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Bail - ~ F1F2li~ -- ~ it' Cases of Mathematics Inst:ruct:ion to Enhance Teaching This case was developed by the collahora- tive team of Margaret Smith, Edward Silver, Mary Kay Stein, Marjorie Henningsen, and Melissa Boston under the auspices of the COMET Project. COMET; funded by the National Science Foundation, is a project aimed at developing case materials for teacher professional develop- ment in mathematics. The project is co- directed by Edward Silver, Margaret Smith, and Mary Kay Stein and is housed at the Learning Research and Development Center at the University of Pittsburgh. For additional information about COMET contact Margaret Smith by phone at (4129 648-7361 or by e-mai! at . OPENING ACTIVITY Solve For the pattern shown below, compute the perimeter for the first four trains, determine the perimeter for the tenth train without constructing it, anti then write a (lescription that coul(1 be use(1 to compute the perimeter of any train in the pattern. (Use the edge length of any pattern block as your unit of measure.) The first train in this pattern consists of one regular hexagon. For each subse- quent train, one a(l(litional hexagon is a(l(le(l. The first four trains in the pattern are shown below.

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Catherine Evans Talks About Her Class ~ have been teaching the new curricu- lum for about six weeks now and ~ have found that my sixth graders are not always prepared for the challenges presented. The tasks in the curriculum generally can't be solved by just using an algorithm, the solution path is not immedi- ately evident and usually involves explor- ing and reasoning through alternatives, and most tasks involve providing a written explanation. If my students can't solve a problem imme(liately, they say, "I (lon't know," and give up. They have had limited experience in elementary school actually engaging actively with mathematics and expressing their thinking and have found this to be very difficult. Seeing students give up has caused me great concern. ~ can't buy the idea that kids don't feel bad starting off with what they perceive to be failure. When they have work they can't do or don't have the confidence to do, then ~ have to intervene. ~ decided to help kids do more verbaliza- tion in class, get to the kids who didn't volunteer and guarantee them success by asking them to do things they couldn't fail to do right. ~ can't ignore the fact that success breeds success. Too many are starting out with what I'm sure they perceive to be failure. In order to ensure student success, ~ have started to make some modifications in the curriculum, at times putting in an extra step or taking out something that seems too hard; rewriting problem instructions so that they are clearer and at an easier reading level; and creating easier problems for homework. In addition, during classroom instruction ~ try to break a task into small subtasks so that students can tackle one part of the task at a time. We have been talking about patterns for a few weeks. The new unit that we started APPE N DIX last week uses trains of pattern blocks arranged in a geometric sequence. The unit is supposed to help students visualize and describe geometric patterns, make conjectures about the patterns, determine the perimeters of trains they build, and ultimately, to develop a generalization for the perimeter of any train in a pattern. This unit really lays the groundwork for developing the algebraic ideas of generali- zation, variable, and function that students will explore in gra(les 6 through 8. Expe- riences like these lay the foundation for more formal work in algebra in eighth gra(le. We spent a lot of time in the beginning of this unit just making observations about the trains the number of pattern blocks in a train, the geometric shapes that comprise a train, and the properties of a train (e.g., each train has four sides, opposite sides of the train are parallel). Students got pretty good at making observations about specific trains once we ha(1 (lone a few, but ~ ha(1 to keep remin(l- ing them that the observations nee(le(1 to be mathematical. For some patterns ~ got some really weird responses like "it looks like a squished pop can" or "it looks like a belt buckle." But once ~ reminded students that the point in making observations was to be able to predict what larger trains were going to look like, they were able to move beyond these fanciful responses. The Class Yesterday for the first time we starte(1 (letermining the perimeters of the trains using the side of the square as the unit of measure. Homework last night ha(1 been to fin(1 the perimeters of the first three trains in the pattern shown below. ~ also aske(1 students to fin(1 the perimeter of the lOth, 20th, anti lOOth trains in this pattern. My plan for class was to begin by (liscuss- ing the homework and then having students explore another pattern.

