AbacabaDabacaba! by Michael Naylor Western Washington University


 Megan Higgins
 2 years ago
 Views:
Transcription
1 AbcbDbcb! by Michel Nylor Western Wshington University The Abcb structure shows up in n mzing vriety of plces. This rticle explores 10 surprising ides which ll shre this pttern, pth tht will tke us through geometry, number systems, rt, music, poetry, higher dimensions, nd more! Did I sy 10 plces? Actully, this one goes to Ruler Tke look t the mrks on ruler in the spce of one inch. There s big mrk dividing the inch in hlf, two shorter mrks dividing those hlves in hlf, nd so on. It s esy to imgine you could keep dividing the ruler gin nd gin infinitely. If you did, you d hve very interesting geometric object clled frctl. 1 2 If the shortest mrks re length 1, the next length 2, nd so on, the pttern of the mrks is This pttern shows up in ll kinds of weird plces; let s give it nme.
2 2. The Nme Insted of using numbers to describe the length of brnches or mrks on the ruler, let s cll the shortest lengths, the next longest b, then c, nd so on. The pttern then becomes: b c b d b c b... or AbcbDbcb! This word sounds very much like the mgicin s phrse brcdbr, nd indeed there re seemingly mgicl properties bout this pttern. To understnd the pttern little better nd see how to continue it, let s see how this pttern grows. Strt with n. 1. To grow the pttern, dd the next letter in the lphbet nd then repet everything tht hs gone before (which is just the letter in this cse.) The next step, then, is b. 2. b Continue by dding the next letter, c, nd repeting the b. 3. bcb The fourth step dds the letter d nd repets the pttern: bcbdbcb! The next few steps re shown: 4. bcbdbcb 5. bcbdbcbebcbdbcb 6. bcbdbcbebcbdbcbfbcbdbcbebcbdbcb It s fun to see how much you cn sy loud. How long would it tke to sy the word ll the wy to z? This word is the bsis for genie nmes in Mggie nd the Abcb Genies (see In the story, Mggie must cll forth genies by sying these nmes, ll the wy to z!
3 3. The Tree Another wy to represent n Abcb ptterns is with binry brnching tree. In this pttern, one object divides into two, then ech of those divides into two, nd so on, like the brnches on this tree: It is not hrd to imgine tht the brnching nd doubling could be continued infinitely, nd if we could only mgnify our view enough times, we would see the sme ptterns continuing forever nd ever... This sme pttern is followed (in reverse) in plyoff schedule where tems or plyers re pired off with the winner of ech round progressing to the next while the loser is eliminted: A B C D Richie Duncn Mikey Jim Pm Shnnon Dvid Tom Duncn Mikey Pm Tom Duncn Pm Pm If we move from the top to the bottom of the plyoff tree, we ll notice tht the nme closest to the top of the chrt ppers in column A, the next highest nme is in column B, then column A, then C, then bck to A, nd so on. The pttern is bcbdbcb.
4 4. Frctls Abcbdbcb is frctl pttern nd it shows up in mny other frctl ptterns. Frctls, like the ruler nd the tree, hve prts tht re similr to the whole if you zoom in nd look closer, you see the ptterns over nd over. Here re three more fmous frctls. The Koch Curve is formed by replcing the center third of line segment with two edges of n equilterl tringle. The Sierpinski Tringle (or Gsket) is formed by removing the center of tringle nd repeting Cntor Dust is mde by successively removing the center third of line segment. Cn you find bcbdbcb in ech of these? Koch Curve Sierpinski Tringle Cntor Dust
5 5. Number Systems The binry number system is the simplest nd most significnt of ll positionl number systems. In binry, only two digits re used, zeroes nd ones. The vlue of written number is determined by the rrngement of zeros nd ones, with the vlue of ech plce being power of 2 rther thn power of 10 s in our fmilir system. The number 19, for exmple, would be written s follows: 19 (bse 10) = digits of binry # vlue of ech digit power of = The count from 0 to 8 in binry is: 0, 1, 10, 11, 100, 101, 110, 111, 1000 The bcbdbcb pttern is repeted t infinite levels in this simple counting pttern. The strt of one such pttern is shown to the left. Cn you find others?
6 6. Philosophy nd Poetry It s fun to think bout how every decision we mke leds us in new direction, s if our lives re n infinite frctl tree. Here s poem tht reflects those decisions... it hs the structure bcbdbcb. And keep my conscience cler nd bright. I ll do wht I know is right Next time, who knows? I just might! I shouldn t do it, so I thought. I guess I m doomed to live this wy. Now the chnce hs slipped wy Should I do it? Should I not? Mybe on some other dy! It wsn t worth it, I would sy. Now I m full of guilty thoughts But tht s tiny price to py! I went nd did it nywy. Tomorrow I will sty wy. I ll just hope I don t get cught They didn t ctch me yesterdy! Decision Tree by Michel Nylor (College Mth Journl, 32, 3, My 2001.)
