after a twelve-game regular season. Data from individual Player Season Totals sheets was referenced to arrive at the shooting percentages and per-game calculations. It’s important to note that finding the Team Averages is not as simple as just adding up the numbers and dividing by the total number of players. You’ll note that some
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players shoot more often than others so all player averages are not created equal. In order to find the accurate team shooting averages, it’s necessary to compile the raw data and find the shooting percentages from this data. For example, the .459 2-pt. FG% noted above comes from the team making a total of 157 2-pt. field goals out of 342 attempts. This process holds true for all of the shooting percentages. Of course, when finding the four per-game team averages, simply adding the individual player averages will produce accurate results.
7.5 Team Standings As League Commissioner, you will want to make sure that league standings are posted on a regular basis. You may elect to assume responsibility for this task or enlist an interested student to take charge. The Team Standings/League Leaders poster included in your Classroom Kit will serve as a useful tool for keeping track of this information. The Team Standings exploration found in Unit 15 of the Coach’s Manual will lead students through the process of keeping accurate track of this information.
7.6 League Leaders Along with Team Standings, League Leaders in a selection of categories can be recorded and updated frequently. The Team Standings/League Leaders poster was designed with this purpose in mind. Once again, you’ll need to decide whether or not you will be the one to keep track of this data or turn it over to a motivated student. The League Leaders exploration found in Unit 15 of the Coach’s Manual provides students with a meaningful, in-depth experience and allows everyone to be involved in the process at some level.
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UNIT
8
Mid-Season Review/ All-Star Game A
round the mid-point of your regular season, you may want take a break to review how league play is going so far. Are the teams competitive? Do any teams appear to have a definitive advantage over some of the others? If you either selected or crafted a program roadmap consisting of 24 or more sessions, the midpoint of the regular season provides a perfect opportunity for student coaches to explore ways in which they might improve their team. After careful analysis of first half player stats, students may look to bolster their team in one of three ways: 1. By participating in a Free Agent Draft 2. By conducting mid-season trades 3. By creating a new or updated Player Card In each case, the primary focus should be on investigating ways to increase offensive production—i.e. score more points!
Scheduling This Unit The table below offers a general time frame you can use to schedule different aspects of this unit. The Estimated Time is an approximation based on 45–60 minute sessions and will vary by academic level of students as well as the level of involvement you want to give your students. Two of the activities below (8.1 Mid-Season Assessment and 8.3 Creating or Updating a New Player Card) have corresponding math explorations that you can find in Part III. Feel free to select activities you feel will best serve your students based on their interest and need. CLASSROOM ACTIVITY
RELATED EXPLORATIONS
8.1 Mid-Season Assessment
14.1 Mid-Season Assessment: Is It Time for a Roster Change?
•Comparing and Operating with Whole Numbers and Decimals •Comparing Theoretical and Experimental Probability •Collecting, Organizing, and Interpreting Data
------------
------------
EST. TIME:
8.2
1–2 sessions
Free Agent Draft
EST. TIME:
MATH CONTENT
1–2 sessions
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CLASSROOM ACTIVITY
RELATED EXPLORATIONS
8.3 Creating an Updated or New Player Card
14.2 Create Your Own Player Card!
•Finding and Using Ratio and Proportions • Converting Fractions, Decimals, and Percents • Constructing Geometric Representations of Numerical Data • Creating Circle Graphs
------------
------------
EST. TIME:
8.4
1–2 sessions
All-Star Game
EST. TIME:
MATH CONTENT
1 session
Before You Begin Depending on the number of teams in your league, you most likely have a collection of Player Cards that went unselected in the initial draft. These cards can be sorted by position, put on display for public viewing and made available for student selection. After carefully considering player performance and perceived needs, student coaches can prepare their desired priority list of draft options. If you’re electing to exercise the different options for students to improve their teams at this point in the season, you may want to limit the draft to one player per team.
8.1 Mid-Season Assessment MATH CONNECTION A more detailed explanation of how to help students analyze their team's performance can be found in 14.1 MID-SEASON ASSESSMENT: IS IT TIME FOR A ROSTER CHANGE? PP. 110–114
With a number of regular season games under their belt, now is an ideal time for your students to conduct a mid-season assessment. Time should be set aside to pull out current Player Cards, completed Game Scoresheets, and Player Season Totals charts. The more data students have available to reference, the better. Instead of running a structured activity, you might try leaving things open-ended and flexible at first. Set aside 20–30 minutes of classroom time for coaches to gather and review their player and team data. Their primary objective should be to identify definitive strengths and areas of concern and determine a viable strategy for addressing any perceived weaknesses. Encourage students to use this time wisely— to engage in meaningful discussion and even seek advice from others. Successful coaches listen as much, if not more, than they talk and are always learning from others. If your students can adopt this philosophy, it’s conceivable that they will see measurable results when they return to coaching in the second half of the season. Once students have had some free-flowing discussion time, present the following four brief statements: • Player data tells a story. • If you like the story your player data is telling you, keep doing what you’re doing.
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• If you don’t like the story your player data is telling you, stories always have the potential to change. • Decide what you can do to change the story from this point on.
From here, students can return to their player data and dig in a little deeper with their analysis. You’ll want to limit the time allowed for this mid-season assessment— probably no more than a classroom session or two. Emphasize to students that desired outcomes for this time are to: a) identify ways to improve team performance, and b) identify which players might be worthy of an All-Star Game nomination!
8.2 Free Agent Draft The option to replace one or more players will generate some excitement from teams who have identified any weaknesses in their teams. Playing half a season of games will give coaches more insight into what numbers they want to see on their Player Cards and what distribution of strengths make a competitive team. The easiest way to provide potential replacement players is to use the unused Player Cards from the original draft. To conduct the Free Agent Draft, you’ll want to bring all students together for a whole “league” event. Gather up all of the Player Cards that were not selected in the initial drafts. Place these on display in a central area of the classroom a day or two in advance of your draft for student coaches to preview. Each team should create their own list of desired draft picks, in priority order, so that they’re prepared on draft day. (This will help things run expeditiously.) As in the initial draft, the question of fairness may be raised. One method you might consider is to allow teams to make their selection based on reverse order of league standings. For example, the team with the lowest winning percentage receives the first selection and continues accordingly until the team with the highest winning percentage makes the final selection. This gives struggling teams new hope and puts the successful teams on guard that they’ll need to stay focused to maintain their foothold in the standings. An alternative method for the draft can be selection by lottery. One easy way to conduct a lottery is by using two decks of cards. For example, if you have 16 teams in your league, select 16 identical cards from each deck. Hold on to one set and randomly distribute the other set—one card per team. Shuffle the remaining deck and draw your first card. The team holding the corresponding card (e.g. ace of hearts) gets the first selection in the draft. Continue this process until all teams have had the opportunity to select a new player.
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Selecting a player in the Free Agent Draft does not require coaches to insert this player in all remaining regular season games. What it does is give coaches another option—someone they can turn to in the event their current player at this position is not performing up to par. Sometimes just the psychological aspect of the draft will inspire new hope and motivation on the part of student coaches.
Mid-Season Trades Is it possible that a player shooting poorly for one team could shoot lights-out in the second half of the season for another team? Mid-season trades provide students with another opportunity to assess player productivity and make changes that could improve their team’s performance. This should not be a mandatory exercise but rather an option for students to consider. Prior to engaging in any trade talks, co-coaches should review their first half player stats to determine areas of strength and areas of need. Once player strengths and needs have been determined and they have a good sense of what they’re in the market for, students should be armed with their collection of Game Summary Stat Sheets and first half Mid-Season Assessment sheets when beginning discussions. All player information should be made readily available as time will be limited and potential trading partners will want to make informed, data-driven decisions. You may want to allow up to a full session for this complete exercise while limiting the actual trade talks and potential moves to half that time. Trades should be limited to one-to-one deals and do not need to be of corresponding positions as long as you are providing other mid-season options such as the Free Agent Draft or allowing student coaches to create an updated or new player card. All trades should be considered binding through the remainder of the regular season and playoffs. MATH CONNECTION Having students create their own Player Card is a mathematically rich activity that touches on several core math topics. Directions for implementing this activity can be found in 14.2 CREATE YOUR OWN PLAYER CARD! PP. 114–119
8.3 Creating an Updated or New Player Card A third option for improving team performance at mid-season that can be presented to students is the opportunity to either update an existing card (based on game play performance) or create a new player card. Step-by-step directions are included in Part III: 14. Improving Your Team for creating an updated version of an existing card. The same process can be used for creating a completely new card using actual player data from a reliable source such as www.nba.com. For purposes of keeping things moving, you may want to limit this function to one player per team at this time. You may elect to allow students to revisit this again later in the season or just prior to the playoffs. Regardless of when students are able to make use of this function, mathematical analysis of game-play data and identification of potential increased offensive productivity should be the impetus for making any changes. The process for this analysis is covered in more detail in Part III: 14. Improving Your Team.
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Creating an Updated or New Player Card
WEB ACTIVITY
Students have access to an exciting feature on the www.nbamathhoops.com website—the ability to create new or updated cards using NBA and/or WNBA statistics. Cards they’re currently using for game play include a player’s most current professional statistics. By visiting this feature on the website, students may find that players currently on their team performed better in a prior year. By scrolling over a given year and comparing the graphs on the website and current player cards, students can obtain a quick visual as to whether or not revamping the card might be advantageous.
8.4 The All-Star Game Professional basketball leagues such as the NBA use all-star games to recognize their best players and to provide their fans with an exhibition of the league’s most talented players. This section explains how to run an all-star game in Math Hoops. The all-star game can be scheduled anytime after the midway point in your league’s season. The date of the game should be announced to team coaches at the beginning of the season, if possible.
Player Nominations Each team is allowed to nominate two players for the all-star game using the form shown below. A copy of the All-Star Nomination Form can be found both in the Coach’s Manual Appendix and on p. 139 at the back of this guide. The nominations should be handed in to you three school days prior to the actual game. In nominating their two players, a team’s coaches must explain their choices. Specifically, they must justify why their two choices were chosen over the three other players on their team. While this justification must be written, it should also be accompanied by an up-to-date Team Analysis Chart (or Mid-Season Assessment sheet) and make reference to data from that sheet. In the game itself, all-stars from the East Division form one team to play against all-stars from the West Division. As a reward for sitting atop the league standings, the coaches of the 1st place teams in each division will have the privilege of serving as lead coaches of the two all-star teams. Once all player nominations are in, copies should be provided to all coaches in each division, as all coaches will participate in
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some fashion in the All-Star Game. Allow some time for all the coaches in a division to get together to discuss the nominations and offer their ideas on which players should constitute their all-star team. All coaches should be prepared to defend a player’s selection to the team based on first half performance. The all-star team itself should consist of 12–15 players and have at least one representative from each team in that division. The final decision regarding all-star team rosters should come from the lead coaches.
Playing the Game The All-Star Game is to be conducted as a full class event and played in 12 –15 minute quarters. While the league-leading coaches are designated as lead coaches, all other coaches should plan to actively participate on some level. Lead coaches can assign duties such as scorekeeper, shot planner calculator, scoreboard operator, shot clock attendant, etc. so that five or six coaches per division are directly involved at all times. If coaches alternate quarters, all division coaches should have the opportunity to directly participate in a minimum of one quarter of the game. An interesting phenomenon that will most likely occur is that the same player will be identified as an all-star for both divisions. This takes place as a result of every set of two teams drafting from the same collection of 32 players. One stipulation you might impose is that the same player cannot be on the floor for both teams at the same time and no player can play more than half of the game. This will force coaches from both sides to map out each player’s floor time prior to the game. You may want to require the lead coaches to show you their game plan, quarter by quarter, in advance of the game to ensure that no player is on the floor for both teams at the same time.
The First NBA All-Star Game The National Basketball Association played its first all-star game on Friday, March 2, 1951. At the time, basketball was reeling from a point-shaving scandal in college game. With fans upset, NBA owners hoped that an exhibition game between its best players would help restore fan enthusiasm. The owner of the Boston Celtics, Walter A. Brown, offered to host the game at the Boston Garden and to cover all expenses, in case the game lost money! It turned out to be a hit, with more
than 10,000 attending the game, eclipsing the Celtics’ average attendance of around 3,500. The final score was Eastern Conference 111, Western Conference 94. Today, the NBA All-Star Game is a mega-media event over the course of a weekend, including slam dunk contests and events that include NBA Legends and WNBA stars. In 2010, more than 108,000 attended the game at Cowboys Stadium in Dallas, Texas.
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The game should be conducted as all other Math Hoops games are, with the exception of extending the time allowed for each quarter or half. You will also need to determine the level of play (Basic or Advanced) and whether or not there will be inclusion of the 35-second or 24-second shot clock. If needed, you can always elect to have two All-Star games at different levels of play going on simultaneously. This is something you will need to determine based on your specific classroom situation. Regardless of the route you choose to go, the results of the game do not become part of players’ season statistics since this is an exhibition game. The coaches of the two teams have one new issue to handle during the All-Star Game. With 12–15 players on each team, this means that coaches must find a way to give every all-star a fair amount of playing time by substituting players at the beginning of every quarter. Coaches will need to have their game plan locked down in advance in order to keep the game moving. They will also want to have access to 2 – 3 scoresheets so that accurate statistics can be kept for each player. At the conclusion of the game, coaches on the winning team will need to review individual player statistics and vote on a Most Valuable Player (MVP). You may want to create a special spot on your Math Hoops bulletin board to recognize this player as the MVP of the All-Star Game. Small touches like this are what bring the whole Math Hoops experience to life!
Preparing for the Second Half With both new and “well-rested” players in hand, co-coaches should be just about ready to dive into the second half of the season. If you’re planning on having an AllStar Game, you may want to have it just prior to the second half so students return on a high note. You may also want to use this time to review general strategies and have coaches summarize their players’ strengths and weaknesses. Refer to Preparing for the Second Half on pages 119–120 of this Teacher’s Guide to find a collection of reflective and analytical questions you can use to prompt students. Responding in writing and verbally processing these questions with others will provide students with a concentrated focus on next steps. With added players, fresh legs, and a whole new perspective—the sky is the limit! It’s time to get rolling with the second half of the regular season!
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UNIT
9
League Playoffs and Championship I
t’s time for students to see if all of the hard work done during the regular season will pay off. The league playoffs are what they have been working towards all along and placement in the playoffs could play a role in how successful they are. The ultimate goal: make it to the championship game and WIN THE WHOLE SHEBANG!
Scheduling This Unit The table below offers one suggestion for a time frame in which to hold League Playoffs and a Championship event. You will need to adjust these times depending on the length of your Math Hoops season, the level of support and interaction you want to have within the school or with parents, and of course, how much time you have available in your schedule. Note that while there are no specific math explorations created especially to go with the League Playoffs, the end of the season can be used to pull together what students have learned throughout the season. If you haven't delved into any of the record-keeping or data analysis investigations found in UNIT 15: WHY DO COACHES NEED STATISTICS?, this would also be an opportune time to introduce them. CLASSROOM ACTIVITY
9.1
The Tournament Series
EST. TIME:
9.2
MATH CONTENT
-----------
-----------
-----------
-----------
3–4 sessions
Creating a School Event
EST. TIME:
RELATED EXPLORATIONS
1–2 sessions
Before You Begin You are about to enter THE most exciting time of the Math Hoops season. Knowing that the playoffs are what students have been working towards all along, you’ll want to make sure you’re well prepared in advance. If you haven't already, pull out the classroom Tournament Poster from your Classroom Kit and post it on the wall. Make sure to map out the schedule of your tournament series so that students know precisely what the format will be. At this point in the season, the last thing they want is an unexpected surprise. Tournament play is a BIG deal so whatever you can do to make it special will be well worth the time and effort. This time presents an ideal opportunity to solicit parent involvement and to showcase your dynamic Math Hoops program.
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9.1 The Tournament Series On the day that your teams finish their Math Hoops season, you can compile the final standings and declare the winning team for each division of your league. (If two or more teams are tied with the best record in their division, all can be declared winners.) Following the regular season, you have the option, if time and interest permits, of conducting a tournament like that used in college basketball. The tournament structure that follows assumes that all students are competing at the same level of play. This may or may not be the case in your particular situation. If all students are at the same level (e.g. Advanced game w/ 35-second shot clock), this process should be fairly easy to follow. If students are playing at different levels, you may want to set up separate playoffs and a championship game for each level of play. Don’t hesitate to modify things to best meet the needs of your students. The focus should be on fair and fun competition regardless of the level(s) of play. In an NCAA-style tournament, a single loss eliminates a team and the last team without a loss is the tournament winner. The organization of your league’s tournament will depend upon the number of teams in the league. The diagram below shows one way to set up a tournament for 8 teams. The first step is to rank the teams in each division according to their winning percentage, with 1 being the team with greatest winning percentage. Then write in the names of the teams corresponding to their ranks. TOURNAMENT FOR 8 TEAMS
WEST DIVISION
EAST DIVISION 1
1
4
4 2 3
CHAMPIONSHIP GAME
2 3
The tournament begins with four games: 1 vs. 4 and 2 vs. 3 in each division. The teams are matched this way so that the best regular season team can be rewarded by having it play the weakest team in its division. At the conclusion of the first round, there are four winning teams that play a second round of games. As in the first round, games are played within each division. The third round matches the winner of the West Division against the winner of the East Division to determine tou nament champion. UNIT 9: LEAGUE PLAYOFFS AND CHAMPIONSHIP • 55
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Ranking Teams with Tie Records
CLASSROOM TIP
The scenario for 8 teams shown above has all teams playing in the first round. If you have 12 teams in your league, the two best teams in each division will receive “byes” in the first round of the tournament. A bye is a privilege whereby a team gets to advance to the second round of a tournament without playing a game.
Here are diagrams showing how to set up a tournament if you have 12 or 16 teams: TOURNAMENT FOR 12 TEAMS
WEST DIVISION
EAST DIVISION
1
4 5
1
5
CHAMPIONSHIP GAME
3 6
2
4 3
2
6
Note that while the tournament with 8 teams has all teams play in the first round, the situation is different with 12 teams. The easiest and fairest way to do this is to assign ranks randomly by a method such as a coin flip. You could also assign rank according to how teams fared when they played against one another. TOURNAMENT FOR 16 TEAMS
WEST DIVISION
EAST DIVISION
1
1
8
8
4
4
5 3
CHAMPIONSHIP GAME
5 3
6
6
7
7
2
2
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March Madness The NCAA Tournament is a rite of spring, aptly dubbed March Madness. In its current incarnation, college men’s basketball teams compete through single-elimination in the hopes of becoming national champion. This tournament began in 1939 with just eight teams competing: Brown, Ohio State, Oklahoma, Oregon, Texas, Utah State, Villanova, and Wake Forest. That year, Oregon beat Ohio State, 4633, to become the first national champion. In the ensuing years, interest grew and the field of
teams expanded, reaching its current spate of 68 colleges represented. The phrase “Final Four” is used to describe the last four teams remaining in the tournament. They play a semifinal round of games to determine the teams that will play for the national championship. As such, the Final Four games have become wildly popular among rabid college basketball fans and even casual observers. Notably, the NCAA owns a trademark on the term “Final Four.”
The Tournament Poster that comes in the Classroom Kit is set up for 16 teams but can be adapted to fit a smaller number of teams. The layouts for 12 and 8 teams above are easily created simply by using only the portions of the larger diagram that are needed.
9.2 Creating a School Event The culmination of an NBA Math Hoops season is always an exciting time. Students diligently review and assess player performance from the regular season and begin to craft their strategic plans for the league playoffs. While the classroom playoffs and league championship leads to some great competition, building on this experience to create a school-wide event can take things to a whole new level. The NBA Math Hoops board game is the perfect vehicle for crafting and conducting a school-wide or multi-school tournament. Taking into account the Basic and Advanced versions of the game, along with integration of the 24-second and 35-second shot clock, it’s possible to run a larger tournament that can be divided into up to six different divisions. This allows students to compete in a tournament setting with other teams performing at a level comparable to their existing skill level. Imagine a full day tournament consisting of 4 classrooms and 12 teams per classroom. The larger tournament can be broken into six smaller tournaments consisting of 8 teams each. Students can either be placed in a specific (mini) tournament based on classroom performance or allowed to register for a level of their choosing.