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train 1 train 2 train 3 As students entered the classroom and got their papers out, ~ made a quick trip to the back of the room to check on the video camera. My colleagues and ~ have decided to videotape some of our classes this year so that we could use the tapes to reflect on how things are going with the new curriculum and to talk about various issues that arise in using the materials. This was my first day of taping, and ~ was a little nervous about being on film. Students asked about the camera as they entered the classroom but seemed unfazed by the idea of being taped. ~ just hoped ~ could forget that it was there. Discussing the Square Pattern Trains In order to get things started, ~ asked students to make observations about the pattern. Shandra said that she had noticed that all of the trains were rectangles. lake said that he noticed that the perimeter of the first train was 4. ~ asked him to come up and show us. When he got to the overhead he took a square tile (black) and laid an edge of the square next to each side of the train as he counted the sides. 1 ~3 This was the procedure we had estab- lishe(1 yesterday, anti ~ was please(1 to see him use it. ~ thanked him anti he returne to his seat. Since lake had started talking about perimeters, ~ (leci(le(1 that we might as well continue in this (Erection. ~ aske(1 Zeke what he foun(1 for the perimeter of the second train. Zeke said he thought it was four. ~ asked him if he would go to the overhead and show us how he got 4. He explaine(l, "the train has four si(les just counted them I, 2, 3, and 4." (See the iagram below.) 2 1= 4 3 ~ saw what Zeke was (loin". He was counting the number of sides, not the number of units in the perimeter. The number of si(les anti number of units were the same in the first figure but not in the secon(1 figure. ~ aske(1 Zeke to stay at the overhea(1 anti ~ aske(1 the class if someone could review what perimeter is. David said that it was the sides all the way around. ~ asked if anyone had another way to say it. David's definition really supported what Zeke had done, and ~ was looking for a definition that would cause students to question Zeke's solution. Finally Nick said that the perimeter would be six. Nick explained, "I used Jake's way and measured all the way around the outside of the train with the square tile. It's not 4 because the top and bottom each have two units." Although this was not the definition ~ was looking for, ~ figured that this explanation would help students see why the perimeter was 6 and not 4. At this point ~ decided to ask Desmond to come up and measure the perimeter of APPE N AX

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the third train for us using the procedure that Nick had just described. I have been trying all year to get him involved. Irately I have been asking him questions that I was sure he could answer. They were not meant to challenge him in any way, just help him feel successful. These experi- ences have had an immediate positive effect on Desmond he would actively participate in class following these epi- sodes. So Desmond came up to the overhead, and I gave him the black square and asked him to measure the third train. I really thought that this would be a simple task, but Desmond did not seem to know what to do. Since this experience was supposed to be about experiencing success, I took his hand and helped him move the square along the outside of the train, counting as we proceeded. ~. 2 3 4 1 ~ ~ ~5. ~ 7 6 ~. I thanked Desmond for his help. I was sure that this would clear up the confu- sion. I told Zeke that a lot of people make the same mistake that he did the first time they do perimeter. Just to be sure that Zeke understood the way to find perim- eter, I asked him if he could build the fourth train in the pattern. He quickly laid four squares side to side. I then asked him if he could find the perimeter by measuring. He proceeded to count the si(les while moving the si(le of the square APPE N DIX along the perimeter of the train 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. He looked up when he finished and announced, "It will be ten!" I thanked him for hanging in there with us, and he returned to his seat. Before moving on to the next part of the assignment, I asked if anybody had noticed anything else about perimeter when they (li(1 just the first three. Angela had her hand up, and I asked her what she had noticed. She explained, "on the third train there are three on the top and three on the bottom, which makes six, and one on each end." I asked her if she would go to the overhead and show us what she meant. She restated, "See there are three up here (pointing to the top of the train) and three down here (pointing to the bottom of the train) and then one on each end." 3 1~L: 1 3 I was surprised by this observation so early on, but knowing that it would be helpful in (letermining the perimeters of larger trains, I asked Angela if she could use her system to find the perimeter of the fourth train. She quickly said "10." I asked her to explain. She proceeded, "four on the bottom and four on the top and one on each end." Class can be pretty fast paced some- times, with individual students, the whole class, and me going back and forth in a rapid exchange. A good example of this happened at this point as I tried to put Angela's observation to the test and see if I could get the whole class involved in

