[metadata]
Title: Lewis_3-22-04
Date: 3-22-04
Time: unknown
Locale: 6th-grade classroom
Activity: a math class of type "discussion"; the topic is "integers"
Recording type: video
Duration: 70 minutes
Principal Investigator: Cathy O'Connor
Researcher: Cathy O'Connor
Transcriber: unknown
Videographer: unknown
Participants:
person: T
name: Greta Lewis
role: teacher
sex: F
age: adult
person: Tonya
role: student
sex: F
age: child
person: James
role: student
sex: M
age: child
person: Kathleen
role: student
sex: F
age: child
person: Monica
role: student
sex: F
age: child
person: Jaime
role: student
sex: M
age: child
person: David
role: student
sex: M
age: child
person: Duane
role: student
sex: M
age: child
person: Margaret
role: student
sex: F
age: child
person: Jacki
role: student
sex: F
age: child
person: Rhonda
role: student
sex: F
age: child
person: Gabi
role: student
sex: F
age: child
person: Esteban
role: student
sex: M
age: child
person: S
name: Student
age: child
person: Intercom
person group: Ss
name: Multiple students
age: child
[/metadata]
[body]
1. T: Okay, we're actually going to do work on something new today.
2. S: [xxxxxxxxxxxxx]
3. T: No, I didn't ask you to get homework, I asked you to take out a piece of paper and head it [5 sec]. And the topic is going to be "Integers". [30 sec] Looks better like that. [12 sec] Ah. I can't get it. [10 sec] Look at the spelling of that word again. Make sure you spelled the word correctly. Integers.
4. Ss: Integers.
5. T: I N T E G E R S. Okay, so we are going to be starting a new topic which is integers, okay? [10 sec] Okay, um and these are numbers that have positive signs and negative signs, you've seen those types of numbers before? No, yes, maybe?
6. Ss: Yes.
7. S: What did you say?
8. T: Positive signs and negative signs in front of the numbers, okay. The first thing we're gonna do is try to get the vocabulary down. So, of course it would help if we plugged it in. [15 sec] Okay? Okay. So, this number here is called "negative seven", okay? Sometimes it looks like it's a minus sign in front of it but we don't say "minus seven", we say "negative seven", okay? so can you write that down?
9. S: Copy that negative seven--
10. T: Uh hum ... and this would be "positive nine". [5 sec] Okay, so I'm gonna ask you to think about his question for a minute why do you think that we have signed numbers? Why do we have signed numbers, numbers with signs in front of them? Why do we need especially negative numbers?
11. S: [xxxx] negative [xxx].
12. T: Just thinking about it for a minute. Okay, I want you to take about thirty seconds and talk to the person next to you, unless you are in a group of three you can talk across the table, okay?
[group work]
13. T: How about, you might want to sit over on side, it'll be easier to see us since no body else is here, okay? Go ahead you can talk while you are moving, of course we're gonna get rid of the gum with a color change.
14. S: Okay.
15. David: I think cause uh when the [xx] seven subtract seven I think it's harder if it's negative to su negative to subtract because you know how we don't know yet, we didn't learn it yet, so we don't know how to do into negative, we cannot subtract seven from nine just like that negative, so I don't think it's harder, it's harder to subtract.
[/group work]
16. T: Okay, so why do you think that we have negative numbers? Any ideas that you talked about in your groups? Only one? No other ideas, never heard the term before, okay. Gabi what do you think?
17. Gabi: Okay, for negative numbers, negative neg-- seven, hold on, negative seven means that seven more wholes are needed to make a whole.
18. T: Say it again.
19. Gabi: Negative seven means seven more wholes are needed to make a whole.
20. T: Seven more wholes are needed to make a whole.
21. Gabi: Yeah, it's kind of like ... if you have one, it's kind of like you owe.
22. T: Okay, you owe what?
23. Gabi: It's say let's like, I'm gonna put it like this, if you need a dollar, dollar and you owe seven dollars, you have to get eight dollars.
24. T: Did we understand what Gabi is talking about?
25. Ss: No.
26. T: Say it again Gabi.
27. Gabi: Negative seven means you need seven more wholes to make a whole. Like how much, like if you owe seven wholes.
28. T: Okay, so if we owe money, we can talk about that as being negative, a negative amount, does that make sense to you.
29. Ss: No.
30. T: If we owe money, do we have the money?
31. Ss: No.
32. T: We don't have the money, but we owe it, and I need to know how much I owe.
33. Monica: So, but that's why I don't understand it, what does that have to do with the negative seven? Like so like if um so like if you're adding uh like a regular number with a negative number and then you get another negative number what, I don't understand.
34. T: Okay, so it's pretty confusing, right? Okay, let's start from the very beginning, negative numbers were started back in the sixteen hundreds when people started doing a lot more trading. Buying and selling, okay? So, if I went into a store and I wanted to buy something that was ten dollars and I only had five dollars in my pocket, would I be able to buy that item?
35. Ss: No.
36. T: I wouldn't be able to buy that item, why not?
37. Rhonda: Because you don't have--
38. S: You only have five dollars and you need five five more dollars
39. T: Okay, once again.
40. Rhonda: You only have five dollars and you need five more dollars--
41. T: And I need five more, so if the shop keeper allowed me to take that item home, I would've paid five dollars for the item, but I would
42. Rhonda: Owe
43. T: Owe five dollars, okay? So, when I owe money, it's called a negative amount of money. Does that make sense? No? David?
44. David: Yeah?
45. T: Can you explain it in your own words?
46. David: Um like let's say it'll be like, can I use your example like put?
47. T: Uh hum, unless you want to use one of your own.
48. David: Alright, let's say I need to buy an item that is fifteen dollars and I only had like fourteen dollars and fifty cents, like let's say I will need fifty more cents, fifty more cents to buy the item, so you owe the the cashier fifty cents, so you will a need fifty cents which is neg-- negative number.
49. T: Okay? Jaime?
50. Jaime: Yeah.
51. T: Can you explain it?
52. Jaime: Um same thing that he said or in a different ?
53. T: It, either one, which would you rather do, you're gonna have to talk loudly cause of the heat, okay?
54. Jaime: Like um if pretend you're gonna buy something, and um it costs like three dollars and then you have two dollars, you're going to owe one dollar.