7 7. Puzzles nd Legends A populr puzzle clled The Towers of Hnoi sks plyers to move stck of disks one t time from one of three pegs to nother peg. Only the top disc of tower my be moved, nd one my never plce lrger disc on top of smller disc. The object is to trnsfer the entire tower to different peg in the fewest number of moves. A version of this gme using crds is t the end of this rticle. The key to solving this puzzle lies in the bcb pttern. There is legend of the Temple of Abcbx, where monks move golden disks on dimond spindles. When they hve moved ll 26, the universe will come to n end. Should we worry? The Temple of Abcbx is high top of tower of stone blocks. The blocks re rrnged s such: Drw hlf squre, cut long the digonl to form right isosceles tringle, nd then drw the lrgest squre possible inside of it. Continue by plcing the lrgest squre possible inside of ll the right tringles creted, nd repet. When you decide to stop, you will hve mde stircse of blocks similr to those t the temple of Abcbx. As you climb these stirs, the size of the blocks you step on mkes the bcbdbcb pttern. In the stircse t the ctul temple, this process is crried out 26 times, the centrl block corresponding to the letter z. b c b d b c b At the end of this rticle you ll lso find templte to mke popup version of this stircse.
8 8. Hyperspce Nvigting higher dimensions? Don t get lost! Cll the leftright direction, the updown direction b, nd the forwrdbck direction c. Moving will get you from one point to nother you trvel line segment. Moving b will move you round the corners of squre. b Try moving bcb on the cube  did you visit ll of its vertices? c b b Let s dd nother direction, the herethere direction, nd cll it d. To trvel to ll of the vertices of this four dimensionl hypercube, just remember the mgic word: Abcbdbcb! A B C D E Here s 4D nd 5D hypercube. Are you up to the chllenge?
9 9. The Complex Plne AbcbDbcb is written ll over the Mndelbrot set. The MSet is fmous frctl with infinite vriety nd complexity, generted by repeting very simple rules. Mny free computer progrms re vilble to explore this fntstic structures. (These were mde with the freewre progrm Xos). Zooming in on the nose to the fr left revels one bcb pttern s shown here there re mny more you cn find.
10 10. Music Abcbdbcb sounds like it could represent series of notes. Indeed, if bcbdbcb is plyed on pino, the result is beutiful nd hunting. Downlod copy from bcbx.com. There is lso fully orchestrted version vilble for free downlod More Abcb vilble online for FREE Red Mggie nd the Abcb Genies Downlod Clssroom Activities, Worksheets, nd Blckline Msters Downlod Abcb Music Cheers! Abcbx
11 RolyAnnie RolyAnnie is solitire crd gme nmed fter the crd queen of Mississippi riverbots, Roly Annie Keim. To ply, use the crds ce through seven. Mke the odd numbered crds blck nd the even numbered crds red, nd stck them in order so tht when the stck is fce up, the ce is on top. You re llowed to mke two dditionl stcks in this gme, right stck nd left stck. You re llowed to move the top crd of ny stck (on the tble or in your hnd) to the top of ny other stck (on the tble or in your hnd). However, you my move only one crd t time, nd you my never plce higher numbered crd on top of lower numbered crd; you my only plce lower number on higher number (the ce hs vlue of 1). You my ply ny crd on n empty stck. The object of the gme is get ll of the crds into either the left or right stck. The strt of gme is shown below. A A 2 A 2 A AA432 A AA43 A AA43 2 A 3 2 A??? 3 A AA4 A AA4 1. Wht is the minimum number of moves required to win this gme? 2. List t lest the first 20 moves. (You my wish to devise your own nottion.) 3. Describe ny ptterns you found while investigting this gme. Describe strtegy tht will llow you to win every time without mking mistke. 4. You nd friend decide to rce. You split deck of crds nd number the crds in ech hlf from 1 to 26. Plying by the sme rules s before, bout how long would it tke to decide winner?
12 POPUP FRACTAL This eyepopping frctl is big hit with kids (nd grownups, too)! It s esy to crete something complicted nd beutiful using ides of frctls. The following pge contins the pttern for mking this work of rt. Supplies: Copies of the hndout on the next pge Scissors optionl: 9 x 12 construction pper nd glue sticks 1. Hve students fold the pge in hlf long the dotted line. Fold so tht the print is on the outside of the folded pge. 2. Cut long the two hevy line segments tht connect to the crese nd fold the flp upwrds. The corners of the flp will fit netly in the L brckets printed on the pge. 3. Open the pge, reverse the fold in the center of the flp nd close the pge so the flp is now completely (nd netly!) inside the folded pper. 4. Two move hevy lines re now seen to rech the creses in the center of the pper. Cut long these lso, then crese, using the L brckets to help. After folding the flps up, you cn flip the pper over nd fold the flps in the opposite direction. This will mke it esier to reverse the flps nd pop them inside. 5. When you ve popped the two smller flps inside, you should now be redy to mke two more cuts long the hevy lines tht now meet the four creses. Continue in this mnner. Point out tht the first cuts mde one flp, the second cut mde two flps, nd the third mde 4. How mny flps re mde by the finl cut? When it is done, you will hve beutiful 3D popup form. This cn be glued to folded sheet of construction pper to mke lovely piece of rt or n eyepopping greeting crd. Questions for discussion: How is this frctl? How is this selfsimilr? How mny of ech size block re there? Wht other properties cn you investigte? Wht do you notice when you look t your model from different perspectives?
