DOCSLIB.ORG

The Four Digits Problem

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Four Digits Problem THE FOUR DIGITS PROBLEM The Four Digits Problem by Andy Pepperdine 2007-01-12 (re-formatted 2010-01-02) based on notes made 50 years ago. This work is released under the Creative Commons Attribution – Share-alike 2.5 license [1]. Make all integers from 0 upwards using the simplest mathematical operators, as defined in some way, and the digits 1, 2, 3 and 4. W. W. Rouse Ball gives this in his book Mathematical Recreations and Essays[2] as a more interesting exercise than the normal Four Fours where all the digits are 4. He states on page 15, that with the digits 1, 2, 3 and 4, and allowing the notation of the denary scale (including decimals), as also algebraic sums, products, and positive integral powers, all numbers from 1 up to 88 can be made. If the use of symbols for square roots and factorials, repeated a finite number of times, is also permitted, then we can get to 264; if negative indices are also permitted, to 276; and if fractional indices are permitted, to 312. NB: I interpret the last comment about “fractional indices” to mean the extraction of roots other than the square root by saying which one to extract. I cannot find any other solution to 277 and 307 otherwise. The 89 values I cannot find up to 1000 are: 313, 331, 391, 407, 419, 421, 437, 439, 443, 446, 454, 455, 457, 461, 467, 493, 557, 559, 586, 587, 614, 617, 633, 653, 659, 661, 662, 667, 772, 773, 779, 787, 788, 791, 807, 813, 815, 817, 823, 831, 833, 850, 851, 853, 857, 859, 861, 867, 871, 874, 877, 879, 881, 883, 885, 886, 887, 893, 897, 906, 907, 908, 911, 913, 914, 915, 917, 919, 921, 922, 923, 927, 929, 931, 932, 933, 935, 941, 942, 946, 947, 949, 951, 953, 956, 979, 983, 986, 989. Examples of each number from 0 upwards are given in the following table. I've tried to keep to the order in which Rouse Ball postulated the use of operators, but also, I've used factorial in preference to the square root sign, and allowed recurring decimals and integral powers in preference to all other powers, factorials and roots. Some interesting alternative methods are noted, too. Five digits Rouse Ball also gives the limits for 5 digits (1, 2, 3, 4 & 5) of 3832 without negative and fractional powers, and of 4282 with. My investigations indicate the same. Digits-1234.odt 1 THE FOUR DIGITS PROBLEM No. Method No. Method 0 14−3−2 33 341−2 1 32−4×1 34 34×2–1 2 123−4 35 342−1 3 2×31−4 36 324×1 4 4×2−3−1 37 2314 5 43−2×1 38 42–3−1 6 431−2 39 42–3×1 7 31−24 40 12×34 8 432−1 41 43–2×1 9 432×1 42 413−2 10 4321 43 43×2– 1 11 42−31 44 4×13 –2 12 4×3×2−1 45 423×1 13 34−21 46 3214 14 21–3−4 47 412×3 15 134−2 