55. T: What did Jaime say? I though you were gonna move over that side so you can see better, you too. Jacki?
56. Jacki: Um he said that if something costs three dollars and you have two dollars, then you have to owe one dollar.
57. T: And we would owe one dollar, okay? So, that doesn't happen so much today when we go in to stores, right? People don't usually say oh if you only have five dollars that's okay, I'll get you the ten dollar item and come back tomorrow and give me a five, doesn't happen so much today, but think way back in the sixteen, seventeen hundreds even into the early eighteen hundreds, shops were in small towns, right most everybody knew each other in the town, so if you, the shop keeper said okay you owe me five dollars, he knew who you were, okay, and he would keep a list, he would keep probably a book, okay, of people who owed money and to show that you owe the money, he would put down a negative sign in front of it so he would know know that you owed him five dollars, okay? Does that make sense?
58. Rhonda: But how come in the batteries there are minus um and plus for what what--
59. T: Okay, so in batteries we also have we have a plus sign and a negative sign okay, that has to do more scientifically with um electrons, I don't know that you talked about those, have you?
60. Ss: [xxxx]
61. T: Mr. Dervin mentioned it. So, you may be doing some work with negative in science as well if you want to talk more about that, okay, we're going to stick with just the mathematical right now, okay? Any other time you might have heard about negative numbers, besides shopping and owing money? [5 sec] Tonya.
62. Tonya: In the temperatures.
63. T: What?
64. Tonya: In the temperatures.
65. T: In the temperatures, good. Have you ever seen it written in the temperature with a a negative sign in front of it?
66. Ss: Yeah.
67. T: And what does that mean? Like remember when we had those really cold days and we didn't have school, what does that mean? Duane.
68. Duane: Like the temperature like below zero.
69. T: Below zero, even colder than zero degrees it got even colder and in order to write that in a way we'd all understand then the meteorologists write a negative sign like a negative ten, okay, that means below zero, good. Okay, so you're getting a little bit better idea of negative numbers are.
70. S: Not yet.
71. T: Not yet, well that's right, we are learning it, right? Okay, so um right now, I am going to put a list up here and I want you to just look at it, don't do any talking, just look. At these two lists integers and not integers. Now, just take a few seconds and study the lists, see what you notice about them [10 sec]. Okay, I want to just talk to the person next to you a minute about these lists. Esteban you need to talk with girls too.
[group work]
72. Monica: If you look in the not integers, they you can't you can't be integer if um if if it's a fraction or decimal.
73. Rhonda: Yeah cause that one doesn't have decimal numbers in them.
74. Monica: Yep, so that so that means that only number can only--
75. Rhonda: Whole numbers whole numbers
76. Monica: Yeah can be integers.
77. Jaime: Uh hum
78. T: So, what are we doing?
79. Jaime: [xxxxx]
80. T: Yeah
81. Jaime: figure it out
82. T: right
83. Jaime: Is that thirty five or three point five?
84. T: Thirty five.
85. Jaime: I think [xxx]
86. T: Talk to Gabi
87. Jaime: [xxxxx]
88. T: David, why don't you um move over here, so Esteban will have somebody to talk to, okay?
[/group work]
89. T: Okay now, I want you to keep that question in your head for a minute, okay and I want you now to look at the list and see if you can add a number to either of these lists for me. Can you add a number, any number, into this list or into this list.
[group work]
90. Tonya: We can add in not integers eight point five cause [xxxxxxxx].
[/group work]
91. T: Okay who thinks they can put new number in one of these lists, and tell me why. Margaret
92. Margaret: Um thirty five in not integers.
93. T: Thirty five?
94. Margaret: Uh hum because um it would in that because it doesn't have a minus before thirty five.
95. T: So, what did Margaret say? Maddie
96. Monica: She said you put in not integers because it doesn't have a minus before thirty five.
97. T: And do you agree with that?
98. Monica: I disagree.
99. T: Why?
100. Monica: Because I think that you should put it in integers because um I think that in integers you can't have a decimal or fraction so I think that you should thirty five in integers and three point five in not integers.
101. T: Margaret, what do you think about that? Did you understand what Maddie said? So what should you do?
102. Margaret: Can you repeat that.
103. Monica: Alright. I disagree, I think you put thirty five in integers because um because because um because um cause if you have an integer number you can't have it can't be a decimal or a fraction, so in if the number is is is not integers it can be a fraction or decimal so I think that you should've put thirty five in integers and three point five in not integers.
104. Margaret: I don't see three point five.
105. Monica: No but, just another fraction, that's just one you can put in not integers.
106. T: So, you are saying the whole number thirty five shouldn't be here, it should be over here? But if she changed it to three point five--
107. Monica: Yeah three point five should go there.
108. T: What do you think about that?
109. Margaret: I disagree, because I think thirty five should be in not integers because um it's not saying minus thirty five, it's just like a whole.
110. T: What did Margaret just say? Can you hear her back there? Tonya?
111. Margaret: I think thirty-five should um go not integers because thirty five is [xxx] thirty five.
112. T: Kathleen?
113. Kathleen: I didn't hear her.
114. T: Come on.
115. Margaret: You should--
116. T: Louder
117. Margaret: You should put thirty-five in not integers because I think because it's not saying minus thirty-five, it's just thirty five wholes.
118. Kathleen: She said that um thirty-five in not integers I think she said, because um it doesn't seem and I couldn't hear with the rest of her.
119. T: Because it doesn't have--
120. S: It doesn't have minus
121. T: A negative sign in front of it. What do you think about that Jaime?
122. Jaime: I disagree with them. I disagree with her because well I agree with Maddie and disagree with her.
123. T: You agree with what Maddie said and disagree with what Margaret said. Okay, go, why?
124. Jaime: Because um if she said, if you look at integers it only has whole numbers it doesn't have decimals or fractions and the not integers it has a decimal and fractions.
125. T: Okay, do you agree with that, whole numbers here, decimals and fractions here? Well there is something else that's confusing them, Rhonda.
126. Rhonda: I agree with them.
127. T: Okay, but Margaret talked about the sign, and she thought thirty-five should go in the not integers because it doesn't have a negative sign, do you agree with that?
