A Math Learning Center publication adapted and arranged by. EUGENE MAIER and LARRY LINNEN

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3 A Math Learig Ceter publicatio adapted ad arraged by EUGENE MAIER ad LARRY LINNEN

4 ALGEBRA THROUGH VISUAL PATTERNS, VOLUME 1 A Math Learig Ceter Resource Copyright 2005, 2004 by The Math Learig Ceter, PO Box 12929, Salem, Orego Tel All rights reserved. QP386 P0405 The Math Learig Ceter is a oprofiit orgaizatio servig the educatio commuity. Our missio is to ispire ad eable idividuals to discover ad develop their mathematical cofidece ad ability. We offer iovative ad stadards-based professioal developmet, curriculum, materials, ad resources to support learig ad teachig. To fid out more visit us at The Math Learig Ceter grats permissio to classroom teachers to reproduce blacklie masters i appropriate quatities for their classroom use. This project was supported, i part, by the Natioal Sciece Foudatio. Opiios expressed are those of the authors ad ot ecessarily those of the Foudatio. Prepared for publicatio o Macitosh Desktop Publishig system. Prited i the Uited States of America. ISBN

5 Eugee Maier is past presidet ad cofouder of The Math Learig Ceter, ad professor emeritus of mathematical scieces at Portlad State Uiversity. Earlier i his career, he was chair of the Departmet of Mathematics at Pacific Luthera Uiversity ad, later, professor of mathematics at the Uiversity of Orego. He has a particular iterest i visual thikig as it relates to the teachig ad learig of mathematics. He is coauthor of the Math ad the Mid s Eye series ad has developed may of the mathematical models ad maipulative that appear i Math Learig Ceter curriculum materials. He has directed fourtee projects i mathematics educatio supported by the Natioal Sciece Foudatio ad other agecies, has made umerous coferece ad iservice presetatios, ad has coducted iservice workshops ad courses for mathematics teachers throughout the Uited States ad i Tazaia. Bor i Tillamook, Orego, he is a lifelog residet of the Pacific Northwest. Larry Lie is the K-12 Mathematics Coordiator for Douglas Couty School District, Castle Rock, Colorado. His mathematics classroom teachig spas over 38 years i public high school ad middle schools i Motaa ad Colorado. He has a Ph.D. from the Uiversity of Colorado at Dever, has made may presetatios at local ad atioal mathematics cofereces, ad has coducted iservice workshops ad courses for teachers throughout the Uited States. Bor i Tyler, Texas, but raised i Billigs, Motaa, he ow calls Colorado his home.

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7 ALGEBRA THROUGH VISUAL PATTERNS VOLUME 1 Itroductio vii LESSON 1 Tile Patters & Graphig 1 LESSON 2 Positive & Negative Itegers 31 LESSON 3 Iteger Additio & Subtractio 47 LESSON 4 Iteger Multiplicatio & Divisio 57 LESSON 5 Coutig Piece Patters & Graphs 73 LESSON 6 Modelig Algebraic Expressios 91 LESSON 7 Seeig & Solvig Equatios 113 LESSON 8 Exteded Coutig Piece Patters 135 VOLUME 2 LESSON 9 Squares & Square Roots 163 LESSON 10 Liear & Quadratic Equatios 185 LESSON 11 Complete Sequeces 217 LESSON 12 Sketchig Solutios 251 LESSON 13 Aalyzig Graphs 281 LESSON 14 Complex Numbers 315 Appedix 333

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9 INTRODUCTION Algebra Through Visual Patters is a series of lessos that comprise a semester-log itroductory algebra course, begiig with the developmet of algebraic patters ad extedig through the solutio of quadratic equatios. I these lessos, studets lear about ad coect algebraic ad geometric cocepts ad processes through the use of maipulatives, sketches, ad diagrams ad the lik these visual developmets to symbolic rules ad procedures. The lessos ca be used with studets who are ivolved i learig first-year algebra wherever their istructio is takig place: i middle school, high school, commuity college, or a adult learig ceter. Sice the Algebra Through Visual Patters lessos are desiged to be accessible to studets whatever their level of uderstadig, the lessos have bee successfully used with studets of varyig backgroud ad ability, icludig Special Educatio studets, studets learig algebra for the first time, those who have struggled with the subject i previous courses, studets who have bee idetified as taleted ad gifted, ad studets of various ages, from middle-schoolers to adult learers. Algebra Through Visual Patters offers a geuie alterative to the usual algebra course. It offers a approach to learig i which teachers ad studets collaborate to create a classroom i which learers explore algebraic cocepts usig maipulatives, models, ad sketches, egage i meaigful discourse o their learig of mathematics, publicly preset their uderstadigs ad solutio to problems, both orally ad i writig, build o their uderstadigs to icrease their learig. The lessos are desiged i such a way as to reder them useful as a stad-aloe curriculum, as replacemet lessos for, or as a supplemet to, a existig curriculum. For example, you might decide to begi with a maipulative approach to factorig quadratic expressios that would lead to symbolic approaches for the same cocept. This approach is built ito Visual Algebra ad thus could be used istead of simply a symbolic approach to factorig quadratics. The likelihood of learig for all studets would be ehaced ad the ed result would be that studets would uderstad factorig as well as icreasig their competecy to factor quadratics. Each lesso icludes a Start-Up, a Focus, ad a Follow-Up. The Focus is the mai lesso, while the Start-Up sets the stage for the Focus or coects it to a previous lesso, ad the Follow-Up is a homework ad/or assessmet activity. Together, Volumes 1 ad 2 of Algebra Through Visual Patters costitute a stad-aloe semester course i algebra or a yearlog course whe used i cojuctio with other text materials. I the latter istace, lessos from Algebra through Visual Patters ca be used to provide a alterative to the purely symbolic developmets of traditioal algebra texts. vii