48 3×142 16 4×32−1 49 4123 17 3×142 50 13×4−2 18 32−14 51 12×43 19 23–4×1 3 52 2× −4 or 43−12 20 213−4 ·1 21 24–3×1 2 53 3 22 43−21 ·1×·4 23 31–2×4 54 4×132 24 4×123 55 4213 3 25 31–4−2 56 41 ·2 26 234−1 4 27 234×1 57 3× −1 ·2 28 32–4×1 58 31×2−4 29 42−13 59 31×3−4 30 13×24 60 4×312 31 43−12 13 61 −4 or 43 –2−1 32 3×12−4 ·2 Digits-1234.odt 2 THE FOUR DIGITS PROBLEM No. Method No. Method 31×4 88 4×23 –1 62 2 13 4! 4! −31 4 89 or 63 23 ·2 ·2 1 · 3×12 90 64 2341 · 4 13×2 65 91 23×4−1 · 4 92 23×4×1 66 31×24 93 23×41 134 67 94 2×413! 2 3 68 34×2×1 2 95 ·4 or 4!×3!−1−1 69 3×24−1 ·1 70 2×314 96 4×213 71 24×3−1 97 14!–23 72 4×21 –3 98 3!–1!–4!2 73 3241 3×4 99 or 123–4! 21 ·1˙ 2˙ 74 4 ·3 3×4 100 or 4−3×·1−2 75 3×214 ·12 76 2×41–3 4 2 101 3 or ·2−3– 4! 31−·2 ·1 77 ·4 102 34×21 78 13×24 103 3!!×·2−41 79 41×2−3 104 13×4×2 32×1 42 80 or 34−21 105 ·4 ·3·1 81 21×4−3 3×4 106 −2 41 ·1˙ 82 ·2·3 4! 107 −13 2×4 ·2 83 3 ·1 108 4×123 84 14×3×2 4!–1 109 –3! 85 41×23 ·2 2×43 86 43×2×1 110 87 21×43 ·1 Digits-1234.odt 3 THE FOUR DIGITS PROBLEM No. Method No. Method 111 14!−32 31–4 135 ·2 3 112 4× −2 ·1 136 1342 3 1 ·4 3 113 !−4−3 137 ·1˙ or 4 ·2 ·2×·1˙ ·2 4 114 3× −2 4 ·1 2 4 138 ·3 or 3!–1! ·2˙ 115 23×41 ·1˙ 116 4×31 –2 139 142−3 117 3×41 –2 42×1 140 3×4 ·3 118 −2 ·1 141 143−2 119 123−4 142 3!×4!–2×1 24 2 120 143 3×4 −1 ·3−·1 144 12×4×3 121 124−3 145 1423 122 31×4−2 3 146 −4 123 3×42–1 ·1×·2 124 4×32 –1 147 21×34 125 41×32 3 −·4 126 42×3×1 148 ·1 3!× 4!1−2 127 1234 ·2 128 132−4 149 3!!×·241 3×2 129 3×412 150 ·1×·4 41–2 130 or 1243! 31 ·3 151 −4 ·2 3 131 −4 3 ·2×·1˙ ·4 152 ·1 132 134−2 ·2 3 −·4 4! 31−·4 133 ·1˙ or 13 153 ·2 ·2 ·2 3 4 154 4 ·2 ·1×·2 134 ·1 or 32!14 ·3 Digits-1234.odt 4 THE FOUR DIGITS PROBLEM No. Method No. Method 31×2 4 155 177 −3 ·4 ·2×·1˙ 156 ! 4! 132 4 −2 178 ˙ 314 ·13 157 2 31 179 4! ·2˙ ·2 158 −4 or 3!!×·214 ·1˙ 3 3×42 180 31 ·1 159 4 ·2 3×2! 181 1 160 32×41 4 4 4! 161 2×3 −1 or 32!41 182 2 ·13˙ 162 1×2×34 4 4 183 3 163 2×3 1 ·2×·1˙ 4 164 2×3 1 184 23×4!−1 34–1 4−·3 165 185 ·2 ·1×·2 ·2˙ 166 4 186 31×24 1˙ 3 · 4!–3 187 −2 167 3!!×·24!−1 ·1˙ 168 42×31 4! 188 3!!×·1˙ 34 ·2˙ 169 −1 ·2 3×14 189 34×1 ·2˙ 170 ·2 41–3 190 34 ·2 171 1 ·2 191 4!×23−1 4 3 172 3!!×·2˙ 192 4 ×21 ·1˙ 193 142−3 173 1324 43·1˙ 3 3!! 194 174 4! or –21! .2˙ ·1×·2 4 4 3 4!–2 314 195 ·1˙ or −3 175 .1˙ ·2 ·2 ·2 – 4! ·2 176 3 196 −4 ·1 ·13 Digits-1234.odt 5 THE FOUR DIGITS PROBLEM No. Method No. Method 4 24 197 −3 216 3 ·1×·2 ·1˙ 413 217 2134 198 ˙ ·2 218 3!4–12 2 199 14 3 24 219 3 4×32 ·1˙ 200 ·1 413 220 4! ·2 201 3 ·1˙ 2˙ 4! 221 32 41 ·1˙ 202 −3 ·2 24 222 3! 4 ·1˙ 203 3 ·1×·2 4!–2 223 3 ·2 ·1 204 3 4 ·1 224 42!×·31˙ 205 41×23 225 3×412 23 23 206 – 4! 226 −4 ·1 ·1 4 3 4! 2 227 231−4 207 ·2 or −3 ·1˙ 4 23 ·1˙ 228 3!