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BASIC
ADVANCED
NO SHOT CLOCK
35 SEC. SHOT CLOCK
24 SEC. SHOT CLOCK
NO SHOT CLOCK
35 SEC. SHOT CLOCK
24 SEC. SHOT CLOCK
# OF TEAMS
8
8
8
8
8
8
# OF GAMES (1st round)
4
4
4
4
4
4
40 minutes
1ST ROUND (all teams)
(15-min. halves)
SEMI-FINALS (top 4 teams)
(20-min. halves)
CHAMPIONSHIP (top 2 teams)
(20-min. halves)
50 minutes
50 minutes
40 minutes
40 minutes
(15-min. halves)
(15-min. halves)
50 minutes
50 minutes
(20-min. halves)
(20-min. halves)
50 minutes
50 minutes
(20-min. halves)
(20-min. halves)
40 minutes
40 minutes
(15-min. halves)
(15-min. halves)
50 minutes
50 minutes
(20-min. halves)
(20-min. halves)
50 minutes
50 minutes
(20-min. halves)
(20-min. halves)
40 minutes
(15-min. halves)
50 minutes
(20-min. halves)
50 minutes
(20-min. halves)
In the 1st round of games, each team is given the opportunity to play 4 teams in their division. A sample schedule with teams A–F is shown below. GAME 1
A vs. B E vs. F
GAME 2
C vs. D G vs. H
B vs. G F vs. C
GAME 3
D vs. E H vs. A
A vs. F E vs. B
GAME 4
C vs. H G vs. D
D vs. A H vs. E
B vs. C F vs. G
The four teams with the best Won/Loss records advance to the semi-finals round where the best team plays the 4th best team and numbers 2 and 3 square off against one another. The winners of the semi-final games would then compete in a final, championship game. Here’s a sample of what such a structure may look like. FINAL 1ST ROUND STANDINGS
Advanced (w/35-second shot clock) TEAM
WINS
LOSSES
PCT.
C
4
0
1.000
F
3
1
.750
A
3
1
.750
G*
2
2
.500
H
2
2
.500
D
1
3
.250
B
1
3
.250
E
0
4
.000
*In this scenario, Team G finished in 4 place ahead of Team H due to a tiebreaker (head-to-head competition). th
In the semi-finals, Team C would play Team G and Team F would play Team A. The winners of these two games would then square off in the Championship Game.
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Creating a school-wide or multi-school tournament using this type of structure serves a number of purposes: • It allows teams to compete in multiple games at the most appropriate skill level.
• Whether multi-classroom or multi-school, it allows students to meet and compete against students they may not know.
• More students are recognized as there is a champion and runner-up in each division.
• A large scale event showcases the program and the high level of student engagement.
• It builds community.
The last item above is key. If you’re going to pull together a school-wide or multischool event, make sure to invite other staff and/or parents to get involved in the planning and also participate in the actual event. You may want to have an adult monitor each game and have others serve as record-keepers for the first round of games. Here are some other things you might incorporate into your event to make it extra special: • Decorate, decorate, decorate! Make the tournament environment as colorful and inviting as possible!
• Encourage students to wear their favorite NBA or WNBA gear to the event— will get them into the tournament spirit!
• Invite a guest speaker. If the opportunity presents itself, having a former or current NBA or WNBA player or coach address the student coaches will add a nice touch to your event.
• Have engraved medals or small trophies for the Champions and Runners-Up in each division.
• Prepare and send out a media release. Students thoroughly engaged in learning and applying important math skills in the context of competing in simulated basketball games will undoubtedly strike a chord with the local media.
Regional and national NBA Math Hoops tournaments are in the preliminary planning phase and will take place in the not-too-distant future. Getting students acclimated to tournament play in any type of tournament setting will be to their advantage should they have the opportunity to participate in one of these future regional or national events.
UNIT 9: LEAGUE PLAYOFFS AND CHAMPIONSHIP • 59
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PA RT
III
Strategies and Math Explorations for Successful Coaches
ood mathematical strategy goes hand in hand with winning in NBA Math Hoops. In order to fully develop sound strategic thinking, a series of relevant mathematical investigations are included in Part III. As students engage in game play, you may identify specific gaps or mathematical misunderstandings that can be effectively addressed through these explorations. While these have all been designed within the context of game play, the mathematical understandings elicited from these explorations are applicable to a number of situations. Opportunities to make these connections will vary—be sure to stay open to the possibilities!
G
One method of ensuring inclusion of these rich mathematical investigations is to build in placeholders when crafting your NBA Math Hoops roadmap. For scheduling purposes, each investigation should take approximately 1 class session, or 45–60 minutes. Carving out time for a targeted mathematical exploration after every two or three games will help provide students with a well-rounded NBA Math Hoops experience.
Focus Questions Each unit in Part III has several explorations related to the general idea of that unit. The title of each exploration serves as a focus question for that particular exploration and is the glue that holds the activity together. Whenever possible, make it a point to redirect students back to the focus question to engage them in either small group or whole group discussion. In most cases, a brief writing exercise is presented at the conclusion of the exploration providing students an opportunity to summarize their learning.
CONNECTING TO COMMON CORE STATE STANDARDS Intentional consideration of the Common Core State Standards for Mathematics informed the development of NBA Math Hoops. Special attention has been given to effective integration of the eight Standards of Mathematical Practice—”processes and proficiencies” with long-standing importance in mathematics education. By focusing on the development of understanding, the strategy and math explorations in Part III provide excellent opportunities to connect the practices to the content.
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Mathematical Practices 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning.
The Standards for Mathematical Practice remain constant for Grades 5–8, but the Standards for Mathematical Content vary by grade level. The Math Hoops experience touches on standards across multiple grade levels. One significant topic embedded throughout the explorations is Ratios and Proportional Relationships— one of the five domains addressed in Grades 6 and 7. Development of the standards in this domain will depend on prior knowledge and instruction along with depth of exploration. Another domain addressed in Grades 6 and 7, Statistics and Probability, sits at the core of Math Hoops game play and can be explored in varying degrees of sophistication. The Standards for Mathematical Content are much more robust than what students encounter solely in the Math Hoops explorations. However, the explorations themselves serve as contextually-engaging springboards into important and relevant content in many areas. For a listing of specific math skills exercised in individual explorations, review the Time Considerations chart in each Unit opener.
In this section Unit 10: Understanding Player Cards Unit 11: Analyzing the Game Unit 12: Creating Teams and a Season Schedule Unit 13: Winning Strategies Unit 14: Improving Your Team Unit 15: Why Do Coaches Need Statistics? 62 • PART III: STRATEGIES AND MATH EXPLORATIONS FOR SUCCESSFUL COACHES
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UNIT
10
Understanding Player Cards E
ach NBA Math Hoops game comes complete with 32 specially designed Player Cards. These Player Cards graphically display the player’s most recent NBA or WNBA season statistics. Two-point and three-point shooting percentages are shown in a circle graph and free throw shooting percentage is shown in a 10 × 10 grid. Understanding the mathematics behind the development of the Player Cards will serve students well as they attempt to field the most competitive team possible.
Time Considerations Each exploration in Unit 10 can be completed in 1–2 classroom sessions depending on the level of students and depth of exploration. It’s suggested that you suspend game play long enough to introduce an activity and then have students complete specific tasks on their own or with their co-coach. Once finished, be sure to allocate an appropriate amount of time to process the exploration with the whole class as this is where meaningful learning is made explicit and connections are made. EXPLORATION
ESTIMATED TIME
MATH CONTENT
10.1 What Are Field Goal Percentages (and Why Are they Written as Decimals)?
1–2 sessions 60–90 minutes
• Relating Fractions, Decimals, and Percents • Comparing and Ordering Percents • Operating with Whole Numbers and Decimals
1–2 sessions Why Use Circle Graphs to Represent Shooting Percentages? 60–90 minutes
• Using Proportional Reasoning • Creating Circle Graphs • Operating with Whole Numbers and Decimals
10.3 Does the Order of Shading Matter on the Free Throw Grids?
• Relating Experimental to Theoretical Probability • Creating Geometric Representations of Numerical Data
10.2
1 session 45–60 minutes
UNIT 10: UNDERSTANDING PLAYER CARDS • 63
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IN THE COACH’S MANUAL On pages 32–35 of the Coach’s Manual, students explore percent-decimal equivalence and how to compare these types of numbers. Students discover that decimal notation is used for FG% because it provides a greater level of precision. When comparing numbers of shots made and shots attempted by two players—first as ratios represented as fractions, and then as decimals—the need for a standardized form of measuring will become apparent.
10.1 What Are Field Goal Percentages (and
Why Are They Written as Decimals)? Each Player Card gives data for three basic statistics for an NBA or WNBA player. For example, the Player Card for LeBron James shows this: 2-PT%: .531 3-PT%: .362 FREE THROW%: .771
If not familiar with sports statistics, you (or a student) may wonder why things are written this way. After all, aren’t percents written with a percent symbol? Of course, they are, but sports statistics have evolved over the last hundred years or so to be written…well… imprecisely (that’s a nice way of saying “incorrectly”). This “imprecision” provides a learning opportunity for your students. Take a close look at the example on p. 32 of the Coach’s Manual. Strictly looking at percentages, the two players seem to be equally proficient as shown in the first table on the page; but by digging a little deeper, the variance in proficiency is uncovered. Students can then apply this understanding as they compute with, compare, and order data given for several different players on the following page in the Coach’s Manual. 1. Complete the chart below. Get creative and make up names for each of the players. The first one has been done for you.
Coach’s Manual, p. 32
PLAYER Bartholomew Dunkman
FGM
FGA
Decimal
FG%
69
145
0.4758621
0.476
143
289
0.4948096
0.495
176
362
0.4861878
0.486
94
179
0.5251396
0.525
259
503
0.5149105
0.515
226
457
0.4945295
0.494
111
216
0.5138888
0.514
179
348
0.5143678
0.514
Coach’s Manual, p. 33
64 • PART III: STRATEGIES AND MATH EXPLORATIONS FOR SUCCESSFUL COACHES
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2. Now rank the players from highest to lowest FG%. You will have to use the “Decimal” column above to correctly rank the players whose FG% are equal when written to three decimal places.
PLAYER
FG% 0.525 0.515 0.514 0.514 0.495
Coach’s Manual, p. 33
0.494 0.486
Bartholomew Dunkman
0.476
3. Complete the chart below by filling in some missing information. Make up a creative name for each player as well.
PLAYER
FGM
FGA
Decimal
FG%
137
271
.5055350
.506
95
182
.5219780
.522
150
306
.4901960
.490
124
239
.5188285
.519 Coach’s Manual, p. 34
Students have learned rules that relate percents and decimals that read something like this: To convert a decimal to a percent, move its decimal point two places to the right and attach a percent symbol. To convert a percent to a decimal, move its decimal point two places to the left and remove its percent symbol.
To some of your students, these rules make perfect sense: Percent, after all, means “per hundred.” So, to convert a decimal to a percent, you multiply the decimal by 100. And you do that by moving its decimal point two places to the right. To reverse the process, you move the decimal point to the left.
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To some of your students, these rules represent one more inexplicable rule in the mysterious land of mathematics: Decimals themselves are not all that easy to understand. And now we are moving decimal points around, two places to the left or to the right. I’ll just memorize the rule and hope to not move the decimal point in the wrong direction.
Math Hoops Player Cards provide an opportunity to explore the percent-decimal relationship. with the relationship can ponder possible explanations for why sports statistics are written the way they are. Somewhere along the way, statistics such as field goal percentage (and, more famously, batting average in baseball) were first written as decimals using three significant digits, and that practice continues to this day. Thus, diehard baseball fans know that Ted Williams hit .406 in 1941. No one writes his batting average as .4057 or 40.6%. In basketball, field goal percentages are written using decimals but you do hear people saying things like “He’s a 50% shooter from the field and a 90% shooter from the foul line.” In baseball, you never hear a .300 hitter being described as a 30% hitter. The availability of the Internet allows students to investigate how other sports represent and refer to the percents used in their data.
A STUDENT ALREADY COMFORTABLE
with the relationship between percents and decimals can focus on Player Cards to help get more comfortable. Each person comes to an understanding in his or her own way, so it is dubious to suggest one route to mastery. But the fact that field goal percentages are written using three decimal places allows a student to focus first on a consistent decimalfraction relationship. For example,
A STUDENT NOT YET FULLY COMFORTABLE
.531 =
531 1,000
If one understands that a fraction can be “reduced” by dividing numerator and denominator by the same number, the fraction above can be rewritten with 100 as denominator: 531 = 1,000
531÷10 1,000÷10
=
53.1 100
Then the final insight comes not from mathematics reasoning, but from verbal understanding. The original decimal, .531, meant 531 per thousand. The rewriting of the fraction converted 531 per thousand to 53.1 per hundred. But that is precisely what percent means. So, you end up with this sequence of understanding: .531 =
531 53.1 = = 53.1% 1,000 100
The student with mastery skips the intermediate steps. Hopefully, reflection on an approach like that shown above can help a struggling student to achieve that mastery. 66 • PART III: STRATEGIES AND MATH EXPLORATIONS FOR SUCCESSFUL COACHES
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Naturally, you’ll need to decide what degree of depth of instruction is most appropriate for your students. Once you feel they have a firm grasp of the shots made/shots attempted conversion to decimal representation, they should be ready to tackle the chart on p. 34 of the Coach’s Manual. Here they’ll not only be looking at 2-pt. shooting percentages, but will also be computing 3-pt. shooting percentages and free throw shooting percentages. 1. Complete the following table and use the data to explain your reasoning in the questions that follow.
PLAYER
Free 3-pt. 2-pt. Free 3-pt. 2-pt. 3-pt. Free Throw Shots Shots Shots Shots Shooting Throws Throws 2-pt. Made Attempted Percentage Made Attempted Percentage Made Attempted Percentage
Player A
90
177
Player B
111
219
Player C
140
267
Player D
99
Player E
151
.508
13
38
.342
64
73
.877
40
97
.412
32
42
.762
.524
24
63
.381
50
55
.909
199
.497
57
142
.401
17
24
.708
288
.524
33
88
.375
49
59
.831
.507
Coach’s Manual, p. 34
To help coaches see the connection between players’ shooting percentages and decisions they’ll make during game play, the Coach’s Manual contains six scenarios with probing questions on pp. 34–35. For each situation, students are asked to name a player and explain the reasoning that supports their response. Referencing the completed chart and specific shooting percentages should confirm that students are making informed, data-driven decisions.
10.2 Why Use Circle Graphs to Represent Field
Goal Percentages?
Coach’s Manual, p. 35
Students have used spinners with games for as long as they can remember. And they are probably increasingly aware of circle graphs as a way to represent data. On each Math Hoops Player Card, a circle graph translates field goal percentages into graph sectors and students then use a game spinner to determine shooting outcomes. Students can deepen their understanding of percents, decimals, and circle graphs by analyzing this aspect of a Player Card. A typical Player Card has about one-half of its total circle graph shaded orange. It could be a little less, or a little more, but the deviation from one-half is usually slight. The explanation for this is simple: most players have a 2-point field goal percentage
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IN THE COACH’S MANUAL On pages 36–39 of the Coach’s Manual, students investigate the use of circle graphs as a means to represent player shooting percentages. Here the connection is made between the decimal representation of the shooting percentage and the number of degrees the central angles would measure to reflect player performance accurately. Students will discover that given the level of precision needed, circle graphs serve as a reasonable and effective tool for representing player data for both 2-pt. and 3-pt. shooting percentages.
close to .500, or 50%. This is the first realization students should have regarding these circle graphs, and it should be well grounded before trying to understand deeper ideas. Of course, there will be anomalies—in fact, one is presented in the Tyson Chandler example on p. 37 of the Coach’s Manual.
Informal Understanding of Circle Graphs Prior to starting this exploration in the Coach’s Manual, students should look through the Player Cards and pull out two players: one with a 2-point field goal percentage greater than .500 and one with a 2-point field goal percentage less than .500. It makes sense that the former has more than half of its circle graph shaded orange, while the latter has less than half shaded orange. Students may question why each graph is divided into four parts and why the 3-pt. shooting percentages are embedded in the orange sectors. You can ask them: What advantages and/or disadvantages would you see to dividing the circle graphs into fewer sectors or a greater number of sectors?
Responses will vary—what’s important is to probe students to share the reasoning behind their responses. Another question you might ask is: What do you notice about the relationship between the orange sectors for 2-pt. shooting percentages and the cross-hatched sectors for 3-pt. shooting percentages?
Students should quickly identify that all of the Math Hoops players have a higher success rate for 2-pt. shots than they do for 3-pt. shots. So embedding the crosshatched sectors in the orange sectors is a consistent way to display success.
Explaining Conversion of Percent to Degrees Typically, if students have difficulty seeing how the circle graphs of the Player Cards are constructed, the trouble lies in understanding how the exact sizes of the orange sectors are determined. This is, unfortunately, where careful step-by-step analysis sometimes hits a wall: a student who gets the idea by working with intuitive concepts (half, more than half, less than half ) may shut down because the jump to an algorithm is not yet intuitive. This exploration and any discussion points you add will help students focus on ideas they can use to help make the process more comprehensible. Coach’s Manual, p. 36
The key idea every student must understand is that circle graphs are formed by drawing radii that break the circle into sectors, just as you would divide a pie into pieces. If four people share a pie equally, each would get one-fourth, or 25%, of the pie. Similarly, if a player had a .250 field-goal percentage, her or his circle graph would have a one-fourth piece (sector) shaded orange.
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The idea of dividing a circle graph, or pie, into four equal sectors can be used to foster understanding because of the convenient central angles that are formed. The diagram on p. 36 of the Coach’s Manual illustrates this point. Most students will recognize that each of the four central angles in the picture is a right angle, the measure of which is 90 degrees. Thus, a circle graph “contains” 4 × 90˚, or 360˚ in all. It is those 360 degrees that must be apportioned among the different sectors of any circle graph. In the case of Tyson Chandler and his .683 2-pt. shooting percentage, the goal is to calculate the rate of success to the nearest degree. The example shown in the Coach’s Manual on p. 37 is important for students to see, because understanding why a circle graph has the EXACT sectors that it shows involves precisely this idea of finding a percent of 360˚. In the case of a circle graph divided into four equal sectors, one-fourth is equivalent to 25% and so: 25% of 360˚ = .25 × 360 = 90˚
The central angle for each sector is therefore 90˚. In the case of a basketball player with a field goal percentage of .683:
MATH CONNECTION Understanding how the construction of the circle graphs relates to a player’s shooting percentage will be especially important if you choose to have students construct their own Player Cards. Turn to 14.2 CREATE YOUR OWN PLAYER CARDS! PP. 114–119
.683 × 360 = 245.88 which can be rounded to 246˚.
If the shooting percentage were being shown as a single orange sector, it would have a central angle of 246˚. Because the Player Cards break up the percentage into two equal sectors, each of the two orange sectors for this player should have a central angle of 123˚. Students can first note that this is 33˚ more than a quarter of a circle. They can also use a protractor to check the precision of the measurement. The exercises and tasks that follow on pp. 37–39 of the Coach’s Manual provide students with the opportunity to compute the appropriate number of degrees for sectors of success which then can be used, by default, to determine the sectors of, well, un-success. Students have the opportunity to put all these concepts together by creating a circle graph reflecting Steve Nash’s career shooting percentages. 2. Complete the chart below. The first one has been done for you. PLAYER
2-pt. FG%
360° × 2-pt. FG% (rounded)
Successful° ÷ 2
Orange Sections
Gray Sections
Player A
.564
360° × .564 = 203°
203° ÷ 2 = 101.5°
101.5°
78.5°
Player B
.483
360˚ × .483 = 174˚
174˚ ÷ 2 = 87˚
87˚
93˚
Player C
.525
360˚ × .525 = 189˚
189˚ ÷ 2 = 94.5˚
94.5˚
85.5˚
Player D
.477
360˚ × .477 = 172˚
172˚ ÷ 2 = 86˚
86˚
94˚
Player E
.542
360˚ × .542 = 195˚
195˚ ÷ 2 = 97.5˚
97.5˚
82.5˚
Player F
.639
360˚ × .639 = 230˚
230˚ ÷ 2 = 115˚
115˚
65˚
Coach’s Manual, p. 37
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3. Which of the players in the table above match the following circle graphs? Player
F
Player
B
1. Think you can create a circle graph for one of the NBA’s premier players? Use the career data below for Steve Nash and give it a try. You will need a calculator to find the degrees for each sector and a protractor to complete your graph. a. First begin with Steve Nash’s 2-pt. shooting percentage data:
Steve Nash CAREER STATS* Coach’s Manual, p. 37
2-PT. FG%
360° × 2-pt. FG% (rounded)
Successful ° ÷ 2
Orange Sections
Gray Sections
.491
360˚ × .491 = 177˚
177˚ ÷ 2 = 88.5˚
88.5˚
92.5˚
b. Now finish your circle graph using Nash’s 3-pt. shooting percentage data: 3-PT. FG%
360° × 3-pt. FG% (rounded)
Successful ° ÷ 2
Crosshatched Sections
.428
360˚ × .428 = 154˚
154˚ ÷ 2 = 77˚
77˚
Steve Nash Coach’s Manual, p. 38
Coach’s Manual, p. 39
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Practice with Protractors
CLASSROOM TIP
If your students have never used a protractor or would benefit from a quick refresher, herein provides a great opportunity for you to do some direct instruction. You may want to conduct a demonstration by using career data from an NBA Hall-of Famer. For example, Earvin “Magic” Johnson who played exclusively for the Los Angeles Lakers, posted a career 2-pt. shooting percentage of .520 and a career 3-pt. shooting percentage of .303.