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11 TILE PATTERNS & GRAPHING LESSON 1 THE BIG IDEA Tile patters provide a meaigful cotext i which to geerate equivalet algebraic expressios ad develop uderstadig of the cocept of a variable. Such patters are a useful cotext for developig techiques for solvig equatios ad for itroducig the cocept of graphig. START-UP FOCUS FOLLOW-UP Overview A tile patter provides the cotext for geeratig equivalet expressios, formulatig equatios, ad creatig bar graphs. Materials Red ad black coutig pieces, 60 per studet. Start-Up Master 1.1, 2 copies per studet ad 1 trasparecy. Start-Up Master 1.2, 1 trasparecy (optioal). Start-Up Master 1.3, 1 copy per studet ad 1 trasparecy. Start-Up Master 1.4, 1 copy per studet ad 1 trasparecy. 1 4 grid paper (see Blacklie Masters), 1 sheet per studet ad 1 trasparecy. Black coutig pieces for the overhead. Overview Tile patters are used to geerate equivalet expressios, formulate equatios, solve equatios, ad itroduce coordiate graphs. Materials Red ad black coutig pieces, 25 per studet. 1 4 grid paper (see Appedix), 2 sheets per studet ad 1 trasparecy. Focus Master , 1 copy of each per studet ad 1 trasparecy. Focus Master 1.3, 1 copy per studet. Black coutig pieces for the overhead. Overview Studets fid patters i arragemets of tile, write equivalet algebraic expressios for the umber of tile i the th arragemet, ad create coordiate graphs to show the umber of tile i certai arragemets. Materials Follow-Up 1, 1 copy per studet. Square tile or coutig pieces (optioal) for studet use at home. 1 4 grid paper (see Appedix). ALGEBRA THROUGH VISUAL PATTERNS 1

12 TEACHER NOTES 2 ALGEBRA THROUGH VISUAL PATTERNS

13 TILE PATTERNS & GRAPHING LESSON 1 START-UP Overview A tile patter provides the cotext for geeratig equivalet expressios, formulatig equatios, ad creatig bar graphs. Materials Red ad black coutig pieces, 60 per studet. Start-Up Master 1.1, 2 copies per studet ad 1 trasparecy. Start-Up Master 1.2, 1 trasparecy (optioal). Start-Up Master , 1 copy of each per studet ad 1 trasparecy. 1 4 grid paper (see Appedix),1 sheet per studet ad 1 trasparecy. Black coutig pieces for the overhead. ACTIONS COMMENTS 1 Distribute coutig pieces to each studet or group of studets. Display the followig sequece of 3 tile arragemets o the overhead. Have the studets form this sequece of arragemets. The have them form the ext arragemet i the sequece. Ask the studets to leave their sequece of arragemets itact so it ca be referred to later. 1 I this ad subsequet activities, the coutig pieces are ofte referred to as tile. I later activities the color of the coutig pieces is relevat; however, color is ot relevat i this activity. May studets will form the 4th arragemet as show to the right. If someoe forms aother arragemet, ackowledge it without judgmet, idicatig there are a umber of ways i which a sequece ca be exteded. I Actio 7 the studets will be asked to covert their sequece of arragemets ito a bar graph. 2 Ask the studets to cosider the sequece of arragemets i which the 4th arragemet is the oe illustrated i Commet 1. Ask them to imagie the 20th arragemet ad to determie the umber of tile required to build it. Ask for a voluteer to describe their method of determiig this umber. Illustrate their method o the overhead, usig a trasparecy of Start-Up Master There are various ways to arrive at the coclusio that 84 tile are required to build the 20th arragemet. Here is oe possible explaatio: There are just as may tile betwee the corers of a arragemet as the umber of the arragemet, for example, i the 3rd arragemet there are 3 tile betwee the corers, i the 4th arragemet there are 4 tile betwee corers, ad so forth. So i the 20th arragemet there will be 20 tile betwee corers o each side. Sice there are 4 sides ad 4 corers, there will be 4 times 20 plus 4 tile. This way of viewig tile arragemets ca be illustrated as show below. cotiued ext page ALGEBRA THROUGH VISUAL PATTERNS 3

14 TILE PATTERNS & GRAPHING LESSON 1 START-UP ACTIONS COMMENTS TILE PATTERNS & GRAPHING LESSON 1 START-UP BLACKLINE MASTER 1.1 Arragemet 1st 2d 3rd 4th 20th 2 cotiued Arragemet 1st 2d 3rd 4th 20th Arragemet 1st 2d 3rd 4th 20th Arragemet 1st 2d 3rd 4th 20th 20 4(1) + 4 4(2) + 4 4(3) + 4 4(4) + 4 4(20) + 4 Some studets may write formulas to represet the umber of tile i ay arragemet. If so, you might ask them to relate their formula to the 20th arragemet. This will be helpful for other studets who eed time to work with specific cases before geeralizig i Actio 4. Note that the sectios of white space i the strips of tile formig the 20th arragemet o Master 1.1 are iteded to suggest there are missig tile i the arragemet. That is, by metally elogatig the strip, oe ca imagie it cotais 20 tile. 3 Distribute 2 copies of Start-Up Master 1.1 to each studet. Have the studets record o a copy of Start-Up Master 1.1 the method of viewig the arragemets described i Commet 1. The ask for voluteers to describe other ways of determiig the umber of tile i the 20th arragemet. Illustrate these methods o the overhead, usig a trasparecy of Start-Up Master 1.1. Have the studets make a record of these methods o their copies of Start-Up Master 1.1. Cotiue util 5 or 6 differet methods have bee recorded. 3 Normally, i a group of 25 to 30 studets, there will be several differet methods proposed for determiig the umber of tile i the 20th arragemet. I order to obtai a variety of ways of viewig the arragemets beyod those suggested by studets, you ca ask the studets to devise additioal ways, or you ca devise other ways. Show o the two followig pages are 5 ways of coutig the umber of tile i the 20th arragemet. 4 ALGEBRA THROUGH VISUAL PATTERNS

15 TILE PATTERNS & GRAPHING LESSON 1 START-UP ACTIONS COMMENTS Arragemet 1st 2d 3rd 4th 20th (2) 4(3) 4(4) 4(5) 4(21) There are 21 tile o each side, startig at oe corer ad edig before the ext corer. Arragemet 1st 2d 3rd 4th 20th (3) 4 4(4) 4 4(5) 4 4(6) 4 4(22) 4 There are 22 tile o each side, coutig each corer twice. Arragemet 1st 2d 3rd 4th 20th (3) + 2(1) 2(4) + 2(2) 2(5) + 2(3) 2(6) + 2(4) 2(22) + 2(20) There are 22 tile o the top ad the bottom ad 20 o each side betwee the top ad the bottom. cotiued ext page ALGEBRA THROUGH VISUAL PATTERNS 5