× −2 or –4 ·1 ·1 41 4! 3 208 3 or −2 ·2 ·1˙ 229 231 –4 4 209 213−4 3 4!–1 230 ·2 or 14×3 ·3−·2 210 ·1 ·2 23 211 214−3 231 4! ·1˙ 4!–3 212 2 3! 4! 3 ·1 232 4× – 2 or −2 .1 ·1 24 213 −3 ·1˙ 233 234−1 43 234 234×1 214 −1 ·2 235 2341 43×1 236 2×3!–1!−4 215 ·2 24 237 −3 ·1 Digits-1234.odt 6 THE FOUR DIGITS PROBLEM No. Method No. Method 238 241−3 264 4!×13−2 1 −2 ·2 ·1 3! 239 3 −4 265 ·4 4×3×2 240 3! ·1 266 −4 ·2×·1˙ 241 41 3 −2 1 4! 3! 267 ·2 or 242 243−1 3 4! ·1 ·2˙ 243 243×1 268 2×134 244 2431 23! 269 −1 41 ˙ 245 3 2 ·4 234 246 41×3×2 270 1 ·1 247 ·2 3 4 23! 271 1 248 31×4×2 ·4˙ 4! 2 4! 249 3 272 32 ·1 ·1 3 4! 250 2×41 273 3 ·2−·1˙ 251 3!×42−1 3! 252 21×4×3 274 4 ·2×·1˙ 253 3!×421 4!31 31 275 254 4 −2 ·2 255 2314! 23×4 276 41!×2·3 or 256 432−1 ·1˙ −·2 4!2 277 34 ·1˙ 257 −3 ·1 − 3!!×·4−·1 2 or 258 4312 278 −4! 4 ·1 –3!!−2 259 3! −1 ×·2 3·4˙ 13×4 279 260 ·1˙ 2 ·2 21×4 4!32 280 261 ·3 ·1˙ ·2−3−·1˙ 262 2 4!×·1˙ 3! or 3!!×·1˙4!−2 281 ·4˙ 4! 263 23 282 4!×12–3! ·1 Digits-1234.odt 7 THE FOUR DIGITS PROBLEM No.
Load more

Recommended publications
  • Beginning Algebra Made Useful Chapter 1
    Beginning Algebra Made Useful Chapter 1 Charlene E. Beckmann, Ph.D. Grand Valley State University © 2020 Table of Contents Chapter 1 Getting Ready to Learn Mathematics ...................................................................... 3 1.1 Learning How to Learn Mathematics .......................................................................................... 3 How to Learn Math for Students ...................................................................................................... 3 Good Questions to Ask Yourself ....................................................................................................... 6 Other Helpful Practices ..................................................................................................................... 7 Mathematical Practices .................................................................................................................... 8 1.1 Homework ................................................................................................................................. 9 1.2 Thinking Together and Mathematical Mindset ........................................................................... 9 Video: Mindset .................................................................................................................................. 9 Activity: Good Group Work (Also from Week 1 of Inspirational Math) .......................................... 10 Activity: Mindset Survey ................................................................................................................