The writing prompt that concludes this exploration—writing a brief note to NBA Commissioner David Stern—allows students to summarize what they’ve just learned about using circle graphs to display shooting percentages. You’ll note that no dedicated space is provided in the Coach’s Manual for this writing exercise. You may elect to create a special NBA Math Hoops bulletin board in your classroom to display student work samples over and above what’s recorded in the Coach’s Manual. A collection of these notes would make an excellent addition to that display.
10.3 Does the Order of Shading Matter on the
Free Throw Grids?
Once students understand that percent means per hundred, you can pose the following question: One player has a free throw percentage of .873 (87.3%) and another .872 (87.2%). How do you “design” the free throw grid on Player Cards for each player?
Here are some steps, ideas, and questions to keep in mind as you let students explore this question. • It is not by chance that the grids used in Math Hoops contain exactly 100 shaded squares. [This is the first realization a student must come to.] • To build a free throw grid, how do you decide exactly how many squares to shade orange? • There will be significantly fewer gray squares than orange squares on each grid. Will they be scattered around the page? Could they be clustered in one corner? Could they be arranged in rows? Would it matter? • Could something other than a grid be used to shoot free throws in Math Hoops? How would it work?
IN THE COACH’S MANUAL On pages 40–42 of the Coach’s Manual, students examine different patterns of shading on the 10 × 10 free throw grids. They compare grids where all the gray squares (representing a missed shot) are grouped together and grids where the gray squares are spread out. By running through a couple of simulated trials—first looking at individual and then group data—students will begin to see that while they may be attracted to certain patterns, it’s the area, or number of squares representing a make or miss that’s the determining factor behind success and failure.
• How does rounding come into play on the free throw grid? Do you think that is OK or does it make the game unfair in some way?
After engaging students in some informal discussion around the questions above, have them take a look at the free throw grids on p. 40 of their Coach’s Manual. In UNIT 10: UNDERSTANDING PLAYER CARDS • 71
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2011–12, Danny Granger’s FT% was .873 and Kyrie Irving’s was .872. You’ll note that on each grid, there are 87 orange-shaded squares and 13 gray-shaded squares. On one grid, the gray-shaded squares are clustered together while on the other, the gray-shaded squares are scattered. Your students may have strong opinions as to which design is better or more favorable and having them run through a simple exercise should help them begin to develop an understanding of the difference between theoretical probability and empirical (experimental) probability.
Coach’s Manual, p. 40
Coach’s Manual, p. 41
Using Simulations to Explore Probability The brief experiment at the top of page 41 of the Coach’s Manual may confirm or conflict with their response to the question at the bottom of the previous page. Because this is such a small sample, it isn’t likely that each pair of students will record an equal number of makes (and misses) for both players. While this may lead students to believe that one design is more favorable than another, compiling data from several groups of students may sway their opinion. Even with the combined data from as many as 16 pairs of students, however, you may still find students leaning in one direction—either based on the cumulative experimental data or purely aesthetic appeal. Taking things a step further, students are then given a chance to design two different free throw grids for Dwight Howard and asked to describe the thinking behind their two designs. Again working with their co-coach or another partner, they roll the dice to simulate 40 attempted shots for each grid and record the results. They can then look at whether or not the variance in their designs made a difference for a larger, yet still relatively small sample. If time allows, create a large chart that allows students to record the results for each set of experimental data. What you and your students will find is that the results will vary from pair to pair. By adding together all of the makes and all of the misses and finding the cumulative free throw percentage for all rolls, regardless of design, it’s highly likely that the rate of success will either match or come very close to matching Dwight Howard’s actual free throw percentage. Because the number of orange (successful) squares is identical for both grids, the theoretical probability is the same. By collecting a significant number of random samples, the experimental probability should come close to matching the theoretical. As the sample size increases, any disparity between the two will continue to shrink.
Coach’s Manual, p. 42
The writing prompt at the bottom of page 42 that concludes this exploration provides students with the opportunity to analyze and synthesize their results and understandings from the trials. When writing their coaching tip, encourage them to reference not only the results from their individual trials but also the cumulative results from all trials. This will help you determine whether or not their level of
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understanding of probability is on par with your expectations or if they may need a refresher a little later in the Math Hoops season.
LOOKING AHEAD When students have a clear understanding of the fundamental math concepts inherent in the Player Cards, they’ll be ready to start delving into the intricacies of the game. In Unit 11, students will investigate the Shot Planner, explore relationships between the even and odd ends of the Game Board, and look at possible benefits to the special division rules included in the Advanced Game. With each step taken, students will become more knowledgeable about the role of mathematics and strategy in the development of the game, which will lead to increased proficiency and confidence.
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UNIT
11
Analyzing the Game A s students begin to get more comfortable with both the Basic and Advanced versions of the game, natural curiosity may lead them to question why certain aspects of the game are the way they are. Despite this natural curiosity, rarely do we take the time to allow students to thoughtfully explore and analyze why particular rules and/or guidelines are implemented. The explorations in this unit allow just that—the chance for students to take a step back and understand the mathematical reasoning behind the organization of specific rules and guidelines. By doing so, student coaches will become more adept at how to make the construction of the game work to their advantage.
Time Considerations The three activities in this unit will each take 1–2 classroom sessions depending on the depth of exploration. This is a good time to take small breaks in game play in order for students to look carefully at specific aspects of the game. Feel free to jump back and forth between games and explorations to best meet your students’ needs and interests, as well as your program implementation schedule. EXPLORATION
ESTIMATED TIME
MATH CONTENT
11.1 Investigating the Shot Planner
1–2 sessions 60–90 minutes
• Operating with Whole Numbers • Investigating Commutative Properties of Arithmetic Operations
11.2 Would You Rather Be Odd or Even?
1–2 sessions 60–90 minutes
• Analyzing Properties of Integers • Exploring Number Patterns to Solve Problems
11.3 How Will the Advanced Division Rules Affect Your Strategy?
1 session 45–60 minutes
• Using Estimation and Rounding to Solve Problems
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IN THE COACH’S MANUAL
11.1 Investigating the Shot Planner NBA and WNBA coaches always keep a dry-erase board with a simple drawing of a basketball court on it nearby for mapping out plays during time outs. When on offense, these plays are designed to show who should get the ball and take the shot. Some very exciting finishes to important games in NBA history have been planned on these boards! In Math Hoops, students will use a special dry-erase Shot Planner to determine their shot options. In both the Basic and Advanced versions of the game, the rules say to record the roll of the dice by writing the largest number first. The focus question you can present to students in this exploration is: “Why do you think this rule exists?”
Basic Game When playing the Basic Game, students first record the roll of the dice and then use those numbers to perform the four basic operations. By comparing results when operating with numbers recorded largest to smallest and also smallest to largest, they should begin to identify mathematical relationships and the rationale behind the existing rules. To get started, students can turn to p. 43 in their Coach’s Manual.
On pp. 43–45 of the Coach’s Manual, students investigate the reasoning behind the rule that states operations should always be performed working from largest number to smallest number. In the Basic Game, this refers strictly to the roll of the decahedra dice. In the Advanced Game, this refers to the roll of the dice and also operations performed with the top row of calculations and operations with those answers and the Ball On number. Once this investigation is complete, students should have a clear understanding of why this specific rule was put in place.
1. Imagine you just rolled the Math Hoops dice and rolled an 8 and a 4. Complete the two charts below, operating on the numbers from left to right as indicated. 8 +4
ROLL
8 4
+
ROLL
4 8
+
12
4+ 8
12
8–4
–
4
4– 8
–
−4
8×4
×
32
4×8
×
32
8÷4
÷
2
4÷8
÷
1 2
The exercise at the bottom of p. 44 of the Coach’s Manual involves a roll of the dice that yields an 8 and 0. Unless the concept of division with zero is something students have previously been introduced to, they will be inclined to record 0 as the quotient regardless of order. The feature at the bottom of Coach’s Manual p. 43 discusses division with zero and can be used to prompt further investigation. You can also find some additional discussion on this topic on p. 18 of this Teacher’s Guide.
Coach’s Manual, p. 43
The questions on pp. 44 – 45 of the Coach’s Manual are intended to help students understand why they should operate with the larger number first. You can guide students to understanding by working through samples with both the Basic and Advanced versions of the game. UNIT 11: ANALYZING THE GAME • 75
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2. Which of your answers are the same regardless of how you recorded the roll of the dice? Why do you think this is so?
Addition and multiplication will be the same due to the Commutative property. Order does not matter. 3. How are the answers for subtraction similar? How are they different?
Both yield the same number—first example is positive integer and second example is negative integer. 4. Do you see a relationship between your division answers? If so, how would you describe this relationship? Coach’s Manual, p. 44
The quotients are reciprocals—when you multiply them, the product is 1.
5. Use the dice rolls below to figure out shot options for the Basic Game. First record the roll from greatest to least, then from least to greatest.
SAMPLE 1
ROLL ROLL
SAMPLE 2
ROLL ROLL
SAMPLE 3
ROLL ROLL
SAMPLE 4
ROLL ROLL
SAMPLE 5
ROLL ROLL
2 6 6 2
9 3 3 9
5 1 1 5 7 4 4 7 8 0 0 8
+ +
8 8
–
×
÷1
–
×
÷
×
÷
×
÷1
−4
+
–
+
–
12 12
+ +
6 6
–
−4
+
–
+ +
8 8
4
– –
11
6
−6
+
11
4
3
−3
– –
8
−8
12 12
27
27
× ×
5 5
3
÷
5
×
÷
× ×
0 0
5
÷1 ÷
21
3 3
×
21
3
÷
—
÷
0
After completing the Basic Game examples, students are given the opportunity to write a summary of what they have discovered. They will be asked to do this again following exploration of samples with the Advanced Game.
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Advanced Game In the Advanced Game, coaches may need to reorder the numbers mentally each time they perform an operation with the Ball On number since it’s possible for it to be greater or less than any of the numbers in the top shaded row. The questions on p. 45 of the Coach’s Manual will reinforce understanding of which operations require a specific order. 1. Both charts below show the identical dice roll and Ball On number — the difference being that the first chart lists the dice roll from largest number to smallest and the second from smallest to largest. Complete the first chart by ww following the Advanced Game rules and always operating from largest number to smallest. Complete the second chart by doing the opposite — operating from smallest number to largest. ROLL
9 2
BALL ON
ROLL
15
2 9
BALL ON
15
+ + –
11 26 4
–
7
+ –
22 8
× +
18
÷
4 or 5
33
+ 19 or 20
–
−3
– 11 or 10
×
×
×
270
× 60 or 75
÷
÷
÷
÷
1 or 2
2 or 3 .8333*
+
–
165
+ –
11 26 4
105
−7
+ –
8 22
× +
18 33
÷
.2222*
+
–
–
−3
×
×
×
×
÷
÷
÷
÷
165 −105 270
1 or 2
−2 or .8333* −3
(4 or 5) or 3
* Dividing a smaller number by a larger number will always yield a fraction, which for the purposes of the game, you should simply disallow. Clever students may suggest that the Advanced Game division rules allow them to round these quotients to 0 or 1. This is a great insight, but one that would result in an unfair advantage to the team playing on the Odd end of the court since it would allow them to take a shot using the number 1 on every turn. Just point out that the Math Hoops rules require the larger number to be divided by the smaller number.
Coach’s Manual, p. 45
After completing the Shot Planner samples, students are once again given the opportunity to explain what they discovered—another chance to reinforce the mathematical relationships presented in the Basic Game examples.
11.2 Would You Rather Be Odd or Even? When students first look at the Math Hoops game board, they will probably see it as just a jumble of numbers, divided into odd and even halves. Little thought will be given to the distinction between odd and even. Playing the game, however, will lead students to perceptions and misperceptions about the ways in which the game plays out on each end of the board. UNIT 11: ANALYZING THE GAME • 77
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IN THE COACH’S MANUAL In the investigation on pp. 46–48 of the Coach’s Manual, students look carefully at outcomes from operating with odd and even integers. Because the game is built around the use of odd and even numbers, students try to make sense of whether or not one yields a definitive advantage over the other. The writing prompt at the conclusion of this exploration provides students with the opportunity to summarize their understandings by selecting one of three options regarding preferences between the odd and even ends of the Game Board and supporting their choice with valid mathematical reasoning.
Suppose a player in the Basic Game rolls the dice and cannot operate with the two numbers to get an answer that fits their end of the court. For example, if the team on the odd end of the court rolls 4 and 2, no operation yields an odd number. Should this happen a disproportionate number of times to one team, it would not be surprising if that team concluded that the game favors the even end of the court. That doesn’t mean that the team is correct, of course. You can use this as an opportunity for students to investigate number properties by examining the universe of possible rolls to determine when the Odd or Even team is left without a shot. They may be surprised by what they find.
Properties of Odd and Even Numbers In the Basic Game, some two-dice rolls lead to the situation in which none of the four operations creates a usable number. For example, rolling 2 and 4 yields no usable numbers for the Odd team while a roll of 0 and 5 yields no usable numbers for the Even team. With two 10-sided dice, there are 10 × 10 = 100 possible rolls. As you might suspect, more rolls yield usable numbers for the Even team than for the Odd team. The table below summarizes the results. Usable Rolls
Unusable Rolls Description of Unusable Rolls
Even Team
90
10
Any roll of 0 and an odd number*
Odd Team
82
18
Any roll of two different even numbers, except for a roll of 2 and 6
* Remember that a roll of 0 and 0 is a usable result because it doesn’t require players to use the Shot Planner. [See rules for Fast Break on p. 6.]
As you can see from the table, the Odd team has nearly twice as many unusable rolls as the Even team. On the Game Board, this is counterbalanced by the extra choices given to the Odd team for three of its shooters: the numbers 15, 35, and 63 can be used as either 2-point or 3-point shooters. This advantage for the Odd team occurs on 10 of the 100 possible two-dice rolls. In this first part of the exploration, students are given the opportunity to identify whether or not an even or odd outcome occurs when looking at the roll of the dice through the lens of even and odd as opposed to specific numbers. Identifying even and odd outcomes resulting from operations of addition, subtraction, and multiplication can be represented in a simple table. Complete the three tables below to see what this means.
+
EVEN
ODD
–
EVEN
ODD
×
EVEN
ODD
EVEN
E
O
EVEN
E
O
EVEN
E
E
ODD
O
E
ODD
O
E
ODD
E
O
Coach’s Manual, p. 46
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Division is a different animal. At the bottom of p. 46 in the Coach’s Manual, students will need to delve a little deeper to list specific examples of a variety of division scenarios. This is a great task for small group work—sharing and comparing results and challenging each other to cite as many examples as possible. Here are some possible responses to the situations posed in Questions 2–5: 2. For each of the following, list one example: a. roll of two odd numbers that can be divided b. roll of two odd numbers that can not be divided
9 and 3 9 and 5
c. roll of one even and one odd number that can be divided d. roll of one even and one odd number that can not be divided e. roll of two even numbers with an even quotient f. roll of two even numbers with an odd quotient
6 and 3 6 and 5
4 and 2 6 and 2
g. roll of two even numbers with a quotient that is not a whole number
6 and 4 Coach’s Manual, p. 46
3. Whatever number is rolled on the first die, what number on the second die guarantees that your quotient is a whole number? 1 4. A player rolled one die first and complained, “That number is the worst for division! You hardly ever get whole numbers.” What number did this player most likely roll? 7 5. Given what you know about properties of even and odd numbers, do you think it’s better to be on the even or odd end of the court? Explain your reasoning.
At this point in the exploration, some students may say that it’s better to be even. That’s fine as long as they can provide mathematical reasons for thinking so. These students may change their minds after completing the second part of the exploration.
A spin-off investigation you may want to consider for your students is one that compares the odd and even ends of the board and looks at the strategy behind choosing which of the four operations to use. Especially when a shot clock is used, time is of the essence and players do not want to waste time on an operation that won’t yield a possible shooter. Students will quickly discover by playing a few games that multiplication and division have a smaller chance of leading to a shooter on the odd end of the board than on the even end of the board. They can investigate and report on why this is so, and quantify the difference. They can also use what they find to help develop effective strategies. For example, teams playing on the odd end of the court can learn to recognize that multiplication can be ignored in the Basic Game unless both numbers rolled are odd.
Coach’s Manual, p. 47
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Investigating Odd and Even Advantages BASIC GAME ONLY The Math Hoops Basic Game Board shows 20 numbers on the Odd end of the court and 20 numbers on the Even end. Each end of the court also has the same number of 2-pt. and 3-pt. shot options. Students might initially think that makes the game fair. If they’ve completed the first half of this exploration, they may also think that there is a slight advantage on the Even end. You may be interested to see what they think after some investigation. In this second part of the exploration, students are asked a number of critical thinking questions that require some investigation and often, an explanation of their thinking. These contextual questions lead students to a final task—determining what side (if any) they would choose if they could be on that side for the entire game. The discussion that will emerge from this activity—explaining and citing evidence to support their thinking—will prove invaluable as students continue to play the game and refine their strategies. These experiences lead to increased confidence, understanding, and contribute to the competitive spirit that makes the Math Hoops classroom emanate with excitement. 1. Suppose you’re the Odd team and you roll a 4 and a 2. To get to your end of the court, you need an odd number. What are your options?
ROLL Coach’s Manual, p. 47
4 2
+
6
–
2
×
8
÷
2
There are no options. This would be considered a turnover.
2. Now suppose you’re the Even team and you roll a 9 and a 6. What are your options in this case? Are you able to get to your end of the court?
ROLL
9
6
+
15
–
3
×
54
÷
3. List all the rolls you can find where there are no shots available on the Odd end of the court. Then do the same for the Even end of the court. Do you think you found them all? Check with other coaches to see if there are any you missed. ODD:
0&2, 0&4, 0&6, 0&8,
EVEN: 0&1, 0&3, 0&5, 0&7, 0&9
2&2, 2&4, 2&8, 4&8 Coach’s Manual, p. 48
5. Suppose you are playing on the Odd team. With less than 30 seconds left in the game and your team down by two points, you roll a 5 and 3. What choices do you have for your shot? What will you do?
The Shot Planner results are 8, 2, 15, /. So there is only one odd number: 15. However, this number is shown twice on the Odd end of the Basic Game Board, as a 2-pt. shot and as a 3-pt. shot.
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11.3 How Will the Advanced Division Rules Affect
Your Strategy?
ADVANCED GAME
When division is used in the Basic Game, the answer or quotient must be a whole number. If the operation leaves a remainder, division gives no options to take a shot. In the Advanced Game it’s a much different story. Using division in the Advanced Game allows students to round up OR down so that they always get a whole number that can be used in the Shot Planner. A roll of 9 and 4 gives a quotient of 2 OR 3 because the exact answer is BETWEEN 2 and 3. The number a coach chooses will most likely depend on whether they’re playing on the ODD or EVEN end of the court.