128. Rhonda: The seven point two three five has um negative sign and it is still in not integers.
129. Margaret: I think your list is mixed up!
130. T: You think my list is mixed up? How dare you? Hehh. Actually, my list isn't mixed up.
131. Monica: Miss Lewis
132. T: So, now let's think about what Rhonda said.
133. Monica: Miss Lewis--
134. T: Not only is there a negative sign here
135. S: It's a decimal
136. T: It's a decimal sum number, with a negative sign, what about this one?
137. Ss: It's a fraction
138. T: With a negative sign?
139. Ss: With a negative sign.
140. T: What about in this column? Kathleen, what about this column with integers, what do you notice about them?
141. Kathleen: That they all regular numbers.
142. T: Pardon me?
143. Kathleen: That they all regular numbers.
144. T: They're all?
145. Kathleen: Regular numbers.
146. T: What do you mean by regular numbers?
147. Kathleen: They don't have, they like they're not like a fraction or um a decimal, they're just normal numbers.
148. T: What numbers.
149. Kathleen: They're just normal numbers
150. T: Normal numbers. Normal numbers, regular numbers, whole numbers, is that what you mean, whole numbers? Okay. Alright, let's try another number, so we don't agree that this should go here?
151. Monica: No, we don't.
152. T: Okay, so let's cross that.
153. S: I think
154. T: What about another number, can you come up with a new number, Esteban.
155. Esteban: Three point five, three point five.
156. T: And which column do you want to put three point five in?
157. Esteban: Not integers.
158. T: Not integers, why did you choose to put it in not integers?
159. Esteban: Because it's a decimal not a regular number,
160. S: Not a whole.
161. Esteban: it's not a regular whole number.
162. T: Not a regular.
163. Ss: Whole number.
164. T: Whole number, what do you think about that James, do you agree or disagree that three point five should be in the not integers column?
165. James: I agree, I agree because um it's like it's kind of like a pattern.
166. T: I am sure they can't hear you cause I can hardly hear you. Say it again.
167. James: I don't I don't understand this.
168. T: Well, what did Esteban just say, why did he put three point five in not integers?
169. James: Because it's not just a whole number, it has a decimal.
170. T: It's got decimals, okay?
171. Monica: Oh miss Lewis?
172. T: Maddie?
173. Monica: And because in in each one like there is a pattern because in in this one it goes decimal number, it goes it goes no dec, it goes no negative number, negative number, no negative number, negative number, no negative number, so in the other one it goes not no negative number, negative number, negative number, no negative number, so I think that it should go in a pattern.
174. T: Do you agree with that, that there is a pattern? Positive, negative, positive, negative, so this one has to be positive?
175. S: Yeah.
176. T: So, so I have to add a positive number in here? Rhonda?
177. Rhonda: It could be, it could be cause it's the um an um decimal number so you could still keep it in the, cause when I did at first is I looked at the columns first and I saw that had wholes, and that had decimals and fractions number, so I think that decimal and fraction numbers go there and--
178. T: And it doesn't matter if it's negative or positive? Do you agree with that David? Can I put another positive number do you think down here now?
179. Ss: Yeah.
180. T: As long as it's what?
181. S: As long as it is a decimal or fraction.
182. T: A decimal or a =
183. Ss: Fraction.
184. T: = fraction, so give me one, Gabi.
185. Gabi: Um I think that's negative thirty-eight point nine.
186. T: Agree or disagree? Jacki thirty-eight point nine, agree or disagree, is that a not integers or integers?
187. Jacki: I think it's in not integers.
188. T: Why?
189. Jacki: Cause it has a decimal point.
190. T: Okay, Duane give me another number.
191. Duane: Like seventy-eight point ... point two.
192. T: Seventy eight point two, okay, where do you want us to put it, in which column?
193. Duane: Not integers.
194. T: Not integers, agree or disagree with that?
195. Monica: I disagree.
196. T: Why?
197. Monica: I think it should be a fraction.
198. T: Do you think that this won't fit here?
199. Monica: Well I think I think it will but I think that it should a fraction.
200. T: Why?
201. Monica: Because--
202. T: Because of the pattern?
203. Monica: Yes
204. T: Okay, it doesn't matter any pattern I just randomly wrote these numbers down, okay? So, it could be a fraction, can you give me a number with a fraction in it?
205. Monica: Um um eighteen and two fifths.
206. T: Eighteen and two-fifths, which column do you think eighteen and two fifth should go in?
207. Monica: Not integers.
208. T: Not integers, okay, but we have nothing in the integer column, any ideas of what we could put in the integers column. Jaime?
209. Jaime: Twenty-seven.
210. T: Twenty-seven, Margaret?
211. Margaret: Thirty-five.
212. S: But we have it.
213. Margaret: Uh.
214. T: Tonya?
215. Tonya: Minus thirty.
216. T: Negative thirty
217. Tonya: Negative thirty.
218. T: Okay, what about negative thirty.
219. S: Yeah?
220. S: That that fits
221. T: Will that fit?
222. S: Yeah.
223. T: How about over here? I see we've got only positives.
224. Monica: You have to oh I know
225. T: Can we only add positive numbers to this list?
226. Ss: No.
227. T: James
228. James: We can add negative.
229. T: Okay, give me a negative number we can put in that list.
230. James: Negative um twenty-eight point two.
231. T: Negative
232. James: Twenty-eight point two
233. T: Twenty-eight point two, okay? Now what I want you to do is go back to the question I asked you originally, okay? Look at these two columns and I want you now to think about this If I were to assign you to write a definition for a an integer, if I asked you to give me a definition for an integer, could you and your partner come up with one.
234. S: Yes.
235. T: Just the person next to you, try to do that now, okay?
[group work]
236. David: Whole numbers, whole numbers that
[/group work]
237. T: When you come up with one write it down, when you and your partner have come up with one, write it down.
[group work]
238. David: Whole numbers that can be positive
239. Esteban: It can't be fraction or decimal number
240. David: Yeah, it can be positive
241. Monica: So we got a number that can be positive or negative but can't be a fraction or decimal.
242. David: Fraction or decimal.
[/group work]
243. T: Okay, now I want you to stop, I want you to stop for a minute and I want you to share your definition with the people across from you, okay? Which means I would like Kathleen and Tonya to come and sit up here with your things, bring your papers too.
