Mathematical Problem Solving for Elementary School Teachers

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  • Mathematical Problem Solving for Elementary

    School Teachers

    Dennis E. White

    April 15, 2013

  • ii

    Copyright

    Copyright c1996, 1997, 1998, 1999, 2000, 2001, 2002, 2003, Dennis White,University of Minnesota. All rights reserved. This document or any portionthereof may not be reproduced without written permission of Dennis White.Any copy of this document or portion thereof must include a copy of this notice.

    Preface

    These notes were written over an ten year period in conjunction with the devel-opment of a mathematics course aimed at elementary education majors. Thecourse had its inception in an ad hoc committee formed to address the Mathe-matical Association of Americas document A Call for Change, which itself wasa response to the National Council of Teachers of Mathematics Curriculum andEvaluation Standards. A Call for Change calls for a restructuring of how weteach and what we teach elementary education students.

    Certain fundamental principles guided the content and pedagogy of thesenotes.

    i. The mathematical content is nontrivial and nonremedial. We assume thestudents have basic manipulative skills. We do not teach remedial skills.We do not teach many topics in a superficial way. The problems are, forthe most part, nontrivial. Some topics are explored to a depth often foundin junior and senior level courses. There is little emphasis placed on drillexercises or memorization.

    ii. The general topics should conform to those described in A Call for Change.Topics include geometry, number theory, algebraic structures, analysis,probability and statistics.

    iii. As also mentioned in A Call for Change, special emphasis is given tothe interconnection of ideas, to the communication of mathematics andto problem solving skills. Material in these notes interconnect in variousways. Many problems emphasize communicating mathematical ideas bothorally and in writing. Many of the problems are open-ended. Some canbe solved using a variety of techniques.

    iv. The course should be given in a non-threatening environment. It is in-tended that these notes be used in a cooperative learning environment.It is also intended that there would be less emphasis on tests and bench-marks. The experience the students using these notes have will be takenback to their own classrooms.

    To the Students

    These notes are substantially different from the mathematics textbooks youmay be familiar with. There are no large bodies of exercises at the end of each

  • iii

    section. There are few drill exercises. There is little repetition. The text isdensely written and requires close and careful reading. Later chapters frequentlyrefer to results, exercises, and ideas from earlier chapters.

    However, these notes are meant to give you a greater understanding of (andmaybe appreciation for) how mathematical problems are really solved. You willoften be led through a series of exercises to an understanding of some fairlydeep mathematical results. It is my belief that most students, with propermathematical skills, can learn some fairly sophisticated mathematics.

    These notes will probably require more effort on your part than perhaps youhave put into other mathematics courses. This is on purpose. I believe thatlearning mathematics takes active participation, including testing hypotheses,constructing examples, forming strategies, and organizing ideas. All these thingsyou must do. The notes cant do them for you; your instructor cant do themfor you.

    Learning mathematics is an active process. It is not possible to learn math-ematics by reading a textbook like a novel. Good mathematics students, fromelementary school to graduate school, read a math book with pencil and paperin hand.

    Mathematics is not a collection of independent topics. It is not Algebra I,Algebra II, Trigonometry, Plane Geometry, etc. All of mathematics isinterconnected in a fundamental way. In these notes, you will find some prob-lems which require methods from several different mathematical areas. Otherproblems have more than one solution, each solution coming from a differentmathematical area. Ideas learned in Chapter 2 will reappear in Chapter 7,Chapter 11 and Chapter 12. Chapter 12 uses ideas from Chapter 2, Chapter 5,Chapter 6, and Chapter 10.

    The topics were chosen because they are related to material that is widelyfound in elementary school curricula. However, these topics are taught at asubstantially deeper level. It is not the purpose of these notes to teach youelementary school mathematics.

    As I mentioned above, it is my belief that most students, with proper mathe-matical skills, can learn the material in these notes. However, students withoutthose prerequisite skills have great difficulty. Those skills include facility athandling fractions, powers, exponents and radicals in both numeric and alge-braic contexts. Students should also understand the basics of analytic geometry:graphing functions, linear and quadratic equations, the quadratic formula, etc.They should also be somewhat familiar with logarithms and with techniques ofcounting.