IN THE COACH’S MANUAL Special division rules have been built into the Advanced Game that add a new dynamic to game play. On pp. 49–51 of the Coach’s Manual, students determine the various outcomes resulting from the game’s division rules and draw conclusions as to how to how they might use them to their competitive advantage during regular season and postseason game play.
In this exploration, students are given the chance to work with a couple of Shot Planner examples where the quotient from the original roll of the dice is first odd and then even. With each, students respond to a series of questions that guide them to identify and compare their options by using the special Advanced Game division rules. The Shot Planner examples and follow-up questions, along with possible responses are provided for your reference. Refer to the Shot Planner below when answering the questions that follow.
ROLL
9
BALL ON
4
12
13 5 36 3 + + + + 25 17 48 15 – – – – 1 7 24 9 × × × × 60 36 ÷ ÷ ÷ ÷ 2 3 3 4
+
–
×
÷
Coach’s Manual, p. 49
1. What options are there for placing the ball on the EVEN end of the court? 2, 4, 24, 36, 48, 60
2. What options are there for placing the ball on the ODD end of the court? 1, 3, 7, 9, 15, 17, 25
3. Which side has more options?
ODD
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4. How do you think things might change if 9 ÷ 4 were recorded as 2 instead of 3? Complete the Shot Planner to find out.
ROLL
9 4
BALL ON
12
13 5 36 2
+
–
+
+
+
–
–
–
25 1
×
17 7
÷
48 24
+ –
×
×
×
×
÷
÷
÷
÷
60
2
2
3
14 10
24 6
5. What options are there for placing the ball on the EVEN end of the court?
2, 6, 10, 14, 24, 48, 60 6. What options are there for placing the ball on the ODD end of the court?
1, 3, 7, 17, 25 7. Which side has more options? How did the number you rounded to affect the number of options for each shot planner? Even end now has more options.
After comparing results from identical dice roll and “Ball On” numbers by altering the quotient on the top row, students are then given an actual game play scenario. Coach’s Manual, p. 50
8. Imagine you are playing on the ODD end of the court. Complete the Shot Planner below so that it gives you the most shot options.
ROLL
7 2
BALL ON
25
+
9
–
5
×
+
+
+
–
–
–
×
×
34 16
÷
3
30 20
÷
14 39 11
×
5
÷
÷ + –
4 29 21
×
1
÷
7
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9. What strategies did you use to find the greatest number of shot options for the ODD side of the court? What would you do differently if you were playing on the EVEN side of the court?
What students will hopefully begin to see is that when the “Ball On” number is odd and their goal is to find the greatest number of odd options, it’s best to have the division answer in the top row be even. When it’s even, the addition, subtraction, and division options in that column will all yield odd numbers and the multiplication option, if 60 or less, will be even. If the goal were to find the greatest number of even options via division, it would be better to have the division answer in the top row be odd. When it’s odd, the addition, subtraction and division options in that column will all yield even numbers and the multiplication option, if less than 60, will be odd.
If students pick up on the even/odd patterns that occur when the “Ball On” number is odd, you can extend this understanding by having them do a similar exercise where the “Ball On” number is even. The pattern with even “Ball On” numbers is that an even quotient in the top row will yield more even numbers in that column and an odd quotient will yield more odd numbers. The concluding game play scenario, presented below, provides the perfect opportunity for students to apply what they’ve learned. 10. You’re in a very close game with less than one minute left. The ball is on 39 when you roll an 8 and a 3. What will be your division answer in the top row of the Shot Planner so that you get the most shot options possible? Create a model to show your results and explain your reasoning.
Coach’s Manual, p. 51
There is a blatant omission in the scenario above. Students are intentionally not told whether they are to look for the most even or odd shot options. Once they figure this out, some will need to construct two Shot Planner grids and work through the possible options to find a reasonable solution. Others will make the connection to the previous scenarios and be able to determine right away that because the “Ball On” number is odd, an even number in the top row will yield the greatest number of odd shot options and an odd number in the top row will yield the greatest number of even shot options. Having this strategy in their coaching arsenal will come in handy, especially in games where the shot clock is being used and efficient use of time is critical.
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UNIT
12
Creating Teams and a Season Schedule I
f it’s your desire to set up a Math Hoops league, the activities in this unit are necessary steps. Middle school students in particular are well aware of the concept of “fairness” and will want to have a say in any kind of player draft they’ll be participating in. Looking closely at player data and determining shooting priorities will aid students in making desirable selections for their respective teams. Involving students in the process of creating the league’s regular season game schedule provides yet another opportunity to garner student buy-in. Use of these activities as a springboard to introduce or reinforce math skills and practices will only enhance the total student experience.
Time Considerations The three activities in this unit will each take approximately 1 classroom session depending on the depth of exploration. Feel free to jump back and forth between practice games and explorations to best meet your students’ needs and interests, as well as your program implementation schedule. EXPLORATION
ESTIMATED TIME
MATH CONTENT
12.1 How Do You Make a Player Draft That’s Fair?
1 session 45–60 minutes
• Designing Math-Based Rules to Create Fairness in Games
12.2 Which Players Should You Choose for Your Team?
1 session 45–60 minutes
• Comparing Decimals and Percents • Using Number Sense to Evaluate Data
12.3 How Do You Design a Creative Season Schedule?
1 session 45–60 minutes
• Identifying Requirements and Constraints for Schedule-Building •Understanding and Using Combinations
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12.1 How Do You Make a Player Draft That’s Fair? It is draft day and each of the teams in your league is undoubtedly eager to select its top choice of players. Some teams are bound to be disappointed if the draft doesn’t go their way and might chirp about how they got cheated in the draft. Such sour grapes are to be expected; but your job, as League Commissioner, is to guarantee that there is no valid point behind the complaining. A draft must be conducted in a fair way so that no team is favored by the inherent process. With Math Hoops, you will actually have several drafts going on simultaneously and each one could potentially take place using a different process. Each game included in the NBA Math Hoops Classroom Kit comes with a set of 32 NBA and WNBA Player Cards. You will want to assign each team to work with another team to create and run a fair player draft. Does this mean more than one team can get the same player? Could Kevin Durant, for example, end up on the roster of half of the teams in your league? It sure does. While this may seem a bit unusual, this process actually allows for a few interesting developments. First, it will quickly become apparent whether student coaches are drawn to particular players because of their name and reputation. Better yet, you will also be able to see if coaches might acknowledge names but instead pay close attention to the numbers and draft players they feel will yield the best results. Every team wants to make the first selection in their draft, but only one team can. No team wants to make the last choice in the draft, but one team must. To be fair, the assignment of order must be random. Students can appreciate this idea and will be asked to devise a method for ensuring a fair first round. For example, choosing numbers blindly from a bag is a fair method. But even there, students may need to discuss why. Suppose the first round is indeed conducted by choosing numbers from a bag. You might propose to your class that all subsequent rounds continue in the same manner, i.e., the team that chose first in the first round also chooses first in all subsequent rounds, and so on. Is this method fair? The team choosing first might nod its approval but the rest of the class should howl in opposition. While this method could be considered “fair” since both teams had an equal chance of choosing first, continuing this advantage through all subsequent rounds would favor the team that selected first.
IN THE COACH’S MANUAL Think your students will want to have any say in the draft process? On pp. 52–53 of the Coach’s Manual, a selection of tasks and questions will help guide them in making sure their voice is heard and that they feel they’re part of the process. After discussing options with their co-coach and then collaboratively designing and agreeing on a draft process they will use with another team in their league, students can begin to prepare for the upcoming draft. The draft itself will take place once students have completed the first half of the exploration and should only be done once you feel students are fully prepared to make informed, educated selections.
CLASSROOM CONNECTION For additional discussion on how to manage the player draft in your classroom, turn to 6.1 CREATING TEAMS AND LEAGUE DIVISIONS PP. 37–38
Using this example, have students discuss ways they might handle the subsequent rounds of the draft that will lead to more equitable results. Here are two ways that might surface in this discussion:
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• Reverse the order of selection in the second round. The team that selected second in the first round now gets the first selection in the second round. In the third round, repeat the order from the first round. In the fourth round, repeat the order from the second round, and so on.
• For each round of the draft, have both teams repeat the blind selection of numbers from a bag (or whatever method was used for the first round).
On page 52 of the Coach’s Manual, coaches are provided space to record up to three different processes for conducting a player draft. Don’t be surprised if students come up with very different approaches that seem fair. Whatever is brought up, have students discuss any new idea so that they understand it and so that they can challenge it. Defending an idea is a wonderful way to understand its strengths and weaknesses more fully. Along with citing the strengths and weaknesses, students should be encouraged to talk about the role of sound mathematical reasoning as it relates to the fairness of their draft processes. Page 53 provides additional thoughtprovoking prompts that focus on the draft process. Coach’s Manual, p. 52
Coach’s Manual, p. 53
Once you’ve assigned each pair of teams to conduct a player draft and they’ve agreed on a fair process, they can proceed to draft players. To help them evaluate their options, you may first want them to work through the investigation in 12.2: Which Players Should You Choose for Your Team? Let them know what to expect when they build their team. They will first want to make sure to draft a player for each position. This should be the focus of each team’s first five picks in the draft. Once this is done, they will have the opportunity to draft three additional players—ones they may use to spell a player that’s not performing as expected or elect to use as a mid-game replacement to (hopefully) provide a much-needed spark that leads their team to victory. A summary writing prompt is included with this exploration that allows students to reflect on their understanding of various draft processes. Here they’ll select the one they feel is most fair and would recommend to two teams about to embark on their own player draft. Select writing samples can be posted on your NBA Math Hoops bulletin board for all to reference and/or used to generate further discussion with your student coaches.
12.2 Which Players Should You Choose for
Your Team?
Before the Draft Now that your students have decided how they will conduct their player draft, it’s time to begin thinking about what it will take to build a competitive team. When
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considering which Player Cards to choose, there are some game-related factors students will want to consider. They will want to: • look for successful 2-point field goal shooters • look for strong 3-point field goal shooters • look for solid free throw shooters • select a different-colored Player Card to fill each position • select reserve players (up to three) that complement their starting five
You may find that students have some ideas of their own that are unrelated to game factors. For example: • They may like a player from watching basketball and want to select that player for that reason alone.
• They may recognize that a player is remarkably talented and want to select that player’s card, but the player’s real-world talent may not be a factor in Math Hoops. (For example, a player that is known as a great rebounder or a great passer will have those talents overlooked by NBA Math Hoops since the Player Cards incorporate only shooting percentages.)
IN THE COACH’S MANUAL On pp. 54–55 of the Coach’s Manual, students are directed to look closely at the data on the Player Cards and even encouraged to play a few practice games to give potential draftees a “test run.” The goal here is to make sure co-coaches are clearly communicating with one another and finding a way to reach consensus on players they hope to draft. Following the draft, students will use p. 56 to carefully examine specific features of the Advanced Game Board design and how they might take full advantage of the strengths of their players during game play.
The tasks and questions presented in the Coach’s Manual on pp. 54–55 are intended to prepare students for draft day. First students will want to think carefully about which of the shooting statistics they value most and why. Once co-coaches have some discussion about priorities, they can begin looking through the Player Cards to determine their desired top choice for each colored card. Students can then record names in the chart at the bottom of page 54. This task is to be done individually. Caution your coaches to be careful not to share their desired picks with the other team they’ll be conducting their draft with! Now it’s time for students to team up with their co-coach. The tasks and questions presented on page 55 of the Coach’s Manual are geared towards carefully preparing for the upcoming draft. Additional preparation may include:
Coach’s Manual, p. 54
• Examining the players that are available and ranking them within each category from top to bottom. The ranking can be based on the three shooting percentages presented on the back of each Player Card. How the percentages are weighed is up to each team. For example, one team might want to emphasize 3-point shooters while another plans to focus primarily on 2-point shooting. A third may decide that a balance of the two is best.
• Agreeing on two or three key players in the draft that they would really like to have on their team. A good goal would be to try and select those players in the first few rounds of the draft.
At least one day before the draft, the assignment of draft order should be determined so that teams can prepare. Remember, each team will be paired with another and be drafting players from the same set of 32 Player Cards. Let each team that earns
Coach’s Manual, p. 55
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the first selection in their particular draft know that at the end of the first round, they will be expected to explain to the entire class the reasoning behind their first selections. The point of this is two-fold: 1. It puts pressure on those teams to think deeply about their important and highly advantageous first picks. Knowing that they will have to expose their reasoning to the entire class will push the team to strategize in a meaningful way. 2. Listening to teams explain their reasoning will help the rest of the class. The listening teams will have a chance to compare other teams’ reasoning with their own and may gain insights into effective strategy.
Draft Strategy It’s possible that your students have already been playing games with an ad hoc group of players. If this is the case, they may have some pre-determined ideas as to particular players they’d like to draft for their “official” team. Students will first need to select five players—one for each color/position on the game board that they’ll consider as their “starting five” for league play. Depending on the amount of time and level of play they’ve experienced with the game to this point, some may not be thinking about strategy at all. You have a choice: • Do you let students draft their teams without ideas from you beforehand? Under this approach, some students might overlook the strategic element entirely, some might intuit how to assess the Player Cards informally, and some might be prepared to truly strategize (having read and/or worked through some of the Coach’s Manual already).
• Do you bring up some formal ideas with the class a day or so before the draft that will help teams strategize? This could involve engaging students in a simple discussion around the following concepts: Success on a shot is determined by working with the graphic display of data on the back of each player’s card and using either a transparent spinner or pair of decahedra (ten-sided) dice. Field goal (2 pt. and 3 pt.) and free throw percentages from the most recent NBA or WNBA season are listed on the back of each player’s card. The higher the number, the greater the likelihood of success for the shooter. Based on foul limitations and the number of circles on the game board that are inside the 3-pt. arc, most shot attempts will come from 2-pt. range. When two players are compared, it’s possible that one might be better in all shooting categories. That makes it easy to prefer this player.
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These are basic ideas. Then strategy becomes a little more delicate. For example: When two players are compared, one might be better in two of the three shooting categories. You might prefer one player or the other depending on how important you think each category is. Having to pick one player of each color affects your strategy. With only 6-7 Player Cards per color, it might be a good idea to prioritize players for each position and whenever possible, pick the better one before it is scooped up. The Game Board does not have an equal number of circles for the five player colors. It’s possible that could affect the number of shooting opportunities a player gets.
After the Draft ADVANCED GAME Undoubtedly your student coaches will be excited to start playing games with their newly-acquired players. If they're playing the Advanced Game, you may want to have them work through page 56 of the Coach’s Manual. Here they’ll first be filling in a valuable chart by referencing the Advanced Game Board and their “starting five”—the first player they drafted for each position. If you have teams start their season with the Basic Game and then progress to the Advanced Game, you may want to wait until just before you begin Advanced Games to go through this activity. The analysis here may give coaches some new ideas about their players. The following chart is an example of how students might respond to the prompts:
Coach’s Manual, p. 56
RED
GOLD
BLUE
PURPLE
GREEN
# of 2-pt. shooting spots
6 3
8 5
5 3
5 4
6 6
# of 3-pt. shooting spots
2
3
EVEN=2 ODD=1
EVEN=1 ODD=2
0
# of passing (assist) opportunities
EVEN=5 ODD=4
8
EVEN=3 ODD=4
4
5
# of 2-pt. shooting opportunities (Think carefully!)
EVEN=7 ODD=6
10
EVEN=5 ODD=7
EVEN=7 ODD=5
11
# of 3-pt. shooting opportunities (Think carefully!)
4
6
EVEN=4 ODD=2
EVEN=2 ODD=4
0
Total # of shooting spots
Preferred order (1–5) for a game-ending 2-pt. shot Preferred order (1–5) for a game-ending 3-pt. shot
After students complete the chart, allow them time to examine and discuss their entries. They can then list three things they discovered by recording this information. The writing prompt that concludes this exploration allows students to communicate how they will apply what they’ve learned to field the most competitive team possible.
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The purpose of this exploration is to get students thinking about and discussing with their co-coach, the interplay between the Game Board and the players they’ve selected with respect to actual game play situations. If they can get a good handle on the number of possible shots for each colored circle on the Game Board AND confidently know who they would elect to take a game-ending 2-pt. or 3-pt. shot, they may just be on their way to a championship Math Hoops season!
12.3 How Can You Design a Creative
Season Schedule? IN THE COACH’S MANUAL Page 57 of the Coach’s Manual can be used to help guide students towards thinking about the rules or parameters for crafting a creative season schedule. The template they’ll find in the Appendix on p. 81 will serve as a useful working tool for mapping out their schedules.
Coach’s Manual, p. 57
Once students conduct their various player drafts and complete some analysis regarding player strengths and game-play strategy, they’ll be anxious to get going on their Math Hoops season. You always have the option of being the one to create the regular season schedule; however, engaging students in the development process will only serve to intensify their investment in the program. This is one of those calls that you as League Commissioner will have to make. If you elect to get students involved in the scheduling process, tools for them to use can be found in the Coach’s Manual. On page 57, three discussion prompts are presented to get student coaches talking with each other. This allows them to begin with a blank slate as they try and figure out a reasonable process that will yield a viable schedule for their particular league. What should emerge from these discussions are clear parameters for proceeding with the creation of a variety of season schedules. The first two prompts on page 57 have students discussing the task with their teammates. Use this part of the activity to make sure students understand that they must initially be aware of how many teams need to be scheduled and how many games they must play. Some groups may come up with their own set of “rules” that they think the league should follow while other groups may exhibit little in the way of rules. That is not a problem, since the third prompt brings the groups together to discuss as a class. Use this discussion to solidify a set of rules that all groups will need to follow when designing a season schedule. For example, your class should answer conclusively: • Do all teams play on a given game day? • How many games will make up your regular season? • Will your league be broken into two conferences? Four divisions? Other options? • Will some teams play each other more than once over the course of the season? • Will all teams play each other the same number of times?
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Undoubtedly you’ll have other questions surface in the discussions and will need to take these into account when determining the scheduling parameters. Once the guidelines have been set, it’s time for students to begin making their own schedules that will be submitted for consideration of league adoption. A season schedule template has been provided in the Coach’s Manual Appendix on page 81. You may want to make multiple copies of this template so that students can play around with variations of the schedule. If they undertake the task earnestly, they may find it difficult to satisfy all of the guidelines. That should be considered part of the learning process. To help students get grounded, you might consider whole-group examination of a schedule for a six-team league in which each pair of opponents meet twice over the course of the season. A schedule shell for this league is presented below. In the shell there are 6 teams called A, B, C, D, E, and F to simplify matters. Since each team has 5 opponents, the schedule is 10 games long. In that way, each team can play each of its opponents exactly twice. Students can work out a schedule in small groups. Day 1
Day 2
Day 3
Game 1
A vs B
A vs C A vs D A vs E A vs F
Game 2
C vs D
B vs D B vs E
B vs F
Game 3
E vs F
E vs F
C vs D D vs E C vs F
C vs F
Day 4
Day 5
Day 6
Day 7
Day 8
Day 9
Day 10
A vs B A vs C A vs D A vs E A vs F
B vs C D vs E B vs E
B vs F
B vs C B vs C
D vs F C vs E D vs F D vs E
Coach’s Manual, p. 81
CLASSROOM CONNECTION To see some sample schedules or some guidelines for bypassing the student activity and preparing a schedule yourself, turn to 6.2 CREATING A SEASON SCHEDULE PP. 38–40
The mathematical idea of combinations is embedded in schedule-making. Specifically, if a league has n teams, there must be n (n−1) games played to have 2 each team play each other team exactly once. In a league of six teams, that number = 30 = 15 games. If you want a schedule that allows each team to play the is 6(5) 2 2 other teams twice (which will allow each team to play a “Home” game and an “Away” game), multiply by 2. The schedule above shows 15 × 2 = 30 games over 10 days so that each team can play each other team exactly twice. Students can rely on the above combinations formula to figure out the necessary number of games in their league, but they can also deduce the same formula intuitively. For example, in an eight-team league, each of the eight teams needs to play each of its seven opponents if you want each team to play each other team once. That leads to the product 8 × 7. But that product includes double-counting of games since there are two teams in each game. Therefore the correct number of = 56 = 28. games in an eight-team league is 8 × 7 2 2 As schedules are being created, free-flowing discussion will permeate the classroom. Encourage students to reference any problems they may have encountered in devising their schedules. Some groups may simply report that the “problem” they encountered was trying to keep track of who plays whom while making sure teams play each other team the same number of times. The kind of organized UNIT 12: CREATING TEAMS AND A SEASON SCHEDULE • 91
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recordkeeping required for making a schedule looks straightforward, but students will soon realize it can be difficult. A more challenging problem arises if your league has an odd number of teams. In that case, not all teams can play on the same day and scheduling requires teams to take turns skipping a game. If there are an odd number of teams, students may want to build in a “Team Stats Update” day for the team left out of a particular day’s game schedule. At the bottom of the schedule template, students are introduced to the notions of Home and Away teams. Since these concepts are used in Math Hoops, students must take account of them in devising their schedules. Ideally, their schedules will allow for teams to share Home and Away status equally, but that can only be guaranteed if teams face each other an equal number of times. They should be prepared to talk about how they addressed this when presenting their completed schedule to the class. Once final schedules have been completed, they should be shared with the entire class and submitted for possible league adoption. One way to do this is to post the schedules on an accessible bulletin board. Students can then carefully review and prioritize their top three preferred schedules. These can be listed on a scratch piece of paper and turned in to you to tabulate results. In the event you don’t have a clear-cut favorite when strictly considering top picks, you can drill down further to determine a winner. With a preferred schedule in hand, you and your student coaches are now ready to begin your Math Hoops regular season!
