Resource Guide to the Arkansas Curriculum Framework for Students with Disabilities for Literacy, Mathematics, and Science
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Digit Problems
Chapter 1 Digit problems 1.1 When can you cancel illegitimately and yet get the correct answer? Let ab and bc be 2-digit numbers. When do such illegitimate cancella- tions as ab ab a bc = bc6 = c , 6 a allowing perhaps further simplifications of c ? 16 1 19 1 26 2 49 4 Answer. 64 = 4 , 95 = 5 , 65 = 5 , 98 = 8 . Solution. We may assume a, b, c not all equal. 10a+b a Suppose a, b, c are positive integers 9 such that 10b+c = c . (10a + b)c = a(10b + c), or (9a + b≤)c = 10ab. If any two of a, b, c are equal, then all three are equal. We shall therefore assume a, b, c all distinct. 9ac = b(10a c). If b is not divisible− by 3, then 9 divides 10a c = 9a +(a c). It follows that a = c, a case we need not consider. − − It remains to consider b =3, 6, 9. Rewriting (*) as (9a + b)c = 10ab. If c is divisible by 5, it must be 5, and we have 9a + b = 2ab. The only possibilities are (b, a)=(6, 2), (9, 1), giving distinct (a, b, c)=(1, 9, 5), (2, 6, 5). 102 Digit problems If c is not divisible by 5, then 9a + b is divisible by 5. The only possibilities of distinct (a, b) are (b, a) = (3, 8), (6, 1), (9, 4). Only the latter two yield (a, b, c)=(1, 6, 4), (4, 9, 8). Exercise 1. Find all possibilities of illegitimate cancellations of each of the fol- lowing types, leading to correct results, allowing perhaps further simplifications. -
Numbers 1 to 100
Numbers 1 to 100 PDF generated using the open source mwlib toolkit. See http://code.pediapress.com/ for more information. PDF generated at: Tue, 30 Nov 2010 02:36:24 UTC Contents Articles −1 (number) 1 0 (number) 3 1 (number) 12 2 (number) 17 3 (number) 23 4 (number) 32 5 (number) 42 6 (number) 50 7 (number) 58 8 (number) 73 9 (number) 77 10 (number) 82 11 (number) 88 12 (number) 94 13 (number) 102 14 (number) 107 15 (number) 111 16 (number) 114 17 (number) 118 18 (number) 124 19 (number) 127 20 (number) 132 21 (number) 136 22 (number) 140 23 (number) 144 24 (number) 148 25 (number) 152 26 (number) 155 27 (number) 158 28 (number) 162 29 (number) 165 30 (number) 168 31 (number) 172 32 (number) 175 33 (number) 179 34 (number) 182 35 (number) 185 36 (number) 188 37 (number) 191 38 (number) 193 39 (number) 196 40 (number) 199 41 (number) 204 42 (number) 207 43 (number) 214 44 (number) 217 45 (number) 220 46 (number) 222 47 (number) 225 48 (number) 229 49 (number) 232 50 (number) 235 51 (number) 238 52 (number) 241 53 (number) 243 54 (number) 246 55 (number) 248 56 (number) 251 57 (number) 255 58 (number) 258 59 (number) 260 60 (number) 263 61 (number) 267 62 (number) 270 63 (number) 272 64 (number) 274 66 (number) 277 67 (number) 280 68 (number) 282 69 (number) 284 70 (number) 286 71 (number) 289 72 (number) 292 73 (number) 296 74 (number) 298 75 (number) 301 77 (number) 302 78 (number) 305 79 (number) 307 80 (number) 309 81 (number) 311 82 (number) 313 83 (number) 315 84 (number) 318 85 (number) 320 86 (number) 323 87 (number) 326 88 (number) -
A Perfect Storm: Variations on an Ancient Theme
A perfect storm: variations on an ancient theme Paul Pollack University of Illinois at Urbana-Champaign February 14, 2011 1 of 32 Three types of natural numbers Among simple even numbers, some are superabundant, others are deficient: these two classes are as two extremes opposed one to the other; as for those that occupy the middle point between the two, they are said to be perfect. { Nicomachus (ca. 100 AD) P Let s(n) = djn;d<n d be the sum of the proper divisors of n. Abundant: s(n) > n, e.g., n = 12. Deficient: s(n) < n, e.g., n = 5. Perfect: s(n) = n, e.g., n = 6. 2 of 32 . In the case of those that are found between the too much and the too little, that is in equality, is produced virtue, just measure, propriety, beauty and things of that sort | of which the most exemplary form is that type of number which is called perfect. The superabundant number is . as if an adult animal was formed from too many parts or members, having \ten tongues", as the poet says, and ten mouths, or nine lips, and provided with three lines of teeth; or with a hundred arms, or having too many fingers on one of its hands. The deficient number is . as if an animal lacked members or natural parts . if he does not have a tongue or something like that. 3 of 32 The superabundant number is . as if an adult animal was formed from too many parts or members, having \ten tongues", as the poet says, and ten mouths, or nine lips, and provided with three lines of teeth; or with a hundred arms, or having too many fingers on one of its hands. -
