Pentominoes. Pentominoes. Bruce Baguley Cascade Math Systems, LLC. The pentominoes are a simple-looking set of objects through which some powerful

Size: px
Start display at page:

Download "Pentominoes. Pentominoes. Bruce Baguley Cascade Math Systems, LLC. The pentominoes are a simple-looking set of objects through which some powerful"

Transcription

1 Pentominoes Bruce Bguley Cscde Mth Systems, LLC Astrct. Pentominoes nd their reltives the polyominoes, polycues, nd polyhypercues will e used to explore nd pply vrious importnt mthemticl concepts. In this pper, the definition of the pentominoes will e exmined. It will e proved tht there re exctly 12 pentominoes. The pentominoes will then e represented s mtrices with the mnipultions of the pentominoes represented s mtrix ddition nd multipliction. The concept of pentomino s mod numers will e introduced nd, in n ppliction of modulr rithmetic, will e used to prove the existence nd uniqueness of pentominoes nd figures mde with pentominoes s well s similr fcts out polyominoes, polycues, nd polyhypercues. The pentominoes re simple-looking set of ojects through which some powerful mthemticl ides cn e introduced, investigted, nd pplied. The pentominoes cn e expressed in n nlytic geometric setting where they re chnged into vectors nd mtrices with integrl entries nd mnipulted s numers. The questions out pentominoes cn e trnsformed into equivlent questions in different mthemticl setting from which some interesting properties of the pentominoes cn e proved. This is similr to the ides in lgeric topology where questions out spheres nd tori cn e nswered y looking t the lgeric structure of pths on their surfces nd converting the topologicl questions into lgeric ones. With pentominoes nd their reltives, the ojects re trnsformed into mtrices nd vectors to which liner lger nd numer theory re pplied. The pentominoes re puzzle tht hs een used y techers to introduce students to importnt mth concepts such s symmetry, re, nd perimeter. Pentominoes re suggested for use y techers on pge 99 of the NCTM Principles nd Stndrds, in the Geometry Stndrd of the Pre-K-2 section. They pper s well in vrious NCTM rticles, such s The Pentomino Squre Prolem in Mthemtics Teching in the Middle School, Mrch 2003 p. 355 nd the 1

2 NCTM Illumintions Exploring Cues Activity Sheet. In these rticles, the following questions re investigted: 1. How mny pentominoes re there? 2. Cn certin figure e mde with pentominoes? 3. Cn the pentomino e folded to mke n open ox? This pper will look t the first two questions ut will go into them in much greter depth using more dvnced mth, nmely liner lger nd modulr rithmetic, to prove some surprising results. First there must e definition of pentomino. The ove ctivities give muddled or incorrect definitions. The usul definition, such s the one given in the NCTM s Principles nd Stndrds is incorrect nd yields n infinite numer of pentominoes. On p.99, it is stted, For exmple, second-grde techer might instruct the clss to find ll the different wys to put five squres together so tht one edge of ech squre coincides with n edge of t lest one other squre (see fig. 4.15). By tht definition, the figure t the left is pentomino ecuse one edge of ech squre coincides with n edge of t lest one other squre. Definitions must e mde crefully so tht ll the squres in the figure re connected. A etter definition would e tht polyomino is plne figure mde of squres such tht two different squres cn touch only on sides which coincide nd for every two squres in the polyomino, there is pth tht goes through djcent squres from the first squre chosen to the lst. The figure t the right shows pth from one squre to nother in lrge polyomino. 2

3 Students could e sked for different tests tht would show this connectedness. Do they hve to find pth etween ny two squres in figure to show tht it is polyomino or cn they find simpler test tht requires fewer pths? The numer of such pths etween ny two squres in polyomino is the hndshke prolem nd requires n(n-1)/2 checks to see if it is true for polyomino with n squres. If one squre is connected y pth to every other squre, only n-1 checks re required. This is sufficient ecuse if there is pth from squre to squre nd pth from squre to squre c, then there is pth from to c going through. Students could e sked to find other tests. For exmple, color squre in polyomino nd color ll the squres in the polyomino tht touch the colored squre. Color ll the squres tht touch colored squre. Continue coloring squres in this mnner in the polyomino s shown elow. If ll the squres in the figure re eventully colored then it is polyomino. This test would require t most n-1 steps in coloring. Students could e sked for proof of this ssertion. The question, Are there exctly 12 pentominoes? cn now e nswered. All the pentominoes cn e found y looking t the tetrminoes, tht is, polyominoes with 4 squres, dding one squre to ech of them, nd throwing out the duplictes. In generl, the n+1-ominoes cn e found y looking t the n-ominoes, dding one squre to ech of them in every possile wy, nd throwing out the duplictes. How is this shown? Tke n n+1-omino. If there is squre in tht n+1-omino tht cn e removed nd the remining n squres re still connected, tht is, is n n-omino, then it hs een shown tht the n+1-omino cn e mde y dding one 3

4 squre to n n-omino. Tke ll the pths from one squre in the n+1-omino to nother squre in the n+1-omino which do not go through the sme squre twice. Since there re finite numer of squres in polyomino, there is longest such pth. The end squres cn e removed with the remining squres of the originl n+1-omino still eing connected, tht is the remining squres form n n-omino for the following reson. If the remining squres did rek into two disconnected pieces, then the longest pth would e in one of the pieces. But there would e longer pth in the originl n+1-omino y connecting the end point tht we removed to squre in the disconnected piece not contining the pth. This is contrdiction, ecuse the pth originlly choosen ws the longest pth in the n+1-omino nd the constructed pth is longer. Thus, the n+1-omino cn e formed from n n-omino y dding one squre. The pth in the polyomino t the right is the longest possile nd oth its endpoints re removle. There re now specific numer of pentominoes to check y dding one squre to ll of the tetrminoes or 4-ominoes nd throwing out duplictes. This process is itertive, there is only one monomino or 1-omino. Generte the one domino or 2-omino from the 1-omino; the two 3-ominoes from the 2-omino; the five 4-ominoes from the two 3-ominoes; nd finlly the twelve pentominoes from the five 4-ominoes. The prolem comes down to eliminting the duplictes. Here re ll the wys to dd squre to the domino or 2-omino. 4

5 Students could show ll these nd then decide which re the sme. Hve the students explin why they think tht two of the figures re the sme y hving them show how they would move one figure to the other. This cn led into discussion of rigid motions in the plne, tht is, rottion nd reflection s well s trnsltion. The students cn continue with this process to otin the 12 pentominoes. Some students might wnt to continue nd find the 35 hexominoes. Here re the 12 pentominoes nd their common letter nmes: T U V W X Y Z F L I P N If liner lger nd numer theory re to e used with the pentominoes, then somehow numers must e ttched to ech pentomino. Let the sides of the squres in the pentominoes e length 1. Look t coordinte plne nd tke the grid of ll points with integrl coordintes. If you plce pentomino on the plne so tht the center of ech of its squres is on grid point, then the pentomino piece could e written s mtrix consisting of 5 ordered pirs. Consider the piece in the figure to the right. This is the F pentomino. You cn consider the piece in numer of wys. First, it is the five squres outlined y the hevy lck line. Second, it is the ( 0, 5) ( 1, 5) ( 2, 5) ( 3, 5) ( 4, 5) ( 5, 5) ( 0, 4) ( 1, 4) ( 2, 4) ( 3, 4) ( 4, 4) ( 5, 4) ( 0, 3) ( 1, 3) ( 2, 3) ( 3, 3) ( 4, 3) ( 5, 3) ( 0, 2) ( 1, 2) ( 2, 2) ( 3, 2) ( 4, 2) ( 5, 2) ( 0, 1) ( 1, 1) ( 2, 1) ( 3, 1) ( 4, 1) ( 5, 1) figure represented y the hevy lue lines. Third, it is the mtrix shown on the left. If we ( 0, 0) ( 1, 0) ( 2, 0) ( 3, 0) ( 4, 0) ( 5, 0) move the piece, we would chnge the mtrix representing the piece, ut we would chnge the mtrix in wy tht ws representle s mtrix opertions. 5