16 TILE PATTERNS & GRAPHING LESSON 1 START-UP ACTIONS COMMENTS 3 cotiued Arragemet 1st 2d 3rd 4th 20th There is a 22 x 22 square with a 20 x 20 square removed from iside it. Arragemet 1st 2d 3rd 4th 20th 2(20) + 1 2(20) + 1 2[2(1) + 1] + 2 2[2(2) + 1] + 2 2[2(3 + 1] + 2 2[2(4) + 1] + 2 2[2(20) + 1] + 2 There are 2 L-shapes, each cotaiig 2 times 20 plus 1 tile, ad 2 corers. 4 Tell the studets that oe of the thigs discovered about the arragemets is that the umber of tile i the bottom row of a arragemet cotais two more tile tha the umber of the arragemet ad, thus, the followig statemets are true: Arragemet 1 cotais tile i the bottom row. Arragemet 2 cotais tile i the bottom row. Arragemet 3 cotais tile i the bottom row. 4 The statemets ca be writte oe at a time o the overhead as you make them or you ca prepare a overhead trasparecy from Start-Up Master 1.2 ad reveal the statemets oe at a time as you make them. A variable is a letter used to desigate a uspecified or ukow umber. Variables allow for great ecoomy i mathematical discourse. I this istace, the use of a variable eables oe to replace a ifiite collectio of statemets with a sigle statemet. Iitially, some studets may ot easily grasp the cocept of variable. However, as variables become a part of classroom discussio, these studets geerally come to uderstad ad use them appropriately. 6 ALGEBRA THROUGH VISUAL PATTERNS

17 TILE PATTERNS & GRAPHING LESSON 1 START-UP ACTIONS COMMENTS Arragemet 4 cotais tile i the bottom row. Arragemet 5 cotais tile i the bottom row. Arragemet 6 cotais tile i the bottom row. Commet that oe could cotiue makig such statemets idefiitely. Poit out that the statemets all have the same form, amely: Arragemet cotais + 2 tile i the bottom row, where the blak is filled by oe of the coutig umbers, 1, 2, 3, 4,. Tell the studets that i mathematical discourse, istead of usig a blak, it is customary to use a letter, for example: Arragemet cotais + 2 tile i the bottom row, where ca be replaced by ay oe of the coutig umbers 1, 2, 3, 4,. Itroduce the term variable. 5 Distribute a copy of Start-Up Master 1.3 to each studet (see followig page). For oe of the methods of viewig arragemets discussed above, illustrate how the th arragemet would be viewed for that method. The write a expressio for the umber of tile i the th arragemet. Ask the studets to do this for the other methods discussed. For each method, ask for a voluteer to show their sketch ad correspodig formula. Discuss. 5 Show o the followig page are illustratios for the 6 methods described earlier, with correspodig formulas. Durig the discussio, you ca poit out otatio covetios which may be ufamiliar to the studets, such as the use of juxtapositio to idicate multiplicatio, e.g., 4, ad the use of groupig symbols such as paretheses to avoid ambiguities, e.g., writig 4( + 1) to idicate that + 1 is to be multiplied by 4 i cotrast to writig which idicates that is to be multiplied by 4 ad the 1 is to be added to that product. cotiued ext page ALGEBRA THROUGH VISUAL PATTERNS 7

18 TILE PATTERNS & GRAPHING LESSON 1 START-UP ACTIONS COMMENTS TILE PATTERNS & GRAPHING LESSON 1 START-UP BLACKLINE MASTER cotiued ( + 1) ( + 2) 4 2( + 2) + 2 ( + 2) ( + 2) ( + 2) 2 2 2(2 + 1) + 2 If desired, the trasparecy ca be cut apart so that, o the overhead, a th arragemet ca be placed alogside its correspodig 20th arragemet. 8 ALGEBRA THROUGH VISUAL PATTERNS

19 TILE PATTERNS & GRAPHING LESSON 1 START-UP ACTIONS COMMENTS 6 List all the expressios obtaied for the umber of tile i the th arragemet. Discuss equivalet expressios. 6 Here are expressios for the umber of tiles i the th arragemet, as illustrated above: 4 + 4, 4( + 1), 4( + 2) 4, 2( + 2) + 2, ( + 2) 2 2, 2(2 + 1) + 2. Expressios, such as those listed, which give the same result whe evaluated for ay possible value of, are said to be equivalet. They are also said to be idetically equal or, simply, equal. Which form of equivalet expressios is preferable depeds upo the situatio. For example, whe is 99, the secod of the above expressios is easy to evaluate, while whe is 98, the third may be preferable. 7 Tell the studets that oe of the arragemets requires 200 tile to build. Ask them to determie which arragemet this is. Discuss the methods the studets use. Relate the studets work to solvig equatios. Repeat, as appropriate, for other umbers of tile. 7 Various methods ca be used. Viewig the arragemet as described i Commet 2 suggests removig the 4 corer tile ad dividig the remaiig 196 by 4. Thus, it is arragemet umber 49 that cotais 200 tile. A sketch such as the oe show below may be helpful tile The above lie of thought ca be give a algebraic cast. The umber of tile i the th arragemet is Thus, oe wats the value of for which = 200. Excludig the 4 corer tiles reduces to 4 ad 200 to 196. Thus, 4 = 196 ad, hece, = 49. A statemet of equality ivolvig a quatity, such as = 200 is called a equatio i. Determiig the quatity is called solvig the equatio. Other ways of viewig the arragemet may lead to other methods of determiig its umber. For example, viewig the arragemet as described i the first method of Commet 3 may lead to dividig 200 by 4 ad otig that the result, 50, is oe more tha the umber of the arragemet. This, i effect, is solvig the equatio 4( + 1) = 200. cotiued ext page ALGEBRA THROUGH VISUAL PATTERNS 9

20 TILE PATTERNS & GRAPHING LESSON 1 START-UP ACTIONS COMMENTS 8 Ask the studets to covert the sequece of 4 arragemets i Actio 1 to portray a bar graph showig the umbers of tile i these 4 arragemets, as illustrated below. The distribute oe sheet of 1 4 grid paper to each studet ad ask the studets to draw a bar graph to illustrate the umber of tile i the first 8 arragemets of this sequece. Discuss the studets observatios about the graph. 8 By rearragig the tile to portray a bar graph, studets are more likely to see the relatioship betwee the sequece ad their bar graph. Show below is a bar graph showig the umber of tile i the first 8 arragemets Number of Tile Number of Arragemet Observatios about the graph may be varied. Here are a few examples: Each bar is 4 squares higher tha the previous bar. The icrease from bar to bar is always the same. The umber of squares i each bar is a multiple of 4, startig with ALGEBRA THROUGH VISUAL PATTERNS

