2008–2009 School Handbook: Volume I
2008–2009School Handbook:Volume IContains 100 creative math problemsthat meet NCTM standards for grades 6-8.For questions about your local MATHCOUNTS program,please contact your local (chapter) coordinator. Coordinator contact informationis available in the Find a Coordinator section of www.mathcounts.org.The printing of this handbook, accompanying registration materialsand their distribution was made possible by 2008 MATHCOUNTS Foundation1420 King Street, Alexandria, VA .orgUnauthorized reproduction of the contents of this publication is a violation of applicable laws.Materials may be duplicated for use by U.S. schools.MATHCOUNTS and Mathlete are registered trademarks of the MATHCOUNTS Foundation.
AcknowledgmentsThe 2007–2008 MATHCOUNTS Question Writing Committee developed the questions for the2008–2009 MATHCOUNTS School Handbook and competitions: Chair: Sandy Powers, College of Charleston, Charleston, S.C.Sam Baethge, San Antonio, TexasChengde Feng, Oklahoma School of Science and Mathematics, Oklahoma City, Okla.Joyce Glatzer, West New York Public Schools, West Patterson, N.J.Greg Murray, Dixie High School, St. George, UtahTom Price, Norris Middle School, Firth, Neb.Susan Wildstrom, Walt Whitman High School, Bethesda, Md.National Judges review competition materials, develop Masters Round questions andserve as arbiters at the National Competition: Richard Case, Computer Consultant, Greenwich, Conn.Flavia Colonna, George Mason University, Fairfax, Va.Peter Kohn, James Madison University, Harrisonburg, Va.Carter Lyons, James Madison University, Harrisonburg, Va.Monica Neagoy, Mathematics Consultant, Washington, D.C.Dave Sundin (STE 84), Statistics and Logistics Consultant, San Mateo, Calif.National Reviewers proofread and edit the MATHCOUNTS School Handbook and/or competition materials:William Aldridge, Springfield, Va.Mady Bauer, Bethel Park, Pa.Susanna Brosonski, Orlando, Fla.Lars Christensen (STE 89), Minneapolis, Minn.Dan Cory (NAT 84, 85), Seattle, Wash.Craig Countryman, San Diego, Calif.Roslyn Denny, Valencia, Calif.Edward Early (STE 92), Austin, TexasNancy English, Glendale, Mo.Barry Friedman (NAT 86), Scotch Plains, N.J.Joan M. Gell, Redondo Beach, Calif.Dennis Hass, Westford, Mass.Bonnie Hayman, St. Louis, Mo.Helga Huntley (STE 91), Seattle, Wash.Doug Keegan (STE 91, NAT 92), Austin, TexasDavid Kung (STE 85, NAT 86), St. Mary’s City, Md.Jane Lataille, Los Alamos, N.M.Stanley Levinson, P.E., Lynchburg, Va.Artie McDonald, P.E. (STE 88), Melbourne, Fla.Paul McNally, Haddon Heights, N.J.Randy Rogers, Cedar Rapids, IowaFrank Salinas, Houston, TexasLaura Taalman (STE 87), Harrisonburg, Va.Craig Volden (NAT 84), Columbus, OhioChaohua Wang, Bloomington, Ill.Deborah Wells, Rockville, Md.Judy White, Littleton, Mass.Yiming Yao (STE 96), Vancouver, British ColumbiaThe Solutions were written by Kent Findell, Diamond Middle School, Lexington, Mass.Editor and Contributing Author: Jessica Liebsch, Manager of EducationMATHCOUNTS FoundationIntroduction and Content Editor: Kristen Chandler, Deputy Director & Director of EducationMATHCOUNTS FoundationExecutive Director: Louis DiGioiaMATHCOUNTS FoundationHonorary Chair: William H. SwansonChairman and CEORaytheon CompanyMathType software for handbook development contributed by Design Science Inc., www.dessci.com, Long Beach, Calif.