13
Student Book SERIES. Fractions. Name
D Student Book Nme Series D Contents Topic Introducing frctions (pp. ) modelling frctions frctions of collection compring nd ordering frctions frction ingo pply Dte completed / / / / / / / / Topic Types
More informationSeries. Teacher. Numbers
Series B Techer Copyright 2009 3P Lerning. All rights reserved. First edition printed 2009 in Austrli. A ctlogue record for this book is vilble from 3P Lerning Ltd. ISBN 9781921860171 Ownership of
More informationStudent Book SERIES. Patterns and Algebra. Name
E Student Book 3 + 7 5 + 5 Nme Contents Series E Topic Ptterns nd functions (pp. ) identifying nd creting ptterns skip counting completing nd descriing ptterns predicting repeting ptterns predicting growing
More informationMath Circles Finite Automata Question Sheet 3 (Solutions)
Mth Circles Finite Automt Question Sheet 3 (Solutions) Nickols Rollick nrollick@uwterloo.c Novemer 2, 28 Note: These solutions my give you the nswers to ll the prolems, ut they usully won t tell you how
More informationLecture 20. Intro to line integrals. Dan Nichols MATH 233, Spring 2018 University of Massachusetts.
Lecture 2 Intro to line integrls Dn Nichols nichols@mth.umss.edu MATH 233, Spring 218 University of Msschusetts April 12, 218 (2) onservtive vector fields We wnt to determine if F P (x, y), Q(x, y) is
More informationFirst Round Solutions Grades 4, 5, and 6
First Round Solutions Grdes 4, 5, nd 1) There re four bsic rectngles not mde up of smller ones There re three more rectngles mde up of two smller ones ech, two rectngles mde up of three smller ones ech,
More informationThe Math Learning Center PO Box 12929, Salem, Oregon Math Learning Center
Resource Overview Quntile Mesure: Skill or Concept: 300Q Model the concept of ddition for sums to 10. (QT N 36) Model the concept of sutrction using numers less thn or equl to 10. (QT N 37) Write ddition
More informationMake Your Math Super Powered
Mke Your Mth Super Powered: Use Gmes, Chllenges, nd Puzzles Where s the fun? Lern Mth Workshop model by prticipting in one nd explore fun nocost/lowcost gmes nd puzzles tht you cn esily bring into your
More informationMisty. Sudnow Dot Songs
Sudnow Dot Songs isty T The Dot Song is nottionl system tht depicts voiced chords in wy where the nonmusic reder cn find these firly redily. But the Dot Song is not intended be red, not s sight reder
More informationMONOCHRONICLE STRAIGHT
UPDATED 092010 HYDROCARBON Hydrocrbon is ponchostyle cowl in bulkyweight yrn, worked in the round. It ws designed to be s prcticl s it is stylish, with shping tht covers the neck nd shoulders nd the
More informationCHAPTER 2 LITERATURE STUDY
CHAPTER LITERATURE STUDY. Introduction Multipliction involves two bsic opertions: the genertion of the prtil products nd their ccumultion. Therefore, there re two possible wys to speed up the multipliction:
More informationSeven Sisters. Visit for video tutorials
Seven Sisters This imge is from www.quiltstudy.org. Plese visit this website for more informtion on Seven Sisters quilt ptterns. Visit www.blocloc.com for video tutorils 1 The Seven Sisters design cn be
More informationSpiral Tilings with Ccurves
Spirl Tilings with curves Using ombintorics to Augment Trdition hris K. Plmer 19 North Albny Avenue hicgo, Illinois, 0 chris@shdowfolds.com www.shdowfolds.com Abstrct Spirl tilings used by rtisns through
More informationDomination and Independence on Square Chessboard
Engineering nd Technology Journl Vol. 5, Prt, No. 1, 017 A.A. Omrn Deprtment of Mthemtics, College of Eduction for Pure Science, University of bylon, bylon, Irq pure.hmed.omrn@uobby lon.edu.iq Domintion
More informationHow to Build Wealth Like Warren Buffett.
Your FREE gift for ordering How to Build Welth Like Wrren Buffett The video ABCTV clled lively, informtive, nd lot of fun. You cn t help but get swept up by the enthusism of this rewrding film. Don t
More informationAddition Mission: Facts Practice
Addition Mission: Fcts Prctice ++ Tble of Contents Addition Mission: Fcts Prctice Mth Counting Apple Mth Hippity Hop Crunchy Celery Addition Superstr Cherry Mth At the Librry Mth Sld 0 Sunny Summer Mth
More informationBirka B22: threaded in variation
Tblet Weving: 4Hole Ptterns Stringcrfter The chrt, fining your wy roun the pttern, n suggestions for viking style bris for rnks in the Drchenwl Acemy of Defence You will nee: 22 crs 1 repet 88 Thres:
More informationExample. Check that the Jacobian of the transformation to spherical coordinates is
lss, given on Feb 3, 2, for Mth 3, Winter 2 Recll tht the fctor which ppers in chnge of vrible formul when integrting is the Jcobin, which is the determinnt of mtrix of first order prtil derivtives. Exmple.
More informationPatterns and Algebra
Student Book Series D Mthletis Instnt Workooks Copyright Series D Contents Topi Ptterns nd funtions identifying nd reting ptterns skip ounting ompleting nd desriing ptterns numer ptterns in tles growing
More informationSPECIAL EDITION. Spring 2012 Ezine. where crafty is contagious
Spring 2012 Ezine SPECIAL EDITION www.clubchiccircle.com where crfty is contgious shmrock sttionery It is so esy to mke these festive homemde crds! Mke homemde stmp from the end of n pencil erser. With
More informationChallenge! 1 Warmup. 2 Conversation. Language box. a Discuss the pictures with a partner.