[group work]
244. Rhonda: Integer is a whole number that can be positive and negative =
245. Monica: Yeah
246. Rhonda: = it can't be a fraction
247. Monica: Positive or negative, it can't be positive and negative =
248. Rhonda: Oh can can be positive--
249. Monica: = or negative, but can't be a fraction or decimal
250. Rhonda: [xxxxx] positive or negative, fraction or a decimal.
251. Monica: That can't be [xxxx]
252. David: Whole numbers that can be positive or negative and it cannot be a fraction or a decimal.
253. T: Okay? Did you write one down?
254. S: Am I supposed to write it?
255. T: Uh hum.
256. Tonya: Miss Lewis, mine just integers have whole numbers and a negative sign.
257. T: Did she agree with that?
258. Kathleen: Yeah, I wrote the other part, I only I only [xxx] that it's integers are whole numbers.
259. T: Integers are whole numbers? So every whole number is an integer, does it matter if it's negative or positive?
260. Kathleen: No.
[/group work]
261. T: Okay, let's go, James, what did your group come up with?
262. James: Um Integer
263. T: Integer
264. James: Integer is a =
265. T: Shhhh
266. James: = integer is a whole number that is either a negative or a positive.
267. T: Number.
268. James: Number.
269. T: Say it one more time.
270. James: A integer is either a whole, I mean is a whole number that is either a negative or a positive number.
271. Monica: I agree.
272. T: Say it again.
273. Monica: I said I agree.
274. T: Say what, say it.
275. Monica: Um that a integer is a whole number that can be positive or negative but can't be a fraction or a decimal number.
276. T: But cannot be a fraction or a decimal number.
277. Monica: Or a decimal number.
278. T: So, you are adding on to James's? Okay. Gabi, what did you have?
279. Gabi: The same thing, exactly the same thing.
280. T: Could you say it please?
281. Gabi: Integers are a whole number that can either be a positive or negative number.
282. T: How about your group?
283. Jacki: Um I wrote a integer is a whole number or a I mean is a negative number or a positive number with the a um negative sign, no wait,
284. Duane: With negative sign before it without a decimal or a fraction. [10 sec]
285. T: Jacki?
286. Jacki: A integer is a whole number with a whole number or without a whole number with the negative sign.
287. Ss: What?
288. T: Try it again.
289. Jacki: A integer is a whole number or without I mean with a whole number with a negative sign.
290. T: Or a positive sign? Could it have either one?
291. Jacki: Yeah.
292. T: Okay, I think yours is a little confused now, see if you can rewrite it, Tonya.
293. Tonya: Int-- integers have whole numbers or negative and positive.
294. T: Okay. So, we we're agreeing that integers need to be whole numbers and they can be either positive or negative, right? Okay, um I have a question, what about the number zero?
295. Monica: It could be, ooh--
296. T: What about the number zero?
297. Monica: Miss Lewis I know I know.
298. T: Jaime?
299. Jaime: It is an integer.
300. T: It's an integer, why why do you say that?
301. Jaime: Because it's it's not a whole number, but um it's it's not a fraction or a decimal number either.
302. T: It's not a whole number? Is zero a whole number?
303. Ss: Yeah.
304. Jaime: It is not a a um a fraction or a decimal.
305. T: Okay, it's not a fraction, it's not a decimal,
306. Jaime: It's a whole number.
307. T: It's a whole number.
308. Jaime: So, that why
309. T: Okay? James
310. James: I think it's both because if if it's a decimal it will be the same, it will be like zero point.
311. T: Zero point zero? But then wouldn't these be the same too twenty seven point zero? Okay, so as long as we can put the point zero after it, then it should stay on the integer column and zero would be zero point zero? Okay, so we're gonna write this definition down um and I want to talk to you for a minute about this word, in in math we would call it the set of whole number, a set , do you have any idea what set means?
312. S: It's a group.
313. T: It's a group that has something in common, okay? So, let's write that down, integers ... Set of
314. Monica: [xxxx]
315. T: whole numbers including zero, right? [5 sec] So it's a set of whole numbers including zero let's put that in parenthesis [8 sec] and integers ... can be either, positive ... or negative. Okay so it's the set of positive and negative whole numbers in other words, including zero, okay? Alright. [8 sec] Have you seen one of these before?
316. S: It's a line.
317. T: It's a line?
318. Monica: It's a number line.
319. T: It's a number line, okay [8 sec] What do you notice about his number line?
320. S: It has a lot of lines
321. T: Hum?
322. David: Intersecting lines.
323. T: Where are the intersecting lines?
324. David: on there.
325. T: What lines are intersecting?
326. David: The little ones?
327. T: The little ones, they intersect because they're vertical
328. David: [xxxxx]
329. T: And the big line is horizontal, okay. What else do you notice about this line? Anything in particular about these little lines?
330. David: Some of them are
331. Tonya: It's divided into fourteen lines.
332. T: It's divided into fourteen there are fourteen of them there, okay. Anything else?
333. S: It has two arrows.
334. T: It has two arrows there. Why does it have two arrows?
335. Monica: Because the numbers you put at the end that means that they could go on.
336. T: They could go on to infinity, right? In either end direction, okay, what else about why did I put these lines where I did? What do you notice about these lines?
337. Monica: Oh because they they have the certain amount of spaces between them.
338. T: Good, they have the certain amount of space between them, those are called "intervals", did we talk about intervals before?
339. Ss: No.
340. T: Okay, so when we're writing numbers on a number line we want to make sure we keep the same intervals or spacing between them, okay. [24 sec] What about this number line now?
341. Tonya: One side it has negative numbers and the other side has positive numbers.
342. T: How do you know these are positive numbers?
343. Tonya: Because they don't have um a negative sign.
344. T: Right, it doesn't have a negative sign and we're so used to using positive numbers that we don't have to write positive in front of them every time we write them, okay? But we do need to write a negative sign in front of numbers when we're talking about negative numbers, okay? What else can you tell me about this number line? ... Rhonda.
345. Rhonda: It goes um it goes from like bigger number and then it goes down like it goes from six to four, so like six, five, four, three, two, one and then they go um past the neg the negatives and then the positives go um zero, one, two, three, four, five, six, seven, so it goes from, negatives go from small from the big to small, then the other positives go from small to big.
