2011-2012 School Handbook - Mission Math Utah

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2011–2012School HandbookContains more than 300 creative math problemsthat meet NCTM standards for grades 6-8.For questions about your local MATHCOUNTS program,please contact your chapter (local) coordinator. Coordinator contactinformation is available through the Find a Coordinator option of theCompetition Program link on www.mathcounts.org. 2011 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 2010–2011 MATHCOUNTS Question Writing Committee developed the questions for the competitions and the2011–2012 MATHCOUNTS School Handbook: Chair: Sam Baethge, San Antonio, TX Tom Butts, University of Texas at Dallas, Richardson, TX Barbara Currier, Greenhill School, Addison, TX Tina Pateracki, Bluffton, SC Tricia Rothenberg, Georgetown, TX Dianna Sopala, Fair Lawn, NJ Patrick Vennebush, Reston, VANational Judges review competition materials and serve as arbiters at the National Competition: Richard Case, Computer Consultant, Greenwich, CTFlavia Colonna, George Mason University, Fairfax, VAPeter Kohn, James Madison University, Harrisonburg, VACarter Lyons, James Madison University, Harrisonburg, VAMonica Neagoy, Mathematics Consultant, Washington, DCHarold Reiter, University of North Carolina-Charlotte, Charlotte, NCDave Sundin (STE 84), Statistics and Logistics Consultant, San Mateo, CANational Reviewers proofread and edit the problems in the MATHCOUNTS School Handbook and/or competitions:William Aldridge, Springfield, VAMady Bauer, Bethel Park, PASusanna Brosonski, Orlando, FLDan Cory (NAT 84, 85), Seattle, WARoslyn Denny, Valencia, CABrian Edwards (STE 99, NAT 00), Evanston, ILNancy English, Glendale, MOChengde Feng, Burlington, MADennis Hass, Newport News, VAHelga Huntley (STE 91), Newark, DEJane Lataille, Los Alamos, NMStanley Levinson, P.E., Lynchburg, VAHoward Ludwig, Ocoee, FLLeon Manelis, Orlando, FLPaul McNally, Haddon Heights, NJRandy Rogers, West Des Moines, IACraig Volden (NAT 84), Columbus, OHChaohua Wang, Bloomington, ILBecky Weiss, Vienna, VADeborah Wells, Rockville, MDAmerican Meteorological Society Writers/Editors: James Brey, Director, AMS Education Program; Elizabeth Mills, AssociateDirector, AMS Education Program; Joseph Moran, Associate Director, AMS Education ProgramThe Solutions to the problems were written by Kent Findell, Diamond Middle School, Lexington, MA.MathType software for handbook development was contributed by Design Science Inc., www.dessci.com, Long Beach, CA.Editor and Contributing Author: Kera Johnson, Manager of EducationMATHCOUNTS FoundationContent Editor: Kristen Chandler, Deputy Director & Program DirectorMATHCOUNTS FoundationIntroduction and Program Information: Archana Mehta, Director of MarketingMATHCOUNTS FoundationExecutive Director: Louis DiGioiaMATHCOUNTS FoundationHonorary Chair: William H. SwansonChairman and CEO, Raytheon Company