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the third train for us using the procedure that Nick had just described. I have been trying all year to get him involved. Irately I have been asking him questions that I was sure he could answer. They were not meant to challenge him in any way, just help him feel successful. These experi- ences have had an immediate positive effect on Desmond he would actively participate in class following these epi- sodes. So Desmond came up to the overhead, and I gave him the black square and asked him to measure the third train. I really thought that this would be a simple task, but Desmond did not seem to know what to do. Since this experience was supposed to be about experiencing success, I took his hand and helped him move the square along the outside of the train, counting as we proceeded. ~. 2 3 4 1 ~ ~ ~5. ~ 7 6 ~. I thanked Desmond for his help. I was sure that this would clear up the confu- sion. I told Zeke that a lot of people make the same mistake that he did the first time they do perimeter. Just to be sure that Zeke understood the way to find perim- eter, I asked him if he could build the fourth train in the pattern. He quickly laid four squares side to side. I then asked him if he could find the perimeter by measuring. He proceeded to count the si(les while moving the si(le of the square APPE N DIX along the perimeter of the train 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. He looked up when he finished and announced, "It will be ten!" I thanked him for hanging in there with us, and he returned to his seat. Before moving on to the next part of the assignment, I asked if anybody had noticed anything else about perimeter when they (li(1 just the first three. Angela had her hand up, and I asked her what she had noticed. She explained, "on the third train there are three on the top and three on the bottom, which makes six, and one on each end." I asked her if she would go to the overhead and show us what she meant. She restated, "See there are three up here (pointing to the top of the train) and three down here (pointing to the bottom of the train) and then one on each end." 3 1~L: 1 3 I was surprised by this observation so early on, but knowing that it would be helpful in (letermining the perimeters of larger trains, I asked Angela if she could use her system to find the perimeter of the fourth train. She quickly said "10." I asked her to explain. She proceeded, "four on the bottom and four on the top and one on each end." Class can be pretty fast paced some- times, with individual students, the whole class, and me going back and forth in a rapid exchange. A good example of this happened at this point as I tried to put Angela's observation to the test and see if I could get the whole class involved in

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using her observation to predict future trains. Once Angela's pattern became obvious to her ~ wanted to make sure that everyone in the class saw it too. So ~ proceeded with the following question and answer exchange: Angela: Me: Angela: Me: Angela: Me: Angela: Me: Angela: Me: Angela: Me: Tamika: Me: CLASS: Me: CLASS: Me: CLASS: Me: CLASS: Using your system, do you think you could do any number say? What would you do for 10? How many on the top and the bottom? 10. How many on the ends? 2. How many all together? 22. Let's do another one. Listen to what she's saying and see if you can do it also. Angela, in train 12, how many will there be on the top and bottom? 12. And then how many will there be on the ends? 2. How many will there be all together? 26. Tamika, what's she doing? She's taking the train number on the top and bottom and adding two. OK, let's everybody try a few. can pick any number. Train 50. How many will there be on the top and bottom? Everybody! 50 [with enthusiasm] How many on the ends? 2. How much all together? 102. Train 100, how many on the top and bottom? 100. [louder and with even more enthusiasm] Me: CLASS: Me: CLASS: Me: CLASS: Me: CLASS: Me: CLASS: How many on the ends? 2. How much all together? 202. Train 1000, how many on the top and bottom? 1000. [loudest of all] How many on the ends? 2. How much all together? 2002. At this point ~ asked if they could describe anything ~ gave them. Another resounding '~ES" answered my question. One of the things that ~ have found is that responding in unison really engages students and helps their confidence. When they respond in unison they feel that they are part of the group. Everyone can participate and feel good about themselves. Angela's observation had really led us to finding the perimeters for any train, so decided to continue on this pathway. asked if anyone had figured out the perimeters using a different way. ~ looked around the room no hands were in the air. ~ wanted them to have at least one other way to think about the pattern so ~ shared with them a method suggested by one of the students in another class. ~ explained that she had noticed that the squares on the ends always have three sides they each lose one on the inside- and that the ones in the middle always have two sides. ~ used train three (shown in the diagram below) as an example and pointed out the three sides on each end and the two sides on the middle square. F: l 1 1 1 1 ~ ~ ~ APPE N DIX

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wanted to see if students understood this so ~ asked how many squares would be in the middle of train 50 with this system. Nick said that there would be 48. ~ then added that there would be 48 two's, referring to the number of sides that would be counted in the perimeter, and three on each end. ~ asked what 48 two's would be. Carmen said it would be 96. ~ then asked what the perimeter would be. Shawntay said that it would 102. She went on to say that that was the same as what we got from train 50 when we did it Angela's way! ~ told the class that was right, there isn't just one way to look at it. Consiclering a New Paffern We had spent nearly 20 minutes on the square pattern, and it was time to move on to another pattern. ~ quickly got out my pattern blocks and built the train shown below on the overhead. ~ told students that ~ wanted them to work with their partners and build the first three trains in the pattern, find the perimeters for these three trains, and then to find the perimeters for the lOth, 20th, and lOOth trains. ~ put the pattern of square trains we had just finished back up on the overhead under- neath the hexagon pattern and suggested that they might want to see if they could find anything that was the same for the hexagon pattern and the square pattern that would help them. Since the generali- zations for the perimeters of these two trains had some similarities, ~ thought this would help them find the perimeters for the larger trains in the hexagon pattern. After about 5 minutes students seemed to be getting restless. Since most seemed to have made progress on the task, ~ decided to call the class together and see what they observed about the pattern. Although this is not exactly what ~ asked them to do make observations ~ felt that it provided a more open opportunity for all students to have something to say. asked Tracy what she had noticed. She said that every time you add one. "Add one what?," ~ asked. "A hexagon," she responded. ~ then asked about the perimeter. Darrel said that he discovered that it was six. 'what was six?," ~ asked. Darrel clarified that six was the amount around the hexagon around the edges on the first train. ~ asked Darrel about the second train. He explaine(l, "the hexagon has six around it anti then you take away one for each side in the middle so it is 5 + 5 or 10. Then on the thir(1 one you still have 5 + 5 for the end ones and you add four more sides for the new hexagon you a(l(le(l." ~ wanted to see if Darrel realize(1 that his observation would lead to a generaliza- tion. ~ aske(1 him if what he ha(1 (liscov- ere(1 would tell him anything about building another train. Darrel sai(l, 'yeah. On train 4 there would be four hexagons. The end ones woul(1 each have five anti the two mi(l(lle ones woul(1 each have four." "If you were to buil(1 train 10," asked, "coul(1 you tell me how many would have four sides and how many would have five si(les?" Darrel appeared to think about it for a few seconds anti train 1 APPE N DIX train 2 train 3