6 For exmple, dding the mtrix on the left to pentomino mtrix would move the pentomino right squres nd up squres. Multiplying pentomino mtrix on the right y these mtrices does the following: ] Do not move the pentomino ] Rotte clockwise Rotte clockwise ] 0 or Reflect through the origin. 0 1 Rotte clockwise ] 0 or Rotte counter-clockwise ] Reflect through the y-xis ] Reflect through the x-xis ] Reflect through the line y = x ] Reflect through the line y = -x. It is the ojective to find some numer or numers tht distinguish ech pentomino. It is not t ll ovious tht the following mtrices ll represent the sme pentomino, the F pentomino: Even shuffling the rows mkes it difficult to see tht the first nd third mtrices contin the sme points. It is importnt to mke it more ovious. Adding the columns to get single vector will give the sme result even if the rows re shuffled. How will this ffect the other mtrices representing the F pentomino? If the pentomino is trnslted over to the right nd up we hve the following: ( 10, 11 ) + ] = ( 10+5, 11+5 ) 6

7 Trnslting the F pentomino over nd up gives vector tht differs from the originl vector y multiple of 5 in oth coordintes. If the vector mod 5 is tken, then the vector of ny trnsltion mod 5 stys the sme. This is clled mod numer for the prticulr pentomino. For the exmple of the F pentmino ove with vector (10,11), the mod numer is (0,1) since (10,11) = (5*2+0, 5*2+1). Any trnslte of this prticulr plcing of the F pentomino would hve mod numer of (0,1). The lck dots on the grid to the right show ll the vector sums of trnsltes tht this prticulr orienttion of the F pentomino could e. The smll red squre in the first qudrnt of the grid shows ll the possile loctions for mod numers of ny pentomino. The red dot is mod numer for this prticulr orienttion of the F pentomino. How do reflections nd rottions ffect the mod numer? Since multiplying vector y mtrix is liner, the mod numer is multiplied y the mtrix. Since we re only deling with interchnging columns nd multiplying y 1 or -1, the mod numer for the new position of the F pentomino will e the mod numer of the originl position multiplied y the mtrix corresponding to the reflection or rottion. Why re the mod numers of ny use in working with the pentominos. The only possile mod numers tht piece cn hve re otined y tking the mod numers for ny orienttion of the piece nd then multiplying those mod numers y the mtrices for the eight reflections nd rottions. Two pentominoes cn e the sme pentomino only if they hve the sme mod 7

8 numers. This is not crucil with the pentominoes ecuse visul inspection cn quickly determine if two pentominoes re the sme except for rottions nd reflections. However when deling with something like polyhypercues, sy with 10 cues in 6 dimensions, then the mod numers will e vectors with 6 coordintes mod 10. There might e some overlp of mod numers etween pieces, ut the mod numers considerly reduce the numer of other polyhypercues needed to e checked for dupliction s well s indicting definite procedure to follow for the check, tht is, orient so tht the mod numers re equl, dictionry order, nd check for equlity. A computer could e used to go rpidly through the possiilities. A second use which hs mny more pplictions with the pentominoes would e showing whether shpe is impossile to mke with the pentominoes. Since ny shpe formed with pentominoes consists of multiple of 5 squres, the mod numer for this shpe is defined nd is equl to the sum of the mod numers of the pieces. Tke the 6 y 10 rectngle, which cn e formed with the 12 pentominoes. Let the pentominoes e plced on the grid. If the sum of the mod numers of the pentominoes does not equl (0,0), the mod numer of the rectngle, the pentominoes in those orienttions cn not form the rectngle y trnsltion without rottion or reflection. It is difficult to stop from physiclly rotting or turning over piece when working with the pentominoes, ut the mod numers would e very useful when solving for shpe using computer progrm. On the remining pges, the mod numers for the P pentomino re shown. One position of the P pentomino is shown in the first qudrnt long with the seven other mod numers tht re otined y reflections nd rottions. The mod numer clsses for the different pentominoes re then shown on the following pge. The pper finishes with severl exmples using mod numers to construct shpes with pentominoes nd prove the impossiility of certin constructions with pentominoes nd dominoes. 8

9 ] = ] = Pentominoes () = (1 3) () = () (-5, 5) (-4, 5) (-3, 5) (-2, 5) (-1, 5) ( 0, 5) ( 1, 5) ( 2, 5) ( 3, 5) ( 4, 5) ( 5, 5) (-5, 4) (-4, 4) (-3, 4) (-2, 4) (-1, 4) ( 0, 4) ( 1, 4) ( 2, 4) ( 3, 4) ( 4, 4) ( 5, 4) (-5, 3) (-4, 3) (-3, 3) (-2, 3) (-1, 3) ( 0, 3) ( 1, 3) ( 2, 3) ( 3, 3) ( 4, 3) ( 5, 3) (-5, 2) (-4, 2) (-3, 2) (-2, 2) (-1, 2) ( 0, 2) ( 1, 2) ( 2, 2) ( 3, 2) ( 4, 2) ( 5, 2) (-5, 1) (-4, 1) (-3, 1) (-2, 1) (-1, 1) ( 0, 1) ( 1, 1) ( 2, 1) ( 3, 1) ( 4, 1) ( 5, 1) (-5, 0) (-4, 0) (-3, 0) (-2, 0) (-1, 0) ( 0, 0) ( 1, 0) ( 2, 0) ( 3, 0) ( 4, 0) ( 5, 0) (-5,-1) (-4,-1) (-3,-1) (-2,-1) (-1,-1) ( 0,-1) ( 1,-1) ( 2,-1) ( 3,-1) ( 4,-1) ( 5,-1) (-5,-2) (-4,-2) (-3,-2) (-2,-2) (-1,-2) ( 0,-2) ( 1,-2) ( 2,-2) ( 3,-2) ( 4,-2) ( 5,-2) (-5,-3) (-4,-3) (-3,-3) (-2,-3) (-1,-3) ( 0,-3) ( 1,-3) ( 2,-3) ( 3,-3) ( 4,-3) ( 5,-3) (-5,-4) (-4,-4) (-3,-4) (-2,-4) (-1,-4) ( 0,-4) ( 1,-4) ( 2,-4) ( 3,-4) ( 4,-4) ( 5,-4) (-5,-5) (-4,-5) (-3,-5) (-2,-5) (-1,-5) ( 0,-5) ( 1,-5) ( 2,-5) ( 3,-5) ( 4,-5) ( 5,-5) -1 0] = ] ] 0-1] = ] () = (2 1) () = (4 2)

