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The ah ha moment A. Problem Solving Courses B. The UNC Statewide Mathematics Contest: 7-12th graders C. Undergraduate Research Projects RICHARD GRASSL - UNC

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Problem Solving Courses -CAPSTONE COURSE FOR ELEMENTARY TEACHERS -MA LEVEL FOR INSERVICE TEACHERS

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An empty mind cannot solve problems -Polya Number themes: 1. Arithmetic growth a.Differencing b.Gauss forward and backward sum 2. Geometric growth a.Geometric ratios b.Shift and subtract 3. Greatest common divisors 4. Least common multiple 5. Special sequences of numbers a.Odds and evens b.Squares c.Triangular numbers d.Prime and composite numbers 6. Parity Algebraic themes: 7. Factoring 8. Factor theorem 9. Remainder theorem 10. Rational root theorem 11. Add-in and subtract-out 12. Telescoping or collapsing sums/products 13. Averages Geometric themes 14. Symmetry 15. Properties of diagonals in polygons 16. Pythagorean theorem 17. Congruent triangles Counting themes 18. Binomial coefficients 19. Permutations 20. Compositions 21. Principle of inclusion-exclusion 22. Pigeonhole principle 23. Mutually exclusive and exhaustive partitions of sets

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The Polya Four Step Understand Restate it, do I need definitions, assumptions, what kind of answer should I get? What skills do I need? Strategy List different types of heuristics to use (data collection-picture- formulas etc), create a plan of attack, list tasks, organize… Implement Execute your plan-keep a record to document successes and failures Tie together Restate the problem, doublecheck, search for essence of problem, create extensions Research suggests that a key difference between novice and expert problem solvers is the amount of time devoted to considering different strategies.

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Across: 1.Square of a prime 4.A prime number 5.A square Down: 1.Square of another prime 2.A square 3.A prime number

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Choose two points. What is the probability that the distance between them is an integer?

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How many fractions can you make if m and n are positive integers and the following hold? m n (a) m < n (b) m + n = 575 (c) Each fraction is reduced Start making them: = 5 2 x = 5 x 7 x 19

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How many positive integers n have divisors? n 2 SOLUTION: The number of divisors d(n) satisfies: Now solve: Only 8, 12 will work

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Overview of problems Find positive integers n and a 1, a 2, a 3, …, a n such that a 1 +a 2 +…+a n =1000 and the product a 1 a 2 a 3 …a n is as large as possible. How many rectangles of all sizes are there in a subdivided 4 by 5 rectangle? How many positive integers have their digits in increasing order? Like 347.

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Find positive integers n and a 1, a 2, a 3 …, a n such that a 1 +a 2 +a 3 +…+a n = 1000 and the product a 1 a 2 a 3 …a n is as large as possible. SUM = = = = 10 PRODUCT 2 * 8 = 16 5 * 5 = 25 2 * 4 * 4 = 32 2 * 2 * 3 * 3 = 36 BEST! CONCLUSION Have as many 3s as possible with a few 2s Replace with Never use any: 4s5s6s7s8s9s EXTEND Allow rational parts Allow real numbers

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How many rectangles are there in a subdivided 4 by 5 rectangle? 4 x 5 2 x 1 1 x 1202 x 1153 x 1104 x 15 1 x 2162 x 2123 x 284 x 24 1 x 3122 x 393 x 364 x 33 1 x 482 x 4 63 x 4 44 x x 542 x 5 33 x 5 24 x 51 ( ) + 2( ) + 3( ) + 4( ) = ( )( ) = 150

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( )( )= () () = 6 2 () 5 2 () What do you hope to hear when a student gets to this stage? ah ha

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How many positive integers have their digits in increasing order? Like 347. Start with an easier problem = = 21 PROOF: Just choose 3 of the 9 digits. Continue: = 84 = 9 3 ()

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…back to the original problem. 9 3 ( ) 9 1 ( ) 2 9 ( ) ( ) 9 4 ( ) 9 5 ( ) 9 6 ( ) 9 8 ( ) ( ) = Ah ha moment Just choose any subset of {1, 2, 3, 4, 5, 6, 7, 8, 9} {7, 4, 5, 2}2457

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m+n 2 ( ) n 2 ( ) m 2 () -- = mn -Algebraically -How many man-woman dancing pairs? -How many lines? m n

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Anatomy of a good problem. Interesting and challenging Open-ended (opportunity for extension) A surprise occurs somewhere A discovery can be made-leads to ah ha Solutions involve understanding of distinct mathematical concepts, skills Problem and solution provides connections Various representations allowed

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Which of these numbers are prime? 101, 10101, , , … STRATEGY: Place in a more general setting x x 4 + x x 6 + x 4 + x 2 + 1= x 4 (x 2 +1)+(x 2 +1) = (x 4 +1)(x 2 +1) x 8 + x 6 + x 4 + x 2 + 1– A geometric sum x x x 5 – 1 x x– 1 x + 1 = = (x 4 + x 3 + x 2 + x + 1)(x 4 – x 3 + x 2 – x + 1) Generalize:1001, , , … X 15 – 1 X (x 5 – 1)(x 10 + x 5 + 1) (x – 1)(x 2 + x + 1) =

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The UNC statewide Mathematics Contest 7 th -12 th graders

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Mathematics Contests Eötvos competitions – Hungary, Polya competitions – Stanford, 1950s Santa Clara Contest – Abe Hillman, 1960s University of New Mexico – Hillman, Grassl University of Northern Colorado –

