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Discrete Math Problem Solvingfor Middle School Students

David Patrick

Art of Problem Solvingwww.artofproblemsolving.com

April 10, 2008NCTM Annual Meeting

Salt Lake City

David Patrick (Art of Problem Solving) Discrete Math for Middle School Students NCTM 2008 — Salt Lake 1 / 18

Outline

1 What is Discrete Math?

2 Why Discrete Math?

3 Some Discrete Math ProblemsCounting & ProbabilityNumber TheoryGraph TheoryTwo-Player Strategy Games

4 ResourcesBooksContestsMath Circles

5 Summary


What is Discrete Math?What discrete math is not

By discrete math, I generally mean things outside of the usual

Pre-algebra↓

Algebra↓

Geometry↓

Advanced Algebra / Trigonometry↓

Precalculus

sequence of coursework (although there is definitely some overlap).


What is Discrete Math?What discrete math is

Discrete math (at the middle-school level) includes the followingsubject areas:

CombinatoricsProbabilityNumber TheoryGraph TheorySet Theory and LogicAlgorithmsTwo-player StrategyGames





CombinatoricsElementary counting problemsPermutations & combinationsPascal’s TriangleInclusion/Exclusion





ProbabilityDiscrete probabilityDependent and independenteventsExpected valueConditional probability





Number TheoryPrimes and divisibilityFactoringEuclidean Algorithm (for GCD)Basic Diophantine equations(equations with integersolutions)





Graph TheoryBasics of finite graphsGraph coloringModeling problems using graphs(e.g. tournaments)Euler tours





Set Theory / LogicUnions and intersectionsTruth tablesInference and logical reasoningElementary proofs





AlgorithmsSimple algorithms for solvingproblemsStandard algorithms (e.g.search, sorts)Estimating run time andtermination





Strategy GamesTic-tac-toeNim and other chip-selectinggamesBoard games (e.g. Chomp, Hex)


Why often overlooked?

Why is discrete math often overlooked or underrepresented at themiddle-school level?

Not a focus of “high-stakes” testingNot a focus of the SATNot perceived as “important” in the same way thatalgebra/geometry are

Arguably, algebra proficiency is the most important feature ofmiddle-school math, and geometry is used to introduce proofs (thoughusually in high school).


Reasons for discrete math

Gateway to highermathematicsGateway to computerprogramming“Real-world”mathematicsMath contestsMathematical reasoningand proofFun!




Discrete math—along withcalculus and abstract algebra—isone of the core components ofmathematics at theundergraduate level, especiallyapplied math. Students seeingdiscrete math in middle or highschool will be at an advantage.




Modern computer science is builtlargely on discrete math, inparticular combinatorics, logic,and graph theory. A course indiscrete mathematics is usually arequired part of pursuing acomputer science degree.




Discrete math allows students tovery quickly explore non-trivial“real world” problems that arechallenging and interesting.




Prominent math competitionssuch as MATHCOUNTS and theAmerican MathematicsCompetitions (AMC 8/10/12)feature discrete math questionsas a significant portion of theircontests.




Relatively few formulas tomemorize; rather, there are anumber of fundamental conceptsto be mastered and applied inmany different ways. Encouragesflexible thinking and developmentof problem-solving skills.




Simply put, many students finddiscrete math more fun thanalgebra or geometry.


Sample ProblemCounting & Probability

2005 AMC 8 #15A five-legged Martian has a drawer full of socks, each of which is red,white, or blue, and there are at least five socks of each color. TheMartian pulls out one sock at a time without looking. How many socksmust the Martian remove from the drawer to be certain there will be 5socks of the same color?(A) 6 (B) 9 (C) 12 (D) 13 (E) 15

The Martian might get unlucky with 12 socks, pulling 4 red, 4 white,and 4 blue. But the 13th sock will guarantee a set of 5.This is an example of the Pigeonhole Principle. This seemingly easyprinciple (which can easily be grasped by middle-school students) isactually quite deep and a key principle of combinatorics.This problem also easily generalizes (# of colors, # of legs, # ofMartians, etc.)