10 = ] ] 1 0 = () = (2 4) () = (4 3) (-5, 5) (-4, 5) (-3, 5) (-2, 5) (-1, 5) ( 0, 5) ( 1, 5) ( 2, 5) ( 3, 5) ( 4, 5) ( 5, 5) (-5, 4) (-4, 4) (-3, 4) (-2, 4) (-1, 4) ( 0, 4) ( 1, 4) ( 2, 4) ( 3, 4) ( 4, 4) ( 5, 4) (-5, 3) (-4, 3) (-3, 3) (-2, 3) (-1, 3) ( 0, 3) ( 1, 3) ( 2, 3) ( 3, 3) ( 4, 3) ( 5, 3) (-5, 2) (-4, 2) (-3, 2) (-2, 2) (-1, 2) ( 0, 2) ( 1, 2) ( 2, 2) ( 3, 2) ( 4, 2) ( 5, 2) (-5, 1) (-4, 1) (-3, 1) (-2, 1) (-1, 1) ( 0, 1) ( 1, 1) ( 2, 1) ( 3, 1) ( 4, 1) ( 5, 1) (-5, 0) (-4, 0) (-3, 0) (-2, 0) (-1, 0) ( 0, 0) ( 1, 0) ( 2, 0) ( 3, 0) ( 4, 0) ( 5, 0) (-5,-1) (-4,-1) (-3,-1) (-2,-1) (-1,-1) ( 0,-1) ( 1,-1) ( 2,-1) ( 3,-1) ( 4,-1) ( 5,-1) (-5,-2) (-4,-2) (-3,-2) (-2,-2) (-1,-2) ( 0,-2) ( 1,-2) ( 2,-2) ( 3,-2) ( 4,-2) ( 5,-2) (-5,-3) (-4,-3) (-3,-3) (-2,-3) (-1,-3) ( 0,-3) ( 1,-3) ( 2,-3) ( 3,-3) ( 4,-3) ( 5,-3) (-5,-4) (-4,-4) (-3,-4) (-2,-4) (-1,-4) ( 0,-4) ( 1,-4) ( 2,-4) ( 3,-4) ( 4,-4) ( 5,-4) (-5,-5) (-4,-5) (-3,-5) (-2,-5) (-1,-5) ( 0,-5) ( 1,-5) ( 2,-5) ( 3,-5) ( 4,-5) ( 5,-5) 0-1] = = 1 0] ] () = (1 2) () = (3 1)

11 ( 0, 4) ( 1, 4) ( 2, 4) ( 3, 4) ( 4, 4) The pentominoes seprte into 6 clsses of pieces. ( 0, 3) ( 1, 3) ( 2, 3) ( 3, 3) ( 4, 3) ( 0, 2) ( 1, 2) ( 2, 2) ( 3, 2) ( 4, 2) The first clss is the clss contining I, X, nd Z. This clss ( 0, 1) ( 1, 1) ( 2, 1) ( 3, 1) ( 4, 1) ( 0, 0) ( 1, 0) ( 2, 0) ( 3, 0) ( 4, 0) hs the mod numers (0 0). ( 0, 4) ( 1, 4) ( 2, 4) ( 3, 4) ( 4, 4) ( 0, 3) ( 1, 3) ( 2, 3) ( 3, 3) ( 4, 3) The second is the clss contining F. This clss hs the mod numers, (0 1), (0 4), (1 0), nd (4 0). ( 0, 4) ( 1, 4) ( 2, 4) ( 3, 4) ( 4, 4) ( 0, 2) ( 1, 2) ( 2, 2) ( 3, 2) ( 4, 2) ( 0, 1) ( 1, 1) ( 2, 1) ( 3, 1) ( 4, 1) ( 0, 0) ( 1, 0) ( 2, 0) ( 3, 0) ( 4, 0) ( 0, 3) ( 1, 3) ( 2, 3) ( 3, 3) ( 4, 3) ( 0, 2) ( 1, 2) ( 2, 2) ( 3, 2) ( 4, 2) The third is the clss contining T nd U. This clss hs the mod numers, (0 2), (0 3), (2 0), nd (3 0). ( 0, 1) ( 1, 1) ( 2, 1) ( 3, 1) ( 4, 1) ( 0, 0) ( 1, 0) ( 2, 0) ( 3, 0) ( 4, 0) The fourth is the clss contining W nd L. This clss hs the mod numers, (1 1), (1 4), (4 1), nd (1 4). ( 0, 4) ( 1, 4) ( 2, 4) ( 3, 4) ( 4, 4) ( 0, 3) ( 1, 3) ( 2, 3) ( 3, 3) ( 4, 3) ( 0, 2) ( 1, 2) ( 2, 2) ( 3, 2) ( 4, 2) ( 0, 1) ( 1, 1) ( 2, 1) ( 3, 1) ( 4, 1) ( 0, 4) ( 1, 4) ( 2, 4) ( 3, 4) ( 4, 4) ( 0, 0) ( 1, 0) ( 2, 0) ( 3, 0) ( 4, 0) ( 0, 3) ( 1, 3) ( 2, 3) ( 3, 3) ( 4, 3) ( 0, 2) ( 1, 2) ( 2, 2) ( 3, 2) ( 4, 2) ( 0, 1) ( 1, 1) ( 2, 1) ( 3, 1) ( 4, 1) ( 0, 0) ( 1, 0) ( 2, 0) ( 3, 0) ( 4, 0) The fifth is the clss contining Y nd P. This clss hs the mod numers, (1 2), (1 3), (2 1), (2 4), (3 1), (), (4 2), nd (4 3). ( 0, 4) ( 1, 4) ( 2, 4) ( 3, 4) ( 4, 4) ( 0, 3) ( 1, 3) ( 2, 3) ( 3, 3) ( 4, 3) The sixth is the clss contining V nd N. This clss hs the mod numers, (), (), (), nd (). ( 0, 2) ( 1, 2) ( 2, 2) ( 3, 2) ( 4, 2) ( 0, 1) ( 1, 1) ( 2, 1) ( 3, 1) ( 4, 1) ( 0, 0) ( 1, 0) ( 2, 0) ( 3, 0) ( 4, 0) 11

12 The mod numers cn e used to see whether certin constuctions with the pentominoes re impossile. It ll depends on the fct tht the sum of the mod numers is the mod numer of the sum, tht is the figure formed y the two pentominoes. Hve students discuss why this is so. Here re some exmples. (0,1) (1 3)+(4 3)=(0 1) Y + P (1 1)+(4 0)=(0 1) L + F (2 4)+()=(0 1) P + N The mod numers cn e used to show tht it is impossile to solve puzzle. Consider this exmple using mod numers to show tht it is impossile to solve certin pentomino prolems. The figure t the right shows tht the figure, 5 x 6 rectngle is constructed of 6 I s. Cn the figure formed y moving one of the squres s shown elow e constructed if ny numer of I s, X s, nd Z s s well s t most one other piece other thn n F is used?. (0 0) Oddly enough, there is no solution to this puzzle. I, X, nd Z ll hve mod numers (0 0). The sum of the mod numers for the finl shpe must e the mod numers of the other piece. The mod numer of the one other piece must e (1 0). Since F is the only piece with this mod numer, the (1 0) shpe on the left cnnot e constructed with one piece tht is not n F nd ll the other five pieces eing I s, X s, or F s. 12

13 As n exercise, tke this solution of the 6 x 10 using the twelve pentominoes. Flip the P pentomino s shown nd prove the twelve pentominoes cnnot e ressemled into the 6 x 10 through trnsltions lone. There is well-known prolem involving checkerord nd dominoes. If ech domino covers exctly two squres on checkerord nd two digonl corners re cut out, cn it e covered using 31 dominoes? The nswer is no ecuse ech domino covers red nd lck squre on the checkerord. The figure on the right hs 30 red squres nd 32 lck squres, so 30 dominoes will lwys leve two lck squres to e covered y domino 31. This will not work. Hence there is no covering. There is similr prolem using mod numers. Cover checkerord with oriented dominoes such s shown on the right. There re two possile orienttions, horizontl nd verticl. Tke one of the horizontl dominoes nd replce it with verticl one s shown elow. Cn the dominoes e put ck without chnging their orienttions nd still cover the checkerord? The nswer is no. Apply the sme rguments (0 0) 13