    [Show full text]
  • Trick Answers in This Task, Students Analyze a Mock Work Sample to Demonstrate and Explain Their Understanding of the Order of Operations
    Georgia Department of Education Common Core Georgia Performance Standards Framework Fifth Grade Mathematics • Unit 1 Constructing Task: Trick Answers In this task, students analyze a mock work sample to demonstrate and explain their understanding of the order of operations. STANDARDS FOR MATHEMATICAL CONTENT MCC5.OA.1 Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols. STANDARDS FOR MATHEMATICAL PRACTICE SMP 2. Reason abstractly and quantitatively. SMP 3. Construct viable arguments and critique the reasoning of others. SMP 6. Attend to precision. SMP 7. Look for and make use of structure. SMP 8. Look for and express regularity in repeated reasoning. BACKGROUND KNOWLEDGE Students have solved two step word problems using the four operations in third grade and multi step equations in 4th grade. Therefore, the understanding of order or operations within the four operations should have been mastered. At the 5th grade level students are now exploring these four operations within parentheses and brackets. COMMON MISCONCEPTIONS: Students may think that all operations must be done in order from left to right. For example, students who are taught to follow PEMDAS may think that multiplication must be done before division. Use of scientific calculator to verify answers may help students to realize the importance of following the conventional order of operations ESSENTIAL QUESTIONS • Why is it important to follow an order of operations? • How can I effectively critique the reasoning of others? MATERIALS • Trick Answer recording sheet • Accessible manipulatives GROUPING: Partner or individual task MATHEMATICS GRADE 5 UNIT 1: Order of Operations and Whole Numbers Georgia Department of Education Dr.
    [Show full text]
  • Missing-Figure Problems Author(S): W
    Missing-Figure Problems Author(s): W. E. H. Berwick Source: The Mathematical Gazette, Vol. 10, No. 155 (Dec., 1921), pp. 359-362 Published by: The Mathematical Association Stable URL: http://www.jstor.org/stable/3604626 . Accessed: 17/01/2015 13:26 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp . JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. The Mathematical Association is collaborating with JSTOR to digitize, preserve and extend access to The Mathematical Gazette. http://www.jstor.org This content downloaded from 198.150.52.9 on Sat, 17 Jan 2015 13:26:08 PM All use subject to JSTOR Terms and Conditions MISSING-FIGURE PROBLEMS. 359 MISSING-FIGURE PROBLEMS. BY W. E. H. BERWICK. IN a recent number of the MathematicalGazette (March 1920) I proposed the following problem for solution: THE FoUR FOURS. x x x)x x x x x x 4( x4 x x (1) xxx (2) x x 4 x (3) x x x x (4) x xxx (5) x 4 x (6) x x x x (7) x x x x (8) Some of the figures in the long division sum have been replaced by crosses, and it is required to restore them.
    [Show full text]
  • {Download PDF} Take Four Ebook
    TAKE FOUR PDF, EPUB, EBOOK Karen Kingsbury | 368 pages | 22 Jun 2010 | ZONDERVAN | 9780310266273 | English | Grand Rapids, United States Take Four: Jazz - Syracuse Additionally, solutions that repeat operators are marked in italics. Note that numbers with values less than one are not usually written with a leading zero. For example, "0. This is because "0" is a digit, and in this puzzle only the digit "4" can be used. A given number will generally have a few possible solutions; any solution that meets the rules is acceptable. Some variations prefer the "fewest" number of operations, or prefer some operations to others. Others simply prefer "interesting" solutions, i. Certain numbers, such as , are particularly difficult to solve under typical rules. This problem and its generalizations like the five fives and the six sixes problem, both shown below may be solved by a simple algorithm. The basic ingredients are hash tables that map rationals to strings. In these tables, the keys are the numbers being represented by some admissible combination of operators and the chosen digit d , e. There is one table for each number n of occurrences of d. Now there are two ways in which new entries may arise, either as a combination of existing ones through a binary operator, or by applying the factorial or square root operators which does not use additional instances of d. The first case is treated by iterating over all pairs of subexpressions that use a total of n instances of d. Memoization is used to ensure that every hash table is only computed once.
