Learner Determination of Content: Experiences in Mathematics Education Outside and Apart from the School Curriculum Dr. Harold Don Allen 6150, avenue Bienville Brossard, Quebec, Canada J4Z 1W8 A Gifted Globe: Tenth World Congress on Gifted and Talented Education Toronto, Canada, 9 August 1993 Selected conjectures, problems, and areas of exploration commonly selected for individual, small-group, and class investigation. SQUARES OF DIFFERENCES. Four numbers (non-negative integers) are chosen and written at corners of a square. Pairs of adjacent numbers are subtracted, smaller from larger, and differences are entered at mid-points of corresponding sides of the square. Joined these mid-points yields a new, rotated square. The process is repeated: "0, 0, 0, 0" is reached in a finite, and surprisingly small, number of steps. Call "Type n" a quadruple which reaches "0, 0, 0, 0" in exactly n steps: thus, "3, 4, 3, 4" is Type 2; "5, 3, 7, 5" is Type 3; "2, 3, 5, 9" is Type 7. Students have found quadruples through Type 7 with little difficulty; higher orders call for luck, intuition, or ingenuity: "1, 21, 58, 126" has been reported to us as Type 15; "60, 181, 402, 810" as Type 16. Experimentation with triples, quintuples, sextuples (and corresponding polygonal configurations) leads to perhaps unexpected results. Students have conjectured that there could be a technique for "reversing" the process, extending to a srequence of "outer squares" and therefore to quadruples of any Type. HAILSTONE NUMBERS. The fanciful name for numbers entering into the rising- falling pattern postulated in the "3x + 1" or Collatz conjecture. Choose a number (a positive integer). Necessarily, the number is even or it is odd. If it is even, divide it by 2. If it is odd, multiply it by 3, then add 1. Using your result, repeat the process. The sequence so obtained tends to rise (due to odd terms), then fall dramatically (due to a multiple factor of 2). Five, for example, rises to 16, then falls: 8, 4, 2, 1. Any number which reaches 1 (that is, in the "hailstone" metaphore, comes down to the ground) is a hailstone number. Which numbers are, and which are not, hailstone numbers? Some numbers rise to astonishing heights, some have long paths of ups and downs. A student of mine chose to investigate "73": three walls of a classroom were covered before "1" was reached. "27" also makes for a rewarding exploration. Which numbers are hailstone numbers? Every number that has been tried--but there is no proof that such must be the case. "Plausible arguments" can be offered on both sides of the question. Any variation on the "hailstone" procedure, for example a "5x + 1" rule, can yield quite different results. An ingenious student version noted that numbers greater than 1 are prime or composite: if prime, double, then add 1; if composite, divide by the smallest prime factor, then subtract 1. Progress of such sequences can be graphed instructively. REVERSALS TO PALINDROMES. "Madam, I'm Adam," "Was it a cat I saw?": palindromes exist for all major languages, and children know them well. Correspondingly, a number palindrome has digits which read the same left-to-right or right-to-left. For this activity, take a number, a positive integer of two or more digits. It is a palindrome (like 30504) or it isn't. If it isn't, then reverse it (write its digits in the opposite order, initial zeros being allowed). Add the number and its reversal. The result may very well be a palindrome (643 + 346 = 989). If it is not, repeat the process. You obtain a palindrome, in most instances, in a very few steps. Other numbers take somewhat longer, and a few (such as 196) offer a challenge to those who wish to be first with a conclusive result. Variations? For one thing, why limit ourselves to our customary number base? THE POLE OF A NUMBER. A concept that rewards thoughtful investigation is that of the pole of a number. By number, understand a multi-digit positive integer, say 6283. Consider its digits, 6, 2, 8, and 3. Write the largest number that can be formed from those digits, 8632, and the smallest such number, 2368. From the largest, subtract the smallest: 8632 - 2368 = 6264. Repeat the process: 6642 - 2466 = 4176. Again: 7641 - 1467 = 6174. Satisfy yourself that 6174 now will endlessly recur. For a four-digit number (digits not all being the same), this inevitable result, 