14 s efore with the mod numers ut now use mod 2 insted of mod 5. The verticl domino hs mod numer (0 1); the horizontl domino hs mod numer (1 0). The overll shpe hs mod numer of (0 0). If we tke out horizontl domino the remining 31 dominos will form shpe tht hs mod numer (1 0), when we dd the verticl domino we get shpe with mod numer (1 0) (0 1) of (1 1). The dominoes with exctly one with chnged orienttion cnnot e put ck onto the checkerord in their new orienttions ecuse the shpe they mke hs mod numers (1 1) nd the mod numers of the checkerord re (0 0). In this rticle, pentominoes nd their close reltives hve een explored. Pentominoes hve een defined more crefully nd the ide of pths through the figures hs een discussed. Using the ide of longest pth, it hs een shown tht n+1-omines re derived from n-ominoes. This ws used to show how to prove tht there re exctly 12 pentominoes. It ws shown how to ssign coordintes to polyomino nd develop mod numers for ech pentomino. Properties of mod numers were developed nd used to prove severl constructions would e impossile. An exmple ws used to show how to develop nd use mod numers for dominoes nd it ws indicted how mod numers could e extended to more dimensions nd numers of squres, cues, hypercues... These investigtions could lso e used to introduce students to different importnt techniques nd rnches of mthemtics. Proof y contrdiction, itertion, nd mth induction were some of the techniques employed. Coordintes from nlyticl geometry, liner lger, 14

15 modulr rithmetic from numer theory, nd group theory figure prominently in the rguments. For exmple, the mod numers could e used to explore symmetry from the point of view of liner lger. The pentominoes lso open n venue into group theory to explore the opertion of group, the eight mtrices for rottion nd reflection, on set, the mod numers, to look t the vrious orits of the mod numers nd the reking down of the mod numers into equivlence clsses. Even prolems in computer science could e investigted. If the pentominoes were to e ssemled into vrious constructions, the mod numers could e used to eliminte mny possile cses to e tried nd improve the efficiency of lgorithms looking for solutions. The use of the mtrices could extend the prolems investigted with more efficient computer progrms from the pentominoes to prolems with polyominoes s well s higher dimension cues nd hypercues. There is more to pentominoes thn re, perimeter, nd simple symmetry nd the techniques used to study them cn extend well eyond wht is tught in middle school. Bio - Bruce Bguley works for Cscde Mth Systems, LLC. He received BA in Mthemtics from Tulne University, n MS in Mthemtics from MIT, nd his techer trining from Heritge College in Toppenish, WA. While teching elementry nd middle school students, he ecme interested in showing mth concepts using mnipultives rther thn relying on memorizing formuls. He hs given numerous workshops t mth conferences over the pst few yers, showing people how to use mnipultives to represent mth concepts from counting, through whole numer, rtionl, nd integer opertions, to solving nd grphing liner equtions s well s proving numer theory prolems. References NCTM Principles nd Stndrds, NCTM, Reston, VA, 2000 The Pentomino Squre Prolem, Mthemtics Teching in the Middle School, NCTM, Mrch 2003 Exploring Cues Activity Sheet, AS-Cues.pdf 15

Square Roots Teacher Notes

Square Roots Teacher Notes Henri Picciotto Squre Roots Techer Notes This unit is intended to help students develop n understnding of squre roots from visul / geometric point of view, nd lso to develop their numer sense round this

More information

LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES

LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES DAVID WEBB CONTENTS Liner trnsformtions 2 The representing mtrix of liner trnsformtion 3 3 An ppliction: reflections in the plne 6 4 The lgebr of

More information

PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY

PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY MAT 0630 INTERNET RESOURCES, REVIEW OF CONCEPTS AND COMMON MISTAKES PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY Contents 1. ACT Compss Prctice Tests 1 2. Common Mistkes 2 3. Distributive

More information

Regular Sets and Expressions

Regular Sets and Expressions Regulr Sets nd Expressions Finite utomt re importnt in science, mthemtics, nd engineering. Engineers like them ecuse they re super models for circuits (And, since the dvent of VLSI systems sometimes finite

More information

The Math Learning Center PO Box 12929, Salem, Oregon 97309 0929 Math Learning Center

The Math Learning Center PO Box 12929, Salem, Oregon 97309 0929  Math Learning Center Resource Overview Quntile Mesure: Skill or Concept: 1010Q Determine perimeter using concrete models, nonstndrd units, nd stndrd units. (QT M 146) Use models to develop formuls for finding res of tringles,

More information

10.5 Graphing Quadratic Functions

10.5 Graphing Quadratic Functions 0.5 Grphing Qudrtic Functions Now tht we cn solve qudrtic equtions, we wnt to lern how to grph the function ssocited with the qudrtic eqution. We cll this the qudrtic function. Grphs of Qudrtic Functions

More information

Reasoning to Solve Equations and Inequalities

Reasoning to Solve Equations and Inequalities Lesson4 Resoning to Solve Equtions nd Inequlities In erlier work in this unit, you modeled situtions with severl vriles nd equtions. For exmple, suppose you were given usiness plns for concert showing

More information

Homework 3 Solutions

Homework 3 Solutions CS 341: Foundtions of Computer Science II Prof. Mrvin Nkym Homework 3 Solutions 1. Give NFAs with the specified numer of sttes recognizing ech of the following lnguges. In ll cses, the lphet is Σ = {,1}.

More information

Chapter 9: Quadratic Equations

Chapter 9: Quadratic Equations Chpter 9: Qudrtic Equtions QUADRATIC EQUATIONS DEFINITION + + c = 0,, c re constnts (generlly integers) ROOTS Synonyms: Solutions or Zeros Cn hve 0, 1, or rel roots Consider the grph of qudrtic equtions.

More information

A.7.1 Trigonometric interpretation of dot product... 324. A.7.2 Geometric interpretation of dot product... 324

A.7.1 Trigonometric interpretation of dot product... 324. A.7.2 Geometric interpretation of dot product... 324 A P P E N D I X A Vectors CONTENTS A.1 Scling vector................................................ 321 A.2 Unit or Direction vectors...................................... 321 A.3 Vector ddition.................................................

More information

Appendix D: Completing the Square and the Quadratic Formula. In Appendix A, two special cases of expanding brackets were considered:

Appendix D: Completing the Square and the Quadratic Formula. In Appendix A, two special cases of expanding brackets were considered: Appendi D: Completing the Squre nd the Qudrtic Formul Fctoring qudrtic epressions such s: + 6 + 8 ws one of the topics introduced in Appendi C. Fctoring qudrtic epressions is useful skill tht cn help you

More information

6.2 Volumes of Revolution: The Disk Method

6.2 Volumes of Revolution: The Disk Method mth ppliction: volumes of revolution, prt ii Volumes of Revolution: The Disk Method One of the simplest pplictions of integrtion (Theorem ) nd the ccumultion process is to determine so-clled volumes of

More information

EQUATIONS OF LINES AND PLANES

EQUATIONS OF LINES AND PLANES EQUATIONS OF LINES AND PLANES MATH 195, SECTION 59 (VIPUL NAIK) Corresponding mteril in the ook: Section 12.5. Wht students should definitely get: Prmetric eqution of line given in point-direction nd twopoint

More information

In the following there are presented four different kinds of simulation games for a given Büchi automaton A = :

In the following there are presented four different kinds of simulation games for a given Büchi automaton A = : Simultion Gmes Motivtion There re t lest two distinct purposes for which it is useful to compute simultion reltionships etween the sttes of utomt. Firstly, with the use of simultion reltions it is possile

More information

Lecture 15 - Curve Fitting Techniques

Lecture 15 - Curve Fitting Techniques Lecture 15 - Curve Fitting Techniques Topics curve fitting motivtion liner regression Curve fitting - motivtion For root finding, we used given function to identify where it crossed zero where does fx

More information

Quadratic Equations - 1

Quadratic Equations - 1 Alger Module A60 Qudrtic Equtions - 1 Copyright This puliction The Northern Alert Institute of Technology 00. All Rights Reserved. LAST REVISED Novemer, 008 Qudrtic Equtions - 1 Sttement of Prerequisite

More information

Section 5-4 Trigonometric Functions

Section 5-4 Trigonometric Functions 5- Trigonometric Functions Section 5- Trigonometric Functions Definition of the Trigonometric Functions Clcultor Evlution of Trigonometric Functions Definition of the Trigonometric Functions Alternte Form

More information

In this section make precise the idea of a matrix inverse and develop a method to find the inverse of a given square matrix when it exists.