    [Show full text]
  • MATHEMATICS GCSE for Edexcel Teacher’S Resource Jess Laffoley, Nathan Barker, Charmayne Bailey, Cecilia Freer, James Freeman, Kathryn Bigg, Susanne Mallett
    Main intro back cover copy text here Rum- Brighter Thinking quo esequos doloreictus et mo volores am, conse la suntum et voloribus. Cerrore voloreriate pa prae es vendipitia diatia necusam ditia aut perrovitam aut eum et im ius dolut exceris et pro maxi- mintum num quatur aut et landese qua- tem. Sedit et am, eum quiassus ius con none eris ne nobis expliquis dolori ne cus, occaest, nam que exped quuntiatur atur additional back cover copy text here Rum- quo esequos doloreictus et mo volores am, conse la suntum et voloribus. Cerrore voloreriate pa prae es vendipitia diatia necusam ditia aut perrovitam aut eum et im ius dolut exceris et pro maxi- mintum num quatur aut et landese qua- tem. Sedit et am, eum quiassus ius con none eris ne nobis expliquis dolori ne cus, occaest, nam que exped quuntiatur atur reprori odi volores tiunto doluptaquis MATHEMATICS GCSE for Edexcel Teacher’s Resource Jess Laffoley, Nathan Barker, Charmayne Bailey, Cecilia Freer, James Freeman, Kathryn Bigg, Susanne Mallett GCSE Mathematics for Edexcel University Printing House, Cambridge CB2 8BS, United Kingdom Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/ukschools/9781107468146 © Cambridge University Press 2015 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press.
    [Show full text]
  • The Generalization from Four 4'S to Four
    Available online www.jsaer.com Journal of Scientific and Engineering Research, 2018, 5(7):107-112 ISSN: 2394-2630 Research Article CODEN(USA): JSERBR The Generalization from Four 4’s to Four n’s Jimson Huang Southwest Career and Technical Academy, Las Vegas, NV 89113 U.S.A Abstract The four 4’s problem is an interesting mathematics puzzle which attracts wide attention from famous mathematicians, college professors, to high school teachers. In this paper, we generalize the four 4’s problem to the general four n’s, which covers all the existing results in the literature. This research may have applications in data management field. Keywords Four 4’s, mathematics puzzle, four n’s, data transmission Introduction The “Four 4’s” is a famous mathematics puzzle and has been discussed widely in mathematics classes. The goal of four 4’s is to find the simplest mathematical expression for every whole number from 1 to some big number, using only common mathematical symbols and the digit 4 (no other digit is allowed). Most versions of the four 4’s require that each expression have exactly four 4’s. The primary difference for different variations of four 4’s is which mathematical symbols are allowed to be used in their formulas. Essentially all variations allow, at least, addition, subtraction, multiplication, division, and parentheses, as well as concatenation (e.g., "44" is allowed as forty-four). Most also allow the factorial ("!"), exponentiation (e.g. "444"), the decimal point (".") and the square root (" ") operation. Other operations allowed by some variations include overline (an infinitely repeated digit), and the greatest integer ([ ]).
    [Show full text]
  • Amsi Schools Lesson Outline
    AMSI SCHOOLS LESSON OUTLINE LEVEL: Year 6 CONTENT: Problem Solving FOCUS: Patterns In the Classroom PURPOSE • Identify the qualities of a good mathematician • Demonstrate the qualities of a good mathematician • Explain and record thinking using a systematic approach • Identity the different mathematical operations • Explain the order of operations • Use brackets to modify the order of operations • Explain the use of brackets • Record and solve equations that involve the use of brackets and different operations WARM UP Good Mathematician Brief discussion about the qualities of good mathematicians - remind students that everyone is a good mathematician and encourage students to use the strategies that they have suggested 24 Game Show students some examples of the 24 game (Open PowerPoint Four fours.ppt) Students need to use the four numbers to make a total of 24 Challenge – How would you accurately record the solution using symbols? INTRODUCTION For this activity students are challenged to create equations using up to four fours and any mathematical operations to make the numbers from 0 to 100; telling the class that the previous class was only able to fin 25 correct solutions encourages them to perform better and to also work collaboratively as a whole class, building on one another’s solutions to quickly increase the overall list. EXPLICIT TEACHING & Four Fours LEARNING • Show students the Four Fours problem • Using up to four fours and any mathematical operation can you make the numbers from 0 to 100 • Introduce the order of operations