21 TILE PATTERNS & GRAPHING LESSON 1 START-UP ACTIONS COMMENTS 9 Give each studet a copy of Start-Up Master 1.4 ad tell them this is a coordiate graph of the first 4 arragemets of the sequece. Ask the studets to discuss their ideas about how the graph was formed ad where o the graph they thik poits for other arragemets i the sequece would lie. Have voluteers show their ideas o a trasparecy of Master 1.4. TILE PATTERNS & GRAPHING LESSON 1 START-UP BLACKLINE MASTER The coordiates of a poit o the graph are a ordered pair of umbers, the first of which tells how may uits to cout from zero alog the horizotal axis (i this case, how may uits to cout to the right of zero, idetifyig the arragemet umber). The secod coordiate i a ordered pair tells how may uits to cout from zero alog the vertical axis (i this case, above zero, idetifyig the umber of tile i the arragemet). It is customary to label the horizotal ad vertical axes by the quatities they represet. (Note that grid lies, ot spaces, are umbered.) Some possible observatios studets may make about the graph iclude: The poits of the graph lie o a straight lie. The poits are equally spaced. To get from oe poit to the ext, go 1 square to the right ad 4 up. The icrease from poit to poit is always the same. Number of Tile (4, 20) (3, 16) (2, 12) (1, 8) There are oly poits o the graph where is a iteger. Some studets may draw a lie coectig the poits of the graph. Note that, while doig so is okay, it does imply there are arragemets for o-itegral values of. That is, it suggests there are arragemets umbered or 4 1 3, for example. The studets may eve suggest ways of costructig such arragemets. However, ote that throughout this lesso, each graph is a set of discrete poits sice is always viewed as a coutig umber Number of Arragemet ALGEBRA THROUGH VISUAL PATTERNS 11

22 TEACHER NOTES 12 ALGEBRA THROUGH VISUAL PATTERNS

23 TILE PATTERNS & GRAPHING LESSON 1 START-UP BLACKLINE MASTER 1.1 Arragemet 1st 2d 3rd 4th 20th Arragemet 1st 2d 3rd 4th 20th Arragemet 1st 2d 3rd 4th 20th 2004, THE MATH LEARNING CENTER ALGEBRA THROUGH VISUAL PATTERNS 13

24 TILE PATTERNS & GRAPHING LESSON 1 START-UP BLACKLINE MASTER 1.2 Arragemet 1 cotais tile i the bottom row. Arragemet 2 cotais tile i the bottom row. Arragemet 3 cotais tile i the bottom row. Arragemet 4 cotais tile i the bottom row. Arragemet 5 cotais tile i the bottom row. Arragemet 6 cotais tile i the bottom row. Arragemet 7 cotais tile i the bottom row. Arragemet 8 cotais tile i the bottom row. Arragemet cotais + 2 tile i the bottom row. Arragemet cotais + 2 tile i the bottom row. 14 ALGEBRA THROUGH VISUAL PATTERNS 2004, THE MATH LEARNING CENTER

25 TILE PATTERNS & GRAPHING LESSON 1 START-UP BLACKLINE MASTER , THE MATH LEARNING CENTER ALGEBRA THROUGH VISUAL PATTERNS 15

26 TILE PATTERNS & GRAPHING LESSON 1 START-UP BLACKLINE MASTER 1.4 Number of Tile (4, 20) (3, 16) (2, 12) (1, 8) Number of Arragemet 16 ALGEBRA THROUGH VISUAL PATTERNS 2004, THE MATH LEARNING CENTER

27 TILE PATTERNS & GRAPHING LESSON 1 FOCUS Overview Tile patters are used to geerate equivalet expressios, formulate equatios, solve equatios, ad itroduce coordiate graphs. Materials Red ad black coutig pieces, 25 per studet. 1 4 grid paper (see Appedix), 2 sheets per studet ad 1 trasparecy. Focus Master , 1 copy of each per studet ad 1 trasparecy. Focus Master 1.3, 1 copy per studet. Black coutig pieces for the overhead. ACTIONS COMMENTS 1 Distribute coutig pieces to each studet or group of studets. Display the followig sequece of 4 tile arragemets o the overhead. Have the studets form this sequece of arragemets. The have them form the ext arragemet i the sequece. 1 Here is the most frequetly suggested 5th arragemet. Ackowledge other ideas suggested by studets. 2 Distribute a copy of Focus Master 1.1 to the studets. Ask the studets to cosider the sequece of arragemets i which the 5th arragemet is the oe illustrated i Commet 1. Ask them to determie a variety of ways to view the 20th arragemet ad to determie the umber of tile required to build it, ad to record their methods o Focus Master 1.1 (see followig page). Place a trasparecy of Focus Master 1.1 o the overhead ad ask for voluteers to describe their methods. 2 The 20th arragemet cotais 401 tile. O the followig page are 4 differet methods of viewig the arragemets. Notice that i Method D some of the tile i the arragemets have bee relocated. The studets may devise other methods of viewig the arragemets. cotiued ext page ALGEBRA THROUGH VISUAL PATTERNS 17

28 TILE PATTERNS & GRAPHING LESSON 1 FOCUS ACTIONS COMMENTS TILE PATTERNS & GRAPHING LESSON 1 FOCUS BLACKLINE MASTER cotiued Arragemet 1st 2d 3rd 4th 5th 20th Method A: A rectagle with a sigle tile attached to each of the loger sides. 1st 2d 3rd 4th 5th 20th (1 3) + 2 (2 4) + 2 (3 5) + 2 (4 6) + 2 (19 21) + 2 Method B: A square with a row of 20 tile o the top ad aother row of 20 tile o the bottom. 1st 2d 3rd 4th 5th 20th (1) (2) (3) (4) (5) (20) 20 Method C: A square with 20 tile removed from the first ad last colums. 1st 2d 3rd 4th 5th 20th (1) 3 2 2(2) 4 2 2(3) 5 2 2(4) 6 2 2(5) (20) 18 ALGEBRA THROUGH VISUAL PATTERNS