    Let me conclude with a word about calculators. For the most part, I havenot discouraged the use of a calculator. Some exercises explicitly call for acalculator. However, besides the mathematical skills described in the precedingparagraph, another prerequisite is an understanding of appropriate calculatoruse. A calculator is not a substitute for mathematical common sense. It shouldnot be used to divide 24 by 6, or to decide if 12 is larger or smaller than

    13 .

  • iv

    To the Instructors

    These notes were designed to be used in a problem-solving environment. Ifeel they work best in a cooperative group setting. They are not designed tobe lectured from, and I dont think they work particularly well in that role.Nevertheless, lectures do have a place in this course, if they are short andappropriate.

    You will surely notice the paucity of drill exercises. While some mathe-matical topics do require drill exercises, my concentration in these notes is onproblem-solving skills. If you feel your class needs extra drill exercises in aparticular area, you are welcome to design your own.

    Because there are few exercises, I expect that most classes will do a sizablepercentage of them. I leave it to your judgment which to omit.

    Many of the exercises are not routine. Many have more than one solutionmethod. Consequently, your classroom role is much expanded over a typicallecture-style course.

    A few of the exercises have been starred. The star means two things. First,the exercise is difficult. Second, the exercise is not in the main flow of ideas,and can be omitted.

    The last chapter, Chapter 13, is an alternative to Chapter 8.

    Acknowledgments

    I would like to thank all the instructors and teaching assistants over the yearswho have helped me in the many revisions of these notes. I want to thankespecially Bert Fristedt (who also provided Chapter 8) and Dennis Stanton fortheir many useful suggestions and comments.