Count Me In!A contribution to theMATHCOUNTS Foundationwill help us continue to makethis worthwhile programavailable to middle school studentsnationwide.The MATHCOUNTS Foundationwill use your contribution forprogramwide support to givethousands of students theopportunity to participate.To become a partner inMATHCOUNTS, sendyour contribution to:MATHCOUNTS FoundationP.O. Box 1338Merrifield, VA 22116-9706Or give online at:www.mathcounts.orgOther ways to give: Ask your employer aboutmatching gifts. Your donationcould double. Remember MATHCOUNTSin your United Way andCombined Federal Campaignat work. Leave a legacy. IncludeMATHCOUNTS in your will.For more information regardingcontributions, call the directorof development at 703-299-9006,ext. 103 or e-mailinfo@mathcounts.org.The MATHCOUNTS Foundation is a 501(c)3organization. Your gift is fully tax deductible.TABLE OF CONTENTSCritical 2008–2009 Dates. 4Introduction. 5New This Year. 5The MATHCOUNTS Web Site. 5The MATHCOUNTS OPLET. 7Building a Competition and/or Club Program . 8Recruiting Mathletes . 8Maintaining a Strong Program. 8MATHCOUNTS Competition Program. 9Preparation Materials. 9Coaching Students. 10Official Rules & Procedures . 12Registration. 12Eligible Participants. 13Levels of Competition. 14Competition Components. 15Additional Rules. 16Scoring . 17Results Distribution. 17Forms of Answers. 18MATHCOUNTS Club Program. 19Club Materials. 19Rules, Procedures & Deadlines. 19Getting Started. 19Attaining Silver Level Status. 20Attaining Gold Level Status. 20Frequently Asked Questions. 21Handbook ProblemsWarm-Ups and Workouts. 23Stretch. 32ResourcesProblem-Solving Strategies. 33Vocabulary and Formulas. 45References. 47Answers to Handbook Problems. 49Solutions to Handbook Problems. 53The National Association of Secondary SchoolPrincipals has placed this program on the NASSPAdvisory List of National Contests and Activitiesfor 2008–2009.Problem Index. 59Request/Registration Form. 63The American Society of Association Executiveshas recognized MATHCOUNTS with a 2001 Awardof Excellence for its innovative, society-enrichingactivities.The MATHCOUNTS Foundation makes its products and services available on a non-discriminatory basis. MATHCOUNTS does not discriminateon the basis of race, religion, color, creed, gender, physical disability or ethnic origin.
Critical 2008–2009 Dates ImmediatelyFor easy reference, write your local coordinator’s address and phone number here.Contact information for coordinators is available in the Find a Coordinator section ofwww.mathcounts.org or from the national office.Send in your school’s Request/Registration Form to receive Volume II of the handbook,the Club in a Box resource kit and/or your copy of the 2008–2009 SchoolCompetition. Items will ship shortly after receipt of your form, with mailing ofthe School Competition Kit following this schedule: September Dec. 12Registration forms postmarked by Oct. 1: Kits mailed in early November.Kits continue mailing every two weeks.Registration forms postmarked by Dec. 12 deadline: Kits mailed in early January.Mail or fax the MATHCOUNTS Request/Registration Form (with payment ifparticipating in the competition) to:MATHCOUNTS Registration, P.O. Box 441, Annapolis Junction, MD 20701Fax: 301-206-9789 (Please fax or mail, but do not do both.)Questions? Call 301-498-6141 or confirm your registration via the Registered Schoolsdatabase and/or 2008–2009 Bronze Level Schools list at www.mathcounts.org in theRegistered Schools or Club Program sections, respectively. Dec. 12(postmarkedby)Competition Registration DeadlineIn some circumstances, late registrations may be accepted at the discretion ofMATHCOUNTS and the local coordinator. Register on time to ensure participationby your students. Mid‑JanuaryIf you have not been contacted with details about your upcoming competition, callyour local or state coordinator!If you have not received your School Competition Kit by the end of January, contactMATHCOUNTS at 703-299-9006. Feb. 1–28Chapter Competitions March 1–28State Competitions March 6Deadline for Math Clubs to reach MATHCOUNTS Silver Level & entry into drawing March 27Deadline for Math Clubs to reach MATHCOUNTS Gold Level & entry into drawing May 7–102009 Raytheon MATHCOUNTS National Competition at the Walt Disney WorldSwan and Dolphin Resort(received by)(received by)Interested in more coaching materials or MATHCOUNTS items?Additional FREE resources are available at www.mathcounts.org.Purchase items from the MATHCOUNTS store at www.mathcounts.org or contact Sports Awards at 800-621-5803.Select items also are available at www.artofproblemsolving.com. MATHCOUNTS 2008–2009