Count Me In!A contribution to theMATHCOUNTS Foundation willhelp us continue to make thisworthwhile programavailable to middle schoolstudents nationwide.The MATHCOUNTS Foundationwill use your contribution forprogramwide support to givethousands of students theopportunity to participate.To become a supporter ofMATHCOUNTS, sendyour contribution to:MATHCOUNTS Foundation1420 King StreetAlexandria, VA 22314-2794Or give online at:www.mathcounts.org/donateOther ways to give: Ask your employer aboutmatching gifts. Your donationcould double. Remember MATHCOUNTSin your United Way andCombined Federal Campaign atwork. 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 2011–2012 Dates . 4Introduction . 5New This Year . 6Reel Math Challenge. . 6Eligibility Update for Homeschoolsand Virtual Schools. 6Mapping to Common Core State Standards. 6Late Registrations . 6Coaches’ Resource Video Series . 7Referral Program . 7MATHCOUNTS and Social Media. . 7The MATHCOUNTS OPLET . 8Building a Competition and/or Club Program . 9Recruiting Mathletes . 9Maintaining a Strong Program . 9MATHCOUNTS Competition Program. 10Preparation Materials . 10Coaching Students . 11Official Rules and Procedures . 13Registration. 13Eligible Participants . 14Levels of Competition . 15Competition Components . 16Additional Rules . 17Scoring . 18Results Distribution . 18Forms of Answers . 19Vocabulary and Formulas . 20MATHCOUNTS Club Program . 22Club Materials . 22Getting Started . 23Attaining Silver Level Status . 23Attaining Gold Level Status . 24Frequently Asked Questions . 25Reel Math Challenge . 27Handbook ProblemsWarm-Ups and Workouts . 29Stretches . 56What About Math? . 61Introduction . 61Problems . 62The National Association of Secondary SchoolPrincipals has placed this program on theNASSP Advisory List of National Contests andActivities for 2011–2012.Answers to Handbook and What About Math? Problems .65Solutions to Handbook and What About Math? Problems .70MATHCOUNTS Problems Mapped to Common Core StateStandards . . 912012 MATHCOUNTSNational Competition SponsorProblem Index . 92Registration Form . 95The MATHCOUNTS Foundation makes its products and services available on a nondiscriminatory basis. MATHCOUNTS does not discriminate on the basis ofrace, religion, color, creed, gender, physical disability or ethnic origin.

2011Sept. 1 Dec. 16CriTiCAL 2011-2012 DATESSend in your school’s Registration Form to receive a hard copy of the MATHCOUNTS SchoolHandbook, the Club in a Box Resource Kit and/or your copy of the 2011-2012 School CompetitionKit. Items will ship shortly after receipt of your form, with the mailing ofthe School Competition Kit following this schedule:Registration Forms postmarked by Oct. 1: Kits mailed in early November.Kits continue mailing every two weeks.Registration Forms postmarked by Dec. 16 deadline: Kits mailed in early January.Mail or fax the MATHCOUNTS Registration Form (with payment if participating in the competition)to:MATHCOUNTS Registration, P.O. Box 441, Annapolis Junction, MD 20701Fax: 240-396-5602 (Please fax or mail, but do not do both.)Questions? Call 301-498-6141 or confirm your registration via the Registered Schools database and/or 2011-2012 Bronze Level Schools list at www.mathcounts.org in the Registered Schools or ClubProgram sections, respectively.Sept. 1Video submission period begins for Reel Math Challenge.Oct. 12-13American Math Challenge, brought to you by MATHCOUNTS and powered by Mathletics Nov. 15General public voting period opens for Reel Math Challenge.Dec. 16Competition Registration Deadline(postmark)2012In some circumstances, late registrations might be accepted, at the discretion of MATHCOUNTS andthe local coordinator. Late fees may also apply. Register on time to ensure participation by yourstudents.Early JanuaryIf you have not been contacted with details about your upcoming competition, call your local orstate coordinator! If you have not received your School Competition Kit by the end of January,contact MATHCOUNTS at 703-299-9006.Feb. 1-29Chapter CompetitionsFeb. 1Video submission period for Reel Math Challenge ends, and general public voting ends. Top 20videos are announced in February.March 1-31State CompetitionsMarch 2Deadline for MATHCOUNTS Club Programs to reach Silver Level. Applications for Silver Level Statusmust be received by MATHCOUNTS as of this date for entry to the Silver drawing.March 15Top four video submissions for Reel Math Challenge are announced.March 30Deadline for MATHCOUNTS Club Programs to reach Gold Level. Applications for Gold Level Status andUltimate Math Challenges must be received by MATHCOUNTS as of this date for entry to the Golddrawing.May 112012 Raytheon MATHCOUNTS National Competition in Orlando, Florida.Interested in more coaching materials or MATHCOUNTS items?Purchase items from the MATHCOUNTS store at www.mathcounts.org or contact Sports Awards at 800-621-5803.Selected items also are available at www.artofproblemsolving.com.4MATHCOUNTS 2011-2012