    [Show full text]
  • Beyond the Four Fours.Pdf, This File
    Beyond The Four Fours an investigation by Hugh Darwen and Gerard Joseph In recreational mathematics the “four fours” problem invites the solver to express every natural number from 1 to 100 using just four 4s and the operators +, −, ×, ÷, √, ! (factorial), and exponentiation. Also admitted are the decimal point (·) and dot-above (˙) to indicate a recurring decimal, and parentheses to specify the order of operations. We investigated similar problems for every combination of four nonzero digits (not necessarily distinct), 495 in all. For added interest, we tried to find what we called clean solutions wherever possible. A clean solution is one that avoids redundant expressions of the following kinds: 1. Multiplication or division of a quantity by 1 (x × 1, x ÷ 1) 2. Addition/subtraction of zero to/from a quantity (x ± 0) 3. Explicit expression of a quantity as the 1st power (x1) or 1st root (1√x) of itself 4. Use of 1 as an explicit power or root of itself (1x or x√1) 5. Explicit expression of the square root of a quantity as its 2nd root (2√x) 6. Repetition of a recurring decimal digit (·ṁṁ) For further added interest, for the twelve integral powers (4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81, 100), we sought power-clean solutions. A solution is power-clean if it is cleanly expressed as xy, where x and y are integers and y has the least value when two or more such expressions are available (i.e., for 16, 64, and 81). For such solutions we strengthen our criteria for cleanness by proscribing the following: 7.
    [Show full text]
  • Intendd for Both
    A DOCUMENT RESUME ED 040 874 SE 008 968 AUTHOR Schaaf, WilliamL. TITLE A Bibli6graphy of RecreationalMathematics, Volume INSTITUTION National Council 2. of Teachers ofMathematics, Inc., Washington, D.C. PUB DATE 70 NOTE 20ap. AVAILABLE FROM National Council of Teachers ofMathematics:, 1201 16th St., N.W., Washington, D.C.20036 ($4.00) EDRS PRICE EDRS Price ME-$1.00 HC Not DESCRIPTORS Available fromEDRS. *Annotated Bibliographies,*Literature Guides, Literature Reviews,*Mathematical Enrichment, *Mathematics Education,Reference Books ABSTRACT This book isa partially annotated books, articles bibliography of and periodicalsconcerned with puzzles, tricks, mathematicalgames, amusements, andparadoxes. Volume2 follows original monographwhich has an gone through threeeditions. Thepresent volume not onlybrings theliterature up to material which date but alsoincludes was omitted in Volume1. The book is the professionaland amateur intendd forboth mathematician. Thisguide canserve as a place to lookfor sourcematerials and will engaged in research. be helpful tostudents Many non-technicalreferences the laymaninterested in are included for mathematicsas a hobby. Oneuseful improvementover Volume 1 is that the number ofsubheadings has more than doubled. (FL) been 113, DEPARTMENT 01 KWH.EDUCATION & WELFARE OffICE 01 EDUCATION N- IN'S DOCUMENT HAS BEEN REPRODUCED EXACILY AS RECEIVEDFROM THE CO PERSON OR ORGANIZATION ORIGINATING IT POINTS Of VIEW OR OPINIONS STATED DO NOT NECESSARILY CD REPRESENT OFFICIAL OFFICE OfEDUCATION INt POSITION OR POLICY. C, C) W A BIBLIOGRAPHY OF recreational mathematics volume 2 Vicature- ligifitt.t. confiling of RECREATIONS F DIVERS KIND S7 VIZ. Numerical, 1Afironomical,I f Antomatical, GeometricallHorometrical, Mechanical,i1Cryptographical, i and Statical, Magnetical, [Htlorical. Publifhed to RecreateIngenious Spirits;andto induce them to make fartherlcruciny into tilde( and the like) Suut.tm2.