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then responded that eight hexagons would have four sides and two hexagons would have five sides. ~ wanted to make sure that students understood what Darrel was saying so ~ asked him where the two with the five sides would be. He looked at me as though ~ were crazy and said, "Mrs. Evans, they would have to be on the ends!" Again, ~ wanted to see if students could use Darrel's method on any train. ~ asked Tommy if he could describe the 20th train. Tommy explained, "For train 20 you'd count the sides and count the ends. You subtract 2 from 20 and that would be IS and then you multiply IS by 4, because all the hexagons in the middle have four and then you would add 10 from the ends." was impressed with his explanation, and he seemed to be pretty proud of himself too. ~ wanted to make sure that everyone had all the steps that tommy had so nicely explaine(l. ~ then asked Jeremy if he could do the 30th train. He said that he didn't know. felt that he could probably do this if ~ provided a little structure for him. ~ asked him how many hexagons would have five sides. He said in a questioning tone, "two?" ~ nodded and said that this was correct. ~ then asked how many hexa- gons would be in the middle. He wrote something down on paper that ~ could not see and indicated that there would be 28. ~ then asked him how many sides each of the 28 hexagons would have on the perimeter. He responded more confi- dently this time with four. ~ then asked the class how we could write 2 five's and 28 four's. No hands shot up immediately anti ~ glanced at the clock. Where ha(1 the time gone the bell was going to ring any minute. ~ told the students that for homework ~ wanted them to come up with a way to calculate the perimeter of the 30th train anti any other train we could come up with. ~ thought that this would push us toward more formal ways of recording calcula- tions an(l, ultimately, generalizations. Reflecting on Class Later That Day The lesson was all I could have aske(1 from the kids! They found the perimeters of the trains and were even making progress on finding generalizations. ~ have ha(1 this kind of a lesson about five times this year, and it is very exciting. ~ want to see the tape as soon as possible to find other things ~ could have done. The kids were very proud of themselves think anti so was I! Reflecting on Class Several Weeks Later A few weeks after this class I ha(1 the opportunity to share a 10-minute segment of a videotaped lesson with my colleagues at one of our staff (levelopment sessions. ~ decided to show a segment from the pattern block lesson since ~ thought it had gone so well. Although they (li(ln't say so directly, ~ think they felt that ~ was too leading. Maybe they were right. Itis easy to be too leading anti feel OK about it because the kids seem happy. After all, many kids are happy with drill and practice. ~ decided to go back and watch the entire tape again anti see if ~ could look at it objectively. The lesson containe(1 too much whole group teacher questioning anti students explaining anti not enough time for students to stretch and discover independently/collaboratively. ~ won- dered, in particular, what most students really un(lerstoo(1 about Angela's method. Sure many of them answere(1 my questions, but were they just mindlessly applying a procedure that they ha(1 rehearsed? Did it mean anything to them? Although choral response might make kids feel good, it really masks what individual students are really thinking anti what it is they un(ler- APPE N AX