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Goals – Educational Value 1. Offer a unique educational challenge to all interested students grades Recognize and reward talented students for their extraordinary achievements 3. Provide an opportunity for university faculty to cooperatively engage in an educational endeavor involving secondary school teachers, parents, and students 4. Recruit talented mathematics students to major in mathematics and the sciences 5. Draw attention to basic themes in the secondary curriculum that we think are important

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What makes this contest different? All students in grades 7-12 in Colorado are eligible. A student need not be selected or prescreened. All students in grades 7-12 take the same exam. The contest is in two rounds: First round (November) – at school site Final round (February) – at UNC First round is jointly graded by secondary teachers and UNC staff Each round consists of 10 or 11 essay type questions Certain problems are paired. A theme is introduced in the FIRST ROUND and is built on in the FINAL ROUND. A solutions seminar for teachers and parents is offered.

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Examples of paired problems… FIRST ROUND How many rectangles? SECOND ROUND How many rectangles? Express 83 as a difference of 2 squares. Example: 7 = (a) Demonstrate that every odd number 2n + 1 can be expressed as a difference of two squares (b) Which even numbers can be so expressed?

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Some data… studentsSchools:6 to over students - Of the top 25 winners about 19% were women - In 2005, Olivia Bishop = First place - In 2007, Hannah Alpert = First place - 38% of the time First Place was achieved by someone in 8th, 9th or 10th grade

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How many positive integers have their digits in strictly increasing order? One of the contest winners zeroed in on the following very succinct and beautiful solution: Since any subset of {1, 2, 3, 4, 5, 6, 7, 8, 9} except the empty set will correspond to an increasing integer, the answer is Our admonition to be creative echoes what Albert Einstein once implied: We all have a brain. Its what we do with it that matters.

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Where are they now? Rice UT Austin Stanford MIT Cornell U. Michigan Harvard Columbia U. Wisconsin CU Boulder ASU U. Chicago AFA Cal Tech Harvey Mudd Lawrence University Wartburg College UC Davis Majors and PhD programs Mathematics Mechanical Engineering Electrical Engineering Chemical Engineering Aerospace Engineering Computer Science Medical School Law School

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Undergraduate Research Projects 1. Leibnitz Harmonic Triangle 2. For which n is V n, the invertibles in Z n, cyclic? 3. What does [f(g(x))] (n) look like?

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Pascal Triangle HOCKEY STICK THEOREM

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Leibnitz Harmonic Triangle

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For which n is V n cyclic? V 9 is but V 8 is not V 9 = {1, 2, 4, 5, 7, 8} is generated by 2 V 8 = {1, 3, 5, 7} is not cyclic since Lots of references – start with Gallian

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Leibnitz Rule for Differentiating a Product (f g) = f g + fg (f g) = f g + 2 fg + fg (f g) = f g + 3fg + 3fg+ fg What happens with [f(g(x))] (n) ? The n-th derivative of a composite function.

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Look at row sums 1, 2, 5, 15, 52, … BELL NUMBERS h=f(g(x))g(x) Let h = f(g(x)) h 1 = f 1 g 1 h 2 = f 1 g 2 + f 2 g 1 2 h 3 = f 1 g 3 +3f 2 g 1 g 2 +f 3 g 1 3 h 4 = f 1 g 4 +f 2 [4g 1 g 3 +3g 2 2 ]+f 3 [6g 1 2 g 2 ]+f 4 [g 1 ] 4 h 5 = f 1 g 5 + f 2 [5g 1 g 4 +10g 2 g 3 ]+ f 3 [10g 1 2 g 3 +15g 1 g 2 2 ] + f 4 [10g 1 3 g 2 ]+ f 5 [g 1 5 ]

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Stirling numbers of 2 nd kind … =1 =2 =5 =15 =52 … … In h 4 4g 1 g 3 +3g 2 2 In h 5 10g 1 2 g 3 +15g 1 g 2 2

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What did we learn? Teachers need experiences constructing the same mathematics that they will be teaching. We should teach through exploration True problem-solving episodes are a rarity in the teaching of school or collegiate mathematics and we should do everything we can to foster them when they do occur. A childs mind is a fire to be ignited, not a pot to be filled. The curriculum should engage students in some problems that demand extended effort to solve so they develop persistance and a strong self-image.

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... Problem solving is not passive – students construct their own solutions, their own problems. A certain amount of struggle and frustration is natural, expected, desired. Care must be taken not to frustrate students to the point where they might become disillusioned and disinterested – students need to be exposed to learning situations where their problem solving ability may be enhanced. Leave questions open enough that students can extend themselves, the problem, and utilize any technology they may think is helpful and appropriate. Allow students to go out and get the tools that they need to solve the problems.

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…continued. Dont impose your idea of a solution on the students, they may come up with a better way. Be ready to provide structure and leading questions when called upon. Be ready to learn in your classroom. Expose students to a variety of technologies so they can pick and choose which works best for them on each problem they are given to solve.

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A final thought. Teaching to solve problems is education of the will. Solving problems which are not too easy for him, the student learns to persevere through unsuccess, to appreciate small advances, to wait for the essential idea, to concentrate with all his might when it appears. If the student had no opportunity in school to familiarize himself with the varying emotions of the struggle for the solution his mathematical education failed in the most vital point. -George Polya

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You can find this presentation at: Grassl/ It will be available after April 20, 2010

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