IntroductionThe mission of MATHCOUNTS is to increase enthusiasm for and enhance achievement in middle schoolmathematics throughout the United States. Currently in our 26th year, MATHCOUNTS meets its missionby providing two separate, but complementary, programs for middle school students: the CompetitionProgram and the MATHCOUNTS Club Program.The MATHCOUNTS Competition Program is a program to excite and challenge middleschool students. With four levels of competition — school, chapter (local), state andnational — the Competition Program provides students with the incentive to preparethroughout the school year to represent their schools at these MATHCOUNTS-hosted*events. MATHCOUNTS provides the preparation materials and the competitionmaterials, and with the leadership of the National Society of Professional Engineers,more than 500 Chapter Competitions, 57 State Competitions and one NationalCompetition are hosted each year. These competitions provide students with the opportunity to go headto-head against their peers from other schools, cities and states; to earn great prizes individually andas part of their school team; and to progress through to the 2009 Raytheon MATHCOUNTS NationalCompetition at the Walt Disney World Swan and Dolphin Resort. There is a registration fee for studentsto participate in the Competition Program, and participation past the School Competition level is limitedto the top eight students per school. A more detailed explanation of the Competition Program follows onpages 9 through 18.The MATHCOUNTS Club Program (MCP) is designed to increase enthusiasm formath by encouraging the formation within schools of math clubs that conduct funmeetings using a variety of math activities. The activities provided for the MCPfoster a social atmosphere, and there is a focus on students working together as aclub to earn recognition and rewards in the MATHCOUNTS Club Program. Somerewards are participation based, while others are achievement based, but all rewardsrequire a minimum of 12 club members. Therefore, there is an emphasis on building a strong club andencouraging more than just the top math students within a school to join. There is no cost to sign up forthe MATHCOUNTS Club Program, but a Request/Registration Form must be submitted to receive theclub materials. (A school that registers for the Competition Program is automatically signed up for theMATHCOUNTS Club Program.) A more detailed explanation of the MATHCOUNTS Club Programfollows on pages 19 through 22.New This Yearthe MATHCOUNTS web site. a new look and great new features designed especially for coachesThe general public still has access to many great materials and resources on www.mathcounts.org,including the Problem of the Week, Volume I of the 2008–2009 MATHCOUNTS School Handbook,general information about the MATHCOUNTS Competition Program and Club Program and archivedmaterials. However, a new Coaches section of the site will provide members-only access to restrictedpages with fantastic resources and networking opportunities.MATHCOUNTS is able to grant coaches access to portions of the web site that are restricted from thegeneral public once coaches have completed two simple steps:1. Create a User Profile on mathcounts.org.2. Either sign up a club with the MATHCOUNTS Club Program or register students for theCompetition Program.*While MATHCOUNTS provides the actual School Competition Booklet with the questions, answers and procedures necessary to run the SchoolCompetition, the administration of the School Competition is up to the MATHCOUNTS coach in the school. The School Competition is notrequired; selection of team and individual competitors for the Chapter Competition is entirely at the discretion of the school coach and need notbe based solely on the School Competition scores.MATHCOUNTS 2008–2009
Once MATHCOUNTS links your User Profile with a Club or Competition school, you will have specialresources available to you each time you log in to mathcounts.org.To create your User Profile:1. Go to www.mathcounts.org.2. Click on the purple Login button at the top,right-hand side of the page (see screen shot tothe left).3. Click on the New User Profile link at thebottom of the form (see screen shot below).4. Complete the online form.The Coaches Home PageShortly after your New User Profile requestis received, MATHCOUNTS will verify yourcoach status and allow you to join the onlinecommunity. (Note: You must (1) have a UserProfile and (2) send in your Request/RegistrationForm for the MATHCOUNTS Club Programand/or Competition Program for us to grant youthis access. The order in which these are done isnot important, but both must be done to gain theextra access.) At that time, you will be able tolog in and view your new MATHCOUNTS homepage. Among other items, there will be a Coachesbutton in the left-hand navigation menu and anMCP Members Only link in the Club Programsection (see screen shot to the left). MATHCOUNTS 2008–2009