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stand. Just because they could come up with answers to my questions doesn't mean that they really understand or that they have any idea how to apply it. ~ am now left wondering what they really learned from this experience. Reflecting at the En cl of the School Year In early rune, at the end of the year retreat, my colleagues and ~ were asked to make a lO-minute presentation regarding the areas which we thought had changed most over the year. ~ began by showing a clip of one of my fall lessons the one in which Desmond went to the overhead to measure the perimeter of the pattern train and in which ~ assumed control, even moving his hands. ~ told my colleagues, "I'd like to start with the first clip because feel it pretty much sums up how ~ taught at the beginning of the year, and I'd like to show you that ~ really have become less directive than this tape." ~ showed the clip without sound. Attention was drawn to two pairs of hands on the overhead, the large pair (mine) which seemed to be moving the smaller pair (Desmond's). ~ explained, 'you'll see Desmond comes up and ~ am very helpful very directive and move the lesson along. That was a big thing with me to move these lessons and if they didn't get it, I'd kinda help them do so." ~ added that ~ asked many yes/no questions very quickly and did not provide time for students to think. In contrast, ~ then showed video clips from the spring, in which ~ walked around the room, asking groups of students questions that would help them focus their efforts rather than telling or showing them what to do. For me, the differences in my actions and interactions with students on these two occasions provided evidence that ~ had changed. APPE N DIX Transition Catherine Evans and her colleagues continued their efforts to improve the mathematics teaching and learning at Quigley Middle School. They met fre- quently to talk about their work and attended professional development sessions once a month and during the summers to support their growth and development. And their efforts were paying off students were showing growth not only in basic skills but also in their ability to think, reason, and commu- nicate mathematically. At the beginning of the third year of the math project a new teacher joined the faculty at Quigley David Young. Catherine and her colleagues welcomed David into their community. From their own experiences they knew how hard it was to teach math "in this way." But David had something that Catherine and her colleagues did not have initially the opportunity to work beside teachers who ha(1 experience with the curriculum. The case of Davi(1 Young picks up at the beginning of David's second year at Quigley. He has been working with Catherine anti others anti has had one year's experience teaching "this new way." Catherine is now beginning her fourth year of the math project. THE PATTERN TRAINS Part 2 David Young David Young has just started his second year at Quigley Middle School. The job at Qu igley was at first overwhelm ing for David. His mathematics teacher colleagues were implementing an instructional program based on a constructivist view of learning Although such approaches had been foundational to his teacher prepara- tion program, his teaching up to this point

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had been fairly traditional. The schools he had been in for student teaching and his first year teaching did not support innova- tion. But at Quigley the students were not passive recipients of what the teacher dished out, and drill and practice seemed to have a fairly limited role in instruction. Students were actually doing mathematics explor- ing, making conjectures, arguing, and justifying their conclusions. The enthusiasm and energy he saw in his colleagues was invigorating hut also scary. His colleagues all had lots of experience, hut he had almost none. He worried about his ability to he a contributing member of the community and whether or not he would he ahie to teach in a new way. David's fears were put to rest early in his first year. His colleagues were very supportive and understanding They told him "war stories" about their initial experiences in teaching "this way" and how they had helped each other through the tough times. They would see him at lunch, in the morning before school, or just in the hall way and ask 'what are you doing today?" and "How is itgoing?" They woul~give him some suggestions based on what had worked well for them, hut they never told him what to do or harshly judged the decisions he made. Mostly they listened and asked a lot of questions. Over time David felt that he could ask or tell them anything It was, he decided, the perfect place to teach. During his tenure at Quigley, David had been working hard to help students develop confidence in their ability to do mathemat- ics which he in turn felt would influence their interest and performance in the subject. Far too many students, he thought, hated math in large measure because they had not been successful in it. He had talked a lot with Catherine Evans about his concerns. Catherine had been quite open about her early experiences in teaching math the new way (just three years ago) and her misstarts in trying to help students feel successful. David came to believe that developing cony dence as a mathematics doer resulted from facing challenges and persevering in the face of them. The key, Catherine had often said, was trying to find a way to support students in solve a chal- lenging task not creating less challenging tasks for students to solve. Davic! Young Talks About His Class This is the beginning of my second year teaching sixth grade with this new curriculum. The first year was rough- both for me and kids as we tried to settle into our new roles in the classroom. Me as the facilitator anti my students as constructors ofknowle(lge. When things did not go well, my colleague Catherine was always there with a sympathetic ear and a word of encouragement. She is such a won(lerful teacher everything in her classroom seems to always go so well. (She is right next (loor, anti we have a connecting door between our rooms. Sometimes during my free period ~ leave the (loor open anti listen in on what is happening over there.) Although she has repeate(lly sai(1 that it was a long anti painful trip from where she started to where she is today, it is har(1 to believe. ~ guess it is comforting though to know that if she ma(le it, ~ can too. Catherine anti ~ are both teaching sixth grade this year, so we touch base nearly everyday about what we are doing. We are only a month into the school year, and so far we have been working with patterns. Up to this point we have focused primarily on numerical patterns. The new unit that we starte(1 yesterday uses trains of pattern blocks arrange(1 in some geometric sequence. The unit is suppose(1 to help students visualize anti (1escribe geometric patterns, make conjectures about the patterns, (letermine the perimeters of APPE N AX