Focus Grmmr Vocbulry Strtegy tlking bout wishes nd possibilities second conditionl I wish + simple pst verbs nd definitions moving to new country showing surprise 1 Wrmup Discuss the pictures with prtner.
More informationPolar Coordinates. July 30, 2014
Polr Coordintes July 3, 4 Sometimes it is more helpful to look t point in the xyplne not in terms of how fr it is horizontlly nd verticlly (this would men looking t the Crtesin, or rectngulr, coordintes
More informationSpotted at APA. Top Points this week. Spring f ro. Year 7. Year 8. Year 9. Year 10. Year 11. Student Newsletter ~
Spring 2017 Top Points this week Yer 7 ly l x Mo Ale Yer 8 ict is Le w ed Be n Yer 9 sh er Ele Du ej on e Yer 10 Yer 11 ri em Me Another very busy week in the cdemy hs just gone by s we hd our APA World
More informationSection 16.3 Double Integrals over General Regions
Section 6.3 Double Integrls over Generl egions Not ever region is rectngle In the lst two sections we considered the problem of integrting function of two vribles over rectngle. This sitution however is
More informationSamantha s Strategies page 1 of 2
Unit 1 Module 2 Session 3 Smnth s Strtegies pge 1 of 2 Smnth hs been working with vriety of multiplition strtegies. 1 Write n expression to desribe eh of the sttements Smnth mde. To solve 18 20, I find
More informationCS2204 DIGITAL LOGIC & STATE MACHINE DESIGN fall 2008
CS224 DIGITAL LOGIC & STATE MACHINE DESIGN fll 28 STAND ALONE XILINX PROJECT 2TO MULTIPLEXER. Gols : Lern how to develop stnd lone 2to multiplexer () Xilinx project during which the following re introduced
More informationPatterns and Relationships
Series Techer Ptterns nd Reltionships opyright 009 3P Lerning. All rights reserved. First edition printed 009 in Austrli. A ctlogue record for this ook is ville from 3P Lerning Ltd. ISBN 97819186034
More informationMEASURE THE CHARACTERISTIC CURVES RELEVANT TO AN NPN TRANSISTOR
Electricity Electronics Bipolr Trnsistors MEASURE THE HARATERISTI URVES RELEVANT TO AN NPN TRANSISTOR Mesure the input chrcteristic, i.e. the bse current IB s function of the bse emitter voltge UBE. Mesure
More informationTRUMBULL PUBLIC SCHOOLS
PS TRUMBULL PUBLC SCHOOLS CURRCULUM DEPARTMENT 6254 Min St., Trumbull, CT 06611 (203) 4524336 FAX (203) 4524342 Lind S. Pslov, Ed.D. Director of Curriculum, nstruction, & Assessments emil: pslovl@trumbullps.org
More informationECE 274 Digital Logic Fall 2009 Digital Design
igitl Logic ll igitl esign MW :PM, IL Romn Lysecky, rlysecky@ece.rizon.edu http://www.ece.rizon.edu/~ece hpter : Introduction Slides to ccompny the textbook igitl esign, irst dition, by rnk Vhid, John
More informationSequential Logic (2) Synchronous vs Asynchronous Sequential Circuit. Clock Signal. Synchronous Sequential Circuits. FSM Overview 9/10/12
9//2 Sequentil (2) ENGG5 st Semester, 22 Dr. Hden So Deprtment of Electricl nd Electronic Engineering http://www.eee.hku.hk/~engg5 Snchronous vs Asnchronous Sequentil Circuit This Course snchronous Sequentil
More informationLECTURE 9: QUADRATIC RESIDUES AND THE LAW OF QUADRATIC RECIPROCITY
LECTURE 9: QUADRATIC RESIDUES AND THE LAW OF QUADRATIC RECIPROCITY 1. Bsic roerties of qudrtic residues We now investigte residues with secil roerties of lgebric tye. Definition 1.1. (i) When (, m) 1 nd
More informationTopic 20: Huffman Coding
Topic 0: Huffmn Coding The uthor should gze t Noh, nd... lern, s they did in the Ark, to crowd gret del of mtter into very smll compss. Sydney Smith, dinburgh Review Agend ncoding Compression Huffmn Coding
More informationSECOND EDITION STUDENT BOOK GRADE
SECOND EDITION STUDENT BOOK GRADE 5 Bridges in Mthemtics Second Edition Grde 5 Student Book Volumes 1 & 2 The Bridges in Mthemtics Grde 5 pckge consists of: Bridges in Mthemtics Grde 5 Techers Guide Units
More informationSection 6.1 Law of Sines. Notes. Oblique Triangles  triangles that have no right angles. A c. A is acute. A is obtuse
Setion 6.1 Lw of Sines Notes. Olique Tringles  tringles tht hve no right ngles h is ute h is otuse Lw of Sines  If is tringle with sides,, nd, then sin = sin = sin or sin = sin = sin The miguous se (SS)
More informationArduino for Model Railroaders
Steve Mssikker Arduino for Model Rilroders Ornge Book Protocol 2 Full Description November 28 Tble of contents Dontors Documenttion Kit V.48 Pge 2 I wnt to tke the time to sincerely thnk you for your
More informationSkills Practice Skills Practice for Lesson 4.1
Skills Prctice Skills Prctice for Lesson.1 Nme Dte Tiling Bthroom Wll Simplifying Squre Root Expressions Vocbulry Mtch ech definition to its corresponding term. 1. n expression tht involves root. rdicnd