The Martian might get unlucky with 12 socks, pulling 4 red, 4 white,and 4 blue. But the 13th sock will guarantee a set of 5.

This is an example of the Pigeonhole Principle. This seemingly easyprinciple (which can easily be grasped by middle-school students) isactually quite deep and a key principle of combinatorics.This problem also easily generalizes (# of colors, # of legs, # ofMartians, etc.)




The Martian might get unlucky with 12 socks, pulling 4 red, 4 white,and 4 blue. But the 13th sock will guarantee a set of 5.This is an example of the Pigeonhole Principle. This seemingly easyprinciple (which can easily be grasped by middle-school students) isactually quite deep and a key principle of combinatorics.

This problem also easily generalizes (# of colors, # of legs, # ofMartians, etc.)







San Diego Math League 2005-06 (Grades 6–8)4 blue marbles and 6 red marbles are in a jar. If you draw 2 marblesout of the jar, what is the probability that you get one of each color?

Answer:815

There are 4 × 6 = 24 ways to draw two marbles of opposite colors(without regard to order).There are

(102)= 45 ways to draw any two marbles (without regard to

order).

So the probability is2445

=8

15.

This is a good example of the danger of “comparing apples andoranges”. Many students counted 24 and 90 getting an answer of4/15.




Answer:815



order).


=8

15.





Answer:815

There are 4 × 6 = 24 ways to draw two marbles of opposite colors(without regard to order).

There are(10

2)= 45 ways to draw any two marbles (without regard to

order).


=8

15.





Answer:815



order).


=8

15.





Answer:815



order).


=8

15.





Answer:815



order).


=8

15.




MOEMS March 2008 Problem of the Month3 dice are rolled and their sum is 7. What is the probability that 2 dice

show the same number?

Answer:35

There are four ways to roll 7 with 3 dice:Rolls: ��

TotalArrangements: 3 6 3 3 15

9 of these arrangements have two dice equal.

Therefore, the probability is915

=35

.

This is a hard, subtle problem involving conditional probability. Onewould generally do some simpler dice problems before working up tothis one.




show the same number? Answer:35





=35

.






There are four ways to roll 7 with 3 dice:

Rolls: ��




=35

.










=35

.






There are four ways to roll 7 with 3 dice:Rolls: �� Total

Arrangements: 3 6 3 3 15



=35

.







Arrangements: 3 6 3 3 159 of these arrangements have two dice equal.


=35

.









=35

.









=35

.



Sample ProblemNumber Theory

2005 AMC 8 #20Alice and Bob play a game involving a circle whose circumference isdivided by 12 equally-spaced points. The points are numberedclockwise, from 1 to 12. Both start on point 12. Alice moves clockwiseand Bob, counterclockwise. In a turn of the game, Alice moves 5 pointsclockwise and Bob moves 9 points counterclockwise. The game endswhen they stop on the same point. How many turns will this take?(A) 6 (B) 8 (C) 12 (D) 14 (E) 24

After each move, Alice is 2 more positions clockwise from Bob thanshe was before the move.So after 6 moves, Alice is 12 positions clockwise from Bob, whichmeans she’s on the same point as Bob.

This is an introduction to modular arithmetic.









After each move, Alice is 2 more positions clockwise from Bob thanshe was before the move.

So after 6 moves, Alice is 12 positions clockwise from Bob, whichmeans she’s on the same point as Bob.













Sample ProblemNumber Theory & Counting

MATHCOUNTS 2008 Chapter Team #6What is the smallest positive integer with exactly 14 positive divisors?

Answer: 192

A number with prime factorization

pe11 pe2

2 . . . pekk

has (e1 + 1)(e2 + 2) · · · (ek + 1) positive divisors.14 only factors as 1 · 14 or 2 · 7.So any number with exactly 14 positive divisors is of the form p13 forsome prime p, or p6q for some primes p and q.The smallest is then 26

· 31 = 192.

This is somewhat harder: it combines several concepts from bothcounting and number theory.