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trains they build, and ultimately, to develop a generalization for the perimeter of any train in a pattern. Last year this unit did not go well. It was too much teacher talk and too little time for students to think. ~ moved them through the entire set of exercises in one period. ~ felt great because ~ had really "covered" the material, but a week later it was clear that the students hadn't gained much from the experience. When ~ talked with Catherine about it she told me about her first time through this unit three years ago. She said that one thing she learned is that kids need time to think, to struggle, and to make sense of things for them- selves. If you make it too easy for them they wall never learn to figure things out for themselves. This made sense to me, but it was hard not to step in and tell them what to do. ~ was determined, however, to do a better job this time around. The CIass Yesterday my sixth grade class spent some time getting familiar with the pattern blocks identifying the shapes and determining the perimeters of the blocks. Today they are going to make observations about trains of pattern blocks and determine the perimeters for the trains. Basically ~ am just going to follow the curriculum here. It suggests giving students a pattern sequence and having them compute the perimeter for the first three or four trains and then to determine the perimeter of a larger train like 10 or 20. Ultimately the curriculum suggests asking students to imagine that they are constructing the lOOth train and to look for ways to find the perimeter. will see how things go, but ~ hope to be able to follow this suggestion and use large numbers like 1000 so there is no way they can build or draw the trains and count the number of si(les. APPE N DIX Geffing Startecl: The Square Tile Pattern ~ started by building the pattern of squares shown below on the overhead and asking students to work with their partners to find the perimeters of the first four trains in the sequence. Emily imme- diately asked for pattern blocks so she could actually build the trains. This of course started a series of requests to use the blocks. ~ hadn't anticipated this, but ~ had no problem with it either. ~ grabbed a few bags of blocks and dropped them off at the tables of students who had requested them. train ~ train 2 Students started building the trains and quickly seemed to realize that the fourth train would have four squares. They then began to determine the perimeter and record their findings. This initial activity seemed to be pretty easy for students. After about five minutes ~ asked Derek to go to the overhead and show us how he found the perimeter for the first three trains. Using a technique that we had use(1 yesterday when we began exploring the perimeter of the blocks, Derek (new line segments parallel to the si(le of the square as he counted (as shown below), in or(ler to show that he hall counte(1 a particular segment. Once he hall com- plete(1 the count, he recor(le(1 the perim- eter on top of the train. ~ asked Derek what the numbers "4," "6," anti "a" repre- sente(l. He respon(le(1 that"these are the (listance around the outside of the train in units." ~ aske(1 what a unit was, anti he explaine(1 that he ha(1 use(1 the si(le of the square as the unit. The previous (lay we train 3

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1 P=4 P=6 P=8 2 2 3 2 3 4 3 1 14 1 4 6 5 8 7 6 had discussed that fact that we were going to be measuring using the side of the square as our unit. That way we could talk about the number of units without worrying about the actual measurement.) ~ then asked the class what they thought the perimeter of the fourth train would be. Crystal said that she thought it would be 10. ~ asked her how she found it. She explained, "I just built the fourth one and counted the way Derek did." lamal said that he got 10 too, but that he just added two more to the third train. ~ asked him to explain. He said, "When you add on one more block to the train the perimeter only gets bigger by two more units cause only the new piece on the top and bottom add to the perimeter." ~ asked the class if they had any questions for lamal. Kirsten said that there were four sides in every square, so how could the perimeter only increase by two? lamal went to the overhead and explained, "See if you look at the second train there are two units on the top and bottom, and one on each side. When to go to train three and add one more square (as shown below) you still only have one unit on each side cause the sides of the new square are on the inside not on the perimeter." 15 ~ then asked students to take a few minutes anti think about what the tenth train woul(1 look like. ~ wanted to be sure that all students ha(1 time to consi(ler this larger train. ~ know that sometimes ~ move too quickly anti (lon't allow enough wait time for students to think about things. This ten(ls to work against the students who have goo(1 ideas but work at a slower pace. Since ~ have been waiting longer, more students have been involved. ~ started by asking Michele what she thought the perimeter would be. She said she got 22. ~ aske(1 her if she coul(1 explain to us how she got this answer. She in(licate(1 that she ha(1 built the tenth train and then counted. Although this was a perfectly good approach for the tenth figure, it was going to be less helpful when we started considering larger trains. I asked if anyone did it another way. Travis said that he got 22 too, but that he just took ten plus ten plus 2. Although his answer was correct, it was not immedi- ately obvious why he added this set of numbers. I asked him why he did this. He explained, "See when I looked at the first four trains I saw that the number of units on the top and bottom were the same as the number of the train. So in FIG new square APPE N DIX