More information10.4 AREAS AND LENGTHS IN POLAR COORDINATES
65 CHAPTER PARAMETRIC EQUATINS AND PLAR CRDINATES.4 AREAS AND LENGTHS IN PLAR CRDINATES In this section we develop the formul for the re of region whose oundry is given y polr eqution. We need to use the
More informationMAXIMUM FLOWS IN FUZZY NETWORKS WITH FUNNELSHAPED NODES
MAXIMUM FLOWS IN FUZZY NETWORKS WITH FUNNELSHAPED NODES Romn V. Tyshchuk Informtion Systems Deprtment, AMI corportion, Donetsk, Ukrine Emil: rt_science@hotmil.com 1 INTRODUCTION During the considertion
More informationPB735 HD DP. Industrial Line. Automatic punch and bind machine for books and calendars
PB735 HD DP Automtic punch nd bind mchine for books nd clendrs A further step for the utomtion of double loop binding. A clever nd flexible mchine ble to punch nd bind in line up to 9/16. Using the best
More informationBUBBLE BOBBLE 1 o C r t n o s l 2 t e G t g n i a t S r d e t 3 a B s c i Pl y a 4 a G m S e r c e n e 5 in r Sco g 6 ms e It 7 a P s s o w d r
BUBBLE BOBBLE 1 Control s 2 Gettin g Strted 3 Bsc i Ply 4 Gm e Sce r en 5 Scoring 6 Items 7 Psswor d 1 Control s Bsic Controls Move left pssword Jump Blow s Confirm/Puse/Resume Menu nd right/enter nvigtion
More informationECE 274 Digital Logic
ECE  Digitl Logic (Textbook  Required) ECE Digitl Logic Instructor: Romn Lysecky, rlysecky@ece.rizon.edu Office Hours: TBA, ECE F Lecture: MWF :: PM, ILC Course Website: http://www.ece.rizon.edu/~ece/
More informationVector Calculus. 1 Line Integrals
Vector lculus 1 Line Integrls Mss problem. Find the mss M of very thin wire whose liner density function (the mss per unit length) is known. We model the wire by smooth curve between two points P nd Q
More informationKirchhoff s Rules. Kirchhoff s Laws. Kirchhoff s Rules. Kirchhoff s Laws. Practice. Understanding SPH4UW. Kirchhoff s Voltage Rule (KVR):
SPH4UW Kirchhoff s ules Kirchhoff s oltge ule (K): Sum of voltge drops round loop is zero. Kirchhoff s Lws Kirchhoff s Current ule (KC): Current going in equls current coming out. Kirchhoff s ules etween
More informationFitting & User Instructions
Issue 03 Alexnder Universl Furniture Risers Note: These frmes require the ddition of n ttchment. (See pges 4  ) Bse Frme Mrk Bse Frme Mrk Fitting & User Instructions These instructions pply to the rising
More informationGeometric quantities for polar curves
Roerto s Notes on Integrl Clculus Chpter 5: Bsic pplictions of integrtion Section 10 Geometric quntities for polr curves Wht you need to know lredy: How to use integrls to compute res nd lengths of regions
More informationEE Controls Lab #2: Implementing StateTransition Logic on a PLC
Objective: EE 44  Controls Lb #2: Implementing Stternsition Logic on PLC ssuming tht speed is not of essence, PLC's cn be used to implement stte trnsition logic. he dvntge of using PLC over using hrdwre
More informationRegular languages can be expressed as regular expressions.
Regulr lnguges cn e expressed s regulr expressions. A generl nondeterministic finite utomton (GNFA) is kind of NFA such tht: There is unique strt stte nd is unique ccept stte. Every pir of nodes re connected
More informationHow To Play Against Stronger Players
How To Ply Aginst Stronger Plyers Ver.. jcs NOV00 Vol. : Illustrtive Teching Gmes SAKAI Michihru Professionl Dn English Lnguge Go Super Book 00 Americn Go Assocition All rights reserved. Reproduction
More informationSOLVING TRIANGLES USING THE SINE AND COSINE RULES
Mthemtics Revision Guides  Solving Generl Tringles  Sine nd Cosine Rules Pge 1 of 17 M.K. HOME TUITION Mthemtics Revision Guides Level: GCSE Higher Tier SOLVING TRIANGLES USING THE SINE AND COSINE RULES
More informationTHE SONGWRITING CLUB SONGS
THE SOGWRITIG LUB SOGS LYRIS AD UKE HORDS BOOK OE It s time to sing with your friends nd fmily! Tips for plying re included re on the lst pge. You cn listen to the recordings (nd the kroke trcks) nd find
More information9.4. ; 65. A family of curves has polar equations. ; 66. The astronomer Giovanni Cassini ( ) studied the family of curves with polar equations
54 CHAPTER 9 PARAMETRIC EQUATINS AND PLAR CRDINATES 49. r, 5. r sin 3, 5 54 Find the points on the given curve where the tngent line is horizontl or verticl. 5. r 3 cos 5. r e 53. r cos 54. r sin 55. Show
More information& Y Connected resistors, Light emitting diode.
& Y Connected resistors, Light emitting diode. Experiment # 02 Ojectives: To get some hndson experience with the physicl instruments. To investigte the equivlent resistors, nd Y connected resistors, nd
More informationSection 17.2: Line Integrals. 1 Objectives. 2 Assignments. 3 Maple Commands. 1. Compute line integrals in IR 2 and IR Read Section 17.