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UNIT
13
Winning Strategies F
ielding a competitive NBA Math Hoops team takes a bit of good fortune and a great deal of planned strategy. In this unit, student coaches have the opportunity to drill down further into specific aspects of the game. Deciding who should get the ball, whether to go for two points or three points, whether to pass or shoot, whether or not to foul—all of these represent decisions students will make in every Math Hoops game. The more students know and the more they can think about possible situations in advance of them actually happening, the better off they’ll be. Astute NBA and WNBA coaches work through a number of possible situations in their head well before their team ever takes the floor. It’s the combination of good planning and sound execution that propels a team to victory. Working through the explorations in this unit will put students in a position to learn, develop, and utilize effective game winning strategies.
Time Considerations Each exploration in this unit can take between 1 and 2 classroom sessions depending on student interest, motivation and depth of investigation. As with the other units, these explorations are intended to enhance the Math Hoops experience and can be integrated into your league by using during planned game-play breaks. EXPLORATION
ESTIMATED TIME
MATH CONTENT
1–2 sessions 60–90 minutes
• Using Probability to Solve Problems
13.2 Go for Two Points or Three Points?
1 session 45–60 minutes
• Using Probability and Expected Value
13.3 Should You Pass or Should You Shoot?
1 session 45–60 minutes
• Comparing Game Strategies Based on Probability
13.4 Can You Beat the Shot Clock?
1–2 sessions 60–90 minutes
• Using Properties of Integers to Solve Problems
13.5 Foul Play: When Is the Best Time to Foul?
1 session 45–60 minutes
• Using Probability and Expected Value
13.6 Lightning Round Strategies
1 session 45–60 minutes
• Analyzing Properties of Integers to Develop Game Strategies
13.1
Who Gets the Ball?
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IN THE COACH’S MANUAL On pp.58–59 of the Coach’s Manual, students begin to look closely at different players’ shooting percentages. Presented with a selection of different game play situations, they make decisions as to who should get the ball, taking into account the theoretical probabilities of success. While references to probability are informal, this exploration can be used as a springboard for discussion around the differences between theoretical and experimental probability.
13.1 Who Gets the Ball? Coaching an NBA Math Hoops team gives students frequent opportunities to make data-driven decisions. One they’ll encounter often is deciding who gets the ball to take a shot once the decahedra dice have been rolled and calculations completed.
Introducing Probability and Basic Strategies Let’s take a look at a specific game-play situation. Suppose the team working on the Odd end of the court rolls 6 and 3 during a Basic Game. The coaches perform the arithmetic and arrive at 9, 3, 18, and 2 for their results. That leaves the team with two choices—9 or 3, both of which would lead to attempting a two-point basket. To choose between 9 and 3, the Odd team coaches must examine the back of their blue Forward and red Guard Player Cards. Samples of these cards are shown below.
If the Odd team knows that its choice of player will attempt a two-point shot, the decision is easy: pick the player with the greater two-point field goal percentage. Coaches can read this number off the top of a Player Card, or they can compare the orange regions. But there is another possibility to consider. What if the defense commits a foul? In the two cards above, the Forward is a much better field goal shooter but a slightly worse free throw shooter than the Guard. Based purely on probabilities of favorable outcomes, one could come to the following conclusions:: • If the Even team is going to call a foul, the Odd team should select the Guard. • If the Even team is not going to call a foul, the Odd team should select the Forward.
Obviously the Odd team cannot read the minds of the Even coaches but it can consider some relevant data. For example:
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• The Forward is a much better field goal shooter than the Guard. • Has the Even team already committed five fouls (its limit) in the half? • Has the Even team’s Forward or Guard already committed three fouls (their limit)?
• Is it the first half of the game where a foul yields a “one and one” situation or the second half where the player fouled automatically gets two foul shots?
On p. 58 of the Coach’s Manual, students are given the opportunity to explore a situation similar to the one outlined above. First they’ll find the possible outcomes for an initial roll of the dice by performing the necessary calculations. They’ll then look closely at two sample Player Cards and respond to game play questions connecting the roll of the dice with the most favorable shooting percentages on the backs of the cards. 1. Imagine you’re playing on the Odd end of the court during a Basic Game and you roll 7 and 4. Perform the arithmetic and record the results in the blank chart below.
ROLL 2. What are your options?
7 4
+
11
–
3
×
21
÷
Coach’s Manual, p. 58
3,11, and 21
3. Which player has the greater probability of making a 2-pt. shot? Who would you select to take the shot? Explain your reasoning. Chris Paul has the greater probability.
The sidebar titled Probability and You on p. 58 is intended to get students thinking about the role of probability when deciding who should get the ball. You may want to use this as a springboard for discussion around the difference between theoretical and empirical (experimental) probabilities.
Advanced Game Situations In the Basic Game there are usually few choices as to who can possess the ball on a given turn. The Advanced Game, however, is a whole different, er…. ball game. With up to sixteen possible outcomes on the Shot Planner and the presence of passing lanes, shooting options abound. On p. 59 in the Coach’s Manual, students encounter more complex decisions by looking at game situations in the Advanced Game. After completing the Shot Planner, students will need to look at the possible shooting options. Two sample Player Cards are again provided that they will refer to when answering situational questions. To wrap up the exploration, students are
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asked to list three things a coach may want to consider when selecting a player to get the ball. 5. Playing the Advanced Game will lead to even more shot choices. Complete the Advanced Shot Planner below but now look for EVEN options. Then use the Advanced Game Board and the Player Cards shown on these two pages to answer the questions.
ROLL
7 2
BALL ON
11
+
9
–
5
×
÷
14
+
+
+
+
–
–
–
–
20 2
16 6
25 3
14
×
×
×
×
÷
÷
÷
÷
55
2
2
3 8
33
2
4
6. Suppose you’ve elected to take a 2 pt. shot. Considering the probability of getting a favorable outcome, which player would be your first choice? Second choice? Why did you select these players?
Of the four cards shown on pages 58–59, the two players with the highest 2-pt. FG% are Dwight Howard with .573 followed by Candice Dupree with .548. Students could bring in additional details, such as the expectation that their player will be fouled, and therefore choose different players. These answers are acceptable as long as they can back up their decisions with mathematical reasoning. 7. You’re down by 2 points and have one chance to try and win the game. Who would you choose to take the final shot? How does probability play into your decision? Coach’s Manual, p. 59
If students declare that they would take a 3-pt. shot, the player with the highest 3-pt. % is Chris Paul with .371. Again, other answers are fine. It’s the reasoning behind the decisions that you want to consider. 8. Knowing that your opponent always has the option to foul, which player would you make sure didn’t get the ball? Explain your reasoning.
The player with the worst free throw shooting percentage is Dwight Howard with a punishingly low .491.
NBA Math Hoops is a fast-paced game. Depending on your specific needs and student interest, you may want to formalize the discussion around these game play scenarios to draw out the mathematical connections. If that’s not of primary concern, you can use this exploration as a means to demonstrate how astute coaches can informally acquire and internalize game strategies.
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IN THE COACH’S MANUAL
13.2 Go for Two Points or Three Points? As previously mentioned, playing Math Hoops provides students with multiple opportunities to make data-driven decisions. An ongoing decision is whether or not to go for two points or go for three points. When faced with this decision, students will find that clearly knowing their players’ strengths and weaknesses will aid them in confidently making the best decision possible. Consider the following game-play situation. Suppose the Even team rolls 7 and 5 during a Basic Game. The coaches perform the arithmetic and must choose between the results 2 and 12. As it turns out, both options result in the gold player (guard) shooting, with 2 being a three-point attempt and 12 being a two-point attempt. Every Math Hoops player has a greater chance of being successful on a 2-pt. field goal than on a 3-pt. field goal. So, on the simplest level, the Even coaches must grapple with this risk/reward dilemma: Do they play it safe and go for two points or do they take a greater risk for the greater reward of three points?
Ever watched a tight basketball game with the final seconds ticking away and wondered if the team with the ball would go for two points to possibly tie the game or three to win? The exploration on pp. 60–61 of the Coach’s Manual serves as a guide for students to make simulated game play decisions based on player performance. The opportunity to informally delve into the concept of expected value is presented as students compare players in possible situational shooting situations.
On p. 60 of the Coach’s Manual, students are asked that very question. They’re given a real game play situation and must refer to a sample Player Card when responding to the questions that follow. Unless they’ve previously had formal instruction on expected value, their methods for assigning a numerical value to possible favorable outcomes will most likely be rather elementary. Here are some likely ways they may respond to the Coach’s Manual copy: 1. If you go for two points, what is the likelihood the outcome will be favorable? Can you think of a way to assign a numerical value to that outcome? Explain your method.
For a 2-pt. attempt, students may say the player has a “better than 50% chance” or “slightly better than 1 out of 2” chance of making a 2-pt. shot. 2. If you go for three points, what is the likelihood the outcome will be favorable? Can you think of a way to assign a numerical value to that outcome? Explain your method.
With regards to a 3-pt. attempt, they may lean towards a “better than 33% chance” or “slightly better than 1 out of 3” chance of making a 3-pt. shot. 3. Have you devised a way to compare values that could help you choose whether or not to take a 2 pt. shot or a 3 pt. shot? If so, explain why you feel your method makes sense.
Coach’s Manual, p. 60
If comparing the possible outcomes shot for shot, most will clearly see advantages to going for two points. It’s only when looking at the data more closely, i.e. taking into account expected value, that they’ll see the relative equity in the attempts.
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The sidebar on Expected Value found in the Coach's Manual on p. 60 is intended to pique student interest in some exploration around the concept of expected value. This can be done by perusing the Internet or available print resources. They may also elect to ask you to explain the concept and how it might apply to situations they’ll encounter in Math Hoops. The explanation provided below may be something you find useful for this purpose. When working through this exploration, most students will not formalize the concept of expected value, but some might light upon it intuitively or formally on their own—possibly with a little prodding from your end. A more likely scenario is that students will compare the orange regions and black crosshatch regions visually and draw some general assumptions from there.
Expected Value There is a mathematical way to analyze the decision facing the Even team. Ignoring other factors (such as score of the game and the likelihood of the shooter being fouled), a number can be calculated for each of the choices using the concept of expected value. To understand this, consider TyrusDax, the shooter shown below. FIELD GOAL
FIELD GOAL PERCENTAGE
two-point
51%
three-point
35%
If this shooter took 100 two-point shots, you do not know exactly how many he would make. But since his field goal percentage represents his historical average, your best guess is that Dax would make 51% of the shots. So, in 100 shots, you would expect Dax to score 51 times, for a total of 51 × 2 = 102 points. Similarly, if Dax were to take 100 three-point shots, you would expect him to score 35 times, for a total of 35 × 3 = 105 points. The results above are for 100 shots. What if Dax takes one shot? Either he scores or he does not, but the concept of expected value allows you to assign a number based on the work shown above. If Dax averages 102 points for every 100 two-point shots,
then he averages 102 ÷ 100 = 1.02 points per twopoint shot. If Dax averages 105 points for every 100 three-point shots, then he averages 105 ÷ 100 = 1.05 points per three-point shot. The answers of 1.02 and 105 are very close, but the expected value is greater for a three-point shot. This means that from a strictly mathematical perspective, the better choice is the three-point shot. Since Math Hoops strategy decisions are not strictly mathematical, a coach in this situation may or may not choose to take the three-point shot. To calculate expected value of any event without going through all the steps shown above, multiply the probability of the event occurring by the value obtained if the event occurs. The letter E is used to represent expected value. Then, following the letter E, the event you are analyzing is put in parentheses. For example, the expected value of making a twopoint field goal can be written as E(2). Here is how to calculate the two expected values for Tyrus Dax. E(2) = (two-point field goal percentage) × (2) = (51%) × 2 = 1.02 E(3) = (three-point field goal percentage) × (3) = (35%) × 3 = 1.05
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Beyond expected value, there are many other variables at play affecting a team’s decision as to whether to choose a two-point or three-point shooter: • Late in a game, a team in the lead might prefer the lower risk of a two-point shot.
• Late in a game, a team that is trailing might need to score points quickly and decide to go for the greater reward of a three-point shot.
• A team must consider the possibility that the defense will foul the shooter. A three-point shooter always gets three foul shots while a two-point shooter is subject to the “one and one” rule in the first half of the game and an automatic two shots in the second half of the game.
The student exploration continues on p. 61 with the presentation of five different game play situations. For each situation, they must refer to two sample Player Cards provided and choose which player they would select to take the shot. While there is no right or wrong answer to each situation, their responses and the discussion that ensues should give you a clear indication of any mathematical reasoning they’re using. The two writing prompts that wrap up the exploration provide another excellent opportunity to assess students’ formal and informal understandings. They also serve as valuable work samples that can be put on display or placed in a student portfolio as evidence of contextual application and understanding of mathematical content and practices.
13.3 Should You Pass or Should You Shoot?
ADVANCED GAME
This exploration allows student coaches to build on understandings derived from the previous exploration—Go for Two Points or Three Points? The Advanced Game expands the shooter choices by allowing passes to be made. No matter what the results of the dice, coaches can try to get the ball to one of their better shooters by making a pass. Most players on both ends of the court have shooting spots with a direct passing lane to one other player. The goal should always be to find a viable shooting location that provides the most favorable options possible. The simple strategy here is for a team to choose its shooter and then decide whether to use that shooter or use a shooter to whom it can make a pass. The decision would then be based on analysis of 2 pt. and 3pt. field goal and free throw percentages as referenced above. On pp. 62-63 in the Coach’s Manual, students are provided with images of the Advanced Game side of the Game Board and a selection of 5 Player Cards—one per position. First students must make a decision whether to pass or shoot given four situations where they’re playing on the ODD end of the court. Next they’re asked to respond to four situations where they’re playing on the EVEN end of the court.
Coach’s Manual, p. 61
IN THE COACH’S MANUAL The concept of “choice” is prevalent in the Advanced Game. On pp. 62–63 of the Coach’s Manual, they’ll encounter several gameplay scenarios that require a split decision—do you pass or do you shoot? This exploration provides students the opportunity to build on understandings gained in the last activity including those dealing with expected value. The extension found at the bottom of p. 63 gives students a chance to stretch their creative side and create a pair of game scenarios— one where they would elect to pass the ball and another where they would elect to shoot.
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When discussing the options they selected, students should be able to reference player shooting percentages and the mathematical reasoning behind their choices. In order to get a little more in-depth, students are then asked to select one situation when they were on the Odd end of the court and chose to pass the ball. They must then support their choice with strong mathematical reasoning. Next they’re asked to select one situation when they were on the Even end of the court and chose to shoot the ball. Again, strong mathematical reasoning should support their decision. You may elect to have students first discuss choices with their co-coach before choosing and supporting their selections. This exercise helps to clarify thinking before putting something more concrete down in writing.
Coach’s Manual, p. 62
A more advanced strategy is for a team to think about the player it wants to get the ball to, before it even rolls the dice. For example, suppose the Even team’s blue player is its best, showing excellent two-point field goal, three-point field goal and free throw percentages. The only circles for the blue player are 18, 42, 46, 50, and 56. But of those five circles, four have passing lines from another player: 20 (gold) to 18, 34 (purple) to 42, 8 (red) to 46, and 16 (green) to 56. When rolling the dice and then filling out the Shot Planner, a coach on the Even team can be looking for any of those nine numbers to pop up, in the hope of having its best player shoot. This is why knowing their players inside and out works to a team’s advantage! An interesting and challenging extension is given at the bottom of p. 63 in the Coach’s Manual. Here coaches are given a clean slate and asked to create two game scenarios—one where they would choose to pass the ball and another where they would elect to shoot. For this exercise, coaches can either refer to the Player Cards at the top of page 63 or create the scenarios around their starting five players. Regardless of which route they go, they should be expected to use player shooting percentages to support their choices with strong mathematical reasoning. This extension is a particularly useful way to see if students have synthesized multiple aspects of the Advanced Game.
Coach’s Manual, p. 63
13.4 Can You Beat the Shot Clock? Introducing the shot clock into game play will generate a great deal of additional excitement. Four shot clocks are included in each Classroom Kit and are to be introduced and used at your discretion. The 24-second and 35-second functions are set by adjusting the toggle switch on the side of the base of the shot clock. And of course, the realistic “NBA buzzer” sound will make your classroom’s simulated games even more lively and competitive!
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There are some important game elements students will need to consider when playing a game using the shot clock. One is deciding when the shot clock should be started. Another is deciding who should assume responsibility for the shot clock for each team. A third and extremely important element is deciding how much time you can actually allow to elapse before safely taking a shot. A team that runs out of time on the shot clock before choosing a shooter and flicking the arrow on the spinner loses possession of the ball. A good Math Hoops team always develops and refines a sound strategy for keeping track of and beating the shot clock. You may elect to introduce the shot clock while students are playing the Basic Game. Because there are only four basic calculations to complete before selecting a shooter, taking a shot before the buzzer sounds may not be difficult for many students. This is a good thing. Developing clear roles and responsibilities should be a primary focus at the Basic Game level. In the Advanced Game, after rolling the dice and completing the calculations for the top row of the Shot Planner, Math Hoops coaches then have up to sixteen additional calculations when operating with those answers and the Ball On number. This is a significantly more difficult task to complete when using the shot clock. Knowing how to deal with this is key to student success with the shot clock.
IN THE COACH’S MANUAL As students get more comfortable with the game, it will be important to gradually step up the pace between shot attempts. When you introduce the Shot Clocks, students will quickly discover the difficulty in completing all calculations on the Shot Planner before the buzzer sounds. The exploration found on pp. 64–65 of the Coach’s Manual has students look at relationships related to operations with even and odd numbers. Without time to complete all operations on the Shot Planner, knowing which ones to focus on and which can be skipped is a crucial coaching strategy.
There is no rule in Math Hoops that requires coaches to complete every square of their Shot Planner before choosing which player gets the ball. A reasonable approach is for one coach to be responsible for quickly and carefully evaluating the numbers as they are written on the Shot Planner. If time is running out on the shot clock and a good shooter is available for the Shot Planner numbers available, a coach can select that player immediately. Coaches will also want to factor in the benefit of using multiplication or division for taking a shot and the possibility of earning an offensive rebound on a missed shot. On pp. 64–65 of the Coach’s Manual, students are guided through a process where they examine the inherent relationships involved in completing mathematical operations with even and odd numbers. The discussion on p. 64 of the Coach’s Manual should help students start thinking about ways to shorten the amount of time spent on math calculations on the Shot Planner. For example, a team playing on the Even end of the court doesn't need to find odd numbers (and vice versa). The goal should be to take the information available and agree on how best to make use of the Shot Planner when the shot clock is in operation. Quickly being able to eliminate numbers that are of no use is a game-play strategy that will serve your Math Hoops coaches well.
Coach’s Manual, p. 64
The questions on p. 65 of the Coach’s Manual will help students analyze the Advanced Game Shot Planner and find patterns they can use to their advantage.
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1. Why does the last row show even OR odd?