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train one there was one unit on the top and one on the bottom. In train two there were two units on the top and on the bottom. In train three there were three units on the top and the bottom. So ~ figured that this would keep going so the tenth train would have 10 units on the top and the bottom. Then for all the trains you have to add on the two sides because they never change." ~ thanked Travis for sharing his strat- egy and asked if anyone had thought about it another way. Joseph said that he multiplied the number of squares in the train by 4, then subtracted the sides that were in the inside. ~ indicated that this was an interesting way to think about it and asked him if he would explain. He began, "Well, each square has 4 sides, so in the tenth train there would be 4 x 10 or 40 sides. But some of these are in the inside, so you have to subtract." "How did he know how many would be on the inside?" ~ asked. He explained, "Well, there are eight squares in the inside of the train, and each of those squares had two sides that didn't count and that gives you 16. Then there are two squares on the outside of the train and that each of those had one side that didn't count, so that gave you 18. So 40 - 18 gives you 22 and that's the answer." As he finished his explanation a few hands shot up around the room. ~ asked the class if they had any questions for Joseph. Kendra asked how he knew that there were eight squares on the inside of the train. Joseph said that he had looked at the first four trains and noticed that the number of squares on the inside was two less than the train number the second train had zero squares on the inside, the third train had one on the inside, and the fourth had two on the inside. ~ thanked Joseph for sharing his thinking about the problem with the class. ~ was really APPE N DIX pleased with the two (different generaliza- tions that had been offered and decided to ask one more question before moving on to a new pattern to see if the class could apply these noncounting approaches to a larger train. ~ asked the class it they could tell me the perimeter of the lOOth train. After waiting about 2 minutes for students to consider the question, ~ asked if anyone had a solution. Katherine said that she thought it would be 202. ~ asked her how she figured it out. She said she needed to draw and came up to the overhead. She drew a rectangle on the overhead and asked us to pretend that it was 100 squares. She then continued, "Like Travis said the number of units on the top and bottom is the train number, and then there are the two on the side. So for the 100 train, it would be 100 + 100 + ~ +~." train number 1 train number I commented that this seemed like a really fast way to do the problem. Rather than ask for additional ways to think about this pattern, ~ decided to move on. ~ passed out a sheet of four patterns (see attached) and asked students to work with their partners on pattern ~ on the sheet. In particular ~ wanted them to sketch the fourth train in the pattern, find the perim- eter of each of the four trains, and then to see if they could find the perimeter of the tenth train without building the train. ~ knew the last condition would be a chal- lenge for some, but ~ wanted them to think harder to find another way. 1

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~ Pattern 1 /\ ~ m\ Pattern 2 O O O O Pattern 3 / / \ \ Pattern 4 A a> ~ an> Continuing Work: The Triangle Pattern ~ walked around visiting the pairs as they worked on the new train. Again students seem to quickly see the pattern add one more triangle and count the sides to find the perimeter. ~ observed several pairs starting to build the tenth train and asked them to try to fin(1 another way. ~ sug- gested that they look at the four trains they had built and see if they could find any patterns that would help them predict the tenth train. In a few cases where the students were really stuck ~ suggested that they try to see if they could find a connection between the train number and the perimeter as a few students hall (lone in the last pattern. Once it appeared that most pairs had ma(le progress on this task, ~ asked James to come up and buil(1 the fourth train and (lescribe the pattern. James quickly assembled the triangles, changing the APPE N AX