Section 7.: Line Integrls Objectives. ompute line integrls in IR nd IR 3. Assignments. Red Section 7.. Problems:,5,9,,3,7,,4 3. hllenge: 6,3,37 4. Red Section 7.3 3 Mple ommnds Mple cn ctully evlute line
More informationTriangles and parallelograms of equal area in an ellipse
1 Tringles nd prllelogrms of equl re in n ellipse Roert Buonpstore nd Thoms J Osler Mthemtics Deprtment RownUniversity Glssoro, NJ 0808 USA uonp0@studentsrownedu osler@rownedu Introduction In the pper
More informationASSEMBLY INSTRUCTIONS
ASSEMBLY INSTRUCTIONS Multi Line 6 x8 255x193x203cm / 100 1 /2 x76 x80 PolyTex, Inc. PO Box 458 27725 Dnville Avenue Cstle Rock, MN 55010 We Site: www.polytex.com English  69717 Hoy Greenhouse Service
More informationAlgorithms for Memory Hierarchies Lecture 14
Algorithms for emory Hierrchies Lecture 4 Lecturer: Nodri Sitchinv Scribe: ichel Hmnn Prllelism nd Cche Obliviousness The combintion of prllelism nd cche obliviousness is n ongoing topic of reserch, in
More information1 tray of toffee 1 bar of toffee. 10 In the decimal number, 0 7, the 7 refers to 7 tenths or
Chpter 3 Deciml Numers Do you know wht DECIMAL is? In chpter, we delt with units, s, 0 s nd 00 s. When you tke single unit nd divide it into (or 0 or 00) its, wht we then hve re deciml frctions of whole
More informationMATH 118 PROBLEM SET 6
MATH 118 PROBLEM SET 6 WASEEM LUTFI, GABRIEL MATSON, AND AMY PIRCHER Section 1 #16: Show tht if is qudrtic residue modulo m, nd b 1 (mod m, then b is lso qudrtic residue Then rove tht the roduct of the
More informationGRADE SECOND EDITION HOME CONNECTIONS ANSWER KEY
HOME CONNECTIONS SECOND EDITION ANSWER KEY GRADE 4 Bridges in Mthemtics Second Edition Grde 4 Home Connections Volumes 1 & 2 The Bridges in Mthemtics Grde 4 pckge consists of: Bridges in Mthemtics Grde
More informationDRAWING TABLE SYSTEM
DRWING TBLE SYSTEM DT31, DT61S, & CDTC71 MODEL VIEWS MY NOT REPRESENT EXCT MODEL PURCHSED : : CRETED NME THIS DRWING IS THE SOLE PROPERTY OF DIVERSIFIED WOOD CRFTS. IT MY NOT BE LTERED OR CHNGED WITHOUT
More informationLittle Strummers. Thanksgiving Songs. Guitar Cours! Supplement. for children ages 6"10. for children ages 610
Little Strummers hnksgiving Sgs Guitr Cours! Supplement children ges 6"10 children ges 610 Sgs complied rrnged by r rbe# hnksgiving Supplement Little Strummers. Since th supplement pcket of music, re no
More informationSo Many Possibilities page 1 of 2
Otober Solving Problems Ativities & So Mny Possibilities pge of Use the blnk spe to solve eh problem. Show ll your work inluding numbers, words, or lbeled skethes. Write omplete sentene below your work
More informationUnit 1: Chapter 4 Roots & Powers
Unit 1: Chpter 4 Roots & Powers Big Ides Any number tht cn be written s the frction mm, nn 0, where m nd n re integers, is nn rtionl. Eponents cn be used to represent roots nd reciprocls of rtionl numbers.
More informationHave u Herd? Herd is the Word. join. toolkit. COOL WAYS to TALK ABOUT BOOKS WITH YOUR
Hve u Herd? Herd is the Word join Th to K nks Wel ir from Libr lingto comin ry L n b g brilli up with for The nt nme the Word o Herd f! toolkit COOL WAYS to TALK ABOUT BOOKS WITH YOUR your friends for
More informationc The scaffold pole EL is 8 m long. How far does it extend beyond the line JK?