In this section make precise the idea of a matrix inverse and develop a method to find the inverse of a given square matrix when it exists. Mth 52 Sec S060/S0602 Notes Mtrices IV 5 Inverse Mtrices 5 Introduction In our erlier work on mtrix multipliction, we sw the ide of the inverse of mtrix Tht is, for squre mtrix A, there my exist mtrix

More information

Solutions to Section 1

Solutions to Section 1 Solutions to Section Exercise. Show tht nd. This follows from the fct tht mx{, } nd mx{, } Exercise. Show tht = { if 0 if < 0 Tht is, the bsolute vlue function is piecewise defined function. Grph this

More information

Use Geometry Expressions to create a more complex locus of points. Find evidence for equivalence using Geometry Expressions.

Use Geometry Expressions to create a more complex locus of points. Find evidence for equivalence using Geometry Expressions. Lerning Objectives Loci nd Conics Lesson 3: The Ellipse Level: Preclculus Time required: 120 minutes In this lesson, students will generlize their knowledge of the circle to the ellipse. The prmetric nd

More information

Basic Math Review. Numbers. Important Properties. Absolute Value PROPERTIES OF ADDITION NATURAL NUMBERS {1, 2, 3, 4, 5, }

Basic Math Review. Numbers. Important Properties. Absolute Value PROPERTIES OF ADDITION NATURAL NUMBERS {1, 2, 3, 4, 5, } ƒ Bsic Mth Review Numers NATURAL NUMBERS {1,, 3, 4, 5, } WHOLE NUMBERS {0, 1,, 3, 4, } INTEGERS {, 3,, 1, 0, 1,, } The Numer Line 5 4 3 1 0 1 3 4 5 Negtive integers Positive integers RATIONAL NUMBERS All

More information

Double Integrals over General Regions

Double Integrals over General Regions Double Integrls over Generl egions. Let be the region in the plne bounded b the lines, x, nd x. Evlute the double integrl x dx d. Solution. We cn either slice the region verticll or horizontll. ( x x Slicing

More information

SCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics. Basic Algebra

SCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics. Basic Algebra SCHOOL OF ENGINEERING & BUILT ENVIRONMENT Mthemtics Bsic Alger. Opertions nd Epressions. Common Mistkes. Division of Algeric Epressions. Eponentil Functions nd Logrithms. Opertions nd their Inverses. Mnipulting

More information

Polynomial Functions. Polynomial functions in one variable can be written in expanded form as ( )

Polynomial Functions. Polynomial functions in one variable can be written in expanded form as ( ) Polynomil Functions Polynomil functions in one vrible cn be written in expnded form s n n 1 n 2 2 f x = x + x + x + + x + x+ n n 1 n 2 2 1 0 Exmples of polynomils in expnded form re nd 3 8 7 4 = 5 4 +

More information

Solving Linear Equations - Formulas

Solving Linear Equations - Formulas 1. Solving Liner Equtions - Formuls Ojective: Solve liner formuls for given vrile. Solving formuls is much like solving generl liner equtions. The only difference is we will hve severl vriles in the prolem

More information

Section 7-4 Translation of Axes

Section 7-4 Translation of Axes 62 7 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY Section 7-4 Trnsltion of Aes Trnsltion of Aes Stndrd Equtions of Trnslted Conics Grphing Equtions of the Form A 2 C 2 D E F 0 Finding Equtions of Conics In the

More information

Assuming all values are initially zero, what are the values of A and B after executing this Verilog code inside an always block? C=1; A <= C; B = C;

Assuming all values are initially zero, what are the values of A and B after executing this Verilog code inside an always block? C=1; A <= C; B = C; B-26 Appendix B The Bsics of Logic Design Check Yourself ALU n [Arthritic Logic Unit or (rre) Arithmetic Logic Unit] A rndom-numer genertor supplied s stndrd with ll computer systems Stn Kelly-Bootle,

More information

Sequences and Series

Sequences and Series Centre for Eduction in Mthemtics nd Computing Euclid eworkshop # 5 Sequences nd Series c 014 UNIVERSITY OF WATERLOO While the vst mjority of Euclid questions in this topic re use formule for rithmetic

More information

Generalized Inverses: How to Invert a Non-Invertible Matrix

Generalized Inverses: How to Invert a Non-Invertible Matrix Generlized Inverses: How to Invert Non-Invertible Mtrix S. Swyer September 7, 2006 rev August 6, 2008. Introduction nd Definition. Let A be generl m n mtrix. Then nturl question is when we cn solve Ax

More information

Operations with Polynomials

Operations with Polynomials 38 Chpter P Prerequisites P.4 Opertions with Polynomils Wht you should lern: Write polynomils in stndrd form nd identify the leding coefficients nd degrees of polynomils Add nd subtrct polynomils Multiply

More information

Vectors 2. 1. Recap of vectors

Vectors 2. 1. Recap of vectors Vectors 2. Recp of vectors Vectors re directed line segments - they cn be represented in component form or by direction nd mgnitude. We cn use trigonometry nd Pythgors theorem to switch between the forms

More information

and thus, they are similar. If k = 3 then the Jordan form of both matrices is

and thus, they are similar. If k = 3 then the Jordan form of both matrices is Homework ssignment 11 Section 7. pp. 249-25 Exercise 1. Let N 1 nd N 2 be nilpotent mtrices over the field F. Prove tht N 1 nd N 2 re similr if nd only if they hve the sme miniml polynomil. Solution: If

More information

1.2 The Integers and Rational Numbers

1.2 The Integers and Rational Numbers .2. THE INTEGERS AND RATIONAL NUMBERS.2 The Integers n Rtionl Numers The elements of the set of integers: consist of three types of numers: Z {..., 5, 4, 3, 2,, 0,, 2, 3, 4, 5,...} I. The (positive) nturl

More information

Or more simply put, when adding or subtracting quantities, their uncertainties add.

Or more simply put, when adding or subtracting quantities, their uncertainties add. Propgtion of Uncertint through Mthemticl Opertions Since the untit of interest in n eperiment is rrel otined mesuring tht untit directl, we must understnd how error propgtes when mthemticl opertions re

More information

Equations between labeled directed graphs

Equations between labeled directed graphs Equtions etween leled directed grphs Existence of solutions Grret-Fontelles A., Misnikov A., Ventur E. My 2013 Motivtionl prolem H 1 nd H 2 two sugroups of the free group generted y X A, F (X, A). H 1

More information

Multiplication and Division - Left to Right. Addition and Subtraction - Left to Right.