The Fourth Power Ebook
THE FOURTH POWER PDF, EPUB, EBOOK Juan Gimenez | 256 pages | 25 Jan 2017 | Humanoids, Inc | 9781594653018 | English | Los Angeles, United States The Fourth Power PDF Book With the emergence of new networks, these networks where struggling to get an audience. By continuing to use this website, you agree to their use. The Complete Works of Michael de Montaigne ed. Download as PDF Printable version. For other uses, see Fourth Estate disambiguation. Fourth-degree equations , which contain a fourth degree but no higher polynomial are, by the Abel—Ruffini theorem , the highest degree equations having a general solution using radicals. It has eaten up the other three. The First Estate consisted of three hundred clergy. Binary numbers Evil Odious Pernicious. And the entire human-like aliens that are much more intelligent, but are losing a war against humanity trope reads as if it was taken out of a Moebius comic, but given a darker treatment. Nov 06, Juju rated it it was ok. London: J Dodsley. Retrieved 17 July Archived from the original PDF on 2 July Graphemics related Strobogrammatic. How do networks make money? Look up fourth estate in Wiktionary, the free dictionary. However, he could work on establishing a frame of reference for his action sequences better, because I had trouble following most of However, if Fables was a letdown, at least I finished it. Notes and Queries. In old days men had the rack. When Benkler was asked to testify in the United States vs. In a parliamentary debate of Thomas Powys, 1st Baron Lilford , MP , demanded of minister William Pitt, 1st Earl of Chatham that he should not allow powers of regency to "a fourth estate: the queen". -
Recreational Mathematics
Recreational Mathematics Paul Yiu Department of Mathematics Florida Atlantic University Summer 2003 Chapters 1–44 Version 031209 ii Contents 1 Lattice polygons 101 1.1 Pick’s Theorem: area of lattice polygon . ....102 1.2 Counting primitive triangles . ...........103 1.3 The Farey sequence . ..................104 2 Lattice points 109 2.1 Counting interior points of a lattice triangle . ....110 2.2 Lattice points on a circle . ...........111 3 Equilateral triangle in a rectangle 117 3.1 Equilateral triangle inscribed in a rectangle . ....118 3.2 Construction of equilateral triangle inscribed in a rect- angle . .........................119 4 Basic geometric constructions 123 4.1 Geometric mean . ..................124 4.2 Harmonic mean . ..................125 4.3 Equal subdivisions of a segment . ...........126 4.4 The Ford circles . ..................127 5 Greatest common divisor 201 5.1 gcd(a, b) as an integer combination of a and b .....202 5.2 Nonnegative integer combinations of a and b ......203 5.3 Cassini formula for Fibonacci numbers . ....204 5.4 gcd of generalized Fibonacci and Lucas numbers ....205 6 Pythagorean triples 209 6.1 Primitive Pythagorean triples . ...........210 6.2 Primitive Pythagorean triangles with square perimeters 211 iv CONTENTS 6.3 Lewis Carroll’s conjecture on triples of equiareal Pythagorean triangles . ........................212 6.4 Points at integer distances from the sides of a primitive Pythagorean triangle . .................213 6.5 Dissecting a rectangle into Pythagorean triangles . 214 7 The tangrams 225 7.1 The Chinese tangram . .................226 7.2 A British tangram . .................227 7.3 Another British tangram .................228 8 The classical triangle centers 231 8.1 The centroid . -
Maths Squared: 100 Concepts You Should Know Pdf