MATHCOUNTS 2008 Chapter Team #6What is the smallest positive integer with exactly 14 positive divisors?Answer: 192


pe11 pe2

2 . . . pekk


· 31 = 192.






pe11 pe2

2 . . . pekk

has (e1 + 1)(e2 + 2) · · · (ek + 1) positive divisors.

14 only factors as 1 · 14 or 2 · 7.So any number with exactly 14 positive divisors is of the form p13 forsome prime p, or p6q for some primes p and q.The smallest is then 26

· 31 = 192.






pe11 pe2

2 . . . pekk

has (e1 + 1)(e2 + 2) · · · (ek + 1) positive divisors.14 only factors as 1 · 14 or 2 · 7.

So any number with exactly 14 positive divisors is of the form p13 forsome prime p, or p6q for some primes p and q.The smallest is then 26

· 31 = 192.






pe11 pe2

2 . . . pekk

has (e1 + 1)(e2 + 2) · · · (ek + 1) positive divisors.14 only factors as 1 · 14 or 2 · 7.So any number with exactly 14 positive divisors is of the form p13 forsome prime p, or p6q for some primes p and q.

The smallest is then 26· 31 = 192.






pe11 pe2

2 . . . pekk


· 31 = 192.






pe11 pe2

2 . . . pekk


· 31 = 192.



Sample ProblemGraph Theory

Mathematical Circles (Chapter 5)In the country of Seven there are 15 towns, each of which is connectedby road to at least 7 others.(a) Is it possible for each town to be connected by road to exactly 7others?(b) Show that it is possible to travel from any town to any other town,either directly or by passing through another town.

(b): A good example of proof by contradiction.Suppose two towns X and Y cannot be travelled between. This meansthat the 7 towns reachable by X and the 7 towns reachable by Y are alldifferent.But this is a total of 16 towns! (X and Y and 7 more for each.) That’stoo many—contradiction!




(a): No.





(a): No. If each of 15 towns has 7 roads leading out of it, that’s a totalof 15 × 7 = 105 roads leading out of towns. But this counts each roadtwice (once for the town at each end of it), so that means 105/2 = 52.5roads. But there’s no such thing as half a road!





(a): No. If each of 15 towns has 7 roads leading out of it, that’s a totalof 15 × 7 = 105 roads leading out of towns. But this counts each roadtwice (once for the town at each end of it), so that means 105/2 = 52.5roads. But there’s no such thing as half a road!

This is a nice example of a parity argument.





(b): A good example of proof by contradiction.

Suppose two towns X and Y cannot be travelled between. This meansthat the 7 towns reachable by X and the 7 towns reachable by Y are alldifferent.But this is a total of 16 towns! (X and Y and 7 more for each.) That’stoo many—contradiction!




(b): A good example of proof by contradiction.Suppose two towns X and Y cannot be travelled between.

This meansthat the 7 towns reachable by X and the 7 towns reachable by Y are alldifferent.But this is a total of 16 towns! (X and Y and 7 more for each.) That’stoo many—contradiction!




(b): A good example of proof by contradiction.Suppose two towns X and Y cannot be travelled between. This meansthat the 7 towns reachable by X and the 7 towns reachable by Y are alldifferent.

But this is a total of 16 towns! (X and Y and 7 more for each.) That’stoo many—contradiction!






Sample ProblemTwo-Player Strategy Games

Pick-Up SticksTina and Val are playing a game of Pick-Up Sticks. The game startswith 27 sticks in a pile. Each player, in turn, removes 1, 2, 3, or 4 sticksfrom the pile. The player who takes the last stick wins. Tina goes first.Who should win and why?

If played correctly, Tina should win.Tina should remove 2 sticks from the pile on her first turn.Then, after Val’s turn, Tina should always remove a number of sticks toleave a multiple of 5.Eventually, Val will have 5 sticks in the pile on her turn, and no matterwhat she takes, Tina can then win.

This is a simple example of a 2-player strategy game, whose strategycan be analyzed by modular arithmetic. It can easily be generalized.

A more complicated type of this game is Nim.




If played correctly, Tina should win.