In the Advanced Game, you can round division answers up or down, so there will always be an even AND odd option. 2. Complete the charts below to identify some patterns that might help you expedite the calculation and shot selection process.
+
O
E
O
E
O
E
O
E
−
O
E
O
E
O
E
O
E
×
O
E
O
O
E
E
O
E
÷
O
E
O
E/O E/O
E
E/O E/O
Now use what you know to find possible even and odd options. There’s no need to do calculations that won’t help you. Remember, the shot clock is running! 3. Looking for EVEN options:
ROLL
7 3
BALL ON
37
10 4 21 2*
+
+ –
–
47
27
+ –
×
41
33
+ –
÷
58
16
+ –
×
×
×
×
÷
÷
÷
÷
370 148 777 4
10
2
39 35
* What other number could this be?
74 18
The Shot Planner above shows all the operation results, but only the answers shaded in blue are truly needed. Coaches who can see patterns that would lead to odd numbers or to numbers that aren’t on the Game Board could skip 10 of the 16 operations.
Coach’s Manual, p. 65
4. Looking for ODD options: ROLL
9 5
BALL ON
8
14 4 45 2*
+
+ –
–
22 6
×
112
÷
1
+ – × ÷
×
12 4
32 2
+ –
÷
53
37
×
360
÷
5
+ – × ÷
10 6
* What other number could this be?
16 4
This Shot Planner also shows all operation results, but only 4 of the 16 possibilities are useful for a team playing on the Odd end of the court. Astute coaches may see that they could get a more favorable set of results if the division answer used in the top row for Question 3 were 3 and if the division answer used in the top row for Question 4 were 1. The insights gleaned from Exploration 11.2: Would You Rather Be Odd or Even? will help some coaches grasp the patterns in this activity more quickly.
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While students work through the steps in this exploration, you’ll want to engage them in informal, small group discussions. Having students verbalize strategies and the reasoning behind them, along with consistent practice, helps lead to automaticity. After a while, many students will develop personalized methods that allow for maximizing the time available on the shot clock. The writing prompt that concludes this exploration gives them the opportunity to record what they’ve learned and any new shot clock strategies that will help them be more competitive in future games. As with all suggested writing assignments, students should be expected to convey their ideas clearly, concisely, and support their thinking with strong mathematical reasoning.
13.5 Foul Play: When Is the Best Time to Foul? While the primary emphasis of the Math Hoops game is offense-driven, there is an element of defense that comes into play. Teams are allowed up to five fouls per half and when used strategically, these can make the difference between victory and defeat. In professional basketball, poor free throw shooters are sometimes fouled in situations where the defense would rather have them attempt foul shots than have their team attempt field goals. Conversely, when a team knows that the defense wants to commit a foul, it will try to keep the ball out of the hands of a poor free throw shooter. These same strategies come into play in Math Hoops. When the team on offense chooses a shooter, the defense must decide whether or not to foul the shooter. It has to weigh many factors in its decision, for example:
IN THE COACH’S MANUAL On pp. 66–67 in the Coach’s Manual, students are provided with a selection of game-play scenarios that involve the use of fouls. A series of questions help students compare the benefits of fouling or not fouling in each scenario. The concept of expected value is revisited.
• How good is the shooter at field goals? • How good is the shooter at free throws? • Is it a “one and one” situation? • How many fouls has the team already committed in the half? • How many fouls has the defensive player assigned to the same position as the shooter already committed in the game?
On pp. 66–67 of the Coach’s Manual, students are given a chance to explore the ins and outs of the foul feature in a Math Hoops game. On p. 66 they’ll find two sample Player Cards—one with an exceptional 2-pt. shooting percentage (Pushy Dixon) and the other with strong 3-pt. and free throw shooting percentages (Sandy Elbows). The first four questions that follow are geared towards hypothetical game play situations. Questions 1 and 2 place students in the defensive position where they’re asked which player they would foul in a given situation. Questions 3 and 4 place students on offense where they determine which player they would want to have the ball in a foul situation.
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Using Expected Value to Analyze Fouls Let’s take a look at the math behind the decision-making process. You may want to use this example with students prior to turning them loose to tackle the questions in this exploration. You already know that the “one and one” situation is used in the first half of each game. When a two-point shooter is fouled, he or she takes one foul shot. If unsuccessful, possession of the ball passes to the other team. If successful, the shooter takes a second foul shot. The “one and one” situation can favor the defense, and shrewd coaches will use it to their advantage. Consider a player whose two-point field goal percentage is a healthy 58% (rounded) while his free throw percentage is a meager 60% (also rounded). If this player attempts a two-point field goal, the expected value is calculated like this:
(FG%) × (No. of Pts.) = Expected Value
.58 × 2 = 1.16 points
If this player instead is sent to the free throw line for two shots, she or he might make or miss either one. If it's the second half of the game, fouling the player will give two shots. To determine the Expected Value, add up the individual Expected Values for each possibility, as in the table below. You use the player’s FT% as the value for making the shot. You subtract the FT% from 1.0 to get the value for missing the shot. Make First Shot?
Make Second Shot?
Probability
Points Scored
Expected Value
Yes
Yes
(0.6) (0.6) = 0.36
2
0.72
Yes
No
(0.6) (0.4) = 0.24
1
0.24
No
Yes
(0.4) (0.6) = 0.24
1
0.24
No
No
(0.4) (0.4) = 0.16
0
0
Final Expected Value 0.72 + 0.24 + 0.24 = 1.2
In this case, the expected value for free throw shooting is greater than field goal shooting, and committing a foul does not seem like a good idea. Change the situation to “one and one” and the defense’s strategy may change. In the table above, the row with blue shading cannot occur in a “one and one” situation: if the player does not make the first free throw attempt, no second shot is taken. That changes the final expected value: 1.2 − 0.24 = 0.96
This is lower than 1.16, the expected value for attempting field goals. In the “one and one” situation, it makes sense for the defense to foul this player. The above scenario may seem unusual in that the player’s 2-pt. field goal and free throw percentages are so close to each other. But players like this do exist in 104 • PART III: STRATEGIES AND MATH EXPLORATIONS FOR SUCCESSFUL COACHES
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professional basketball and in Math Hoops. Astute coaches will look for them when a game is played. How might you use the information above to help students respond to the questions found on pp. 66–67? For one, having them replicate the chart above with new numbers is a good start. Pushy Dixon
2-pt FG%: 56% (rounded)
FT%: 64% (rounded)
Make First Shot?
Make Second Shot?
Probability
Points Scored
Yes
Yes
(0.64) (0.64) = 0.41
2
Expected Value 0.82
Yes
No
(0.64) (0.36) = 0.23
1
0.23
No
Yes
(0.36) (0.64) = 0.23
1
0.23
No
No
(0.36) (0.36) = 0.13
0
0
Final Expected Value 0.82 + 0.23 + 0.23 = 1.28
Sandy Elbows
2-pt FG%: 51% (rounded)
FT%: 71% (rounded)
Make First Shot?
Make Second Shot?
Probability
Points Scored
Yes
Yes
(0.71) (0.71) = 0.50
2
Expected Value 1.01
Yes
No
(0.71) (0.29) = 0.21
1
0.21
No
Yes
(0.29) (0.71) = 0.21
1
0.21
No
No
(0.29) (0.29) = 0.08
0
0
Final Expected Value 1.01 + 0.21 + 0.21 = 1.43
While it’s not necessary for students to use expected value, having this information available will aid them in formulating reasonable responses to the first four questions. 1. If you were going to foul a player about to take a two-point shot in the first half of a game, which of these two would you want it to be? Why? Explain your reasoning.
Better to foul Player A in the first half of a game, because the “one and one” rule is applied: Player A: 0.82 + 0.23 = 1.05 points Player B: 1.01 + 0.21 = 1.22 points 2. If you were going to foul a player about to take a two-point shot in the second half of a game, which of these two would you want it to be? Why? Explain your reasoning.
Coach’s Manual, p. 66
Better also to foul Player A in the second half of the game. Player A: 0.82 + 0.23 + 0.23 = 1.28 points Player B: 1.01 + 0.21 + 0.21 = 1.43 points
3. If these two players were on your team, which player would you prefer your opponent foul in the first half of a game? Why? Explain your reasoning.
Prefer to have Player B fouled (better free throw shooter) 4. If these two players were on your team, which player would you prefer your opponent foul in the second half of a game? Why? Explain your reasoning.
Prefer to have Player B fouled (better free throw shooter) Coach’s Manual, p. 67
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5. Ten seconds left in the half and a tie score. Pushy Dixon is on your team and about to take a 3-point shot. Would you rather he take the shot or be fouled by your opponent? Why? Explain your reasoning.
Pushy Dixon has a .341 3-pt. shooting percentage and .64 free throw percentage, so you can calculate the expected values as follows: Make First Shot?
Make Second Shot?
Make Third Shot?
Probability
Points Scored
Expected Value
Yes
Yes
Yes
(0.64) (0.64) (0.64) = 0.26
3
0.79
Yes
Yes
No
(0.64) (0.64) (0.36) = 0.15
2
0.29
Yes
No
Yes
(0.64) (0.36) (0.64) = 0.15
2
0.29
No
Yes
Yes
(0.36) (0.64) (0.64) = 0.15
2
0.29
Yes
No
No
(0.64) (0.36) (0.36) = 0.08
1
0.08
No
Yes
No
(0.36) (0.64) (0.36) = 0.08
1
0.08
No
No
Yes
(0.36) (0.36) (0.64) = 0.08
1
0.08
No
No
No
(0.36) (0.36) (0.36) = 0.05
0
0.00
Final Expected Value 0.79 + 0.29 + 0.29 + 0.29 + 0.08 + 0.08 + 0.08 = 1.90
Expected value for a 3-pt. field goal: 0.34 × 3 = 1.02 points Expected value for three free throw attempts: 1.90 points In this case, a 3-pt. field goal attempt and three free throws yield very different expected values! 6. Last shot of the game and you’re up by two points. Sandy Elbows is on your opponent’s team and about to take a 3-point shot. Will you let her shoot or call a foul? Explain your reasoning. Coach’s Manual, p. 67
Here are calculations for Sandy Elbows with her .405 3-pt.% and .71 FT%. Make First Shot?
Make Second Shot?
Make Third Shot?
Probability
Points Scored
Expected Value
Yes
Yes
Yes
(0.71) (0.71) (0.71) = 0.36
3
1.07
Yes
Yes
No
(0.71) (0.71) (0.29) = 0.15
2
0.29
Yes
No
Yes
(0.71) (0.29) (0.71) = 0.15
2
0.29
No
Yes
Yes
(0.29) (0.71) (0.71) = 0.15
2
0.29
Yes
No
No
(0.71) (0.29) (0.29) = 0.06
1
0.06
No
Yes
No
(0.29) (0.71) (0.29) = 0.06
1
0.06
No
No
Yes
(0.29) (0.29) (0.71) = 0.06
1
0.06
No
No
No
(0.29) (0.29) (0.29) = 0.02
0
0.00
Final Expected Value 1.07 + 0.29 + 0.29 + 0.29 + 0.06 + 0.06 + 0.06 = 2.13
Expected value for a 3-pt. field goal: 0.41 x 3 = 1.23 points Expected value for three free throw attempts: 2.13 points
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If your students take time exploring and comparing the expected values of a 3-pt. field goal attempt versus three free throws for these two players, they may conclude that fouling (or getting fouled) on a 3-pt. field goal attempt gives a big advantage to the shooting team. On the other hand, fouling is also a guaranteed way to slow down the game. That’s a creative coaching strategy in and of itself!
13.6 Lightning Round Strategies The lightning round (i.e. overtime) comes into play when the score is tied at the end of regulation. Starting with the team that ended the game on the Even end of the court, teams take turns having three different players on their team attempt a 2-point or 3-point field goal. No fouls are allowed in overtime and teams can select any players to shoot, as long as three different players are used. At the conclusion of the lightning round, the team with the higher score is the winner. Should the game remain tied at the end of the lightning round, it is declared a tie. While there is no formal exploration of strategies for the lightning round in the Coach’s Manual, you may want to facilitate some informal exploration using the information that follows. Since a team must select three different players to take three shots in the lightning round, it will undoubtedly go with the players it feels are its best shooters. But a team must also take into account whether two-point or three-point field goals are attempted. You can prompt students to consider strategy for lightning rounds prior to playing games, but the best way for them to gain insight is to get some experience with games that end in lightning rounds first. For this reason, you may want to wait until all teams have played 5 or 6 games before calling attention to specific lightning round strategies. One way to do this is to have coaches imagine a lightning round in which they will shoot second and ask: • What would you do if your opponent misses the first shot? • What would you do if your opponent hits the first (2 point) shot? • What would you do if your opponent hits the first (3 point) shot?
If teams haven’t encountered lightning rounds in their games yet, you may want to set up an exercise where coaches play lightning rounds, alternating between going first and going second. Besides the questions above, you can then use the following scenarios and accompanying questions to generate some interesting student discussion: • Suppose you are the EVEN team and shooting first. Do you attempt a threepoint shot in the hopes of making it and putting serious pressure on your opponent? Or are you wary of missing a three-point attempt and then allowing your opponent to gain the upper hand with its first shot?
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• Suppose you are the ODD team, taking your second shot with the score tied. Do you attempt a three-point shot in the hopes of making it and forcing the EVEN team to do the same in its last shot? • Suppose you are the EVEN team, taking your last shot with the score tied. Do you attempt a two-point or three-point shot? What is the reasoning behind your selection? • Suppose you are the ODD team, taking your last shot trailing by two points. Do you attempt a two-point shot to tie or a three-point shot to win? What factors other than the score might you take into consideration?
While there are no definitive correct/incorrect responses, it will be useful for students to think about in advance, how they might handle various lightning round situations. Listening to other perspectives and participating in some meaningful and spirited discussion can help students formulate and/or solidify their own unique strategies.
FOCUS QUESTIONS REVISITED Throughout these unit explorations and again at the conclusion of the final one, be sure to refer students back to the focus questions. Taken together, these questions should help coaches think strategically and raise their level of play. You may even wish to pose a summary focus question at the end of these explorations: What game-play strategies have you and your co-coach learned that you can use to put the most competitive team on the floor?
Students should constantly be encouraged to reflect on and critique the effectiveness of the game-play strategies they’ve adopted. Successful coaches are constantly learning and open to new ideas. What level of success will your Math Hoops coaches attain?
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UNIT
14
Improving Your Team I Use the following units f you either selected or crafted a program in Part II to help create a roadmap consisting of 32 or more sessions, classroom experience: the midpoint of the regular season provides UNIT 8: MID-SEASON REVIEW / ALL-STAR GAME a perfect opportunity for student coaches to PP. 47-53 explore ways in which they might improve their team. After careful analysis of first half player stats, students may look to bolster their team in one of three ways: • Drafting a new player from the set of Player Cards • Creating their own Player Card • Making a trade with another team In each case, the primary focus should be on investigating ways to increase offensive production—i.e. score more points!
Time Considerations This unit can take 2–4 classroom sessions depending on depth of exploration. You may elect to break things up over a week or two and continue game play while facilitating this investigation. EXPLORATION
ESTIMATED TIME
MATH CONTENT
14.1 Mid-Season Assessment: Is It Time for a Roster Change?
2–3 sessions 90–150 minutes
• Comparing and Operating with Whole Numbers and Decimals • Comparing Theoretical and Experimental Probability • Collecting, Organizing, and Interpreting Data
14.2 Create Your Own Player Cards!
1–2 sessions 60–90 minutes
• Finding and Using Ratio and Proportions • Converting Fractions, Decimals, and Percents • Constructing Geometric Representations of Numerical Data • Creating Circle Graphs
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IN THE COACH’S MANUAL Midway through the regular season is an excellent time for briefly suspending game play and having students reflect on first half performance. On pages 68–69 of the Coach’s Manual, students are instructed to use tools provided in the Appendix of their manual to collate player and team data. A series of thoughtprovoking questions are provided so students can assess player and team performance and decide whether or not a roster change would be of potential benefit.
14.1 Mid-Season Assessment: Is It Time for a
Roster Change?
At the conclusion of the first half of the regular season (minimum of 5 games), students should fill out a Mid-Season Assessment form found on p. 84 of the Coach’s Manual. To complete this task, student coaches will need to have ready access to the Game Summary Stat Sheet (Coach’s Manual p. 82) for each of their regular season games and a Player Season Totals form (Coach’s Manual p. 83) for each of their players. The information gathered will be used to help make decisions about possible personnel changes for the second half of the season. Before having students analyze their own team, you may want to engage them in a whole-class exercise as a model for analysis. Start by showing students the summary data on a Mid-Season Assessment form as it might look after 6 games. You can make copies of the form below or reproduce the information to show to students. You will also want to make the Player Cards for these players available to all students.
Mid-Season Assessment Team Coaches
CLASSROOM CONNECTION Suggested mid-season activities and advice on how to implement them can be found in PART II: UNIT 8: MID-SEASON REVIEW/ ALL-STAR GAME PP. 47–53
Indiana Fever Tyra Wilson and Steve Lee
Player Name/Position
6
Games Played
2-Pt FGM
2-Pt FGA
2-pt. FG%
3-pt. FGM
3-pt. FGA
3-pt. FG%
FTM
FTA
FT%
1.
Sylvia Fowles (Center)
9
15
.600
N/A
N/A
N/A
4
7
.571
2.
Kevin Durant (Forward)
7
13
.538
3
6
.500
2
3
.667
3.
Blake Griffin (Forward)
5
9
.556
0
2
.000
2
2
1.000
4.
Becky Hammon (Guard)
3
9
.333
3
4
.750
3
4
.750
5.
James Harden (Guard)
6
13
.462
1
3
.333
3
6
.500
6.
Maya Moore (Forward)
1
2
.500
0
0
N/A
1
2
.500
7.
Dwyane Wade (Guard)
2
4
.500
0
1
.000
0
0
N/A
8.
Chris Paul (Guard)
1
1
1.000
1
2
.500
1
2
.500
TEAM TOTALS Note: You will need the information from your Player Season Totals sheet to calculate the statistics in this sheet.
ABBREVIATIONS FG%: Field Goal Percentage FT%: Free Throw Percentage
PPG: Points Per Game SPG: Steals Per Game
RPG: Rebounds Per Game APG: Assists Per Game
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This form provides a summary of a team’s performance for the first half of the season. So, how can it be best used to gauge future performance and inform decisions for increasing productivity? The questions below can be used to help facilitate either small or whole group discussion. This will set the stage for students to decide whether they want to sit tight or make changes to their team. As a basis for comparison, here are the statistics on the actual Player Cards for each player in the form on the opposite page.
Who is the Fever’s most productive player? What factors might account for the high productivity?
Students may have differing ideas on what “productive” means. Is the most productive player the one who has scored the most points? Or the player with the highest scoring percentage? In most cases, students will identify Sylvia Fowles or Blake Griffin as the most productive. Make sure they justify their choices. Some students will point to one of the players in the 6th, 7th, or 8th row in the chart. These are the “bench” players for this team, and as such, have taken very few shots. Since the
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sample set of shots for these players is so small, their statistics would usually be considered too skewed to count as valid for comparison. You can focus students’ attention on this by requiring that players have played in a minimum number of games or taken a minimum number of shots. Who is the Fever’s least productive player? What factors might account for the low productivity?
Depending on how they’ve defined “productive”, students will select either the player with the lowest shooting percentage or the player who scored the fewest points. An interesting thing to note is that the lowest FG% shooter in the Fever’s starting lineup (Becky Hammon: .333 FG%) actually scored the same number of points as the next lowest (James Harden: .462 FG%) due to the number of 3-point shots she made. How do the shooting percentages from game play compare with the shooting percentages on the Player Cards? What might be attributed to either higher or lower shooting percentages as a result of game play?
As might be expected, players who have taken more shots will have statistical data closer to the numbers on their Player Cards. Where there are anomalies (Blake Griffin’s FT%, Becky Hammon’s 3-pt. %, the shooting percentages of the “bench” players), you can engage students in a discussion of theoretical probability (the Player Card stats) vs. experimental probability (the stats from the Math Hoops games themselves). Does it appear that the Fever’s opponents have picked up on any “free throw liabilities?” If so, how can you tell?