The mathcounts OPLET(Online Problem Library and Extraction Tool). a database of 10,000 MATHCOUNTS problems and the ability to generateworksheets, flash cards and Problems of the DayThrough www.mathcounts.org, MATHCOUNTS is offering the MATHCOUNTS OPLET — a databaseof 10,000 problems and the ability to create personalized worksheets, flash cards and Problems of the Daywith these problems. After purchasing a 12-month subscription to this online resource, the user will haveaccess to MATHCOUNTS School Handbook and MATHCOUNTS competition problems from the past10 years and the ability to extract the problems in personalized formats. (Each format is generated into apdf to be printed.)Imagine the time that can be saved preparing for club meetings,practice sessions or classroom teaching!Once the subscription is purchased, the user can access the MATHCOUNTS OPLET each time s/he goesto www.mathcounts.org and logs in. Once on the MATHCOUNTS OPLET page, the user can tailor theoutput to his/her needs by answering a few questions. Among the options that can be personalized are: Format of the output: Worksheet, Flash cards or Problems of the Day Number of questions to include Math concept: Arithmetic, Algebra, Geometry, Counting and Probability, Number Theory, Otheror Random Sampling MATHCOUNTS usage: Problems without calculator usage (Sprint Round/Warm-Up), Problemswith calculator usage (Target Round/Workout/Stretch), Team problems with calculator usage(Team Round), Quick problems without calculator usage (Countdown Round) or RandomSampling Difficulty Level: Easy, Easy/Medium, Medium, Medium/Difficult, Difficult or Random Sampling Year range from which problem was originally used in MATHCOUNTS materials: Problems aregrouped in five-year blocks in the systemOnce these criteria have been selected, the user either (1) can opt to have the computer select theproblems at random from an appropriate pool of problems or (2) can select the problems from thisappropriate pool of problems him/herself.How does a person gain access to this incredible resource as soon as possible?A 12-month subscription to the MATHCOUNTS OPLET can be purchased at www.mathcounts.org. Thecost of a subscription is 275; however, schools registering students in the MATHCOUNTS CompetitionProgram will receive a 5 discount per registered student.Once MATHCOUNTS processes your subscription payment for the MATHCOUNTS OPLET, you willbe given access to the MATHCOUNTS OPLET page and sent notification that you are a registered user.Note: You first must create a User Profile on www.mathcounts.org in order for us to grant you access tothis resource.If you would like to get a sneak peek at this invaluable resource before making your purchase, you cancheck out screen shots of the MATHCOUNTS OPLET at www.mathcounts.org. You will see the easewith which you can create countless materials for your Mathletes, club members and classroom students.MATHCOUNTS 2008–2009
Building a competition and/orclub ProgramRecruiting Mathletes Ideally, the materials in this handbook will be incorporated into the regular classroom curriculum sothat all students learn problem-solving techniques and develop critical thinking skills. When a school’sMATHCOUNTS Competition Program and/or Club Program is limited to extracurricular sessions, allinterested students should be invited to participate regardless of their academic standing. Because thegreatest benefits of the MATHCOUNTS program are realized at the school level, the more Mathletesinvolved, the better. Students should view their experience with MATHCOUNTS as fun, as well aschallenging, so let them know from the very first meeting that the goal is to have a good time whilelearning.Some suggestions from successful competition and club coaches on how to stimulate interest at the beginningof the school year: Build a display case using MATHCOUNTS shirts and posters. Include trophies and photos fromprevious years’ club sessions or competitions. Post intriguing math questions (involving specific school activities and situations) in hallways, thelibrary and the cafeteria. Refer students to the first meeting for answers. Make a presentation at the first pep rally or student assembly. Approach students through other extracurricular clubs (e.g., science club, computer club, chess club). Inform parents of the benefits of MATHCOUNTS participation via the school newsletter or PTA. Create a MATHCOUNTS display for Back-to-School Night. Have former Mathletes speak to students about the rewards of the program. Incorporate the Problem of the Week from the MATHCOUNTS web site (www.mathcounts.org) intothe weekly class schedule. Organize a MATHCOUNTS Math Club.Maintaining a Strong ProgramKeep the school program strong by soliciting local support and focusing attention on the rewards ofMATHCOUNTS. Publicize success stories. Let the rest of the student body see how much fun Mathleteshave. Remember, the more this year’s students get from the experience, the easier recruiting will be nextyear. Here are some suggestions: Publicize MATHCOUNTS meetings and events in the school newspaper and local media. Inform parents of meetings and events through the PTA, open houses and the school newsletter. Schedule a special pep rally for the Mathletes. Recognize the achievements of club members at a school awards program
The MATHCOUNTS Competition Program is a program to excite and challenge middle school students. With four levels of competition — school, chapter (local), state and national — the Competition Program provides students with the incentive to prepare throughout the school year to represent their schools at these MATHCOUNTS-hosted* events.
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