Multiplication and Division - Left to Right. Addition and Subtraction - Left to Right. Order of Opertions r of Opertions Alger P lese Prenthesis - Do ll grouped opertions first. E cuse Eponents - Second M D er Multipliction nd Division - Left to Right. A unt S hniqu Addition nd Sutrction

More information

Quadratic Equations. Math 99 N1 Chapter 8

Quadratic Equations. Math 99 N1 Chapter 8 Qudrtic Equtions Mth 99 N1 Chpter 8 1 Introduction A qudrtic eqution is n eqution where the unknown ppers rised to the second power t most. In other words, it looks for the vlues of x such tht second degree

More information

4 Geometry: Shapes. 4.1 Circumference and area of a circle. FM Functional Maths AU (AO2) Assessing Understanding PS (AO3) Problem Solving HOMEWORK 4A

4 Geometry: Shapes. 4.1 Circumference and area of a circle. FM Functional Maths AU (AO2) Assessing Understanding PS (AO3) Problem Solving HOMEWORK 4A Geometry: Shpes. Circumference nd re of circle HOMEWORK D C 3 5 6 7 8 9 0 3 U Find the circumference of ech of the following circles, round off your nswers to dp. Dimeter 3 cm Rdius c Rdius 8 m d Dimeter

More information

r 2 F ds W = r 1 qe ds = q

r 2 F ds W = r 1 qe ds = q Chpter 4 The Electric Potentil 4.1 The Importnt Stuff 4.1.1 Electricl Potentil Energy A chrge q moving in constnt electric field E experiences force F = qe from tht field. Also, s we know from our study

More information

Example 27.1 Draw a Venn diagram to show the relationship between counting numbers, whole numbers, integers, and rational numbers.

Example 27.1 Draw a Venn diagram to show the relationship between counting numbers, whole numbers, integers, and rational numbers. 2 Rtionl Numbers Integers such s 5 were importnt when solving the eqution x+5 = 0. In similr wy, frctions re importnt for solving equtions like 2x = 1. Wht bout equtions like 2x + 1 = 0? Equtions of this

More information

4.0 5-Minute Review: Rational Functions

4.0 5-Minute Review: Rational Functions mth 130 dy 4: working with limits 1 40 5-Minute Review: Rtionl Functions DEFINITION A rtionl function 1 is function of the form y = r(x) = p(x) q(x), 1 Here the term rtionl mens rtio s in the rtio of two

More information

Math 135 Circles and Completing the Square Examples

Math 135 Circles and Completing the Square Examples Mth 135 Circles nd Completing the Squre Exmples A perfect squre is number such tht = b 2 for some rel number b. Some exmples of perfect squres re 4 = 2 2, 16 = 4 2, 169 = 13 2. We wish to hve method for

More information

Chapter 6 Solving equations

Chapter 6 Solving equations Chpter 6 Solving equtions Defining n eqution 6.1 Up to now we hve looked minly t epressions. An epression is n incomplete sttement nd hs no equl sign. Now we wnt to look t equtions. An eqution hs n = sign

More information

Factoring Polynomials

Factoring Polynomials Fctoring Polynomils Some definitions (not necessrily ll for secondry school mthemtics): A polynomil is the sum of one or more terms, in which ech term consists of product of constnt nd one or more vribles

More information

Content Objectives: After completing the activity, students will gain experience of informally proving Pythagoras Theorem

Content Objectives: After completing the activity, students will gain experience of informally proving Pythagoras Theorem Pythgors Theorem S Topic 1 Level: Key Stge 3 Dimension: Mesures, Shpe nd Spce Module: Lerning Geometry through Deductive Approch Unit: Pythgors Theorem Student ility: Averge Content Ojectives: After completing

More information

9.3. The Scalar Product. Introduction. Prerequisites. Learning Outcomes

9.3. The Scalar Product. Introduction. Prerequisites. Learning Outcomes The Sclr Product 9.3 Introduction There re two kinds of multipliction involving vectors. The first is known s the sclr product or dot product. This is so-clled becuse when the sclr product of two vectors

More information

Matrix Algebra CHAPTER 1 PREAMBLE 1.1 MATRIX ALGEBRA

Matrix Algebra CHAPTER 1 PREAMBLE 1.1 MATRIX ALGEBRA CHAPTER 1 Mtrix Algebr PREAMBLE Tody, the importnce of mtrix lgebr is of utmost importnce in the field of physics nd engineering in more thn one wy, wheres before 1925, the mtrices were rrely used by the

More information

. At first sight a! b seems an unwieldy formula but use of the following mnemonic will possibly help. a 1 a 2 a 3 a 1 a 2

. At first sight a! b seems an unwieldy formula but use of the following mnemonic will possibly help. a 1 a 2 a 3 a 1 a 2 7 CHAPTER THREE. Cross Product Given two vectors = (,, nd = (,, in R, the cross product of nd written! is defined to e: " = (!,!,! Note! clled cross is VECTOR (unlike which is sclr. Exmple (,, " (4,5,6

More information

P.3 Polynomials and Factoring. P.3 an 1. Polynomial STUDY TIP. Example 1 Writing Polynomials in Standard Form. What you should learn

P.3 Polynomials and Factoring. P.3 an 1. Polynomial STUDY TIP. Example 1 Writing Polynomials in Standard Form. What you should learn 33337_0P03.qp 2/27/06 24 9:3 AM Chpter P Pge 24 Prerequisites P.3 Polynomils nd Fctoring Wht you should lern Polynomils An lgeric epression is collection of vriles nd rel numers. The most common type of

More information

Graphs on Logarithmic and Semilogarithmic Paper

Graphs on Logarithmic and Semilogarithmic Paper 0CH_PHClter_TMSETE_ 3//00 :3 PM Pge Grphs on Logrithmic nd Semilogrithmic Pper OBJECTIVES When ou hve completed this chpter, ou should be ble to: Mke grphs on logrithmic nd semilogrithmic pper. Grph empiricl

More information

Anti-derivatives/Indefinite Integrals of Basic Functions

Anti-derivatives/Indefinite Integrals of Basic Functions Anti-derivtives/Indefinite Integrls of Bsic Functions Power Rule: x n+ x n n + + C, dx = ln x + C, if n if n = In prticulr, this mens tht dx = ln x + C x nd x 0 dx = dx = dx = x + C Integrl of Constnt:

More information

CONIC SECTIONS. Chapter 11

CONIC SECTIONS. Chapter 11 CONIC SECTIONS Chpter 11 11.1 Overview 11.1.1 Sections of cone Let l e fied verticl line nd m e nother line intersecting it t fied point V nd inclined to it t n ngle α (Fig. 11.1). Fig. 11.1 Suppose we

More information

Binary Representation of Numbers Autar Kaw

Binary Representation of Numbers Autar Kaw Binry Representtion of Numbers Autr Kw After reding this chpter, you should be ble to: 1. convert bse- rel number to its binry representtion,. convert binry number to n equivlent bse- number. In everydy

More information

Variable Dry Run (for Python)

Variable Dry Run (for Python) Vrile Dr Run (for Pthon) Age group: Ailities ssumed: Time: Size of group: Focus Vriles Assignment Sequencing Progrmming 7 dult Ver simple progrmming, sic understnding of ssignment nd vriles 20-50 minutes

More information

Geometry and Measure. 12am 1am 2am 3am 4am 5am 6am 7am 8am 9am 10am 11am 12pm

Geometry and Measure. 12am 1am 2am 3am 4am 5am 6am 7am 8am 9am 10am 11am 12pm Reding Scles There re two things to do when reding scle. 1. Mke sure you know wht ech division on the scle represents. 2. Mke sure you red in the right direction. Mesure Length metres (m), kilometres (km),

More information

0.1 Basic Set Theory and Interval Notation

0.1 Basic Set Theory and Interval Notation 0.1 Bsic Set Theory nd Intervl Nottion 3 0.1 Bsic Set Theory nd Intervl Nottion 0.1.1 Some Bsic Set Theory Notions Like ll good Mth ooks, we egin with definition. Definition 0.1. A set is well-defined

More information

Exponentiation: Theorems, Proofs, Problems Pre/Calculus 11, Veritas Prep.

