1. SCRIPT OF EXCERPTS FROM THE BORDER PROBLEM 2. 1:12:21 3. …and can anybody tell me how many tiles you would need to put a border around the outside of this pool. Lulu.
4. [Lulu] I think you’d need 24
5. How’d you get that?
6. [Lulu] You need 5 tiles for each side and you need 4 for all the corners
7. OK, you want to come to the overhead use the overhead and show how you did that? ….What do you guys think? You agree? You disagree? Mike
8. [Mike] I disagree. I think it’s 20.
9. You think it’s 20? Why?
10. [Mike] Cause 5 x 4 is 20.
11. OK. Let’s comment on what Mike said. Leo, what do you think of what Mike said ?
12. [Leo] Lulu was saying that you also have to have the tiles to put at the corners, so that’s another 4 tiles so that makes 24.
13. Did you get that [to Mike]? What’s different about what Leo said than how you were thinking about it?
14. [Mike] He added the corners.
15. [Tiffany] Why did you multiply 5 by 4?
16. [Mike] cause it’s 5………there
17. [Tiffany] Why did you multiply the length by width……….…do you understand what I’m saying? a. [pause]
18. Do you have a comment Mile?
Metamorphosis Teaching Learning Communities: www.metatlc.com 1 Lucy West: [email protected] 19. [Tiffany] I guess I didn’t think you were finding the area, I mean I don’t know, something like that.
20. Yes Mile?
21. [Mike] It wouldn’t be 25; there aren’t 5 sides, there are 4 sides.
22. She’s bringing up the idea that area, length times width…were you thinking of area? … 23. 24. [Mike] I…guess.
25. You think you were? But you just said area would be 5 times 5 would be 25?
26. [Mike] Oh wait, no, no.
27. Keep thinking about it. I think you were thinking about something we are going to talk about anyway. So see if you can make a connection. Does anybody else have a comment about the number of tiles you would need? [notices what Lulu has been doing on the overhead] Oh Lulu, you did exactly what I hoped you would do. Look how she’s setting these up 28. Amber?
29. [Amber] She’s putting the four corner squares…um… in different colors to show that the tiles on each side representing the five tiles on each side plus the corners.
30. 1:15:10
31. What do you think is the part of the border that people might forget? 32. Pascal?
a. [Pascal] The corners
33. Why do you think people might forget the corners? 34. Claire
35. [Claire] cause when you look at the drawing you only think about the inside and you don’t really think about how to create the outside. So you only think about what’s inside.
36. What’s inside the square?
Metamorphosis Teaching Learning Communities: www.metatlc.com 2 Lucy West: [email protected] 37. [Claire]You only think about the inside, but you don’t think about, like, if you were actually to create a border. You just think it would be the way it looks. You don’t think about how you actually create the outside.
38. OK. Who wants to add on to that idea? Chloe
39. [Chloe] I think people think more about the perimeter…
40. Aha…
41. [Chloe] and they don’t include the corners because when you count up a perimeter…so I think people take border and put it with perimeter and don’t include the corners, because when you only think perimeter you only think how many units across and down and how many units are around it, not surrounding it.
42. Can you connect your idea about perimeters to something else someone in here already said?
43. [Chloe] Mike? He wasn’t including the corners because he figured if you were just going to border the sides of a pool you wouldn’t include the corners because you don’t count that in a perimeter.
44. OK, then Mike does that clarify does that clarify what you were thinking of in the first place?
45. [Mike] Yeah.
46. So which measurement were you thinking of?
47. [Mike] Just 5 times 4
48. Which is equal to the area, the perimeter..??
49. [Mike] The perimeter
50. OK
51. 1:17:08 [4:45 elapsed]
Metamorphosis Teaching Learning Communities: www.metatlc.com 3 Lucy West: [email protected] 52. 53. Border Problem SEGMENT 2 54. 1:18:57 55. Is the number of tiles for the border greater or less than the area of the pool?
56. Greater.
57. OK. So now what is it going to be?
58. [Kristen] 36?
59. How’d you get that
60. [Kristen] Cause I counted the amount going down and the amount going across and it was 6.
61. It was 6? You want to come up and count for us? 62. Let’s watch him and see.
63. [Kristen] Oops
64. What’d you get?
65. [Kristen] 7 and 6.
66. 7 and 6?
67. [Kristen] Um hmm.
68. Let’s think about I again. Show us how you counted. Point to each square and count out loud so we can hear you.
69. [Krsten] 1,2,3,4,5,6,7..
70. OK, let’s stop there. The length of this side here is seven units. 71. Now you want to take the length of this side?
72. [Kristen] 1, 2….start from here?
73. What do you think?
Metamorphosis Teaching Learning Communities: www.metatlc.com 4 Lucy West: [email protected] 74. [Kristen] Yeah.
75. OK.
76. [Kristen] 1,2,3,4,5,6.
77. So what’s the length of this side?
78. [Kristen] 7
79. So how you going to find your area? 80. [wait time] 81. If you need help you can ask somebody in class. 82. [wait time]
83. [Kristen[ It’s 49.
84. (Turns to the class) What do you guys think? Would it be 49?
85. [yes]
86. 1:20:44
87. So you want to tell us what you did the first time that was different than how you figured out the second time?
88. [Kristen] Oh, when I was looking on there I guess I got mixed up. Like I counted one yellow for two like you know from far away when you’re counting sometimes you skip, so I guess I skipped a cube…
89. So you just miscounted?
90. [Kristen] Yeah
91. OK
92. [Kristen] And then I counted this way, but then I didn’t count this one for going this way.
93. Aha, so you counted one side of this red-corner one, but not the other side of it when you were going down?
Metamorphosis Teaching Learning Communities: www.metatlc.com 5 Lucy West: [email protected] 94. [Kristen] Yeah.