29 TILE PATTERNS & GRAPHING LESSON 1 FOCUS ACTIONS COMMENTS Method D: A square with a sigle tile attached to the lower left corer. 1st 2d 3rd 4th 5th 20th Distribute a copy of Focus Master 1.2 to each studet. Ask the studets to devise methods of viewig the th arragemet of the sequece. For each method, ask the studets to illustrate that method of viewig the arragemet ad write a formula for the umber of tile i the th arragemet which reflects that method of viewig the arragemet. Place a trasparecy of Focus Master 1.2 o the overhead ad ask for voluteers to show their methods. 3 Show below are the th arragemets correspodig to the 4 ways of viewig the arragemets show above. ( + 1) Method A Method B ( 1) ( 1)( + 1) + 2 ( 1) ( 1) ( 1) Method C Method D ( + 1) ( + 1) TILE PATTERNS & GRAPHING LESSON 1 FOCUS BLACKLINE MASTER 1.2 ( + 1) I additio to the methods show above. Other methods are possible. Show below are methods which few view the th arragemet as a cofiguratio from which tile have bee removed. The regios from which tile have bee removed are shaded. ( 2 ( 1)) + ( + 1) 2 2 (( + 1) ) + 1 ( + 1) ( 1) ALGEBRA THROUGH VISUAL PATTERNS 19

30 TILE PATTERNS & GRAPHING LESSON 1 FOCUS ACTIONS COMMENTS 4 Ask the studets to determie which arragemet cotais 170 tile. Discuss the methods they use. 4 If the arragemet is thought of i the maer of Method D o the previous page, oe of the 170 tile would be attached to a square formed with the remaiig 169. The side of this square, 13, is the umber of the arragemet. A solutio to the equatio = 170 has bee foud. 169 tile Thikig about the arragemet i the maer of Method A o the previous page, 2 of the 170 tile are attached to a rectagle formed by the remaiig 168. The dimesios of this rectagle differ by 2. Examiig factors of 168, oe fids the dimesios are 12 ad 14. Sice the umber of the arragemet is 1 more tha the smaller of these umbers (or 1 less tha the greater), it is 13. Note that a umber has bee foud, amely 13, such that ( 1)( + 1) + 2 = tile ALGEBRA THROUGH VISUAL PATTERNS

31 TILE PATTERNS & GRAPHING LESSON 1 FOCUS ACTIONS COMMENTS 5 Distribute 1 4 grid paper to the studets. Ask them to costruct ad label a coordiate graph which shows the umber of tile i the first 5 arragemets. Ask the studets for their observatios. 5 A coordiate graph of the first 5 arragemets is show below. Here are some possible observatios. Number of Tile The poits do ot lie o a lie. The vertical distat betwee poits icreases as the umber of arragemet icreases. The vertical distace betwee poits goes up by 2 as we move from poit to poit; at first, it s 3, the it s 5, the 7, ad so forth Number of Arragemet 6 Distribute a copy of Focus Master 1.3 to each studet (see followig page). Ask the studets to complete parts a) ad b). Discuss the studets resposes, i particular, ask for voluteers to show the sketches they made i part b). The ask the studets to complete the remaiig parts. Discuss their results ad the methods used to arrive at them. 6 Parts of this activity could be assiged as homework. a) Here is the most frequetly suggested 4th arragemet: cotiued ext page ALGEBRA THROUGH VISUAL PATTERNS 21

32 TILE PATTERNS & GRAPHING LESSON 1 FOCUS ACTIONS COMMENTS TILE PATTERNS & GRAPHING LESSON 1 FOCUS BLACKLINE MASTER 1.3 a) Draw the ext arragemet i the followig sequece: 6 cotiued b) There are 122 tile i the 40th arragemet. The studets may have difficulty drawig simple sketches which depict their method of viewig their arragemet. Iitially, they may iclude more detail tha ecessary. Here are some possibilities: 1st 2d 3rd 4th b) How may tile does the 40th arragemet cotai? Draw a rough sketch or diagram that shows how you arrived at your aswer. c) Fid at least 2 differet expressios for the umber of tile i the th arragemet. For each expressio, draw a rough sketch or diagram that shows how you arrived at that expressio. d) Which arragemet cotais exactly 500 tile? Draw a rough sketch or diagram of this arragemet. e) O a sheet of 1 4 grid paper, costruct ad label a coordiate graph showig the umber of tile i each of the first 8 arragemets. f) (Challege) Two arragemets together cotai 160 tile. Oe of the arragemets cotais 30 more tile tha the other. Draw a rough sketch or diagram of these 2 arragemets. Which arragemets are these? x x x 41 1 missig tile c) Here are some sketches of the th arragemet: ( + 1) + 1 missig tile 3( + 1) 1 22 ALGEBRA THROUGH VISUAL PATTERNS

33 TILE PATTERNS & GRAPHING LESSON 1 FOCUS ACTIONS COMMENTS d) If a arragemet cotais 500 tile, the top row cotais 498 3, or 166, tile. The umber of tile i the top row is the same as the umber of the arragemet. e) Below is a graph showig the umber of tile i each of the first 8 arragemets. Number of Tile Number of Arragemet f) As show i the figure below, the smaller arragemet cotais 130 2, or 65, tile. A arragemet with 65 tile has 21 tile i the top row ad hece is the 21st arragemet. The larger arragemet has 30 3 = 10 more tile i the top row ad, hece, is the 31st arragemet. 30 tile 130 tile ALGEBRA THROUGH VISUAL PATTERNS 23

34 TEACHER NOTES 24 ALGEBRA THROUGH VISUAL PATTERNS

35 TILE PATTERNS & GRAPHING LESSON 1 FOCUS BLACKLINE MASTER 1.1 Arragemet 1st 2d 3rd 4th 5th 20th Arragemet 1st 2d 3rd 4th 5th 20th Arragemet 1st 2d 3rd 4th 5th 20th Arragemet 1st 2d 3rd 4th 5th 20th 2004, THE MATH LEARNING CENTER ALGEBRA THROUGH VISUAL PATTERNS 25

36 TILE PATTERNS & GRAPHING LESSON 1 FOCUS BLACKLINE MASTER ALGEBRA THROUGH VISUAL PATTERNS 2004, THE MATH LEARNING CENTER

37 TILE PATTERNS & GRAPHING LESSON 1 FOCUS BLACKLINE MASTER 1.3 a) Draw the ext arragemet i the followig sequece: 1st 2d 3rd 4th b) How may tile does the 40th arragemet cotai? Draw a rough sketch or diagram that shows how you arrived at your aswer. c) Fid at least 2 differet expressios for the umber of tile i the th arragemet. For each expressio, draw a rough sketch or diagram that shows how you arrived at that expressio. d) Which arragemet cotais exactly 500 tile? Draw a rough sketch or diagram of this arragemet. e) O a sheet of 1 4 grid paper, costruct ad label a coordiate graph showig the umber of tile i each of the first 8 arragemets. f) (Challege) Two arragemets together cotai 160 tile. Oe of the arragemets cotais 30 more tile tha the other. Draw a rough sketch or diagram of these 2 arragemets. Which arragemets are these? 2004, THE MATH LEARNING CENTER ALGEBRA THROUGH VISUAL PATTERNS 27