iNTrODuCTiONThe mission of MATHCOUNTS is to inspire excellence, confidence and curiosity in U.S. middle school studentsthrough fun and challenging math programs. Currently in its 29th year, MATHCOUNTS meets its missionby providing three separate but complementary programs for middle school students: the MATHCOUNTSCompetition Program, the MATHCOUNTS Club Program and the newest program, Reel Math Challenge.The MATHCOUNTS Competition Program is designed to excite and challenge middle schoolstudents. With four levels of competition - school, chapter (local), state and national - theCompetition Program provides students with the incentive to prepare throughout the schoolyear to represent their schools at these MATHCOUNTS-hosted* events. MATHCOUNTSprovides the preparation materials and the competition materials, and with the leadershipof the National Society of Professional Engineers, more than 500 Chapter Competitions, 56State Competitions and one National Competition are hosted each year. These competitionsprovide students with the opportunity to go head-to-head against their peers from other schools, cities andstates; to earn great prizes individually and as members of their school team; and to progress to the 2012Raytheon MATHCOUNTS National Competition in Orlando, Florida. There is a registration fee for students toparticipate in the Competition Program, and participation past the School Competition level is limited to the top10 students per school. A more detailed explanation of the Competition Program follows on pages 10 through 21.The MATHCOUNTS Club Program (MCP) is designed to increase enthusiasm for mathby encouraging the formation within schools of math clubs that conduct meetings thatare fun and include a variety of math activities. The resources provided through theMCP are also a great supplement for classroom teaching. The activities provided for theMCP foster a positive social atmosphere, with a focus on students working together asa club to earn recognition and rewards in the MATHCOUNTS Club Program. Some rewardsare participation based, and others are achievement based, but all rewards require a minimum number of clubmembers (based on school size). Therefore, there is an emphasis on building a strong club and encouraging morethan just the top math students within a school to join. There is no cost to sign up for the MATHCOUNTS ClubProgram, but a Registration Form must be submitted to receive the free club materials. (A school that registersfor the Competition Program is automatically signed up for the MATHCOUNTS Club Program.) A more detailedexplanation of the MATHCOUNTS Club Program follows on pages 23 through 26.Sponsored in part by the Department of Defense, the Reel Math Challenge is an innovativeprogram involving teams of students using cutting-edge technology to create videos aboutmath problems and their associated concepts. This new competition is meant to excite studentsabout math while allowing them to hone their creativity and communication skills. Studentswill form teams consisting of four students and will create a video based on one of the 300problems included in this handbook. In addition, students will be able to form teams with peersfrom around the country. As long as a student is a sixth-, seventh- or eighth- grader, he or shecan participate. Each video must teach the solution to the selected math problem as well as demonstrate thereal-world application of the math concept used in the problem. All videos will be posted to www.reelmath.org,where the general public will vote on the best videos. The 20 videos with the highest vote totals will advance tothe semifinals of the competition, after which a panel of MATHCOUNTS judges will review them and select fourfinalists. Each of the four finalist teams will receive an all-expense-paid trip to the 2012 Raytheon MATHCOUNTSNational Competition, where they will present their videos to the 224 students competing in that event.The competitors will then vote for one of the four videos to be the winner of the Reel Math Challenge. Eachmember of the winning team will receive a 1000 college scholarship. A more detailed explanation of Reel MathChallenge follows on page 27.*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 not required;selection of team and individual competitors for the Chapter Competition is entirely at the discretion of the school coach and need not be based solely onSchool Competition scores.MATHCOUNTS 2011-20125