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orientation each time he added one. He explained, "you just add one more triangle each time and every new one is turned the opposite way of the last one." ~ then asked Catherine what she found for the perimeter of each train. She said that the first one was 3, the second one was 4, the third one was 5, and the fourth one was 6. ~ asked her what the fifth one would be. She quickly said "7." ~ asked her how she did it so fast, and she responded, "After the first one you just add one every time. The fourth train is 6 so the fifth train would be one more." ~ then asked if anyone could tell me what the perimeter of the lOth train would be. lanelle said she thought it would be 12. { asked her how she found it. She said she made a table and looked for a pattern. Since this was the first time anyone had mentioned making a table, thought it would be worth having her explain this strategy to the class. She came up and constructed the table shown below. She explained, "I looked in the table and ~ saw that the perimeter kept going up by one, but that the perimeter was always two more than the train number. So that for train number TO the perimeter would be two more or 12." Train # Perimeter 3 2 3 5 6 Before ~ could even ask if anyone had done it another way, Joseph was waving his hand. He announced that he got 12 too, but that he did it another way. He said that the train number was the same as the number of triangles, just like the squares. He went on, "Since each triangle has three sides, ~ multiplied the number of APPE N DIX triangles by 3. So 3 x TO = 30. But then you have to subtract the sides that are in the inside. It's like the square. You take the number of triangles on the inside. For the tenth train that would be 8. Each of those triangles has two sides that don't count and that give you 16. Then there are two triangles on the outside of the train and that each of those had one side that didn't count, so that makes 18. 30 - ~ = 12." ' Avow," ~ said, "there are lots of different ways to look at these trains aren't there?" ~ was ready to move on, but Darrell was trying to get my attention. He said, "Aren't you gonna ask us to find the lOOth?" That hadn't been my plan, but if he wanted to find the lOOth ~ was happy to oblige. ~ aske(1 Darrell if he wanted to tell us what the perimeter of the lOOth train would be. He said, "It'll be 102. Cause like Janelle sai(l, it will always be two more." ~ asked the class if they agreed with Darrell. ~ saw lots of nodding heads that convinced me that we were indeed making progress. Exploring Three New Pafferns ~ told the class that they would have 15 minutes to work with their partners on patterns 2, 3, and 4. For each pattern they needed to sketch the next train, find the perimeter for all trains, anti (1etermine the perimeter for the lOth train without buil(ling the train. ~ wanted students to have a longer perio(1 of time for exploring the patterns without interruption. ~ figure(1 that in 15 minutes everyone would at least get the first one (lone, and pattern four would be a challenge for those who got that far since it was less straight- forwar(1 than the previous patterns anti that the o(l(1 anti even trains woul(1 be described differently. As students worked on the patterns, ~ again walked around the room observing

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what they were doing, listening in on their conversations, occasionally asking a question, and reminding them that they would need to be able to justify their methods to the rest of the class. The most challenging aspect of the task for most students was finding the perimeter to the tenth train without drawing it. For pattern 2, ~ encouraged them to try to find a way to talk about the perimeter of a figure in terms of the train number. "How are those two numbers related?," ~ asked as ~ moved from group to group. Discussing the Hexagon Pattern After 15 minutes all students had completed patterns 2 and 3. Since there were only 10 minutes left in class ~ thought ~ would have them talk about pattern 2 before the bell rang. ~ started by asking lungsen to describe the pattern and give the perimeter for the first four. She explained that each train had the same number of hexagons as the train number and that the perimeters were 6, 10, 14, and IS. 'what would the perimeter of the next one be?," ~ asked. lames said he thought it would be 24 because the hexagon had six sides and it would be six more. Michelle said that she thought it would 23, because it would only be 5 more because all sides didn't count. ~ asked if anyone had a different guess. Derek said that he thought it would be 22. A number of students chimed in with "I agree!" ~ asked Derek to tell us how he got 22. He said that every time you added a new hexagon, you only added on four more sides. '~he perimeters were 6, 10, 14, and IS. You just keep adding four." ~ asked if anyone could explain it another way. Kirsten said that she thought she could. "Every time you add another hex", she explained, "you just a(l(1 two si(les on the top and two on the bottom." She pointed to the trains on the overhead and continued, "If you look at train two, you have four si(les on the top, four on the bottom, and the two on the ends. If you look at train three you added one more hex which gives you two more si(les on the top and the bottom. That gives you just four more." . . ;` , APPE N AX

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~ asked if anyone had found the perim- eter of the tenth train. Carmen said that she thought it would be 42. ~ asked how she got this. She said that the tenth train would have 20 sides on the top, 20 sides on the bottom, and one on each end. ~ asked how she knew it would be 20. She went on to explain, '~he number on top is double the train number. See, the second train has four, the third train has six, so the tenth train would have 20." ~ thanked Carmen for sharing her solution and asked if anyone had another way. Joseph was again waving his hand. asked Joseph if he used his method on this problem too. He said he did and explained that since each hexagon had six sides you needed to multiply the train number by 6 to get 60. Then you needed to subtract the inside sides which would be 18. So it would be 60 - 18 which was APPE N DIX 42. Kirsten asked Joseph if you always subtracted IS for the tenth train. Joseph said that so far that seemed to work for the squares and the hexagons, but he wasn't sure if it always worked. Kirsten's question was a good one. ~ made a note to be sure to include a pattern for which it would not work, just to push Joseph to consider what was generalizable about his approach and what wasn't. ~ finally asked about the perimeter of the lOOth train. It seemed as though everyone thought they had it this time. ~ took a quick look at the clock. The beD was going to ring any minute. ~ told students for homework to write down what they thought the perimeter of the 100th one would be and to explain how they figured it out. We would start there the next day and then jump right in and try pattern 4.