Tina should remove 2 sticks from the pile on her first turn.Then, after Val’s turn, Tina should always remove a number of sticks toleave a multiple of 5.Eventually, Val will have 5 sticks in the pile on her turn, and no matterwhat she takes, Tina can then win.






If played correctly, Tina should win.Tina should remove 2 sticks from the pile on her first turn.

Then, after Val’s turn, Tina should always remove a number of sticks toleave a multiple of 5.Eventually, Val will have 5 sticks in the pile on her turn, and no matterwhat she takes, Tina can then win.






If played correctly, Tina should win.Tina should remove 2 sticks from the pile on her first turn.Then, after Val’s turn, Tina should always remove a number of sticks toleave a multiple of 5.

Eventually, Val will have 5 sticks in the pile on her turn, and no matterwhat she takes, Tina can then win.




















A more complicated type of this game is Nim.David Patrick (Art of Problem Solving) Discrete Math for Middle School Students NCTM 2008 — Salt Lake 13 / 18

Discrete Math ResourcesBooks

Art of Problem Solving booksIntroduction to Counting & Probability by D. PatrickIntroduction to Number Theory by M. Crawford

Written specifically for high-performing students in grades 6-10Mathematical Circles (Russian Experience) by D. Fomin, S.Genkin, I. Itenberg [AMS]

Problem Solving Through Recreational Mathematics by B.Averbach and O. Chein [Dover]MATHCOUNTS School Handbook (annual) [MATHCOUNTSFoundation]

Contains worksheets to be used progressively throughout the year.Available for free at http://www.mathcounts.org


Discrete Math ResourcesContests

MATHCOUNTSOver 6,300 schools and 500,000 students at the local levelBoth individual and team componentsHandbooks provide a year-round curriculumCompetition Jan (in-school), Feb (local), Mar (state), May (national)

American Mathematics Competitions (AMC 8)Late November in schoolsOver 2,200 schools and 147,000 students25 question, 40 minute, multiple-choice contestPrizes at the school, state, and national levels

Math Olympiads for Elementary and Middle Schools (MOEMS)Two divisions (grades 4–6 and grades 7–8)Over 5,000 teams and 150,000 students5 monthly contests (Nov-Mar)

Visit these organizations’ booths in the Exhibit Hall!




















Discrete Math ResourcesMath Circles

A math circle is a meeting of students and teachers in an informal(usually out-of-school) setting.

Map of math circles in the US and Canada (courtesy MSRI)David Patrick (Art of Problem Solving) Discrete Math for Middle School Students NCTM 2008 — Salt Lake 16 / 18


From www.mathcircles.org:

Mathematical Circles are a form of education enrichment and outreachthat bring mathematicians and mathematical scientists into directcontact with pre-college students. . . The goal is to get the studentsexcited about the mathematics, giving them a setting that encouragesthem to become passionate about mathematics.

Don’t take the phrase “mathematicians and mathematical scientists”too seriously: many good math circles are organized by teachers,parents, or college students.

If there’s a math circle near you, encourage your most avid students toattend. Most don’t require a long-term commitment. Some wantteachers to come too!

If there’s not a math circle near you, consider starting one! A greatresource—Circle in a Box—is available from www.mathcircles.org.





















If there’s not a math circle near you, consider starting one! A greatresource—Circle in a Box—is available from www.mathcircles.org.David Patrick (Art of Problem Solving) Discrete Math for Middle School Students NCTM 2008 — Salt Lake 17 / 18

Summary

Discrete math should be an important, useful, and fun part of themiddle-school math curriculumNo “magic wand” or “one size fits all” approachGood way to develop problem-solving skillsLots of resources available

Learn MoreWant to discuss more or see some resources? Visit us online at:

www.artofproblemsolving.comor stop by Booth #2524 in the Exhibit Hall


Summary

Discrete math should be an important, useful, and fun part of themiddle-school math curriculumNo “magic wand” or “one size fits all” approachGood way to develop problem-solving skillsLots of resources available

Learn MoreWant to discuss more or see some resources? Visit us online at:

www.artofproblemsolving.comor stop by Booth #2524 in the Exhibit Hall