6174, is called the pole of the number. Is there a pole for five-digit numbers, six-digit numbers? Might there be some instructive way to alter the rules? A RACE BETWEEN SETS. Polya defined a "number of the even type" (not necessarily what we think of as an even number) as a number (positive integer greater than 1) which, when written as a product of prime factors, has an even number of such factors. Thus, 441 (7x7x3x3, 4 prime factors) and 1500 (5x5x5x3x2x2, 6 prime factors) are numbers of the even type. A "number of the odd type," correspond- ingly, when written as a product of prime factors, has an odd number of such factors. Thus, 48 (3x2x2x2x2, 5 prime factors), 127 (127, 1 prime factor), and 51975 (11x7x5x5x3x3x3, 7 prime factors) are numbers of the odd type. Note that 2, 3, 5, 7, and 8 are numbers of the odd type, while 4, 6, 9, and 10 are numbers of the even type. Through 10, numbers of the odd type lead numbers of the even type 5 to 4. Polya's 1919 conjecture: starting with 2, counting to any number, however high, "odd type" numbers always exceed "even type" numbers. Through 96, "odds" lead "evens" only by 48 to 47 (but 97, 98, 99 all are "odd type"), so the race can be close, and instructive to follow. "Intuition" somehow favours "odds" as natural winners: no doubt to the delight of Polya, whose love was heuristics, the conjecture has been demonstrated to be false. GEOBOARD SOPHISTICATION. Polygons having lattice-point vertices are what we usually investigate on the geoboard, but we tend to back off when the going gets tough--and the challenge worthwhile. Limit oneself, initially, to the 3x3 geoboard, nine points (or nails) in the usual square array. Define polygon--to rule out non-simple configurations. Define congruency (rule out look-alikes obtainable by translation, rotation, reflection). List obtainable polygons: the 8 incongruent triangles, 16 quadrilaterals, other polygons. Define units of length, area. Classify the figures by area, perimeter. Extend to 4x4 and larger grids. Extend to grids other than rectangular. Reconsider your defini- tion of polygon, your criteria for classification. STRUCTURE IN MATHEMATICS. A rich source of important mathematical insights is any finite system whose properties parallel those which we associate with numbers and number operations. I like "braids," exotic combinatoric objects which are easy to draw, instructive to list, and straightforward to combine under a simple, binary operation. "Braids" may turn up in college as a one- term exercise in group theory--but they can be enjoyed at a much earlier age. Three-line braids, the simplest useful form, are drawn by placing three dots in a row above, three below, and making one-to-one linear connections between. The connections identify the braid. Six permutations make for six different three-line braids, and a combining operation yields the 6x6 table of a closed, noncommutative system. Systematic investigation yields the 24 corresponding four-line braids. Identity elements stand out, but inverses call for some thought. The order of elements is a worthwhile concept. Other "hands-on" structures include polygonal isometries and clock arithmetics. POLYOMINO SOPHISTICATION. The usual polyomino is a plane geometric shape compounded of congruent squares compounded by linking along wholly shared sides. Resulting shades are classified by numbers of squares. Polyominoes are deemed different if one cannot be obtained from another by translation, rotation or reflection (essentially, "turning over"). Five different 4-square polyominoes, 12 different 5-square polyominoes, and 35 different 6-square polyominoes exist, and complete sets often can be sketched on the chalkboard in a single session. Quadrille paper suffices in the planning stage. Pieces can be cut out of heavier stock to permit their properties to be investigated. A double set of the 4-squares (10 pieces) will make an 8x5 rectangle, a relatively simple task. One set of 5-squares will make a 10x6 rectangle, or an 8x8 square with a 2x2 centre "hole." Much space is devoted to polyominoes in the literature of recreational mathematics. Instructive variations on the multiple squares are multiple triangles (equilateral) aand multiple hexagons (regular); also multiples of the isosceles right triangle. A logical step into another dimension gives "polycubes," multiple cubes sharing