    [Show full text]
  • 5Th-Math-Unit-1.Pdf
    Georgia Standards of Excellence Curriculum Frameworks Mathematics GSE Fifth Grade Unit 1: Order of Operations & Whole Numbers These materials areMathematics for nonprofit GSE educational Fifth Grade purposes Unit One only. Unit Any 1: Order other of use Operations may constitute and Whole copyright Numbers infringement. Richard Woods, State School Superintendent July 2017 Page 1 of 89 All Rights Reserved Georgia Department of Education Georgia Standards of Excellence Framework GSE Order of Operations and Whole Numbers • Unit 1 Unit 1: Order of Operations and Whole Numbers TABLE OF CONTENTS Overview .............................................................................................................................3 Standards for Mathematical Practice ...................................................................................4 Standards for Mathematical Content ...................................................................................5 Big Ideas ..............................................................................................................................6 Essential Questions ..............................................................................................................6 Concepts and Skills to Maintain ..........................................................................................7 Strategies for Teaching and Learning ..................................................................................8 Selected Terms and Symbols .............................................................................................10
    [Show full text]
  • 1 Star Problems
    1 Star Problems ADD T O O NE -THOUSAND Problem There are exactly three different pairs of positive integers that add to make six. 1 + 5 = 6 2 + 4 = 6 3 + 3 = 6 How many different pairs of positive integers add to make one-thousand? Solution By writing the sums as: 1 + 999, 2 + 998, 3 + 997, ... , 499 + 501, 500 + 500. It is clear that the number of different pairs is five hundred. How many different triplets of positive integers add to make one-thousand? Problem ID: 137 (Dec 2003) Difficulty: 1 Star [ ] ADD T O S IX Problem By using positive integers, how many different ways can you make a sum that is equal to six? For example you could use: 3 + 1 + 1 + 1 = 6 4 + 2 = 6 1 + 2 + 3 = 6 (Consider 4 + 2 to be the same as 2 + 4) Solution Considering the number of 1's used in the sum. 6x1's: 1+1+1+1+1+1 5x1's: None 4x1's: 1+1+1+1+2 3x1's: 1+1+1+3 2x1's: 1+1+2+2 and 1+1+4 1x1: 1+2+3 and 1+5 0x1's: 2+2+2, 2+4 and 3+3 Giving 10 solutions. What if 2+4 is considered to be different to 4+2? Investigate the number of different sums to make all the integers from 1 to 100. Problem ID: 30 (Jan 2001) Difficulty: 1 Star [ ] ANCIENT R IDDLE Problem The end of topic tests in mathematics are looming and you realise that you have spent one too many lunch times surfing the internet and not revising.
    [Show full text]
  • Mathematics in School: Searchable Index and Archive Via JSTOR
    Mathematics in School: Searchable Index and Archive via JSTOR This is a start of what might turn out to be a much larger project. It covers the first 25 Volumes (1971-1997). Its structure is • Column 1 is the author. • Column 2 is a reference: volume. issue. page • Column 3 is the title All MA members have access to all of these via the JSTOR archive which can be found in the Members' Area. To do so, a member will need to log on (see top right). The username is the e-mail registered with the MA and the password was chosen by you when you first signed in. Once in the Members' Area, JSTOR is an option on the left. For the items listed below (and the next 15 or so volumes) select: Mathematics in School. We would welcome comments. If it is useless, say so. For typos, give the reference. I am willing to continue but only if it is worthwhile. Bill Richardson (Editor in Chief) e-mail: [email protected] Last update: 22/7/18 Abas, Jan 20.2.2 Algorithms for the Seven Distinct Frieze Types Abas, Jan 20.3.24 The Seven Frieze Types: Some Classroom Experiences Abbott, Steve 21.2.12 Algebra with Multilink Cubes Abdelnoor, Jason 07.4.4 How Slatyford School Uses SMP Books in the Organisation of Its Mathematics Teaching Achiman, Gad 10.3.2 Rules in Mathematics Adamson, Ann 13.5.2 Old Squares New Faces Adamson, Beryl 07.5.29 Understanding Fibonacci Numbers Ahmed, Afzal 14.2.5 Low Attainers Mathematics Project Albany, Eric 01.5.2 Times Ancient and Modern Albany, Eric 06.3.26 Mathematics 5-11: A Revised and Extended Version of the 'Nuffield Mathematics Teaching Project' Alderman, Robert 06.5.5 Constructions Based on the Triangle Alderman, Robert 07.4.15 A Circle Problem Alfonsín, J.
    [Show full text]