3 7. 7.2 Trigonometry in three dimensions Questions re trgeted t the grdes indicted The digrm shows the ck of truck used to crry scffold poles. L K G m J F C 0.8 m H E 3 m D 6.5 m Use Pythgors Theorem
More informationSECOND EDITION HOME CONNECTIONS GRADE
SECOND EDITION HOME COECTIONS GRADE 4 Bridges in Mthemtics Second Edition Grde 4 Home Connections Volumes 1 & 2 The Bridges in Mthemtics Grde 4 pckge consists of: Bridges in Mthemtics Grde 4 Techers Guide
More informationCS2204 DIGITAL LOGIC & STATE MACHINE DESIGN SPRING 2005
CS2204 DIGITAL LOGIC & STATE MACHINE DESIGN SPRING 2005 EXPERIMENT 1 FUNDAMENTALS 1. GOALS : Lern how to develop cr lrm digitl circuit during which the following re introduced : CS2204 l fundmentls, nd
More informationarxiv: v1 [cs.cc] 29 Mar 2012
Solving Mhjong Solitire ords with peeking Michiel de Bondt rxiv:1203.6559v1 [cs.cc] 29 Mr 2012 Decemer 22, 2013 Astrct We first prove tht solving Mhjong Solitire ords with peeking is NPcomplete, even if
More informationBoo to You Mystery Quilt FORT WORTH FABRIC STUDIO  DESIGNED BY: LINDSEY WEIGHT
his sew long will lst five weeks. Boo to You Mystery Quilt R WR BR U  GN BY: NY WG will include P downlod ech week so you cn esily print off the instructions for ech block. www.fortworthfbricstudio.blogspot.com
More informationDigital Design. Chapter 1: Introduction
Digitl Design Chpter : Introduction Slides to ccompny the textbook Digitl Design, with RTL Design, VHDL, nd Verilog, 2nd Edition, by, John Wiley nd Sons Publishers, 2. http://www.ddvhid.com Copyright 2
More informationExercise 11. The Sine Wave EXERCISE OBJECTIVE DISCUSSION OUTLINE. Relationship between a rotating phasor and a sine wave DISCUSSION
Exercise 11 The Sine Wve EXERCISE OBJECTIVE When you hve completed this exercise, you will be fmilir with the notion of sine wve nd how it cn be expressed s phsor rotting round the center of circle. You
More informationCS 135: Computer Architecture I. Boolean Algebra. Basic Logic Gates
Bsic Logic Gtes : Computer Architecture I Boolen Algebr Instructor: Prof. Bhgi Nrhri Dept. of Computer Science Course URL: www.ses.gwu.edu/~bhgiweb/cs35/ Digitl Logic Circuits We sw how we cn build the
More informationMETHOD OF LOCATION USING SIGNALS OF UNKNOWN ORIGIN. Inventor: Brian L. Baskin
METHOD OF LOCATION USING SIGNALS OF UNKNOWN ORIGIN Inventor: Brin L. Bskin 1 ABSTRACT The present invention encompsses method of loction comprising: using plurlity of signl trnsceivers to receive one or
More informationThe Discussion of this exercise covers the following points:
Exercise 4 Bttery Chrging Methods EXERCISE OBJECTIVE When you hve completed this exercise, you will be fmilir with the different chrging methods nd chrgecontrol techniques commonly used when chrging NiMI
More informationSECOND EDITION HOME CONNECTIONS ANSWER KEY GRADE
SECOND EDITION HOME CONNECTIONS ANSWER KEY GRADE 5 Bridges in Mthemtics Second Edition Grde 5 Home Connections Volumes & 2 The Bridges in Mthemtics Grde 5 pckge consists of: Bridges in Mthemtics Grde 5
More informationREVIEW, pages
REVIEW, pges 510 515 6.1 1. Point P(10, 4) is on the terminl rm of n ngle u in stndrd position. ) Determine the distnce of P from the origin. The distnce of P from the origin is r. r x 2 y 2 Substitute:
More informationTheme: Don t get mad. Learn mod.
FERURY When 1 is divided by 5, the reminder is. nother wy to sy this is opyright 015 The Ntionl ouncil of Techers of Mthemtics, Inc. www.nctm.org. ll rights reserved. This mteril my not be copied or distributed
More informationSECOND EDITION HOME CONNECTIONS GRADE
SECOND EDITION HOME CONNECTIONS GRADE 5 Bridges in Mthemtics Second Edition Grde 5 Home Connections Volumes & 2 The Bridges in Mthemtics Grde 5 pckge consists of: Bridges in Mthemtics Grde 5 Techers Guide
More informationGeneral Augmented Rook Boards & Product Formulas
Forml Power Series nd Algebric Combintorics Séries Formelles et Combintoire Algébriue Sn Diego, Cliforni 006 Generl Augmented Rook Bords & Product Formuls Brin K Miceli Abstrct There re number of soclled
More informationFP2 POLAR COORDINATES: PAST QUESTIONS
FP POLAR COORDINATES: PAST QUESTIONS. The curve C hs polr eqution r = cosθ, () Sketch the curve C. () (b) Find the polr coordintes of the points where tngents to C re prllel to the initil line. (6) (c)
More informationA Development of EarthingResistanceEstimation Instrument
A Development of ErthingResistnceEstimtion Instrument HITOSHI KIJIMA Abstrct:  Whenever erth construction work is done, the implnted number nd depth of electrodes hve to be estimted in order to obtin
More informationOperation Manual. Addendum. Embroidery Machine. Product Code: 884T13
Emroidery Mchine Opertion Mnul Addendum Product Code: 884T13 Be sure to red this document efore using the mchine. We recommend tht you keep this document nery for future reference. ABOUT THIS MANUAL ABOUT
More informationFrom OffTheShelf to MarketReady New Age Enclosures is your Single Source Solution. Let us quote modifiying our Stock Enclosures to meet your
From OffTheShelf to MarketReady New ge Enclosures is your Single Source Solution. Let us quote modifiying our Stock Enclosures to meet your enduse. visit newageenclosures.com/services or call 8554NENCL
More informationEnergy Harvesting TwoWay Channels With Decoding and Processing Costs
IEEE TRANSACTIONS ON GREEN COMMUNICATIONS AND NETWORKING, VOL., NO., MARCH 07 3 Energy Hrvesting TwoWy Chnnels With Decoding nd Processing Costs Ahmed Arf, Student Member, IEEE, Abdulrhmn Bknin, Student
More informationCAL. NX15 DUODISPLAY QUARTZ
L. NX15 UOISPLY QURTZ l nlogue time disply l igitl time nd clendr l hronogrph l Tchymeter l t recll function l lrm l Illuminting light (Electroluminescent pnel) ENGLISH Illuminting light (TIME/LENR mode
More informationREVIEW QUESTIONS TOPIC 5 TRIGONOMETRY I FLUENCY
TOPIC 5 TRIGONOMETRY I REVIEW QUESTIONS FLUENCY The most urte mesure for the length of the third side in the tringle elow is: A 4.83 m B 23.3 m C 3.94 m D 2330 mm E 4826 mm 2 Wht is the vlue of x in this
More information(1) Primary Trigonometric Ratios (SOH CAH TOA): Given a right triangle OPQ with acute angle, we have the following trig ratios: ADJ
Tringles nd Trigonometry Prepred y: S diyy Hendrikson Nme: Dte: Suppose we were sked to solve the following tringles: Notie tht eh tringle hs missing informtion, whih inludes side lengths nd ngles. When
More informationABB STOTZKONTAKT. ABB ibus EIB Current Module SM/S Intelligent Installation Systems. User Manual SM/S In = 16 A AC Un = 230 V AC
User Mnul ntelligent nstlltion Systems A B 1 2 3 4 5 6 7 8 30 ma 30 ma n = AC Un = 230 V AC 30 ma 9 10 11 12 C ABB STOTZKONTAKT Appliction Softwre Current Vlue Threshold/1 Contents Pge 1 Device Chrcteristics...