346. T: So, from the left the negative numbers the numbers are are bigger and they go smaller till then get to zero and then
347. Rhonda: They get um then the neg the positives get bigger, I mean get smaller and negatives get bigger.
348. T: Okay, Duane.
349. Duane: I've got a question though.
350. T: Okay.
351. Duane: You know how you put the zero right there, doesn't it have to have a negative sign?
352. T: The zero?
353. Duane: Yeah.
354. T: Good question, do you think zero needs a negative sign?
355. Ss: No.
356. T: Jacki.
357. Jacki: Um I think it doesn't because it's in the middle, so it could be positive or a negative.
358. T: What did Jacki just say?
359. S: I couldn't hear her.
360. T: What did Jacki just say James?
361. James: She was saying we don't need a negative sign because it's in the middle, it could be in the middle.
362. T: Cause it's in the middle and it's it can be either positive or negative, yes?
363. Kathleen: I think that that um the number line it's like it's like um like a temperature on the on the news because um on the temperatures, the the weather guy, he it's like a normal number like forty degrees then after it gets below zero, you see like minus one minus two, minus three.
364. T: Good, okay, so that means that the temperature is getting?
365. Kathleen: Smaller below zero.
366. Ss: Warmer.
367. T: So how does it feel?
368. Ss: Colder.
369. T: Colder, as you get away from zero, as you get away from zero going to the left it gets colder, as you get away to from zero going to the right it gets?
370. Ss: Hotter.
371. T: Hotter, or warmer, James.
372. James: That that looks like the one from the test.
373. T: I can't hear you.
374. James: It looks like the one on the test.
375. T: From the pretest, yeah, it does look like it okay, um as we move to the right from zero to the right we have positive number, and from zero to the left are negative numbers, and as Kathleen just said these numbers are, it's getting hotter, these numbers are growing in size, okay, positive numbers are moving to the right and they're getting larger, okay. What do you think about negative numbers then?
376. Esteban: Getting smaller.
377. T: They're what? Getting smaller?
378. Esteban: Colder.
379. T: They're it's getting colder, okay, or if you think about didn't we talk about money too, we said we talked about negative numbers, does that make sense Jaime? If we get further away from zero and we put a negative number in front of it, what does that mean?
380. Jaime: [xxx] that you owe.
381. T: We owe more so we actually have less, is that what you? Okay, and what did you just talked about Esteban?
382. Esteban: That the numbers get smaller, that the numbers gets smaller.
383. T: The numbers themselves are getting?
384. Esteban: Smaller.
385. S: Bigger.
386. T: The number itself is getting to be a bigger number, but the amount is =
387. S: Colder.
388. T: = colder, okay, does that make sense, we're gonna keep talking about this cause this is a really tough concept to get, okay. What I want you to do right now is to draw your own number line and extend it from negative ten to positive ten, okay? If you didn't, you don't have room? Okay do another one then, just draw another one from negative ten to positive ten, okay, and think about what we talked about here, intervals.
[group work]
389. T: Margaret, what do you know about the intervals?
390. Margaret: [xxxx]--
391. T: Let's be careful, draw these carefully.
392. Margaret: [xxxx] number line?
393. T: Pardon me?
394. Margaret: It's gonna be twenty [xxxxx] or twenty one?
395. T: Why don't you figure that out?
396. Monica: It has to go from it has to go zero, one, two, three, four, five, six, seven, eight, nine, ten or ten and then ten nine, eight, whatever?
397. T: Right. [60 sec]
[/group work]
398. T: Okay, now look at the person next to you and I want you to check each others', change books, check each others' and I want you to look for things like intervals, for arrows at the end of the line, to the person next to you.
[group work]
399. T: You have a warning, you have a warning. If you think there's anything that needs to be changed, I want you to talk to the person about that.
400. S: [xxxxxxxx] bigger.
401. Monica: [xxxxxxxx] zero.
402. Rhonda: I [xxx] my zero.
403. T: What about Margaret's?
404. James: The spaces get smaller.
405. T: The spaces get smaller.
406. S: I don't have a space.
407. T: So, what should you do?
408. Margaret: [xxxxx]
409. T: What should you do? Can you just squash the numbers in when you don't have enough space? You need to extend the line and make sure the spacing is accurate. The intervals need to be the same, so let's make another one and let's make it carefully.
410. Margaret: [xxxxx]
411. T: Because it needs to be the same distance, look.
[/group work]
412. T: Oh and one more thing, one more thing, please look up here ... let's look up here ... not only do the numbers, uh not only do the intervals have to be even, the space between the lines has to be even, but when we are marking them, Margaret, let's look up here please, when we're marking them, we put the number right below the mark, not in the middle of the space, right below the mark, okay? Right here the two goes right below the mark that I made, the vertical mark that I made, okay? So, if you haven't done that, please take a minute and to it now.
[group work]
413. Monica: Miss Lewis I have a question, is it okay that I have spaces that are all like that?
414. T: It should still be the same amount of space between zero and one =
415. Monica: My number line is too long, my number line is too long.
416. T: = this is too much. Well, then what could you put on there?
417. Monica: Eleven?
418. T: Yeah. It can either, because it's extending in either direction, remember? But the spacing has to be even.
419. Rhonda: How do we know if it's even?
420. T: How do we know if it's even? Well this--
421. David: [xxxx]
422. T: That was a good idea, good idea, um these we know are even, right? Look at Maddie's.
423. David: How do you know they're even?
424. T: Does this look even?
425. Monica: Just for one space.
426. T: Right, so, not just redo it, why don't you do it on the back? Come on, let's go.
427. Monica: [xxx] it's gonna be messy, oh [xxxxx]
[/group work]
428. T: Okay? Yours are okay? Even spaces? Okay. We're gonna move on now ... James.
429. James: Uh if you have [xxx] number line--
430. T: Pardon me?
431. James: You know how we made the number line [xxx] the lines both have to end with ten or this one could end with ten and this one--
432. T: Well, what does the arrow mean?
433. James: It goes on.
434. T: It goes on into infinity, so you could keep adding on to the right and not an equal number on the left, is that what you mean? Well, we're gonna talk about that, okay? Okay. I want you to look up here, okay, one, a word that you are going to hear a lot about when we're talking about negatives is opposite, okay, so I want to review some examples of opposites, okay. Let's let's look here, if I have, let's start with an easy one, how about east? If I have east, what would the opposite be?