NEw ThiS YEArREEL MATH CHALLENGEThe Reel Math Challenge is a brand new program involving teams of students using technology to create videosabout math problems and their associated concepts. Reel Math Challenge is meant to excite students aboutmath while allowing them to hone their creativity and communication skills. Students will form teams consistingof four students and will create a video based on one of the 300 problems included in the MATHCOUNTS SchoolHandbook. Each video must teach the solution to the selected math problem as well as demonstrate thereal-world application of the math concept used in the problem.All videos will be posted to www.reelmath.org, where the general public will vote on the best videos. The 20videos with the highest vote totals will advance to the semifinals of the competition, after which a panel ofMATHCOUNTS judges will review them and select four finalists.Each of the four finalist teams will receive an all-expense-paid trip to the 2012 MATHCOUNTS NationalCompetition, where they will present their videos to the 224 students competing in that event. The competitorswill then vote for one of the four videos to be the winner of the Reel Math Challenge. Each member of thewinning team will receive a 1000 college scholarship. Once the contest has concluded, the websitecontaining the archive of videos will remain available for public use at no charge. Because these videos will becross-referenced with specific MATHCOUNTS problems and each MATHCOUNTS problem is indexed to specificmath concepts and standards, this library of videos will be a powerful teaching tool for teachers and students toutilize for years.The submission period for videos opens September 1, 2011 and will remain open through February 1, 2012.Public voting begins November 15, 2011 and will remain open through February 1, 2012. For more information,visit www.reelmath.org.ELIGIBILITY UPDATE FOR HOMESCHOOLS AND vIRTUAL SCHOOLSHomeschools and virtual schools are now eligible to participate as individuals and as members of a team.However, prior to participation in a MATHCOUNTS Chapter Competition, each registering homeschool,homeschool group or virtual school must complete and return the Homeschool Participation Attestation Form,which can be downloaded from the right navigation bar by visiting www.mathcounts.org/competition.Homeschools: The definition of homeschool and the requirements for participation are outlined on page 14, andthese have not changed since last year.virtual Schools: Any virtual school interested in registering students must contact the MATHCOUNTS nationaloffice at 703-299-9006 before December 16, 2011 for registration details.MAPPING TO COMMON CORE STATE STANDARDSWe are pleased to announce that this year, the MATHCOUNTS School Handbook problems will be categorized notonly by topic and difficulty level but also by Common Core State Standard. The Common Core StateStandards Initiative is a state-led effort coordinated by the National Governors Association Center for BestPractices (NGA Center) and the Council of Chief State School Officers (CCSSO). The Standards were developed incollaboration with teachers, school administrators and experts in the field of mathematics. Currently, 43 statesand the District of Columbia have adopted the Standards. We hope this additional resource will allow you tofurther utilize handbook problems by supplementing your classroom teaching.LATE REGISTRATIONSThis program year, an additional fee of 20 will be assessed to process late registrations. Registrationspostmarked after December 16, 2011 will be considered late. (This fee is for each late registration form and notfor each student being registered.) See the Registration Form on page 95 for additional details.6MATHCOUNTS 2011-2012