38 TILE PATTERNS & GRAPHING LESSON 1 FOLLOW-UP BLACKLINE MASTER 1 1 Show here are the first 3 arragemets i a sequece of tile arragemets. a) Describe, i words oly, the 50th arragemet so ayoe who reads your descriptio could build it. b) Determie the umber of tile i the 50th arragemet. Draw a rough sketch or diagram that shows how you determied the umber. c) Fid at least 2 differet expressios for the umber of tile i the th arragemet. Draw rough sketches or diagrams to show how you obtaied these expressios. d) O a sheet of 1 4 grid paper, draw a graph showig the umber of tile i the first several arragemets of the above sequece. 2 Repeat parts b), c), ad d) above for each of the followig sequeces of tile arragemet. I II III 28 ALGEBRA THROUGH VISUAL PATTERNS 2004, THE MATH LEARNING CENTER

39 TILE PATTERNS & GRAPHING LESSON 1 ANSWERS TO FOLLOW-UP 1 1 a) Some possible descriptios of the 50th arragemet: A row of 53 tile with colums of 50 tile added uder the first ad last tiles i the row. 2 I The 50th arragemet cotais 5050 tile. b) Possible ways of viewig 50th arragemet: colums of 51 tile with a row of 51 tile added betwee the top tiles of the 2 colums b) The 50th arragemet cotais 153 tile (50 2 ) + 50 c) Possible ways of viewig th arragemet: (50) 3 51 c) Possible ways of viewig th arragemet: (2 + 1) ( + 3) + 2 3( + 1) d) 20 d) Number of Tile Number of Tile Number of Arragemet Number of Arragemet cotiued THE MATH LEARNING CENTER ALGEBRA THROUGH VISUAL PATTERNS 29

40 TILE PATTERNS & GRAPHING LESSON 1 ANSWERS TO FOLLOW-UP 1 (CONT.) 2 cotiued II The 50th arragemet cotais 203 tile. b) Possible ways of viewig 50th arragemet: 1 III The 50th arragemet cotais 2702 tile. b) Possible ways of viewig 50th arragemet: ( ) (50) + 1 (53)(51) 1 c) Possible ways of viewig th arragemet: c) Possible ways of viewig th arragemet: ( + 1) ( ) + 1 ( + 1) ( + 3)( + 1) 1 d) 20 d) Number of Tile Number of Tile Number of Arragemet Number of Arragemet 30 ALGEBRA THROUGH VISUAL PATTERNS 2004, THE MATH LEARNING CENTER

41 POSITIVE & NEGATIVE INTEGERS LESSON 2 THE BIG IDEA Collectios of red ad black coutig pieces serve as a model of the itegers. By combiig, observig, ad discussig the coutig pieces, studets develop metal pictures that help them to uderstad ad retai the meaig of itegers ad termiology associated with itegers. These experieces lay groudwork for uderstadig arithmetic operatios with itegers. START-UP FOCUS FOLLOW-UP Overview Studets write equatios suggested by questios about collectios of tile. They cosider whether or ot these equatios ca be solved if oly coutig umbers are available. Materials Tile for overhead Overview Red ad black coutig pieces are used to itroduce siged umbers ad provide a model for the itegers. Studets use the pieces to model situatios that ivolve itegers. Materials Red ad black coutig pieces (see Focus Commet 1), 25 per studet. Red ad black coutig pieces for the overhead. Focus Master 2.1, 1 copy per studet. Focus Master 2.2, 1 trasparecy. Overview Studets form or sketch collectios of bicolored coutig pieces ad determie their et values. Materials Follow-Up 2, 2 pages ru back-to-back, 1 copy per studet. Red ad black coutig pieces, 25 or more per studet (see Appedix; you could make a paper or cardstock copy of a quarter sheet of coutig pieces for each studet to cut out ad keep at home). ALGEBRA THROUGH VISUAL PATTERNS 31

42 TEACHER NOTES 32 ALGEBRA THROUGH VISUAL PATTERNS

43 POSITIVE & NEGATIVE INTEGERS LESSON 2 START-UP Overview Studets write equatios suggested by questios about collectios of tile. They cosider whether or ot these equatios ca be solved if oly coutig umbers are available. Materials Tile for overhead ACTIONS 1 Place several small collectios of tile o the overhead as show. COMMENTS 1 The collectios with the umber of pieces i each collectio: Cout the pieces i the first collectio ad record the umber of pieces it cotais uder the collectio. Repeat for the remaiig collectios. The usage of the term coutig umber varies. Some sources iclude 0 i the coutig umbers. Tell the studets that the umbers 1, 2, 3, we use to cout the umber of pieces i a collectio are called the coutig umbers or the atural umbers. Poit out that for every coutig umber there correspods a set of tile cotaiig that umber of pieces. 2 Place a row of 4 tile o the overhead followed by a row of 7 tile. Beeath the rows write the questio, How may tile must be added to the top row so that it has the same umber of tile as the secod row? 2 The questio might be phrased, What must be added to 4 to get 7? or, i algebraic form, What is the solutio of the equatio 4 + = 7? The studets may suggest aother equatio, for example, = 7 4. How may tile must be added to the top row so it has the same umber of tile as the bottom row? Ask the studets to write a equatio whose solutio provides a aswer to this questio. ALGEBRA THROUGH VISUAL PATTERNS 33