435. Ss: West.
436. T: West, the opposite would be west, we need to get another color pen [14 sec] Oops, so the opposite of east would be?
437. Ss: West.
438. T: West. The opposite of hot would be?
439. Ss: Cold.
440. T: Cold. A five yard gain in football, the opposite of a five yard gain, what does that mean a five yard gain? Anybody play football?
441. Ss: Yeah.
442. T: James, what is a five yard gain?
443. James: It mean you went all five yards.
444. T: You went five yards from where you the ball was when you started, you went five yards toward the goal line, right? What would be the opposite of a five yard gain.
445. James: Loss
446. T: A five yard loss would mean that you were pushed back away from the goal line, so this would be a five yard?
447. Ss: Loss.
448. S: [xxx] the negative sign.
449. T: How about a check for ten dollars?
450. S: Check?
451. Ss: A bill.
452. T: A bill for ten dollars, good.
453. S: I've got a bill for ten dollars.
454. T: Three feet above see level?
455. Ss: Three feet under.
456. T: Three feet under or below. A bill for a hundred dollars?
457. Ss: A check for a hundred.
458. T: A check for a hundred ... your left hand?
459. Ss: Your right hand.
460. T: Will be your right, wrong?
461. Ss: Right.
462. T: Right, or?
463. Ss: Correct.
464. T: Correct ... A rise of twelve degrees in the thermometer?
465. S: A loss.
466. Ss: A decrease.
467. T: A decrease or a decline, okay. A decrease?
468. S: Yeah.
469. T: That little symbol means deg-- degrees, right? An elevator going up five floors.
470. Ss: An elevator going down five floors.
471. T: Okay. So, some of these are more mathematical than others, okay. So, let's talk about those. If I say I have a bill for ten dollars right, David you mentioned we could say?
472. David: Ten bucks?
473. S: Ten negative.
474. T: What sign will we put in front?
475. S: Negative.
476. T: A bill for ten dollars, would that be negative? Yeah, you're right you're right, sorry would be negative. How about another one that has to do with numbers? The five yard loss?
477. Monica: Would be a negative five.
478. T: Three feet below?
479. Ss: Is a negative three.
480. T: A check for a hundred?
481. Ss: Negative hundred.
482. Ss: Positive.
483. T: Positive a hundred, you get a check, somebody gives you money, right? How about a decrease in twelve degrees?
484. Ss: Negative twelve.
485. T: Min-- negative twelve and down five floors?
486. Ss: Negative five.
487. T: Negative five, okay? How about the opposite then, what would the opposite of negative ten be?
488. Ss: Positive ten.
489. T: Positive ten. So the opposite of a bill for ten dollars would be?
490. S: A check.
491. T: A check for ten dollars. What would the opposite of negative five doll--five be?
492. Ss: Positive five.
493. T: Positive five.
494. Ss: Positive three.
495. T: Positive three.
496. Ss: Negative one hundred.
497. T: Negative one hundred.
498. Ss: Positive twelve.
499. T: Positive twelve.
500. Ss: Positive five.
501. T: Positive five. Brad isn't here either, okay? Okay. Alright, what I want you to do right now is let's see ... let's think about a number, let's think about our number line, and let's take our, the number lines you just made and if I say a positive three on the number line, what would the opposite of a positive three on the number line?
502. Ss: Negative three.
503. T: Would be negative three. If I have a positive six on my number line?
504. Ss: Negative six.
505. T: Negative six, okay. How are we determining opposites here with our number line then?
506. Monica: Cause of the signs.
507. T: There are signs, right, but you said positive three was the opposite of negative three.
508. Monica: Cause and also I think because um each of them on different sides.
509. T: Different sides of what?
510. Monica: Of the zero.
511. T: Of the zero, absolutely right, okay. To be an opposite in math, you have equally as far away from zero on opposite sides of the number line. Okay? So, a positive three is one, two, three intervals to the right of zero, and a negative three is one =
512. Ss: Two
513. T: = two =
514. Ss: Three.
515. T: = three intervals to the left of zero, okay? So, on opposites sides of the zero one is to the right and it's opposite is to the left, okay? So, if I start at zero and I go three places to the right =
516. Monica: That'll be positive three.
517. T: = that's positive three, to find it's opposite, I would have to again go,
518. Ss: Three.
519. T: Three places to the left, okay? What do you think is the opposite of five? Talk to the person next to you, the opposite of five?
520. T: [xxxxxx]
521. S: Positive five?
522. T: Yeah the opposite of positive five and you need to tell me why.
[group work]
523. Monica: Because it's =
524. T: Talk to the person next to you.
525. Monica: = it's five spaces away from zero but it's it's uh
526. Rhonda: It's five away from zero and five
527. T: Okay, Gabi and Jaime you're supposed to be talking together ... Jaime, what did she say?
528. Jaime: She said it's a negative five because ... I couldn't hear her.
529. S: I said it's negative five because it's five spaces from zero.
530. Jaime: Five spaces from the zero, cause they're equal.
531. T: What's equal?
532. Jaime: Um the numbers you know how you said that in order to be in order to be [xxxx] there has to be um =
533. T: Hey, no get down.
534. Jaime: = five spaces to the right of zero and five spaces to the left of zero.
[/group work]
535. T: What did he just say? Okay, let's stop for a minute. When I asked you to talk to each other, what are you supposed to do?
536. Ss: Talk to each other.
537. T: Talk to each other, listen to each other, right? Are you supposed to come up and ask me if you can go to the bathroom?
538. Ss: No.
539. T: No, are you supposed to be talking about anything else?
540. Ss: No.
541. T: No, okay. What is the opposite of five? Jacki.
542. Jacki: Negative five.
543. T: And why did you and Duane determine that?
544. Jacki: Uh because you know how three you moved um three spaces to the right and it's equals three, so if you did the same thing on the left equals negative three.
545. T: Okay, to the right of what? We moved three three places to the right of zero would be positive three, three places to the left of zero would be?