FREE MATHS SQUARED: 100 CONCEPTS YOU SHOULD KNOW PDF Rachel Thomas,Marianne Freiberger | 256 pages | 16 Jun 2016 | Apple Press | 9781845436452 | English | London, United Kingdom Maths Squared: Concepts You Should Know by Rachel Thomas & Marianne Freiberger - In mathematicsa square number or perfect square is an Maths Squared: 100 Concepts You Should Know that is the square of an integer; [1] in other words, it is the product of some integer with itself. The name square number Maths Squared: 100 Concepts You Should Know from the name of the shape. Hence, a square with side length n has area n 2. In other words, if a square number is represented by n points, the points can be arranged in rows as a square each side of which has the same number of points as the square root of n ; thus, square numbers are a type of figurate numbers other examples being cube numbers and triangular numbers. Square numbers are non-negative. Another way of saying that a non-negative integer is a square number is that its square root is again an integer. A positive integer that has no perfect square divisors except 1 is called square-free. The concept of square can be extended to some other number systems. If rational numbers are included, then a square is the ratio of two square integers, and, conversely, the ratio of two square integers is a square, e. The number m is a square number if and only if one can arrange m points in a square:. The expression for the n th square number is n 2. -
Recreational Mathematics
Recreational Mathematics Paul Yiu Department of Mathematics Florida Atlantic University Summer 2003 Chapters 13–16 Version 0300709 Chapter 1 Digit problems 1 When can you cancel illegitimately and yet get the correct answer? 2 A multiplication problem 3 A division problem 4 The most famous Monthly problem Appendix: Squares with 9 distinct nonzero digits Appendix: Squares with 10 distinct digits Exercise Charles Twigg’s square number puzzle Cross number puzzle xx8xx xxx)xxxxxxxx xxx xxxx xxx xxxx xxxx 402 Digit problems 1.1 When can you cancel illegitimately and yet get the correct answer? Let ab and bc be 2-digit numbers. When do such illegimate cancellations as ab a b a = = bc bc c give correct results in lowest terms? How about a b c a = , b cd d a allowing perhaps further simplifications of d ? 1.1 When can you cancel illegitimately and yet get the correct answer? 403 A Multiplication problem A multiplication of a three-digit number by 2-digit place number has the form ppp × pp pppp pppp p pppp in which all digits involved are prime numbers. Reconstruct the multi- plication. 404 Digit problems 1.2 A division problem Reconstruct the division problem ∗∗∗∗∗ ∗∗) ∗∗∗∗2 ∗ ∗∗ ∗∗∗ ∗∗ ∗∗∗ ∗∗∗ ∗∗ ∗∗ Charles Twigg: If the digit 2 is replaced by 9, the answer is also unique. If the dividend is prefixed by another asterisk, then there are two so- lutions. 1.3 The most popular Monthly problem 405 1.3 The most popular Monthly problem The following problem, E 1111, is said to be the most popular problem in the American Mathematical Monthly. -
Appendix a Useful Notions and Facts
Appendix A Useful notions and facts In this appendix we introduce most of the statistics on permutations and words ap- pearing in this book, as well as several well-known sequences of numbers and certain lattice paths. Also, we state in Table A.7 several classes of restricted permutations together with references and initial values of the corresponding sequences. A.1 Statistics on permutations and words Definition A.1.1. In a permutation π = π1π2 : : : πn, πi is a left-to-right maximum (resp., left-to-right minimum) if πi ≥ πj (resp., πi ≤ πj) for 1 ≤ j ≤ i. Also, πi is a right-to-left maximum (resp., right-to-left minimum) if πi ≥ πj (resp., πi ≤ πj) for i ≤ j ≤ n. Example A.1.2. If π = 3152647 then the left-to-right maxima are 3, 5, 6 and 7, the left-to-right minima are 3 and 1, the right-to-left maximum is 7, and the right-to-left minima are 7, 4, 2 and 1. In Table A.1 we define all permutation statistics appearing in the book. The list of statistics is essentially taken from [259]. Note that many of the statistics defined make sense for words, though not all. For example, the statistic head :i, the position of the smallest letter, is not defined for words as we may have several instances of the smallest letter. The expression π1 appearing in Table A.1 means the permutation beginning with π and ending with a letter that is larger that any letter in π. Remark A.1.3.