44 POSITIVE & NEGATIVE INTEGERS LESSON 2 START-UP ACTIONS COMMENTS 3 Iterchage the two rows of tile ad raise the same questio. 3 The studets reactios to the questio may vary. Some may say it s ot possible. Some may say the questio does t make sese. Oe equatio is 7 + = 4. Ask the studets to write a equatio whose solutio provides a aswer to this questio. Some studets who are familiar with egatives, may say that a solutio to the equatio is 3. However, if we limit the umbers at our disposal to the atural umbers, there is o solutio to the equatio. Solicit the studets reactios. Discuss whether or ot this equatio has a solutio. 4 Tell the studets that by itroducig red tile, which have the effect of egatig black tile, oe ca create a tile model for dealig with the equatio 7 + = 4, ad similar equatios. Describe to the studets how this will be doe. 4 This will be doe by turig attetio from the umber of tile i a collectio to the value of the tile i a collectio, ad agreeig that a red tile egates the value of a black tile ad coversely. So, istead of askig, What must be added to a collectio of 7 tile to get a collectio of 4 tile? we ca ask, What must be added to a collectio whose value is 7 black i order to get a collectio whose value is 4 black? The aswer to the latter questio is 3 red tile. The shift from talkig about the umber of tile to the value of the tile is similar to chagig the questio How may cois are i your pocket? to What is the value of the cois i your pocket? There are 6 cois i a collectio of 1 pey, 3 ickels, ad 2 dimes, but the value of the collectio is 36 cets. 34 ALGEBRA THROUGH VISUAL PATTERNS

45 POSITIVE & NEGATIVE INTEGERS LESSON 2 FOCUS Overview Red ad black coutig pieces are used to itroduce siged umbers ad provide a model for the itegers. Studets use the pieces to model situatios that ivolve itegers. Materials Red ad black coutig pieces (see Focus Commet 1), 25 per studet. Red ad black coutig pieces for the overhead. Focus Master 2.1, 1 copy per studet. Focus Master 2.2, 1 trasparecy. ACTIONS COMMENTS 1 Place the studets i groups of 2-4 ad give red ad black coutig pieces to each studet. Draw a chart like the oe show below o the overhead or chalkboard. Drop a small hadful of coutig pieces o the overhead. Discuss the meaig of et value ad the record the iformatio about this collectio o the first lie of the chart. Total No. of Pieces No. of Red No. of Black Net Value 1 Coutig pieces are red o oe side ad black o the other. They ca be made from red cardstock usig the masters i the Appedix (or bicolored plastic pieces ca be purchased from The Math Learig Ceter). Copy the Coutig Piece Master/Frot o oe side of red cardstock ad the Coutig Piece Master/Back o the other side; the cut o the lies. Oe sheet of cardstock will provide eough coutig pieces for four studets. Ay cardstock or plastic coutig piece will appear as a black piece o the overhead; to make red overhead pieces copy the Coutig Piece Master/Frot o red trasparecy film ad cut o the lies. Note that throughout this lesso a umber of ew terms are itroduced i referece to the red ad black coutig pieces. It is t iteded or ecessary to emphasize memorizatio of this vocabulary. Rather, terms will become familiar through use durig this ad the followig lessos. Red ad black pieces are said to be of opposite color. The et value of a collectio of coutig pieces is the umber of red or black pieces i the collectio that ca ot be matched with a piece of the opposite color. A collectio i which all pieces ca be matched with a piece of the opposite color has a et value of 0. Collectio 1 Collectio 2 Collectio 1 at left cotais 12 pieces, 5 black ad 7 red. Its et value is 2 red. Collectio 2 at the left cotais 8 pieces, 4 red ad 4 black. Its et value is 0. This iformatio is recorded i the followig table. Total No. of Pieces No. of Red No. of Black Net Value R ALGEBRA THROUGH VISUAL PATTERNS 35

46 POSITIVE & NEGATIVE INTEGERS LESSON 2 FOCUS ACTIONS COMMENTS 2 Ask a voluteer to take a modest collectio of coutig pieces (a doze or so) from their set ad drop them o their desktop. Record iformatio about this collectio o the chart. Repeat this Actio with differet studets util there are several etries o the chart. 2 You could have a studet report the umber of pieces of each color i their collectio ad the ask the class for the et value of the collectio. 3 Discuss the iformatio cotaied i the chart. I particular, draw out the studets observatios cocerig et values. If discussio of each of the followig questios is ot iitiated by studets, ask the groups to explore them ow: a) What is the effect of addig/ removig a equal umber of red ad black pieces to/from a collectio? b) For ay give ozero et value, what is the collectio with the fewest pieces that has that et value? c) What is the collectio with the fewest pieces that has et value 0? 3 To clarify the itet of these questios, it may help to begi discussio usig specific examples. The ask for geeralizatios. For example, before posig b), you might ask the followig: What are several collectios that have a et value of 2 red? What is the collectio cotaiig the fewest umber of pieces that has this value? If a collectio has a et value of 2 red ad cotais 10 black pieces, how may red pieces are i the collectio? What geeral observatios ca you make about collectios with et value 2? a) Addig or removig a equal umber of red ad black pieces from a collectio does ot chage its et value. b) For a give ozero et value, the collectio with the fewest pieces that has that et value cotais either all red pieces or all black pieces. For example, the collectio with the fewest pieces that has a et value of 3 red is a collectio of 3 red pieces. c) The collectio cotaiig 1 black piece ad 1 red piece has et value 0. It is also coveiet to say that the empty collectio, that is, the collectio cotaiig o pieces, has et value 0. 4 Discuss with the studets how plus ad mius sigs will be used to desigate et values. Total No. of Pieces No. of Red No. of Black Net Value 2R 0 5B Some studets may have already suggested usig positive ad egative umbers to idicate red ad black et values. I these materials a mius sig idicates a red et value ad a plus sig idicates a black et value. For example, a et value of 3 red is writte 3 (read egative three ); a et value of 2 black is writte + 2 (read positive two ). Note that the mius ad plus sigs are writte i superscript positio. Numbers to which a plus or mius sig are attached are called siged umbers. You might have a voluteer write the appropriate siged umber alogside the et values i the chart developed earlier, as show to the left. The colors red ad black were selected because of the covetio i bookkeepig to refer to beig "i the red" ad "i the black" to mea, respectively, owig more moey tha oe has or havig more moey tha oe owes. 36 ALGEBRA THROUGH VISUAL PATTERNS