546. Ss: Negative.
547. T: Negative three, so when I asked for five--
548. Intercom: Ms. Lewis?
549. T: Yes?
550. Intercom: Do you have Mr. Dervin's class?
551. T: Yes.
552. Intercom: Can you send Dante to the main office please?
553. T: Sure. And so Jacki positive five is?
554. Jacki: Five I mean yeah five.
555. T: Five places to the?
556. Jacki: Zero.
557. T: Right of zero and negative five?
558. Jacki: Is left.
559. T: Five places to the left of zero? Five spaces, five intervals to the left of zero. Okay, um. I want you to find three numbers, and then I want your partner to give me give you the opposite of those three numbers. So, right now you write three numbers down ... and trade papers with your partner and have them write the opposite down next to it [6 sec] Okay?
[Note: 30 sec group work]
560. T: Okay, so what was one of your numbers, Kathleen?
561. Kathleen: Three.
562. T: Could we come up with a different number since we've kind of exhausted three?
563. Kathleen: Nine.
564. T: Nine, so, what should be the opposite of Kathleen's number nine?
565. Ss: Negative nine.
566. T: Negative nine, agree with that?
567. Ss: Yes.
568. T: Another number.
569. S: Nine ninety-nine.
570. T: Nine hundred ninety-nine? What should be the opposite of nine hundred ninety nine?
571. Ss: Negative nine hundred ninety-nine.
572. T: Another number, David.
573. David: Three hundred.
574. T: Three hundred, the opposite?
575. Ss: Negative three hundred.
576. T: Do we have to start with positives?
577. Ss: No.
578. S: Negative three billion.
579. T: Negative three billion?
580. Monica: Positive three billion.
581. T: Okay so what don't you make that number line now.
582. Monica: What?
583. T: Make the number to show the opposite.
584. S: It's too big.
585. Monica: For three billion?
586. T: Yeah.
587. S: The number is too big.
588. T: You couldn't do that?
589. Monica: [xx] make it and put put two lines [xx] three billion and put the zero in [xxx].
590. T: There you go, say what what would you do Maddie.
591. Monica: I would just write the number line, put a line, put three billion then put a zero, then put on to the left, I would put um positive
592. S: Oh, positive.
593. T: And what would you just have to make sure that did?
594. Monica: That they were equal amount away from zero.
595. T: So, the zero is exactly in the middle. Good job Maddie, very good.
596. David: You get a sticker.
597. T: Okay, you got a sticker Maddie. There you go David. Okay, who can explain what they think the opposite of zero is?
598. S: Oh I know
599. T: What do you think the opposite of zero is?
600. S: Nothing.
601. T: Kathleen?
602. Kathleen: Negative zero.
603. T: Negative zero?
604. S: No.
605. Ss: Positive zero.
606. T: Positive zero
607. James: Just zero.
608. T: What?
609. James: Just zero.
610. T: Just zero, why?
611. James: Cause we can't move it to other places.
612. T: Okay, so we can't, if I said what's positive zero away from zero, can you move it?
613. Ss: No.
614. T: No, what's negative zero spaces away from zero, can we move it?
615. Ss: No.
616. T: No, so the opposite of zero is?
617. Ss: Zero.
618. T: Zero, just zero, okay? Um alright, I want you to write that down, so we don't forget it the opposite of zero is zero. [10 sec] Okay. Now I want you to look at your number line, find number negative two, find negative two on your number line, okay?
619. Monica: Negative,
620. T: Now, I want you to find two numbers that are the same distance away from num-- negative two.
621. S: Zero
622. Ss: What?
623. T: Find num negative two, choose two numbers that are the same distance away from negative two.
624. S: Okay.
625. T: Are those numbers are going to be opposites?
626. S: No.
627. T: Tonya.
628. Tonya: They are both going to be negative.
629. T: They both be negative, what numbers did you talk about?
630. Tonya: Two and two and four
631. Monica: And then it will be
632. T: We're gonna use negative two, now tell me a number away from negative two, any number.
633. Ss: Negative four.
634. T: Negative four, so we would have to move in what direction?
635. Ss: To left.
636. T: From here we would move to the left two so then we would have to move to the?
637. S: Left.
638. Ss: Right.
639. T: Right two, alright so we end at negative four and zero, are those numbers opposite?
640. Ss: No.
641. T: Are those numbers opposite? Jacki
642. Ss: Yeah.
643. Jacki: Uh no.
644. T: Why not?
645. Jacki: Cause they're still negative.
646. T: Okay, what did we say about opposites? In order to be opposites
647. Jacki: Um in order to be opposites, it has to be, if it's a negative number, it has to be positive number.
648. T: Does the negative number's opposite have to be positive number? It's a good question.
649. Monica: No, it could be
650. T: Does a negative number's opposite have to be a positive number?
651. Monica: No, I don't think so.
652. T: Talk to your your partner about that,
653. Monica: I don't think so.
654. T: Just the person next to you.
655. Monica: Because
656. T: Does the negative number's positive have to be a positive number?
657. Monica: The opposite of zero is zero if okay, if you have negative seven and negative seven again I think that [5 sec]. I think that four minus, negative four.
658. T: Why? Does it have to be positive? Why not?
659. Jaime: Cause uh [xxxxxxxxxxxxxx]
660. T: [xxxx] come on
661. Gabi: Negative four and zero are not opposites because, let's see, yeah they are because they're same distance away from each other.
662. T: So as long as it is the same distance away from each other they are opposites?
663. Gabi: Yes.
664. T: So you agree with this then negative four and zero are opposites. James.
665. James: Um it it has to be like a multiple, like one times two equals two and like two times two equals four [xxxxx].
666. T: We seem to have some confusing talk here about opposites, okay. My question was do you think if you have a negative number, does its opposite have to be positive? What did you think about that? Esteban.
667. Esteban: [xxxxx]
668. T: You can't think about it?
669. S: I was thinking
670. T: What do you think Rhonda.
671. Rhonda: Um I think that it does have to be positive.
672. T: Why?
673. Rhonda: Because like minus ten and like cause you're going below zero and the opposite of minus ten would be um um just positive ten, so it would just be ten, it's like you can use that for um for money, like minus like no nothing was the opposite of like, I don't know how to explain it, like what's the opposite of no ten dollars.
674. T: What's the opposite of ten dollars? What's the opposite of having ten dollars?
675. Rhonda: Not having ten dollars.
676. T: Is that the opposite of ten dollars?
677. Rhonda: Negative.
678. T: A negative ten dollars or owing ten dollars, right? Having ten dollars the opposite of having ten dollars would be?