More informationPLANNING & LAYOUT BENCH
MODEL VIEWS MY NOT REPRESENT EXCT MODEL PURCHSED : : : LL VLUES ± OVERLL CRETED 1 OF 5 K:\DIVERSIFIED\\CD\.dwg, 11/30/2015 9:58:11 M, DWG To PDF.pc3 TOOLS REQUIRED 1 2 " WRENCH 1 2 " SOCKET ND TORQUE WRENCH
More informationABOUT THIS MANUAL ABOUT THIS MANUAL
ABOUT THIS MANUAL ABOUT THIS MANUAL This mnul provides detils on IQ Designer, which is ville with the upgrde. Mke sure tht the mchine hs een upgrded to the most recent version. When you find this icon
More informationUnderstanding Basic Analog Ideal Op Amps
Appliction Report SLAA068A  April 2000 Understnding Bsic Anlog Idel Op Amps Ron Mncini Mixed Signl Products ABSTRACT This ppliction report develops the equtions for the idel opertionl mplifier (op mp).
More informationDigital Design. Sequential Logic Design  Controllers. Copyright 2007 Frank Vahid
Digitl Design Sequentil Logic Design  Controllers Slides to ccompny the tetook Digitl Design, First Edition, y, John Wiley nd Sons Pulishers, 27. http://www.ddvhid.com Copyright 27 Instructors of courses
More informationChess. DUMmIES 2ND EDITION. by James Eade FOR
Chess FOR DUMmIES ND EDITION by Jmes Ede Chess FOR DUMmIES ND EDITION Chess FOR DUMmIES ND EDITION by Jmes Ede Chess For Dummies, nd Edition Published by Wiley Publishing, Inc. River St. Hoboken, NJ
More informationClick Here to BEGIN: Independent Agent Contracting Link
LOA Guide : Getting Contrcted with AllWell Centene Medicre Advntge This guide is for ll gents who ssign commissions to n gency. You will NOT need to provide ny gency informtion in this contrcting; s ll
More informationmac profile Configuration Guide Adobe Photoshop CS/CC Sawgrass Virtuoso SG400/SG800 Macintosh v
mc profile Mcintosh 10.510.10 Configurtion Guide Adoe Photoshop CS/CC Swgrss Virtuoso SG400/SG800 v20150427 Configurtion Guide  Photoshop CS/CC Swgrss SG400/800 Before proceeding, ensure the correct
More informationDataflow Language Model. DataFlow Models. Applications of Dataflow. Dataflow Languages. Kahn process networks. A Kahn Process (1)
The slides contin revisited mterils from: Peter Mrwedel, TU Dortmund Lothr Thiele, ETH Zurich Frnk Vhid, University of liforni, Riverside Dtflow Lnguge Model Drsticlly different wy of looking t computtion:
More informationProbability and Statistics P(A) Mathletics Instant Workbooks. Copyright
Proility nd Sttistis Student Book  Series K P(A) Mthletis Instnt Workooks Copyright Student Book  Series K Contents Topis Topi  Review of simple proility Topi  Tree digrms Topi  Proility trees Topi
More informationFrom OffTheShelf to MarketReady New Age Enclosures is your Single Source Solution. Let us quote modifiying our Stock Enclosures to meet your
From OffTheShelf to MarketReady New ge Enclosures is your Single Source Solution. Let us quote modifiying our Stock Enclosures to meet your enduse. visit newageenclosures.com/services or call 8554NENCL
More informationDRAWER BASE DRAWING TABLE
MODEL VIEWS MY NOT REPRESENT EXCT MODEL PURCHSED B UPD STEP 1 NME ECN NO. : : : DIMENSION RE IN INCHES 1 OF 10 K:\DIVERSIFIED\\CD\.dwg, 9/25/2018 9:04:08 M, DWG To PDF.pc3 TOOLS REQUIRED 1 2 " WRENCH 1
More information