95. OK. Natalie you want to add on?
96. [Natalie] Yeah, I got the same answer by adding 24 plus 25 because you knew that there were 24 tiles around the square of 5 by 5, so 25 plus 24 is 49 so I agree with Kristin.
97. Where’d you get the 24 from? [Natalie] The 24 is the number of tiles around the like, the border, around the square in the middle.
98. Ah. What do you guys think of that? Could somebody add on to Natalie’s thought? 99. Claire.
100. [Claire] Natalie found the outside of the inside pool, and then she took that, that was 24, she counted the number of tiles and she got 24. Then she counted the outside amount of tiles needed to build the outside pool border, and she added 24 and 25 and she got 49.
101. OK, ____, you want to say something about that?
102. [shakes head no]
103. OK I thought you were putting your hand up.
104. [Girl} No I just wanted to add why Kevin would have miscounted, but…
105. Go ahead
106. [Girl] No, it’s OK.
107. OK, so we’ve figured out….yes, Tiffany?
108. [Tiffany] Would the…don’t you know that the length and the width are the same would you still use that to see if they were the same, like since the length with the border, 7 and 7. No, because he put 7 and 6. It’s so easy to think that it’s the same, length and width? …no because, he made a mistake. Kristen said the length and width were 7 and 6.
109. Yes
110. [Tiffany} He could use 5 and 5 just to say that the border would still be the same, length and width, 7 and 7. So…All right. It’s hard to explain. Metamorphosis Teaching Learning Communities: www.metatlc.com 6 Lucy West: [email protected] 111. You want somebody to try to add on?
a. [Tiffany] Yeah. Lulu
b. [Lulu]…saying that you know that the pool is a square, so and since it’s a square the sides are all the same, 5 by 5, and so you’d know that since the pool’s a square then the tiles would have to make a square too. So you’d know that the sides couldn’t be 7 and 6, they’d have to be 7 and 7 or a number that’s the same. 112. 1:24:05 113. 1:24:17 114. …to get at what you were talking about. (speaking to Natalie) You were saying that if you know that the number of the tiles on the border is 24 and the number of tiles in the area is 25, you can add those two things together. How did you know that the number of tiles in the border was the same thing as the area of the border? 115. [pause] 116. …because to add them together to get the total area you have to think they’re both area.
117. [Natalie} I sort of looked at the diagram. It’s filling up the space, it’s taking up the space, so it was taking up that area, like the border.
118. So each square has a measurement of what
119. [Natalie] One unit
120. By
121. [Natalie]One unit
122. One unit by one unit. So one square here, one tile has an area of one. That’s an excellent strategy that Natalie came up with. When you’re doing you’re group work I want you to keep in mind the idea of dividing the area of the whole into different parts and seeing if you could do anything with that. 123. 1:25:16 [6:00 elapsed for segment 2]
Metamorphosis Teaching Learning Communities: www.metatlc.com 7 Lucy West: [email protected] 124. 125. SEGMENT 3 126. 1:27:39
127. Is the perimeter the same as the number of tiles we got for the border?
128. Here’s the number of tiles we need to make the border of the pool. And here’s the number of tiles in the perimeter. Are they equal to each other?
129. [Nicole] I don’t think they would be equal because Mike first said that the perimeter, I mean the border would be 20. He got that because the formula for the perimeter is adding up all the sides. So what Lulu did, she had to like put in for the corners. It’s not the same. It’s extra.
130. Which one has some extra?
131. [Nicole] The border.
132. Why?
133. [Nicole] you have to put units at the corners.
134. It that what happened?
135. [Nicole] Yeah.
136. The number of tiles we needed was what?
137. [Nicole] 24
138. But the perimeter of the border and the pool was what?
139. [Nicole] 28
140. So which one has an extra?
141. [Nicole] Oh, the perimeter of the pool.
142. How’d that happen?
143. [NicoleI have no idea.
Metamorphosis Teaching Learning Communities: www.metatlc.com 8 Lucy West: [email protected] 144. Well, think about that. How’d that happen? [pause] 145. You know. Just look at it.
146. Only one person has an idea [there’s a hand up]
147. [Girl] it be because of the corners and you have two sides. So you have to add an extra for …uh, to the perimeter.
148. Lulu, do you agree with what she just said?
149. [Lulu] Basically; I was just going to say that since the reason that it’s 24 tiles and 28 units is because for 20 of the tiles only one side is showing, like on the very outside. And then for all of the corners two sides are showing. And since there are four corners, four times two is eight, so 20 plus 8 is 28.
150. Excellent. Anyone want to add on to that? 151. [no takers] 152. When you’re counting the number of tiles, you’re counting each tile once. Correct? When you’re counting the perimeter, does each tile only get counted for one length? [wait time]
153. [Leo] It’s two times.
154. Exactly. The corners get counted twice. And how many corners are there?
155. [Leo] Four
156. So how many extra units are there?
157. [Leo] Eight
158. Well…
159. [Leo] Four
160. Four. So would be why we use 24 or the number of tiles and plus four more because you count four of them twice.
161. So, we calculated the number of tiles we need to border a pool with a side length of five. Now what I want to do – see if we can move from the specific example to something more general. Can we find an efficient way to calculate the number of tiles
Metamorphosis Teaching Learning Communities: www.metatlc.com 9 Lucy West: [email protected] you would need to border a square pool no matter what the side lengths of the pool were? How would we go about doing that? 162. Why don’t you talk to the person next to you for one second to see if you could figure out a way we could come up with something that would tell us the number of tiles no matter what…
163. 1:34:15 [6:24 elapsed for segment 3]
Metamorphosis Teaching Learning Communities: www.metatlc.com 10 Lucy West: [email protected]