  • Contents

    1 Number Sequences 1

    1.1 Recursions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

    1.2 Explicit Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

    1.3 Summing Arithmetic Sequences . . . . . . . . . . . . . . . . . . . 12

    1.4 Summing Geometric Sequences . . . . . . . . . . . . . . . . . . . 17

    1.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

    1.6 Fibonacci Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . 22

    1.7 Tower of Hanoi . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

    1.8 Divisions of a plane . . . . . . . . . . . . . . . . . . . . . . . . . . 29

    2 Counting 31

    2.1 When to Add . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

    2.2 Permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

    2.3 Combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

    2.4 Selections with Repetitions . . . . . . . . . . . . . . . . . . . . . 43

    2.5 Card Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

    3 Catalan Numbers 49

    3.1 Several Counting Problems . . . . . . . . . . . . . . . . . . . . . 49

    3.2 One-to-One Correspondences . . . . . . . . . . . . . . . . . . . . 55

    3.3 The Recursion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

    3.4 The Explicit Formula . . . . . . . . . . . . . . . . . . . . . . . . . 68

    4 Graphs 77

    4.1 Eulerian Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

    4.2 Special Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

    4.3 Planar Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

    4.4 Polyhedra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

    4.5 Tessellations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

    4.6 Torus Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

    4.7 Coloring Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

    4.8 Tournaments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

    v

  • vi CONTENTS

    5 Integers and Rational Numbers 107

    5.1 Primes and Prime Factorization . . . . . . . . . . . . . . . . . . . 107

    5.2 The Euclidean Algorithm and the GCD . . . . . . . . . . . . . . 110

    5.3 Number Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

    5.4 Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

    5.5 Rational Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . 121

    5.6 Countability of the Rational Numbers . . . . . . . . . . . . . . . 125

    6 Modular Arithmetic 127

    6.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

    6.2 Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

    6.3 Solving Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 132

    6.4 Divisibility Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

    6.5 Nim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

    7 Probability and Statistics, I 145

    7.1 Equiprobable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

    7.2 The General Model . . . . . . . . . . . . . . . . . . . . . . . . . . 148

    7.3 Binomial Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

    7.4 Misuse of Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . 152

    7.5 Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

    7.6 Conditional Probabilities . . . . . . . . . . . . . . . . . . . . . . . 156

    8 Vector Geometry 165

    8.1 Review of some plane geometry . . . . . . . . . . . . . . . . . . . 165

    8.2 Parametric representations of lines . . . . . . . . . . . . . . . . . 168

    8.3 Distances and norms . . . . . . . . . . . . . . . . . . . . . . . . . 175

    8.4 Orthogonality and perpendicularity . . . . . . . . . . . . . . . . . 176

    8.5 Three-dimensional space . . . . . . . . . . . . . . . . . . . . . . . 179

    8.6 Pythagorean triples . . . . . . . . . . . . . . . . . . . . . . . . . . 180

    9 Trees 185

    9.1 Counting Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

    9.2 Minimum Spanning Trees . . . . . . . . . . . . . . . . . . . . . . 198

    9.3 Rooted Trees and Forests . . . . . . . . . . . . . . . . . . . . . . 202

    10 Real and Complex Numbers 215

    10.1 Irrational Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . 215

    10.2 Rational Approximations . . . . . . . . . . . . . . . . . . . . . . 218

    10.3 The Intermediate Value Theorem . . . . . . . . . . . . . . . . . . 220

    10.4 Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . 222

    10.5 Zeros of Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . 225

    10.6 Algebraic Extensions and Zeros of Polynomials . . . . . . . . . . 229

    10.7 Infinities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234

    10.8 Constructible Numbers . . . . . . . . . . . . . . . . . . . . . . . . 235

  • CONTENTS vii

    11 Probability and Statistics, II 23911.1 Expectation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23911.2 Central Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . 24411.3 Measures of the Spread . . . . . . . . . . . . . . . . . . . . . . . 24911.4 The Central Limit Theorem . . . . . . . . . . . . . . . . . . . . . 25111.5 Applying the Central Limit Theorem . . . . . . . . . . . . . . . . 25511.6 Odds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259

    12 Finite Fields 26112.1 A Review of Modular Arithmetic . . . . . . . . . . . . . . . . . . 26112.2 A Tournament . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26312.3 A Field with Four Elements . . . . . . . . . . . . . . . . . . . . . 26612.4 Constructing Finite Fields . . . . . . . . . . . . . . . . . . . . . . 26712.5 Tournaments Revisited . . . . . . . . . . . . . . . . . . . . . . . . 269

    13 Areas and Triangles 27113.1 Simple Areas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27113.2 Similar Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27313.3 Pythagorean theorem . . . . . . . . . . . . . . . . . . . . . . . . 27813.4 Pythagorean triples . . . . . . . . . . . . . . . . . . . . . . . . . . 28413.5 Circles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28813.6 Volumes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293

  • List of Figures

    1.1 Triangular numbers . . . . . . . . . . . . . . . . . . . . . . . . . . 131.2 Triangular, square and pentagonal numbers . . . . . . . . . . . . 161.3 Puff pastry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201.4 Three fractal snowflakes . . . . . . . . . . . . . . . . . . . . . . . 221.5 Fractal snowflake . . . . . . . . . . . . . . . . . . . . . . . . . . . 231.6 Domino tilings of 2 1, 2 2, 2 3 and 2 4 rectangles . . . . . 241.7 Construction of f3 . . . . . . . . . . . . . . . . . . . . . . . . . . 251.8 Construction of f4 . . . . . . . . . . . . . . . . . . . . . . . . . . 251.9 Tower of Hanoi . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281.10 Dividing a plane . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

    2.1 Bills block walk . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

    3.1 Triangulations of a quadrilateral . . . . . . . . . . . . . . . . . . 493.2 Triangulations of a pentagon . . . . . . . . . . . . . . . . . . . . 503.3 Triangulation of a hexagon . . . . . . . . . . . . . . . . . . . . . 503.4 Blockwalking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.5 Outlines with three headings . . . . . . . . . . . . . . . . . . . . 533.6 Tableaux of children . . . . . . . . . . . . . . . . . . . . . . . . . 533.7 A 2 8 tableau . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553.8 Another 2 8 tableau . . . . . . . . . . . . . . . . . . . . . . . . 563.9 Hint: count the shaded blocks in each row . . . . . . . . . . . . . 573.10 16 people, one handshake . . . . . . . . . . . . . . . . . . . . . . 583.11 20 people, one handshake . . . . . . . . . . . . . . . . . . . . . . 593.12 40 people, two handshakes . . . . . . . . . . . . . . . . . . . . . . 603....

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