COACHES’ RESOURCE vIDEO SERIESMATHCOUNTS has created a new video series to help you understand the free resources available to you in yourClub in a Box Resource Kit. To date, there are three videos, but the library will continue to grow as we see a need.Introduction to the Club in a Box Resource Kit - This short video will provide you with an understanding of thematerials included in your Club in a Box Resource Kit and how to best utilize them.Achieving Silver Level and Gold Level Status in the MATHCOUNTS Club Program - This video explains the stepsnecessary for your school to achieve Silver Level and Gold Level Status in the Club Program. Requirements,incentives and prizes are discussed.Introduction to the MATHCOUNTS School Handbook - Have you received your handbook but are unsure abouthow to best utilize the problems? The School Handbook video discusses the variety of problems that can befound in the handbook as well as how to utilize the problems for MATHCOUNTS competition preparation orclassroom use.All of these videos can be accessed at www.mathcounts.org/videos or on the MATHCOUNTS YouTube page,which can be found at www.youtube.com/mathcountsfdn.REFERRAL PROGRAMOn a recent survey, over 90% of MATHCOUNTS coaches said they would refer MATHCOUNTS to a colleague. Yourreferral can be rewarded by having a portion of your registration fees reimbursed! MATHCOUNTS is continuing itsreferral and registration incentives for the Competition Program.Simply refer a middle school that is not currently participating in MATHCOUNTS*. If the school you referregisters a team of four students for the MATHCOUNTS Competition Program this school year, your school andthe referred school each will receive a refund of 10. With enough successful referrals, your school could bereimbursed for its entire registration fee! The new Registration Form on page 95 has a section for you to indicatethe school(s) you wish to refer.Incentive for MATHCOUNTS Club Program SchoolsSchools that previously have participated only in the MATHCOUNTS Club Program (and not in the CompetitionProgram) may also take advantage of the incentive program. If you choose to register a team of four for theMATHCOUNTS Competition Program this school year, you will receive a refund of 10. Additional details may befound on the Registration Form on page 95.MATHCOUNTS AND SOCIAL MEDIADid you know that MATHCOUNTS is now on Facebook, Twitter and YouTube? Stay updated on importantMATHCOUNTS news and math happenings through posts and tweets. Watch highlights from the NationalCompetition, access the Coaches’ Video Series, watch a promotional video about the Club Program and view allof the newest MATHCOUNTS Minis online!Experience MATHCOUNTS on social media at the following m/mathcountswww.youtube.com/mathcountsfdn*To qualify as a referred school, a school may not have participated in the MATHCOUNTS Competition Program for the previous twoschool years.MATHCOUNTS 2011-20127

ThE mAThCOuNTS OPLET(Online Problem Library and Extraction Tool). . . a database of thousands of MATHCOUNTS problems AND step-by-step solutions,giving you the ability to generate worksheets, flash cards and Problems of the DayThrough www.mathcounts.org, MATHCOUNTS is offering the MATHCOUNTS OPLET - a database of 10,000problems and over 4,000 step-by-step solutions, with the ability to create personalized worksheets, flash cardsand Problems of the Day. After purchasing a 12-month subscription to this online resource, the user will haveaccess to MATHCOUNTS School Handbook problems and MATHCOUNTS competition problems from the past 11years and the ability to extract the problems and solutions in personalized formats. (Each format is presented ina pdf file to be printed.)Once the subscription is purchased, the user can access the MATHCOUNTS OPLET each time he or she goes towww.mathcounts.org/oplet and logs in. Once on the MATHCOUNTS OPLET page, the user can tailor the outputto his or her needs by answering a few questions. These are some options that can be personalized: Format of the output: Worksheet, Flash Cards orProblems of the Day Number of questions to include Solutions (whether to include or not for selectedproblems) Math concept: Arithmetic, Algebra, Geometry, Countingand Probability, Number Theory, Other or a RandomSampling MATHCOUNTS usage: Problems without calculator usage(Sprint Round/Warm-Up), Problems with calculator usage(Target Round/Workout/Stretch), Team problems withcalculator usage (Team Round), Quick problems withoutcalculator usage (Countdown Round) or a RandomSampling Difficulty level: Easy, Easy/Medium, Medium, Medium/Difficult, Difficult or a Random Sampling Year range from which problems were originally used inMATHCOUNTS materials: Problems are grouped in fiveyear blocks in the system.Once these criteria have been selected, the user either (1) can opt to have the computer select the problemsat random from an appropriate pool of problems or (2) can manually select the problems from this appropriatepool of problems.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. The cost ofa subscription is 275; however, schools registering students in the MATHCOUNTS Competition Program willreceive a 5 discount per registered student. If you purchase OPLET before October 14, 2011, you can save atotal of 75* off your subscription. Please refer to the coupon above for specific details.If you would like to get a sneak peek at this invaluable resource before making your purchase, you can check outscreen shots of the MATHCOUNTS OPLET at www.mathcounts.org/oplet. You will see the ease with which youcan create countless materials for your Mathletes, club members and classroom students.*The 75 savings is calculated using the special 25 offer plus an additional 5 discount per student registered for the MATHCOUNTSCompetition Program.8MATHCOUNTS 2011-2012