Exponentiation: Theorems, Proofs, Problems Pre/Calculus 11, Veritas Prep. Exponentition: Theorems, Proofs, Problems Pre/Clculus, Verits Prep. Our Exponentition Theorems Theorem A: n+m = n m Theorem B: ( n ) m = nm Theorem C: (b) n = n b n ( ) n n Theorem D: = b b n Theorem E:

More information

MATH 150 HOMEWORK 4 SOLUTIONS

MATH 150 HOMEWORK 4 SOLUTIONS MATH 150 HOMEWORK 4 SOLUTIONS Section 1.8 Show tht the product of two of the numbers 65 1000 8 2001 + 3 177, 79 1212 9 2399 + 2 2001, nd 24 4493 5 8192 + 7 1777 is nonnegtive. Is your proof constructive

More information

Pythagoras theorem and trigonometry (2)

Pythagoras theorem and trigonometry (2) HPTR 10 Pythgors theorem nd trigonometry (2) 31 HPTR Liner equtions In hpter 19, Pythgors theorem nd trigonometry were used to find the lengths of sides nd the sizes of ngles in right-ngled tringles. These

More information

Answer, Key Homework 8 David McIntyre 1

Answer, Key Homework 8 David McIntyre 1 Answer, Key Homework 8 Dvid McIntyre 1 This print-out should hve 17 questions, check tht it is complete. Multiple-choice questions my continue on the net column or pge: find ll choices before mking your

More information

Algebra Review. How well do you remember your algebra?

Algebra Review. How well do you remember your algebra? Algebr Review How well do you remember your lgebr? 1 The Order of Opertions Wht do we men when we write + 4? If we multiply we get 6 nd dding 4 gives 10. But, if we dd + 4 = 7 first, then multiply by then

More information

Section A-4 Rational Expressions: Basic Operations

Section A-4 Rational Expressions: Basic Operations A- Appendi A A BASIC ALGEBRA REVIEW 7. Construction. A rectngulr open-topped bo is to be constructed out of 9- by 6-inch sheets of thin crdbord by cutting -inch squres out of ech corner nd bending the

More information

Novel Methods of Generating Self-Invertible Matrix for Hill Cipher Algorithm

Novel Methods of Generating Self-Invertible Matrix for Hill Cipher Algorithm Bibhudendr chry, Girij Snkr Rth, Srt Kumr Ptr, nd Sroj Kumr Pnigrhy Novel Methods of Generting Self-Invertible Mtrix for Hill Cipher lgorithm Bibhudendr chry Deprtment of Electronics & Communiction Engineering

More information

5.2. LINE INTEGRALS 265. Let us quickly review the kind of integrals we have studied so far before we introduce a new one.

5.2. LINE INTEGRALS 265. Let us quickly review the kind of integrals we have studied so far before we introduce a new one. 5.2. LINE INTEGRALS 265 5.2 Line Integrls 5.2.1 Introduction Let us quickly review the kind of integrls we hve studied so fr before we introduce new one. 1. Definite integrl. Given continuous rel-vlued

More information

1 Numerical Solution to Quadratic Equations

1 Numerical Solution to Quadratic Equations cs42: introduction to numericl nlysis 09/4/0 Lecture 2: Introduction Prt II nd Solving Equtions Instructor: Professor Amos Ron Scribes: Yunpeng Li, Mrk Cowlishw Numericl Solution to Qudrtic Equtions Recll

More information

Formal Languages and Automata Exam

Formal Languages and Automata Exam Forml Lnguges nd Automt Exm Fculty of Computers & Informtion Deprtment: Computer Science Grde: Third Course code: CSC 34 Totl Mrk: 8 Dte: 23//2 Time: 3 hours Answer the following questions: ) Consider

More information

Math 314, Homework Assignment 1. 1. Prove that two nonvertical lines are perpendicular if and only if the product of their slopes is 1.

Math 314, Homework Assignment 1. 1. Prove that two nonvertical lines are perpendicular if and only if the product of their slopes is 1. Mth 4, Homework Assignment. Prove tht two nonverticl lines re perpendiculr if nd only if the product of their slopes is. Proof. Let l nd l e nonverticl lines in R of slopes m nd m, respectively. Suppose

More information

Pure C4. Revision Notes

Pure C4. Revision Notes Pure C4 Revision Notes Mrch 0 Contents Core 4 Alger Prtil frctions Coordinte Geometry 5 Prmetric equtions 5 Conversion from prmetric to Crtesin form 6 Are under curve given prmetriclly 7 Sequences nd

More information

Mathematics. Vectors. hsn.uk.net. Higher. Contents. Vectors 128 HSN23100

Mathematics. Vectors. hsn.uk.net. Higher. Contents. Vectors 128 HSN23100 hsn.uk.net Higher Mthemtics UNIT 3 OUTCOME 1 Vectors Contents Vectors 18 1 Vectors nd Sclrs 18 Components 18 3 Mgnitude 130 4 Equl Vectors 131 5 Addition nd Subtrction of Vectors 13 6 Multipliction by

More information

Review guide for the final exam in Math 233

Review guide for the final exam in Math 233 Review guide for the finl exm in Mth 33 1 Bsic mteril. This review includes the reminder of the mteril for mth 33. The finl exm will be cumultive exm with mny of the problems coming from the mteril covered

More information

11. PYTHAGORAS THEOREM

11. PYTHAGORAS THEOREM 11. PYTHAGORAS THEOREM 11-1 Along the Nile 2 11-2 Proofs of Pythgors theorem 3 11-3 Finding sides nd ngles 5 11-4 Semiirles 7 11-5 Surds 8 11-6 Chlking hndll ourt 9 11-7 Pythgors prolems 10 11-8 Designing

More information

Mathematics Higher Level

Mathematics Higher Level Mthemtics Higher Level Higher Mthemtics Exmintion Section : The Exmintion Mthemtics Higher Level. Structure of the exmintion pper The Higher Mthemtics Exmintion is divided into two ppers s detiled below:

More information

Uniform convergence and its consequences

Uniform convergence and its consequences Uniform convergence nd its consequences The following issue is centrl in mthemtics: On some domin D, we hve sequence of functions {f n }. This mens tht we relly hve n uncountble set of ordinry sequences,

More information

AREA OF A SURFACE OF REVOLUTION

AREA OF A SURFACE OF REVOLUTION AREA OF A SURFACE OF REVOLUTION h cut r πr h A surfce of revolution is formed when curve is rotted bout line. Such surfce is the lterl boundr of solid of revolution of the tpe discussed in Sections 7.