679. Rhonda: Owing ten dollars.
680. T: Owing tend dollars so the opposite of plus ten dollars would be
681. Rhonda: Owing.
682. T: Negative ten dollars, okay, anybody else?
683. Monica: I think that I think that it doesn't have to be a positive number because um I don't know why I just think that the opposite has to be a positive.
684. T: Well, let's go back to our definition, what's our definition of opposites?
685. Monica: You didn't say.
686. S: [xxxx] negative or positive.
687. T: Is that our definition of opposites?
688. Monica: You didn't say.
689. T: Okay, then let's come up with a definition and write it down. What do you think a definition of opposites should be?
690. Monica: The number--
691. T: James?
692. James: A number that has to move the same spaces to the left and the right of--
693. T: Of zero? What did James say?
694. Jacki: That um um wait I forgot um the number of spaces away for the right or the left of the zero.
695. T: So, an opposite, opposite numbers are numbers that are equally distant from zero?
696. S: From zero.
697. T: One is to the right of zero and one is to the?
698. Ss: Left.
699. T: Left of zero, okay, then we need to write that down so we don't forget it. Okay, opposites, write down opposites [9 sec] a positive number and a negative number that are each the same distance away from zero. Let's go back to our example of three, okay. [13 sec] Okay, if I add the opposite of three, which would be what?
700. S: Six
701. T: The opposite of three? Negative three, if I move back to the left in a negative direction three, what do I end up at?
702. S: Zero.
703. T: Zero, so if I move three in a positive and if I move three in a negative direction, I should end up at?
704. Ss: Zero.
705. T: Zero, if I have seven, positive seven if I move seven in a positive direction and I end up with seven then it's negative if I move that same number back to the left it a negative direction, then I should end up at zero. Does that make sense? So, if I move in a positive direction, I am moving to the right, If I am moving in a negative direction, I am moving to the?
706. Ss: Left.
707. T: Left, so if I move in a positive direction from zero seven places to the right, if I move in a negative direction back I should end up at zero, okay? So, I want to add that down here to the definition, okay? Opposites are a positive number and a negative number that are each the same distance away from zero so [5 sec] their sum equals zero ... Okay? Alright, we're going to um wrap this up after I just get back to this one question are negative four and zero opposites? Let's go back to that question, come on I know it's hot in here, but let's just try to make through this and we'll open up the windows afterwards. Are negative four and zero opposites?
708. Ss: Yes.
709. T: Are negative four and zero opposites? Maddie, what do you think?
710. Monica: Uh-uh.
711. T: Why not?
712. Monica: Becuase zero zero is um for both the negative and positive sign so I don't think it's that's the negative in that case.
713. T: So, can a negative number be the opposite of zero? Rhonda.
714. Rhonda: I think it can, I think it can.
715. T: Let's go back to the definition of opposites, everybody read the definition of opposites.
716. Ss: A positive and negative number that are the same distance away from zero and that's why their sum equals zero.
717. T: So, a negative number which is?
718. Ss: Four.
719. T: Negative four, is that the same distance away from zero
720. Ss: Yes.
721. T: As zero. Are these opposites? Gabi?
722. Gabi: No.
723. T: Why not?
724. Gabi: Because zero is no distance away from zero at all.
725. T: I thought we already said that, right? Remember that? Zero is no distance away from zero, is it?
726. Ss: No.
727. T: Negative four is how far away from zero?
728. Ss: Four spaces
729. T: Four spaces. Let me ask you this look at the number four, okay, we're gonna make number four our guiding point, okay? Here is number four, I'm going to move two places to the right of four, what do I get?
730. Ss: Six.
731. T: I'm going to places to the left of four?
732. Ss: Two.
733. T: Two, does that mean two and six are opposites?
734. Ss: No.
735. T: Does that mean two and six are opposites?
736. Ss: No.
737. T: I started at four, I moved two places to the right and I got six, I moved two places to the left from four and I got two
738. Monica: Oh miss Lewis
739. T: Does that mean six and two are opposites? Margaret
740. Margaret: No.
741. T: Why not?
742. Margaret: Because it because they are both the same amount of spaces from number four.
743. T: Okay, they are both the same amount of spaces away from four, does that make them opposites?
744. MichelleL: Wait, wait, I don't.
745. T: Go back to the definition, go back to the definition, this is a grouchy table today.
746. S: Why?
747. T: I don't know why, maybe it's Monday.
748. Monica: Miss Lewis
749. T: What do you think?
750. Monica: I think that I don't that they are negative
751. T: Why, that they're opposites you mean?
752. Monica: They're opposites cause they both they both um to the right of zero and to the right of zero are positive numbers
753. T: What did Maddie just say Gabi?
754. Gabi: That.
755. T: You have a warning, well then you better pay attention the first she says it, say it again.
756. Monica: Two two and six are not opposites because they're both to the right of zero and to the right of zero are all positive numbers.
757. T: And what did we say about opposites, one has to be?
758. Ss: A negative.
759. T: And one has to be?
760. Ss: Positive.
761. T: A positive, just because these are two away from four, does they-- does that make them opposites?
762. Ss: No.
763. T: No, it has one has to be negative one has to be positive and they have to be equally distanced from what number?
764. Ss: Zero.
765. T: From zero, it doesn't matter that they are two away from four, okay, opposites we have to look at zero, make sure Gabi, stop please, make sure they're the same distance away from zero. Okay, we're gonna have to talk about that some more tomorrow, but I'm just going to summarize what we learned today, okay, see if you agree or disagree we'll pick it up tomorrow. We learned that integers are a set of positive whole numbers, right, that includes zero, and the set of a whole numbers that include positive number and negative numbers is called the set of?
766. Ss: Numbers, whole numbers.
767. T: What do we call them? The set of whole numbers both positive and negative and including zero
768. S: Integers.
769. T: It's a set of integers, so that's an idea, we got to get that term down, okay, integers. We've learned that the concept of opposites involves a positive number and a
770. Ss: A negative number.
771. T: Negative number and they have to be the same distance away from zero and if we add two opposites together, we end up with?
772. S: Zero.
773. T: Zero, a sum of zero. Okay, and we used a number line to try to make some sense of integers, right? Okay, so that was a lot for today, good job, except for the grouchy table. And we will take this up tomorrow.
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