BuiLDiNg A COmPETiTiON AND/OrCLuB PrOgrAmRECRUITING MATHLETES Ideally, the materials in this handbook will be incorporated into the regular classroom curriculum so that allstudents learn problem-solving techniques and develop critical thinking skills. When a school’s MATHCOUNTSCompetition Program or Club Program is limited to extracurricular sessions, all interested students should beinvited to participate regardless of their academic standing. Because the greatest benefits of the MATHCOUNTSprograms are realized at the school level, the more Mathletes involved, the better. Students should view theirexperience with MATHCOUNTS as fun, as well as challenging, so let them know from the very first meeting thatthe goal is to have a good time while learning.Here are some suggestions from successful competition and club coaches on how to stimulate interest at thebeginning of the school year: Build a display case showing MATHCOUNTS shirts and posters. Include trophies and photos from previousyears’ club sessions or competitions. Post intriguing math questions (involving specific school activities and situations) in hallways, the library andthe cafeteria. Refer students to the first meeting for answers. Use the MATHCOUNTS poster from your Clubin a Box Resource Kit! Make a presentation at the first pep rally or student assembly. Approach students through other extracurricular clubs (e.g., honor society, science 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 website (www.mathcounts.org/potw) into theweekly class schedule.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 Mathletes have.Remember, the more this year’s students get from the experience, the easier recruiting will be next year. Hereare 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. Have a students-versus-teachers Countdown Round, and invite the student body to watch. Solicit donations from local businesses to be used as prizes in practice competitions. Plan retreats or field trips for the Mathletes to area college campuses or hold an annual reunion. Take photos at club meetings, coaching sessions or competitions and keep a scrapbook. Distribute MATHCOUNTS shirts to participating students. Start a MATHCOUNTS summer-school program. Encourage teachers of students in lower grades to participate in mathematics enrichment programs.MATHCOUNTS 2011-20129

mAThCOuNTS COmPETiTiON PrOgrAm . . .A mOrE DETAiLED LOOkThe MATHCOUNTS Foundation administers its math enrichment, coaching andcompetition program with a grassroots network of more than 17,000 volunteers whoorganize MATHCOUNTS competitions nationwide. Each year more than 500 localcompetitions and 56 “state” competitions are conducted, primarily by chapter and statesocieties of the National Society of Professional Engineers. All 50 states, the Districtof Columbia, Puerto Rico, Guam, Virgin Islands, U.S. Department of Defense schoolsand U.S. State Department schools worldwide participate in MATHCOUNTS. Here’severything you need to know to get involved.PREPARATION MATERIALSThe annual MATHCOUNTS School Handbook provides the basis for coaches and volunteers to coach studentMathletes on problem-solving and mathematical skills. Coaches are encouraged to make maximum use ofMATHCOUNTS materials by incorporating them into their classrooms or by using them with extracurricular mathclubs. Coaches also are encouraged to share this material with other teachers at their schools, as well as withparents.MATHCOUNTS has created a new video series to help you understand the free resources available to you. Todate, there are three videos, but the library will continue to grow as we see a need. The Introduction to theMATHCOUNTS School Handbook video discusses the variety of problems that can be found in the handbook aswell as how to utilize the problems for MATHCOUNTS competition preparation or classroom use. Please takeadvantage of these great new resources. All of these videos can be accessed at www.mathcounts.org/videos oron the MATHCOUNTS YouTube page, which can be found at www.youtube.com/mathcountsfdn.The 2011–2012 MATHCOUNTS School Handbook, formerly in two volumes, has been consolidated

The MATHCOUNTS Competition Program is designed 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. MATHCOUNTS

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