a common face. Classical puzzles exist of this type. RESIDUE FIGURES AND PATTERN EXPLORATION. Patterns of straight lines (which are tangent lines enveloping mathematical curves) yield an attractive union of number sequences and geometric representations. The student begins with a circle, already divided by n equally spaced, numbered points, 1 to n. Attract- ive results are obtained with n = 60, 72, or 96. The student chooses a multi- plier, an integer between 2 and n - 1. The multiplier determines the curve. The number of points determines the resolution, the detail. Numbered points are joined, in pairs. If the multiplier is 2, then 1 is joined to 2x1, or 2; 2 is joined to 2x2, or 4; 3 is joined to 3x2, or 6; and so on. When the obtained product exceeds n, one subtracts n + l as often as is necessary to obtain a result between 1 and n: thus, with n = 72, 36 is joined to 72, but 37 is joined to 74 - 73, or 1. The completed pattern consists of n segments, which (for a given multiplier, m) envelope a characteristic curve. A Valentine heart, a cardioid, results for m = 2; a kidney-shaped nephroid for n = 3; a three-leaf clover (curve of three cusps) for n = 4. Rules can be varied, and the results can be attractive. This becomes "curve stitching" when theory is combined with craftsmanship. Place black velvet on soft pine. Lay the completed diagram on the velvet. Hammer brass finishing nails through each of the n points. Wind coloured cord around the nails, one continuous path, along a route determined by the lines. Remove and dicard the drawing. The stitched curve, with a single knot, now should be ready for framing. Small values of n such as 12, 16, or 18 will not define the curve as attractively, but may yield interesting patterns for colouring of regions. In every instance, n should be so selected that n + 1, the number subtracted, is prime. NUMBER TYPES, DIVISORS, AND SOME RELATED CONJECTURES. Incredibly challenging conjectures can be expressed in extremely simple terms in some areas of number theory, and intuitive exploration has much to commend it. We restrict our- selves to positive integers. Prime numbers have exactly two divisors, 1 and themselves: 2, 3, 5, 7, 11, ..., accordingly, are prime. 1, the unit of this system, is unique. Other numbers have more than two divisors, and are termed composite: 4, 6, 8, 9, 10, ..., are composite. Pairs of consecutive odd integers which both are prime are called twin primes: 11, 13; 59, 61; 71, 73 are twin primes. Is there a largest, therefore last, prime number? The Greeks asked this ... and answered it. Is there a last pair of twin primes? We don't know. Goldbach asked: Can every even number, starting with 4, be written in at least one way as the sum of two primes? We don't know. De Bouvelles noted: Take any multiple of 6. Add 1 to it; subtract 1 from it. The pair of numbers that you get includes at least one prime. Thus, 7x6 = 42, a mul- tiple of 6, and 41 and 43 both are prime. Is De Bouvelles' conjecture true? How would you feel if you showed a three century old conjecture to be false? Divisors of a number which are less than the number itself are said to be proper divisors. Early mathematicians, who were heavily into lucky and un- lucky numbers which implied numerology, spoke of deficient, abundant, and perfect numbers, according to whether proper divisors summed to less than, more than, or equal to the number: 6, 28, and 496 were among the first recog- nized perfect numbers. Prime numbers can be listed readily by an elementary but clever approach, the Sieve of Eratosthenes. As the sequence develops, remarkable properties are likely to be observed. Slight variation in the sifting process will produce not primes but quite possibly an equally interest- ing set. OTHER PRODUCTIVE AREAS. Random walks, intuitive probability, lottery simula- tion, St. Petersburg paradox, dice-based chaos games. Recreational topology, including classic problems, network tracing, map colouring. Rational and ir- rational numbers, decimal representations, repetend lengths, other bases. Prime-rich expressions, representation on Ulam's "square spiral." Construction of skeletal polyhedra, from first principles. Solitaire and competitive math- related games: Sprouts, Hex, Conway's Life. Estimations and other group com- petitions. Harold Don Allen, Ed.D., F.C.C.T., Toronto, Canada, 9 August 1993.