The Player Card that shows the worst FT% for the Fever is Blake Griffin with .521. However, Blake’s game data is 1.000 because he’s 2 for 2. In the meantime, the data shows that Blake has made 11 field goal attempts. With 11 regular shot attempts and only 1 foul, it looks like the Fever’s opponents are ignoring their best opportunity to use fouls successfully. The Fever has been given the option to draft a new player to replace one of their current players. What position would you recommend they draft for and what information should they consider when making their selection?
There is no right or wrong answer to this question, so you will want to focus on students’ reasoning when they respond. Are they looking at the Player Card stats or the Mid-Season Assessment stats? If the latter,
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are they reasoning that a player with better stats than their Player Card won’t continue the streak they’re on? Or that a player with lower game stats than their Player Card stats will continue to perform poorly because of their performance in the first half of the season? Another key element in making a determination to replace a player is knowing what statistics could be improved to begin with. For example, the Fever has a couple of very high 2-pt. scorers (at least from a theoretical probability perspective) in Sylvia Fowles and Blake Griffin (.591 and .549 respectively). If the options for replacement players are to come from the Math Hoops set of Player Cards, it will be difficult to find players who will improve on these statistics. On the other hand, if you allow students to create their own Player Cards using data from any current NBA or WNBA player, they may well be able to make a card with a higher 2-pt. % or a card with both a high 2-pt. AND 3-pt. % (since neither Sylvia Fowles nor Blake Griffin will be helpful when it comes to 3-pt. shots). Following the draft, the Fever is also given the opportunity to update a current Player Card using recorded game-play data. Whose card should they update and what new strategies might they employ to maximize the benefits of this option?
Make sure students understand that all the game-play data would be replaced if they choose this option. Some students may suggest updating Kevin Durant’s card, thereby gaining a greater probability for both 2-pt. % (from .492 to .538) and 3-pt. % (from .387 to .500). This will then come with a decrease in FT% (from .860 to .667). Students should be cognizant of this consequence and address this in justifying their response. Once you’ve engaged in discussion with students using the sample Indiana Fever Mid-Season Assessment form, students should perform a similar analysis with their own team. The questions on page 68 of the Coach’s Manual are similar to the sample questions above. Students will be able to answer these after completing their own Mid-Season Assessment sheets and analyzing the data for their own players.
Coach’s Manual, p. 68
Question 4 on page 69 of the Coach’s Manual asks students to consider the impact of passing the ball in the Advanced Game. Assists are not part of the Mid-Season Assessment but students can pick up this data easily by reviewing their Game Summary Stat Sheets or Player Season Totals sheets. Coach’s Manual, p. 69
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Extension: Mid-Season Trades The Extension activity at the bottom of p. 69 in the Coach’s Manual offers an interesting twist when considering mid-season replacements. Instead of pulling new players from whatever Player Cards were not selected in the initial draft, students can engage in trade talks with other teams. This will require them not only to do the analysis for their own players but also to make that analysis available to other teams. Have students provide the Mid-Season Assessment form and Game Scoresheets for any
game the player they’re looking to trade has been in. If there are Game Summary Stat Sheets and Player Season Totals for the player, those should be available as well. The idea in the trade talks is to ensure that all parties have the same statistical information prior to a trade. Trades are especially useful when teams have a bench player with the same strengths as one of their starting players. While the bench player may not offer anything new to the original team, he or she could be just what another team
is looking to find for one of their starting players. There will usually be very little experimental data on a bench player—perhaps even none at all! If the player took only a handful of shots, the game statistics may be quite unlike the numbers on the Player Card itself. For this reason, pay attention to whether coaches are using the game data or the Player Card data when deciding if they want to pick up a player from another team.
14.2 Create Your Own Player Cards! MATH CONNECTION The explorations in UNIT 10 (PP. 63-73) provide the background in proportional reasoning and decimalpercent equivalence that lie at the heart of this exercise. You may want to review those explorations first.
One of the options for improving team performance that can be presented to students is to let them update an existing card (based on game play performance) or create a new Player Card. This will take more time than drafting one of the existing Player Cards, but the mathematical richness of the activity will be well worth it. Creating a Player Card offers students a concrete model for understanding abstract ideas of ratio and proportion. The act of translating numerical data (such as “.476 FG %” or “4 out of 11 three‑pt. shots”) to a pictorial representation in the form of circle graphs and shaded grids will bring these concepts to life much more forcefully than just using formulas. The information on page 70 of the Coach’s Manual gives students several ideas for creating a Player Card and the questions that follow will help them plan out what they need to do to complete this activity. In the back of this Teacher’s Guide (p. 141), there is a blank Player Card Template you can copy and pass out to students. The Template looks like a Player Card but without the measurements for the circle graph or shading for the 10x10 grid filled in. You will need to provide students with compasses and protractors so they can create accurate circle graphs.
Coach’s Manual, p. 70
You may wish to have students work through a sample Player Card before setting them loose to create replacement players for their team. Question 5 on page 71 of the Coach’s Manual presents statistics for a fictional player that can be used for this
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IN THE COACH’S MANUAL While a free agent draft and mid-season trade are effective ways to make a roster change, creating your own Player Cards trumps both when it comes to personalization and student buy-in. On pages 70–71 of the Coach’s Manual, students are given suggestions for accessing resources to gather player data. They are then guided through a series of questions designed to get them to think about how they will use the data to create their own, personalized Player Cards. Doing a test run with player data provided gives them a chance to try out various design elements and refine the process in preparation of creating new Player Cards for their team.
task. Having all students work on the same Player Card and sharing their work will uncover which parts of this activity are challenging and would benefit from additional instruction from you. Seeing how the same statistical information leads to different Player Card designs will reinforce how numbers can be represented in multiple ways. Walk through the step-by-step directions below as coaches work on Question 5 from the Coach’s Manual or an actual replacement Player Card. These directions can be used for creating an updated version of an existing card or creating a completely new card using actual player data from a reliable source such as www.nba.com. Of course, you can always allow students to create a fictional player as well—just make sure the shooting percentages fall within a “reasonable” range of performance.
Activity: How to Make a Player Card Using the information in Question 5 on p. 71 of the Coach’s Manual, create a Player Card with the following statistics: 2-Pt. % = .543
3-Pt. % = .328
FT % = .845
STEP 1: Determining Area for 2-pt. and 3-pt. Shots (Makes and Misses)
Coach’s Manual, p. 71
To find the appropriate area of a circle graph for successful 2-pt. shots, you will use the shooting percentage (recorded to three decimal places) and the number UNIT 14: IMPROVING YOUR TEAM • 115
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of degrees in a circle (360). Multiply 360 by the shooting percentage to get the number of degrees to be shaded to denote a successful shot. EXAMPLE 360 × .543 = 195.48 Round this to 195°. 195° of the circle will be shaded orange to denote a successful 2-pt. shot.
Subtract this number from 360 to get the number of degrees that would denote an unsuccessful shot. 360 − 195 = 165 165° of the circle will be unshaded to denote an unsuccessful 2-pt. shot.
All current Player Cards consist of circle graphs that are divided into four sectors— two equal sectors representing a “Make” and two equal sectors representing a “Miss.” To stay consistent with this practice, divide the measurement of the two sectors by 2 to get four sectors in all. MAKE
MISS
195° ÷ 2 = 97.5°
165° ÷ 2 = 82.5°
Round this to 98°.
Round this to 82°.
Each “Make” sector will measure 98°.
Each “Miss” sector will measure 82°.
(SHADED ORANGE)
(NOT SHADED)
The same exercise can be completed for 3-pt. shots although you will only need to find the area for successful shots. Successful 3-pt. shots are denoted by a crosshatch within the successful 2-pt. shot area. EXAMPLE 360 × .328 = 118.08 Round this to 118°. 118° of the circle will be crosshatched to denote a successful 3-pt. shot. 118° ÷ 2 = 59° Each crosshatched “Make” sector will measure 59° within an orangeshaded area.
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Extension: Varying the Circle Graph Designs When coaches are making their own Player Cards, they may question why they need to break up the circle graph into 4 sectors. The truth is they don’t. An interesting extension exercise is to have some coaches make circle graphs with 2 sectors only, some make typical 4-sector graphs, and perhaps even some make graphs with 6 or 8 sectors. Would any of these be a better way to design the cards than the others? Ask your students what they think and have them compare spin results to see if it makes a difference.
Theoretically, it makes no difference. The probability is the same as long as the proportions are the same. In actuality, coaches may see a difference for a variety of non-mathematical reasons: Dividing
into a greater number of sectors creates more opportunities for imprecision in measurement.
Increasing
the number of lines on a circle graph may result in more spins that are “liners.”
STEP 2: Graphing the Sectors for 2-pt. and 3-pt. shots
To graph each sector, draw a vertical radius to use as a starting point. Find the appropriate number of degrees for a successful 2-pt. shot—in this case, 98°, and draw aInitial new radius. radius Initial radius
98° 98°
82° 82°
This 98° sector should be shaded orange.
Using the new radius drawn as the starting point, find the appropriate number of degrees for an unsuccessful 2-pt. shot—in this case, 82°, and draw a new radius. This 82° sector should be left unshaded.
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82°
Repeat the process for a successful 2-pt. shot. Once the third radius is drawn and the second “successful” sector is shaded, you will have a completed circle graph representing the 2-pt. shooting performance of the given player.
98°
To graph the sectors for successful 3-pt. shots you will only need to work within the successful, orange-shaded regions of the 2-pt. shots. You’ll notice that on the predesigned NBA Math Hoops Player Cards, the area for successful 3-pt. shots, denoted by a crosshatch, is centered in the orange-shaded “successful” area for 2 pt. shots. When creating new Player Cards, centering may present a challenge to students. What we suggest is that instead of centering, students follow the same process they used when graphing the sectors for successful 2-pt. shots—i.e. start from an existing drawn radius. 59°
59°
Begin with an initial radius already drawn as the starting point. Find the appropriate number of degrees for a successful 3-pt. shot—in this case, 59°, and draw a new radius. This 59° sector should be in the first orange-shaded sector and crosshatched.
Move to the next orange-shaded sector and repeat the process.
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STEP 3: Determining & Graphing Area for Free Throws
The graph for free throws is a 10 × 10 grid that lies to the right of the 2-pt. and 3-pt. circle graph. Orange-shaded squares represent a “make” and grey-shaded squares represent a “miss.” Once students have determined th hich ones to shade in the two colors. To find the appropriate area of the 10 × 10 grid for successful free throws, you will use the shooting percentage (recorded to three decimal places) and the number of squares in the grid (100). Multiply 100 by the free throw shooting percentage to get the number of squares to be shaded to denote a successful shot. EXAMPLE 100 × .845 = 84.5 Round this to 85°. Eighty-five (85) of the 100 available squares will need to be shaded orange. The choice of which squares to shade is left up to individual coaches. (NOTE: Since free throw percentages tend to be high, it’s usually easier to subtract the FT% from 100 to determine the number of gray-shaded squares there should be.) 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9
0 1 2 3 4 5 6 7 8 9
0 1 2 3 4 5 6 7 8 9
You may want to ask students if it makes any difference as to which of the squares are shaded orange. While students may have aesthetic preferences, mathematically the likelihood of making or missing a free throw remains the same regardless of which ones are shaded orange.
Preparing for the Second Half Once students have assessed their teams—and possibly made some replacements— you can turn their attention to the second half of the Math Hoops season. How have students used the mid-season activities to improve their teams? Use questions
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like the following to get students thinking about new and modified strategies for how best to maximize performance from new or existing players: CLASSROOM CONNECTION Directions for running a successful second half of a regular season ending in a tournament event are given in UNIT 9: LEAGUE PLAYOFFS AND CHAMPIONSHIP PP. 54–59
• Who is your go-to player if you need a quick 2 points? What is the likelihood that she/he will be successful? Who is your second choice if your first choice is not available? • Who is your go-to player if you need 3 points? Who is your second choice if your first choice is not available? • How many shots per game did your top three performers average in the first half of the season? What was their combined shooting percentage for 2-pt. shots? 3-pt. shots? • What strategies will you employ to increase the point output of your top three performers? How will you measure success? • Did your opponents identify and take advantage of any free throw liabilities you have on your team? If so, what steps might you take to minimize the effectiveness of this strategic move? • Now that you’re well-versed in how the game is played, what is the target point total you would shoot for in a fast-paced game? On average, how many shots (2-pt. and 3-pt.) do you feel your players would have to take to have a good shot at achieving this total?
You can ask students to respond verbally or in writing to the questions above. To make the most of this exercise, here are a few possible options to consider: • Assign co-coaches the task of preparing written responses to some or all of the questions and discuss responses with another team of co-coaches. Perhaps allow students to select questions they feel would be of most value to them.
• Assign (or have students self select) 1–2 questions per team to first respond to in writing, then lead a whole group, round-robin discussion launched by their respective responses.
• Select any number of questions and facilitate either small or large group discussions based on perceived need and value. You may want to first have your students record brief written responses so that they have notes to refer to during the discussion.
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UNIT
15
Why Do Coaches Need Statistics? T
he various forms that are found in the Appendix of the Coach’s Manual—and the Appendix in this Teacher’s Guide—provide students with a way to evaluate and compare players. Many of these sheets can be used to track player and team data on a regular basis as coaches play games in a regular Math Hoops season. The extent to which this data is then analyzed will vary depending on your class and your goals for Math Hoops. There is much value in coaches engaging deeply in data analysis: • Coaches learn how straightforward mathematics can help them analyze and compare their players in ways that help them form winning strategies. For example, coaches may discover that a “favorite” player on their team does not measure up when compared to a “less favorite” player using field goal percentage and free throw percentage. • Coaches practice basic mathematics skills and see arithmetic as leading to meaningful and useful results instead of as just cold and disconnected numbers on a page. For example, finding and ordering quotients is a standard arithmetic task that is sometimes assigned outside of meaningful context. The same task in Math Hoops serves a coach’s purpose of ranking his team’s players according to points per game or field goal percentage. • Immersion in data can open a coach’s eyes to new patterns that would not be observed by just playing the game. For example, coaches can compare field goal and free throw percentages (empirical probabilities) with theoretical probabilities (given on Player Cards).
Time Considerations The individual activities in this unit will take 1 or 2 classroom sessions each depending on depth of exploration. Note however that keeping track of the statistics described in the activities will be an ongoing responsibility throughout the regular season. In cases where you want to track data on the classroom posters (Team Standings/League
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Leaders), you may want to assign this as a shared responsibility amongst several teams. These charts can be updated after each round of games or on a regular schedule (e.g., at the end of the week). EXPLORATION
IN THE COACH’S MANUAL On pages 72–73 of the Coach’s Manual, students are introduced to a method for computing winning percentages. Winning percentages are used to rank teams from best to worst based on won/loss records where teams may or may not have played an equal number of games. Practice with computing these percentages will help prepare students for maintaining Team Standings for your Math Hoops league.
ESTIMATED TIME
MATH CONTENT
• Comparing and Ordering Decimals and Percents • Collecting, Analyzing, and Displaying Data
15.1
Team Standings
1 session 45–60 minutes
15.2
Per-Game Statistics
1 session 45–60 minutes
• Finding Rates • Computing Averages of Data Sets • Collecting, Analyzing, and Displaying Data
15.3
Team Analysis
1–2 sessions 60–90 minutes
• Collecting, Analyzing, and Displaying Data • Comparing and Operating with Whole Numbers and Decimals
15.4
League Leaders
1-2 sessions 60–90 minutes
• Collecting and Organizing Data in Tables
15.1 Team Standings The presentation of Team Standings is an important aspect of the NBA Math Hoops experience for students. Seeing how their team compares with others in the league can serve as motivation to work harder and smarter. On page 72 of the Coach’s Manual, students are presented with basic information about the utility of Team Standings and how they work in the NBA. The NBA and WNBA have a clear-cut system for determining which teams earn the right to compete in the playoffs. Throughout the regular season, teams are ranked by winning percentage in their respective conferences and divisions. In the NBA, the top 8 teams in the Western Conference and the top 8 teams in the Eastern Conference are rewarded with a playoff spot. Several playoff series eventually lead to the NBA Finals where two teams compete for the title of World Champion. Team standings are kept throughout the season so that each team knows exactly where they stand at all times. The NBA divides their 30 teams into two conferences—the Western Conference and the Eastern Conference. Both conferences are broken into three divisions of five teams with each division having a distinct name. The Western Conference consists of the Pacific, Northwest, and Southwest divisions. The Eastern Conference consists of the Atlantic, Southeast, and Central divisions. To compile team standings, each team’s winning percentage (WP) must first be calculated. The winning percentage is a ratio of wins (W) to games played (total of wins and losses):
Coach’s Manual, p. 72
WP =
Wins Wins + Losses
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So, a team with 9 wins and 6 losses has a winning percentage of .600. 9 9 = = 60% 9+6 15
Note that winning percentages, like other basketball percentages, are written as decimals to three places. So this team’s WP is .600. (It would be a very bad idea to round WPs to one decimal place. A team that won 8 of its 15 games would appear to have the same winning percentage as a team that won 8 of 14 games.) After calculating each team’s WP, league standings rank the teams from highest to lowest WP. If two teams’ have identical win-loss records, the teams are listed alphabetically. On page 73 of the Coach’s Manual, students are given the standings for each division in the Western Conference after 20 games of the 2011–12 NBA season. They are then instructed to find each team’s winning percentage (Pct.). These percentages are provided below for easy reference. WESTERN CONFERENCE
PACIFIC
W
L
PCT.
L.A. Clippers
10
6
.625
L.A. Lakers
11
8
.579
Phoenix
6
12
.333
Golden State
6
12
.333
Sacramento
6
13
.316
NORTHWEST
W
L
PCT.
Oklahoma City
16
3
.842
Denver
14
5
.737
Portland
12
8
.600
Utah
10
7
.588
9
10
.474
SOUTHWEST
W
L
PCT.
San Antonio
12
8
.600
Dallas
12
8
.600
Houston
11
8
.579
Memphis
10
8
.556
4
15
.211
Minnesota
New Orleans
Coach’s Manual, p. 73
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You may want to ask your students what similarities they see between finding winning percentages and shooting percentages. When calculating winning percentages, the “whole” represents the total number of games played, with the “parts” being the number of games won and the number of games lost. When calculating shooting percentages, the “whole” represents the total number of shots taken, with the “parts” being the number of shots made and the number of shots missed. Both percentages are found by setting up ratios of favorable outcomes/total outcomes and then using division to convert the ratios into decimal representation. Students should be able to note that the highest percentage possible, assuming the favorable outcomes match the possible outcomes, is 1 or 1.00. A good mental math exercise to use with students is to have them make rough estimates of winning percentages. Give them two numbers—the number of games won and the total number of games. Establishing benchmarks will help students with their estimations. For example, present your students with the following information: Number of games won: 12 Number of games played: 27
Here are some questions you might ask to help them with their estimation. Each question serves as a benchmark to narrow the focus. • Is the winning percentage greater than or less than .500?
Since 12 is less than ½ of 27, the winning percentage will be less than .500. • Is the winning percentage greater or less than .250?
With 27 total games played, add one more to get to 28 and you have a multiple of 4. ¼ of 28 (or .250 × 28) would be 7. Since 12 games were won, the winning percentage would be greater than .250. • Is the winning percentage greater or less than .333?
⅓ of 27 is equal to 9 so 9 games won would yield a .333 winning percentage. Given that the team had 12 wins, the winning percentage would be greater than .333. You can see that with only three questions, it’s easy to determine that the winning percentage will fall somewhere between .333 and .500. Ask additional questions if you want to help students establish another benchmark and narrow the focus even further. By the way, the winning percentage for 12 games won out of 27 games played is .444. Both numbers are divisible by 3 and so the ratio represented as a fraction can be reduced to 4/9. When dividing any positive integer less than 9 by 9, the numerator is repeated indefinitely. By encouraging students to memorize the
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decimal equivalent of simple fractions, these can be used as benchmarks for making reasonable estimates. Team standings should be updated after each game played with winning percentages used to determine the order of the standings. Be sure to take full advantage of the Team Standings poster in your NBA Math Hoops Classroom Kit. You may want to have students take turns keeping track of the standings for your classroom league, however, all students can be held responsible for calculating their team’s winning percentage following every game.