More information

Unit 6: Exponents and Radicals

Unit 6: Exponents and Radicals Eponents nd Rdicls -: The Rel Numer Sstem Unit : Eponents nd Rdicls Pure Mth 0 Notes Nturl Numers (N): - counting numers. {,,,,, } Whole Numers (W): - counting numers with 0. {0,,,,,, } Integers (I): -

More information

Integration by Substitution

Integration by Substitution Integrtion by Substitution Dr. Philippe B. Lvl Kennesw Stte University August, 8 Abstrct This hndout contins mteril on very importnt integrtion method clled integrtion by substitution. Substitution is

More information

Numerical Solutions of Linear Systems of Equations

Numerical Solutions of Linear Systems of Equations EE 6 Clss Notes Numericl Solutions of Liner Systems of Equtions Liner Dependence nd Independence An eqution in set of equtions is linerly independent if it cnnot e generted y ny liner comintion of the

More information

A new algorithm for generating Pythagorean triples

A new algorithm for generating Pythagorean triples A new lgorithm for generting Pythgoren triples RH Dye 1 nd RWD Nicklls 2 The Mthemticl Gzette (1998); 82 (Mrch, No. 493), p. 86 91 (JSTOR rchive) http://www.nicklls.org/dick/ppers/mths/pythgtriples1998.pdf

More information

Introduction to Mathematical Reasoning, Saylor 111

Introduction to Mathematical Reasoning, Saylor 111 Frction versus rtionl number. Wht s the difference? It s not n esy question. In fct, the difference is somewht like the difference between set of words on one hnd nd sentence on the other. A symbol is

More information

Section 2.3. Motion Along a Curve. The Calculus of Functions of Several Variables

Section 2.3. Motion Along a Curve. The Calculus of Functions of Several Variables The Clculus of Functions of Severl Vribles Section 2.3 Motion Along Curve Velocity ccelertion Consider prticle moving in spce so tht its position t time t is given by x(t. We think of x(t s moving long

More information

Experiment 6: Friction

Experiment 6: Friction Experiment 6: Friction In previous lbs we studied Newton s lws in n idel setting, tht is, one where friction nd ir resistnce were ignored. However, from our everydy experience with motion, we know tht

More information

SPECIAL PRODUCTS AND FACTORIZATION

SPECIAL PRODUCTS AND FACTORIZATION MODULE - Specil Products nd Fctoriztion 4 SPECIAL PRODUCTS AND FACTORIZATION In n erlier lesson you hve lernt multipliction of lgebric epressions, prticulrly polynomils. In the study of lgebr, we come

More information

4.11 Inner Product Spaces

4.11 Inner Product Spaces 314 CHAPTER 4 Vector Spces 9. A mtrix of the form 0 0 b c 0 d 0 0 e 0 f g 0 h 0 cnnot be invertible. 10. A mtrix of the form bc d e f ghi such tht e bd = 0 cnnot be invertible. 4.11 Inner Product Spces

More information

Babylonian Method of Computing the Square Root: Justifications Based on Fuzzy Techniques and on Computational Complexity

Babylonian Method of Computing the Square Root: Justifications Based on Fuzzy Techniques and on Computational Complexity Bbylonin Method of Computing the Squre Root: Justifictions Bsed on Fuzzy Techniques nd on Computtionl Complexity Olg Koshelev Deprtment of Mthemtics Eduction University of Texs t El Pso 500 W. University

More information

Lecture 2: Matrix Algebra. General

Lecture 2: Matrix Algebra. General Lecture 2: Mtrix Algebr Generl Definitions Algebric Opertions Vector Spces, Liner Independence nd Rnk of Mtrix Inverse Mtrix Liner Eqution Systems, the Inverse Mtrix nd Crmer s Rule Chrcteristic Roots

More information

Quadrilaterals Here are some examples using quadrilaterals

Quadrilaterals Here are some examples using quadrilaterals Qudrilterls Here re some exmples using qudrilterls Exmple 30: igonls of rhomus rhomus hs sides length nd one digonl length, wht is the length of the other digonl? 4 - Exmple 31: igonls of prllelogrm Given

More information

Curve Sketching. 96 Chapter 5 Curve Sketching

Curve Sketching. 96 Chapter 5 Curve Sketching 96 Chpter 5 Curve Sketching 5 Curve Sketching A B A B A Figure 51 Some locl mximum points (A) nd minimum points (B) If (x, f(x)) is point where f(x) reches locl mximum or minimum, nd if the derivtive of

More information

Know the sum of angles at a point, on a straight line and in a triangle

Know the sum of angles at a point, on a straight line and in a triangle 2.1 ngle sums Know the sum of ngles t point, on stright line n in tringle Key wors ngle egree ngle sum n ngle is mesure of turn. ngles re usully mesure in egrees, or for short. ngles tht meet t point mke

More information

Lines and angles. Name. Use a ruler and pencil to draw: a 2 parallel lines. c 2 perpendicular lines. b 2 intersecting lines. Complete the following:

Lines and angles. Name. Use a ruler and pencil to draw: a 2 parallel lines. c 2 perpendicular lines. b 2 intersecting lines. Complete the following: Lines nd s 1 Use ruler nd pencil to drw: 2 prllel lines 2 intersecting lines c 2 perpendiculr lines 2 Complete the following: drw in the digonls on this shpe mrk the interior s on this shpe c mrk equl

More information

2 DIODE CLIPPING and CLAMPING CIRCUITS

2 DIODE CLIPPING and CLAMPING CIRCUITS 2 DIODE CLIPPING nd CLAMPING CIRCUITS 2.1 Ojectives Understnding the operting principle of diode clipping circuit Understnding the operting principle of clmping circuit Understnding the wveform chnge of

More information

Volumes of solids of revolution

Volumes of solids of revolution Volumes of solids of revolution We sometimes need to clculte the volume of solid which cn be obtined by rotting curve bout the x-xis. There is strightforwrd technique which enbles this to be done, using

More information

MATLAB Workshop 13 - Linear Systems of Equations

MATLAB Workshop 13 - Linear Systems of Equations MATLAB: Workshop - Liner Systems of Equtions pge MATLAB Workshop - Liner Systems of Equtions Objectives: Crete script to solve commonly occurring problem in engineering: liner systems of equtions. MATLAB

More information

Well say we were dealing with a weak acid K a = 1x10, and had a formal concentration of.1m. What is the % dissociation of the acid?

Well say we were dealing with a weak acid K a = 1x10, and had a formal concentration of.1m. What is the % dissociation of the acid? Chpter 9 Buffers Problems 2, 5, 7, 8, 9, 12, 15, 17,19 A Buffer is solution tht resists chnges in ph when cids or bses re dded or when the solution is diluted. Buffers re importnt in Biochemistry becuse

More information

Example A rectangular box without lid is to be made from a square cardboard of sides 18 cm by cutting equal squares from each corner and then folding

Example A rectangular box without lid is to be made from a square cardboard of sides 18 cm by cutting equal squares from each corner and then folding 1 Exmple A rectngulr box without lid is to be mde from squre crdbord of sides 18 cm by cutting equl squres from ech corner nd then folding up the sides. 1 Exmple A rectngulr box without lid is to be mde

More information

APPLICATION OF INTEGRALS

APPLICATION OF INTEGRALS APPLICATION OF INTEGRALS 59 Chpter 8 APPLICATION OF INTEGRALS One should study Mthemtics ecuse it is only through Mthemtics tht nture cn e conceived in hrmonious form. BIRKHOFF 8. Introduction In geometry,

More information

One Minute To Learn Programming: Finite Automata

One Minute To Learn Programming: Finite Automata Gret Theoreticl Ides In Computer Science Steven Rudich CS 15-251 Spring 2005 Lecture 9 Fe 8 2005 Crnegie Mellon University One Minute To Lern Progrmming: Finite Automt Let me tech you progrmming lnguge

More information

Vectors. The magnitude of a vector is its length, which can be determined by Pythagoras Theorem. The magnitude of a is written as a.

Vectors. The magnitude of a vector is its length, which can be determined by Pythagoras Theorem. The magnitude of a is written as a. Vectors mesurement which onl descries the mgnitude (i.e. size) of the oject is clled sclr quntit, e.g. Glsgow is 11 miles from irdrie. vector is quntit with mgnitude nd direction, e.g. Glsgow is 11 miles

More information

Union, Intersection and Complement. Formal Foundations Computer Theory

Union, Intersection and Complement. Formal Foundations Computer Theory Union, Intersection nd Complement FAs Union, Intersection nd Complement FAs Forml Foundtions Computer Theory Ferury 21, 2013 This hndout shows (y exmples) how to construct FAs for the union, intersection

More information