47 POSITIVE & NEGATIVE INTEGERS LESSON 2 FOCUS ACTIONS COMMENTS 5 Drop a small hadful of coutig pieces o the overhead. Ask the studets for the et value of the resultig collectio. The ask the studets for the et value of the collectio that would be obtaied if all of the coutig pieces were tured over. Repeat this Actio for two or three other collectios. 5 Turig over all pieces i a collectio chages the sig (or, color) of its et value. Thus, if all pieces i a collectio whose et value is + 3 (or 3 black) are tured over, the resultig collectio will have value 3 (or 3 red). For example, see collectios A ad B i Commet 6. 6 Referrig to the results of Actio 5, itroduce the terms opposite collectios ad opposite et values to the studets. 6 Two collectios are called opposites of each other if oe ca be obtaied from the other by turig over all of its pieces. The et values of opposite collectios are opposite et values. Collectios A ad B, show below, are opposite collectios. Their et values, + 3 ad 3, are opposite et values, that is, + 3 is the opposite of 3, ad 3 is the opposite of + 3. Collectio A Net Value = +3 Collectio B Net Value = 3 Note that a collectio which has the same umber of red ad black pieces is its ow opposite. The et value of such a collectio is 0. Thus the opposite of 0 is 0. If this is ot suggested by a studet, brig it up by placig a collectio with et value 0 o the overhead. Ask the studets to fid its opposite ad to determie if there are other umbers that have the same value as their opposite (there are oe). ALGEBRA THROUGH VISUAL PATTERNS 37

48 POSITIVE & NEGATIVE INTEGERS LESSON 2 FOCUS ACTIONS COMMENTS 7 Distribute a copy of Focus Master 2.1 to each studet or pair of studets. Ask the studets to use their coutig pieces (or to imagie collectios of pieces) to help them fill i the missig umbers. Discuss the methods studets used to arrive at their aswers. 7 You may eed to remid the studets that a mius sig idicates a red et value ad a plus sig idicates a black et value. Some studets may arrive at correct aswers by imagiig the pieces. Urge studets who have difficulty to form each collectio of pieces. POSITIVE & NEGATIVE INTEGERS LESSON 2 FOCUS BLACKLINE MASTER Fill i the missig umbers: Total No. of Pieces No. of Red Pieces No. of Black Pieces Net Value a) b) c) d) e) f) g) h) Suppose that: Collectio X cotais 2 red ad 7 black pieces; Collectio Y cotais 8 red ad 5 black pieces; ad Collectio Z cotais 7 red ad 3 black pieces. a) Record the et value of collectio X:, Y:, Z:. b) Record the et value if collectios X ad Y are combied:. c) Record the et value if collectios Y ad Z are combied:. d) Record the et value if collectio X ad the opposite of collectio Y are combied:. 8 Poit out the fact that the et values of collectios of coutig pieces whe expressed as siged umbers serve as a model of the set of umbers called the itegers. Discuss the studets ideas about which umbers are i the set of itegers. Poit out that the positive itegers ca be idetified with the atural umbers. 8 The collectio of black et values, + 1, + 2, + 3,, represets the positive itegers ad the collectio of red et values, 1, 2, 3,, represets the egative itegers. A 0 et value represets the zero iteger. The set of positive itegers, + 1, + 2, + 3,, ca be idetified with the set of atural umbers, 1, 2, 3,, sometimes also referred to as the coutig umbers. (I some textbooks, 0 is icluded i the coutig umbers.) Cosequetly, the + sig is ofte omitted whe referrig to a positive iteger, i.e., 3 is writte i place of + 3. The idetificatio of the positive itegers with the atural umbers works because the umber of tile i a all-black collectio is umerically equal to its et value. 38 ALGEBRA THROUGH VISUAL PATTERNS

49 POSITIVE & NEGATIVE INTEGERS LESSON 2 FOCUS ACTIONS COMMENTS 9 Place a trasparecy of Focus Master 2.2 o the overhead ad ask the studets to use red ad black coutig pieces to model Situatio a). Have voluteers share their models at the overhead ad discuss questios which could be aswered usig their models. Repeat for oe or more of b)-d). POSITIVE & NEGATIVE INTEGERS LESSON 2 FOCUS BLACKLINE MASTER 2.2 Use black ad/or red coutig pieces to model each of the followig situatios: a) Durig two plays, the football team gaied 5 yards ad lost 3 yards. b) Durig the moth of November, Kerry eared $17 baby-sittig, spet $13 o a ew shirt, received a gift of $5 from her gradma, paid $7 for her little brother s birthday preset. The ew book she wats to buy costs $4. c) Mel the elevator operator decided to keep track of his elevator trips durig oe half hour period last Moday. Whe he started keepig track, he was t o the groud floor. Here is a list of his elevator trips that half hour: up 3 floors, dow 2 floors, up 5 floors to the top floor, dow 4 floors, up 2 floors, up 1 floor, dow 3 floors, ad dow 4 more floors where he eded o the groud floor. d) Celia iveted a ew game: place 3 gree, 1 red, 1 blue, ad 1 yellow game marker i a bag; radomly draw oe marker, record its color, replace the marker, ad shake the bag; repeat util you ear 7 poits (for each gree marker you lose 1 poit ad for each red, blue, or yellow marker you ear 1 poit). 9 Itegers ca be used to describe situatios that ivolve opposite words such as, above ad below, before ad after, orth ad south, gaiig ad losig, etc. For example, istead of sayig The high temperature for the day was 18 degrees above zero ad the low was 7 degrees below zero, oe ca use itegers ad say, The high temperature was 18 degrees ad the low was 7 degrees. Red ad black coutig pieces ca be used to model situatios ivolvig itegers ad reveal aswers to may questios about those situatios. For example, a model of the example give i the precedig paragraph might suggest the questio, What was the differece betwee the high ad low temperature? This actio ad Actio 10 are iteded to develop readiess for additio ad subtractio of itegers, which are explored i Lesso 6. Some questios prompted by a model of each situatio are listed below: a) A model of this situatio is show here: Gaied Lost What was the et yardage gaied or lost durig the two plays? How may more yards were gaied tha lost? b) Does Kerry have eough to buy her ew book? What was Kerry s et gai or loss for the moth of November? c) How may floors are i the buildig? What floor did Mel start o? Did Mel travel farther goig up or goig dow? How much farther? d) Studets may wish to carry out a short experimet. Some may use red ad black pieces to record their poits lost or gaied. Other studets may suggest replacig the 3 gree markers with 3 red coutig pieces ad the other 3 markers with 3 black coutig pieces, based o the poits assiged to each marker. Some questios they may pose iclude: What is the theoretical probability Celia will lose/wi oe poit o ay sigle draw? What are the fewest draws it would take to ear 7 poits? Is it possible to ear 7 poits i a eve umber of draws? a odd umber? (Oe studet suggested this as a great game to give childre they are baby-sittig, sice theoretically the game would ever ed.) ALGEBRA THROUGH VISUAL PATTERNS 39

50 TEACHER NOTES 40 ALGEBRA THROUGH VISUAL PATTERNS