15.2 Per-Game Statistics The concept of per-game statistics can be introduced to students after they have recorded data for several games. The goal for this type of data collection is to give students a new tool for comparing players. It’s important they realize that per-game stats are not exact numbers. You can start by posing the following situation: Suppose one player takes 4 two-point shots and makes 3 of them. A second player makes 5 of 12 two-point shots. Who had the better game?
To compare these two players, you can look at their point totals and at their field goal percentage. Field goal percentage is found by dividing field goals made (FGM) by field goals attempted (FGA): FGM FGA
FG% =
Here’s how you would compare the two players mentioned above. PLAYER
Points
FG%
1 2
6 10
.750 .417
IN THE COACH’S MANUAL On pages 74–75 of the Coach’s Manual, students are introduced to a process for computing the average or approximate per-game performance for a selection of game statistics. By dividing the total number of a particular statistic (e.g. assists) by the number of games, students will be able to compute approximate per-game performance. An important concept here is that per-game statistics provide a general “average” overview of player performance while game-by-game statistics provide much more specificity.
Player 2 scored more points but had a much lower field goal percentage. Player 1, on the other hand, did not score as many points but only missed 1 shot, leading to a stellar field goal percentage of .750. The above analysis is one way that percentages can be used by Math Hoops coaches to evaluate their players. These three percentages can be calculated from the Game Scoresheets: 2-pt FG% 3-pt FG% FT%
Additionally, these percentages can be found for teams as a whole.
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Another way to analyze and compare players is by using a “per game” approach. If Player 1 has scored 200 points in 12 games and Player 2 has scored 164 points in 8 games, you can tell immediately that Player 1 has scored more points. But the fact that Player 2 has played in fewer games has to be accounted for when you compare the two players. To do this, you can find each player’s points per game, or PPG: PPG =
Total Points Number of Games Played
The table below compares points per game for the two players. Typically, you round PPG to one decimal place as that usually provides the level of precision you’ll need to compare players accurately. PLAYER
Points
Games
PPG
1 2
200 164
12 8
16.7 20.5
Here are the different “per game” statistics that coaches can use to evaluate their players. Points per Game Steals per Game Rebounds per Game Assists per Game
As with percentages, these “per game” statistics can also be found for teams as a whole.
Coach’s Manual, p. 74
In this exploration, students learn how to calculate per-game statistics in order to compare players with different sets of data. On p. 74 of the Coach’s Manual, they’ll see examples for finding both points-per-game (PPG) and assists-per-game (APG). To use per-game statistics effectively, students must first understand how to use decimals to provide a desired level of precision. In the assists-per-game example on page 74, if decimals weren’t used, both players would average 3 assists per game. By carrying the division out to the tenths place, it’s clear that Player C’s 2.8 assists-per-
All-Time Leaders Sitting atop the NBA’s career leader board with 30.1 points per game are Michael Jordan and Wilt Chamberlain, two of the league’s greatest players ever. Their averages are rounded to one decimal place. Have your coaches do the math to see whether Jordan or Chamberlain is the true leader. Michael Jordan Wilt Chamberlain
GAMES
POINTS
1,072 1,045
32,292 31,419
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game is greater than Player B’s 2.6 assists-per-game. Of course, now you’ll need to have the discussion to describe what .8 or .6 of an assist looks like! On page 75, students are given the opportunity to practice calculating points-pergame, assists-per-game, rebounds-per-game, and steals-per-game. This practice will prepare them for keeping track of their players’ per-game statistics as they progress through the NBA Math Hoops season. Answers are provided in the charts below for easy reference. 1. POINTS PLAYER
Total Points
Games
Points/Games
PPG
Player A
57
7
57/6
8.1
Player B
59
8
59/8
7.4
Player C
63
5
63/5
12.6
C
TOP PLAYER IN THIS CATEGORY: Player
2. ASSISTS PLAYER
Total Assists
Games
Assists/Games
APG
Player A
23
6
23/6
3.8
Player B
16
5
16/5
3.2
Player C
17
6
17/6
2.8
Games
Rebounds/Games
RPG
A
TOP PLAYER IN THIS CATEGORY: Player
3. REBOUNDS PLAYER
Total Rebounds
Player A
13
3
13/3
4.3
Player B
8
5
8/5
1.6
Player C
19
4
19/4
4.8
Coach’s Manual, p. 75
C
TOP PLAYER IN THIS CATEGORY: Player
4. STEALS PLAYER
Games
Steals/Games
SPG
Player A
Total Steals 22
9
22/9
2.4
Player B
6
6
6/6
1.0
Player C
10
7
10/7
1.4
TOP PLAYER IN THIS CATEGORY: Player
A
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At the conclusion of this exploration, students are asked to list in priority order, the per-game statistics they feel a Math Hoops coach should know. They’re also asked to write a few sentences explaining why they chose the particular order they did. There is no one correct response here although students should be able to clearly explain the reasoning behind their particular order of importance. A follow-up discussion you’ll want to have with your students is around the similarities and differences between finding per-game statistics and shooting percentages. In both cases, data are represented as ratios and compared by using division. With per-game statistics, the particular statistic serves as the numerator and the number of games as the denominator. When divided, the quotient is rounded to the nearest tenth for comparison. The quotient can be (and often is) greater than 1. With shooting percentages, the number of shots made serves as the numerator and the number of shots attempted as the denominator. When dividing the numerator by the denominator, the quotient is rounded to the nearest thousandth for comparison. With regards to shooting percentages, it’s never possible for the quotient to be greater than 1. IN THE COACH’S MANUAL Comprehensive team analysis can take place once students enter player data into some of the forms provided in the Coach’s Manual Appendix. On pages 76–77 in the Coach’s Manual, students are given instruction on using a Team Analysis Chart to gather data. Students are then guided through a series of questions intended to analyze player and team performance. This type of analysis is best done at mid-season and at the end of the regular season, in preparation for league playoffs and hopefully, the championship game!
15.3 Team Analysis Students have been fully engaged in game play and while they’ve been collecting player and team data along the way, most will not have spent much time thinking about or knowing what to do with it. This exploration provides an opportunity for student coaches to look closely at the concept of “team” and analyze team data. Decisions they make moving forward should build on identified team strengths.
Team Analysis Chart TEAM: New York Knickerbockers GAMES PLAYED: 10 COACHES: Natalia Sanchez & Marcus Williams Name
D. Howard K. Durant K. Bryant B. Griffin C. Pondexter TEAM TOTALS
Total Points
52 83 132 63 113 443
PPG
2-pt. FG%
3-pt. FG%
5.2 8.3 13.2 6.3 11.3 44.3
.528 .491 .511 .531 .474 .503
.000 .368 .364 .250 .333 .343
FT%
APG
.556 3.7 .833 2.1 .923 1.4 .727 2.5 .706 1.8 .787 11.5
KEY PPG: Points Per Game FT%: Free Throw Percentage FG%: Field Goal Percentage APG: Assists Per Game
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Reviewing and Analyzing Sample Team Stats On pp. 76 of the Coach’s Manual, students are provided a sample (modified) Team Analysis Chart. Data on this chart is calculated by regularly recording and updating individual player data on the Player Season Totals form found in the Coach’s Manual Appendix (p. 83). Transferring this information to the Team Analysis Chart allows for easy access to data useful in making coaching decisions. After reviewing the chart (as seen on the opposite page), students are asked to respond to a set of questions. Below each question you will find information that may help you guide students through this process. 1. Who has the “hot hand” from 2-pt. range after 10 games? What coaching strategy could you use to take better advantage of their success?
The data tells us that both D. Howard and B. Griffin have the “hot hand” from 2-pt. range—both shooting well above the team average. An immediate coaching strategy students might employ would be to get the ball to these two players whenever possible. One thing to note, however, is that based on the limited data, we can determine that these two players have also taken the least number of shots. The player taking the greatest number of shots (K. Bryant) also has a shooting percentage that exceeds 50% from 2-pt. range. A more sophisticated strategy might include getting the ball to K. Bryant more often depending on what the circle graphs on the Player Cards look like. You might encourage students to do this comparison while considering their response.
Coach’s Manual, p. 76
2. How important is the passing game (assists) for this team? How can you tell based on the information given?
Assists are awarded when a player passes the ball to a teammate for a 2-pt. or 3-pt. shot and the shot is made. Based on the data given, the team averages 11.5 assists per game. A question you may want to pose to students is “What does .5 of an assist look like?” They’ll want to see that the 11.5 is an average and actual assists are recorded exclusively as whole numbers. Students can deduce that the 11.5 assists per game lead to anywhere from 23 (11.5 × 2 = 23) to approximately 35 (11.5 × 3 = 34.5) points per game. With the team averaging slightly over 44 points per game, assists contribute to over half of the team’s points. In other words, the coaches managing this team are consciously making decisions to get the ball into the hands of their higher percentage shooters. For this team, the passing game is an important p art of their coaching strategy.
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3. What do you feel is the team’s greatest strength? Explain your reasoning.
This question is intentionally very open-ended and you’ll want to pay close attention to the reasoning that students use as they could make a valid argument for several different areas based on the data given. For example, the team is successful on over 50% of their 2-pt. shots, makes one out of three 3-pt. shots and close to eight out of ten free throws. And then there is the strength of utilizing assists. Students may also choose to focus on an individual player’s data and cite that player as the team’s greatest strength.
Coach’s Manual, p. 76
4. What do you feel is the team’s Achilles heel (greatest weakness)? Explain your reasoning.
As with the greatest strength, determining the greatest weakness is open to interpretation. Again, look carefully at the reasoning behind the decision. If looking exclusively at individual player data, D. Howard is definitely the team’s greatest liability at the free throw line. An astute opposing team would take advantage of their fouls by fouling this player the maximum of three times in a close game. 5. What information would help you get a more detailed snapshot of the team’s shooting performance? How could you access this information?
Coach’s Manual, p. 77
What students are given is the total points, points per game and shooting percentages. What’s not provided is the detail—the number of shots attempted and the number of shots made by each player for each type of shot. This data can be gathered from the game scoresheets and recorded/compiled on the full Team Stat Sheet. One way to get at the importance of having this data is to have students find the average of the individual player shooting percentages. Students will find that the averages shown do not match their calculations. That’s because the shooting averages shown are based on compiled data—total number of shots made/total number of shots attempted.
New coaching strategies that student coaches record should reflect an understanding of the questions they’ve just considered.
When to Do the Math?
CLASSROOM TIP
In professional basketball, per-game statistics and percentages are updated following each game played. You can follow this rule but shouldn’t feel obliged to do so. The decision of when to expect coaches to prepare Team Analysis Charts is strictly a function of your preferences and goals, as instructor and as League Commissioner. Our suggestion is that coaches prepare and submit their Team Analysis Charts following their last game played each week. The one occasion for which coaches must prepare and submit Team Analysis Charts is the All-Star Game. [See Unit 8: Mid-Season Review/All-Star Game for more information.]
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Once you’ve engaged students in discussion using the sample New York Knickerbockers Team Analysis Chart, you’ll want them to consider these same questions (and any others you or they may add) while referring to the data on their own comprehensive Team Analysis Chart. This is a worthwhile exercise to do after 5 games, 10 games, and at the conclusion of the regular season. Coaches that take the time to do this type of team analysis will be much better prepared for playoff competition!
15.4 League Leaders During the regular season, the NBA and WNBA track the top players on an ongoing basis, ranking them according to various statistics. For example, if you go to http:// www.nba.com/statistics/ or http://www.wnba.com/statistics/ you can get the most up-to-date league statistics in Scoring, Rebounds, Assists, Steals, 2-pt. Field Goal %, 3-pt. Field Goal % and Free Throw %. What statistics do you think would be useful to keep track of in your NBA Math Hoops league?
Tracking League Leaders League leaders in various offensive categories can be posted on a weekly basis on the League Leaders chart provided in the NBA Math Hoops classroom kit. You may want to have students volunteer to keep track of the leaders in a category of particular interest to them. On p. 78 in the Coach’s Manual, students are shown a sample chart that shows the League Leaders (top 8 players) in 2-pt. Field Goal % after 5 games in a season. You’ll note that the team names include both real NBA and WNBA teams and some that students created. 2-PT. FIELD GOAL % Name
Team
Position
FGM
FGA
2-pt. FG %
Candace Parker
Bank Shots
F
24
37
.649
Andrew Bynum
Pistons
C
41
65
.631
LeBron James
Fever
F
15
26
.577
Katie Douglas
Netsters
G
25
44
.568
Chris Paul
Lakers
G
22
39
.564
Dwight Howard
Desert Hawks
C
28
51
.549
Pau Gasol
Timberwolves
F
17
32
.531
Russell Westbrook
Celtics
G
26
49
.531
IN THE COACH’S MANUAL Keeping track of League Leaders during a Math Hoops season creates a more complete experience and adds to the excitement. Pages 78–79 of the Coach’s Manual focus on tracking leaders based on shooting percentages. This exercise provides practice with ratio/decimal conversions and ordering numbers. Students are encourage to track shooting percentages along with two per-game statistics (points-per game and assists-per game) on the Team Standings/League Leaders poster found in each NBA Math Hoops Classroom Kit.
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Looking at the Data Throughout the Math Hoops experience you’ve seen a consistent message of driving students towards the data in order to promote data-driven decision making. The following questions are presented in the Coach’s Manual to prompt students to look closely at the data. 1. What does this League Leaders chart tell you about these players and their respective teams?
These players represent the “best of the best” from eight different teams with regards to 2-pt. shooting percentage. It also tells you that the second best shooting percentage for each team is less than the lowest shooting percentage listed on this chart. Otherwise, there would be multiple players listed for the same team. 2. Look at the last two players. What do you notice about their statistics? Why would Pau Gasol be placed above Russell Westbrook? Coach’s Manual, p. 78
The shooting percentages are identical but the number of field goals made and field goals attempted are different. Pau Gasol has made 17 of 32 2-pt. shots attempted (17/32) and this ratio converted to a decimal is .53125. Russell Westbrook has made 26 of 49 2-pt. shots attempted (26/49) and this ratio converted to a decimal is .5306122. While both of these shooting percentages can be rounded to .531, Pau Gasol’s is actually slightly higher. This example can be used to demonstrate how decimal equivalence helps to compare the variance in player data and the role precision plays (or can play) when comparing what appear to be similar shooting percentages.
Transferring and Practicing What’s Been Learned Now that students have a grasp of how to chart League Leaders with regards to shooting percentages, they can be instructed to practice what they’ve learned—first in small groups and then as a whole class.
Coach’s Manual, p. 79
On p. 79 of the Coach’s Manual, students are given three tables they can use to find the top 5 League Leaders in 2-pt. %, 3-pt. %, and Free Throw %. Teams will need to share information about their top players with each other to determine the top players in the league. Wherever percentages are tied between two players, coaches should check the specific ratios that resulted in those percentages. Carrying the division out to a greater number of decimals may reveal one statistic to be greater than the other. It’s also possible that some coaches will have “bench” players with high numbers based on very little data. Allow students to wrestle with a situation such as a player who’s 2 for 2 and therefore has a 1.000% 2-pt. percentage. Most coaches will say that it’s unfair to allow these types of players to be counted amongst league leaders.
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If this happens, students should suggest possible definitions for which players are eligible—for example, only players with a certain number of shots attempted, or only players who are part of the “starting 5” on a team. A well-informed coach will refer to the League Leader boards often. Statistics tell an important story. Encourage teams to get all they can from their statistics to help them be even more competitive.
Wrapping Up Having students keep track of League Leaders serves several purposes. First, it provides students with extra practice in computing and verifying the accuracy of shooting percentages and other basketball-related statistical measures. It also provides rich fodder for discussion and can lead to in-depth questioning and help your students develop a better understanding of how data is used in real-world contexts. What’s probably most important to many students is that it adds yet another layer to the total experience and allows them bragging rights—if only temporarily. A rich, meaningful learning experience coupled with appropriate celebration of the competitive spirit is what NBA Math Hoops is all about!
UNIT 15: WHY DO COACHES NEED STATISTICS? • 133
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Appendix SEASON SCHEDULE TEMPLATE GAME SUMMARY STAT SHEET* PLAYER SEASON TOTALS* MID-SEASON ASSESSMENT ALL-STAR NOMINATION FORM TEAM ANALYSIS CHART PLAYER CARD TEMPLATE* PROGRAM ROADMAP
A copy of most of the forms in this Appendix can also be found in the Coach’s Manual Appendix. They are included here for easy reference. In addition, you will want to take note of the forms that are marked with an asterisk. Each team will need multiple copies of these sheets.
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Season Schedule Template The grid below is a blank template that shows one way to lay out a season schedule. This sample lists 10 days and allows for up to 8 games each day, but it is up to you to decide how long the schedule needs to be and how many games will be played each day. GAME 1
GAME 2
GAME 3
GAME 4
GAME 5
GAME 6
GAME 7
GAME 8
Day 1 DATE
Day 2 DATE
Day 3 DATE
Day 4 DATE
Day 5 DATE
Day 6 DATE
Day 7 DATE
Day 8 DATE
Day 9 DATE
Day 10 DATE
When two teams play basketball, one is the Home team and one is the Away team. Figure out a way you can indicate who the home team is in each game.
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Game Summary Stat Sheet FINAL SCORE
Date of Game Your Team Opponent
2-Pt Player/Position
FGM
3-Pt FGA
FGM
TEAM
TEAM
SCORE
SCORE
Free Throws FGA
FTM
FTA
Total Points
S
F
R
A
1. 2. 3. 4. 5. *6. *7. *8.
TEAM TOTALS Note: When finished, transfer player stats to the Player Season Totals sheets. *Use these rows for any mid-game player replacements.
ABBREVIATIONS FGM: Field Goals Made FGA: Field Goals Attempted
Coach's Signature
FTM: Free Throws Made FTA: Free Throws Attempted
S: Steals F: Fouls
R: Rebounds A: Assists
Date
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Player Season Totals Player
Position
Team Coaches 2-Pt
Date of Game
Opponent
FGM
3-Pt
FGA
FGM
FGA
Free Throws FTM
FTA
Total Points
S
F
R
A
SEASON TOTALS TOTAL GAMES PLAYED
Note: You will need a copy of this sheet for each of your players.
ABBREVIATIONS FGM: Field Goals Made FGA: Field Goals Attempted
FTM: Free Throws Made FTA: Free Throws Attempted
S: Steals F: Fouls
R: Rebounds A: Assists 137
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2-Pt FGM
2-Pt FGA
Mid-Season Assessment Team Coaches
Player Name/Position 1. 2. 3. 4. 5. 6. 7. 8. TEAM TOTALS
2-pt. FG%
ABBREVIATIONS PPG: Points Per Game SPG: Steals Per Game
3-pt. FGM
3-pt. FGA
Games Played
3-pt. FG%
RPG: Rebounds Per Game APG: Assists Per Game
FTM
FTA
FT%
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Note: You will need the information from your Player Season Totals sheets to calculate the statistics in this sheet.
FG%: Field Goal Percentage FT%: Free Throw Percentage
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All-Star Nomination Form Team Coaches
All-Star 1
窶アll-Star 2
Explain why you chose these two players:
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2-Pt FG%
Team Analysis Chart Team Coaches
Player Name/Position 1. 2. 3. 4. 5. 6. 7. 8. TEAM TOTALS
3-Pt FG%
ABBREVIATIONS
FT%
PPG: Points Per Game SPG: Steals Per Game
PPG
Games Played
SPG
RPG: Rebounds Per Game APG: Assists Per Game
RPG
APG
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Note: You will need the information from your Player Season Totals sheets to calculate the statistics in this sheet.
FG%: Field Goal Percentage FT%: Free Throw Percentage
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Player Card Template
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NBA Math Hoops: Program Roadmap School/Organization
Facilitator
Grade Level(s)
Venue (circle one): IN SCHOOL AFTER SCHOOL YOUTH CENTER OTHER Duration: TIME PER SESSION: Date(s):
minutes/hours
# of Sessions:
# of Students:
# of weeks:
to
Session 1
Session 2
Session 3
Session 4
Session 5
Session 6
Session 7
Session 8
Session 9
Session 10
Session 11
Session 12
Session 13
Session 14
Session 15
Session 16
Session 17
Session 18
Session 19
Session 20
Session 21
Session 22
Session 23
Session 24
Session 25
Session 26
Session 27
Session 28
Session 29
Session 30
Session 31
Session 32
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