APPENDIX B: A Protocol of an Experienced Tutor

In the following transcription the student’s written remarks are indicted in brackets. The tutor is an experienced current middle school math teacher. The student is a seventh grade male student who is a student in the tutor’s classroom. The student had a list of problems in front of him and each problem’s text is reprinted (__underlined__) when the student reads the problem. The student had a blank sheet of paper on which he wrote his answers. Generally, his paper included only attempts at symbolization with a few accompanying words possibly indicating the units. The session lasted approximately one hour. The session consisted of 17 problems, 8 of which the student answered correctly on the first try. This transcription was made from a video tape of the session. Pauses are indicated with colons and one colon indicates about one half of a second pause.

2. TUTOR [Opening remarks and asks student to read out aloud and begin]

3. STUD [Reads problem] __Mike starts a job at McDonald's that will pay him 5 dollars an hour. Mike gets dropped off by his parents at the start of his shift. Mike works a "h" hour shift. Write an expression for how much he makes in one night?__

4. [writes "h*5=how much he makes"]

5. TUTOR That’s' right number. #2

6. STUD [reads problem]* *__Mary opened a new music store. She got CDs delivered on her first day. She got 5 truck loads of CDs delivered. Each truck that arrived dropped off 12 boxes. Each box she received had "c" CDs. She sold CDs for 11 dollars each. How many CDs were delivered that first day?__

7. [writes "5*12*c=# of CD’s"]

8. TUTOR OK

9. STUD [reads problem] __Ann is in a rowboat in a lake that is 2400 yards wide. She is 800 yards from the dock. She then rows for "m" minutes back towards the dock. Ann rows at a speed of 40 yards per minute. Write an expression for Ann's distance from the dock.__

10. [writes "800-11*40"]

11. TUTOR OK I know there are some harder ones in here. :::::

12. TUTOR Read out aloud so we know what's wrong.

14. STUD [reads problem] __Ms. Lindquist is a math teacher. Ms. Lindquist teaches 62 girls. Ms. Lindquist teaches "f" fewer boys than girls. Write an expression for how many students Ms. Lindquist teaches.__

15. [writes "62+(62-f)= # of students"]

16. TUTOR Good. All right. Number of students. Cruising!

17. STUD [reads problem] __Mark went to the store to buy 4 jackets that cost "d" dollars each. When he got there the store was having a sale of 1/3 off the usual prices. How much did each jacket cost him?__::::::

18. STUD Are the jackets all the same price?

19. TUTOR Yup. Each.:::::

[writes "(d-2/3)*4="]::::::::::::

20. TUTOR What's that?[points at the subtraction sign] Is that a subtraction?

21. STUD That's two thirds

22. TUTOR You mean that subtraction?

23. STUD Yeah, that’s a subtraction of two thirds.

24. TUTOR OK. Can you.. Is two thirds.. What does two-thirds represent?

25. STUD 33 percent

26. TUTOR Right.. Or.. I mean. Not, Not, what is it in percent but what does it represent?

27. STUD 2/3 of the whole price.

28. TUTOR Is that what you are subtracting?

29. STUD Yeah

30. TUTOR Are you subtracting two thirds of the whole price?

31. STUD Yes

32. TUTOR What if.. What if the jacket costs 9 dollars, How much would you be subtracting?

33. STUD Wait hold on ::: Oh that would be wrong.

34. [crosses out previous line and writes "(d-33%)*4 = money spent"]:::::::::::::::::::::::::::::::

35. TUTOR Um. So if instead of "d" dollars it cost ten dollars, what would the answer be? So if it cost 10 dollars what would the answer be? Not ten dollars. Ten dollars is bad; what if it cost 9 dollars? What would the answer be.

36. STUD Three. No- it would be six dollars.

37. TUTOR It would cost six dollars. Is your expression gonna get that? Really, what are they actually asking for?

38. STUD Um.. Home much did each jacket cost?

39. TUTOR Right.

40. STUD Then this would be...

41. TUTOR That's not necessary [ Tutor points to the "4"]. So would your expression work?

42. STUD [crosses out the incorrect "4" from his answer of "(d-33%)*4" ] Yeah, I think so

43. TUTOR Try it. How would you get 6 using your expression. Because your expression should give me that answer that you know it’s right.

44. STUD It would be nine minus 33% equals ::::[writes "9-33%=6"]:::: 33% is three, yeah, so the number would be 6.

45. TUTOR So how did you get that three?

46. STUD Well one third of nine is three.

47. TUTOR Right. You have to tell your reader that. That's what your expression is all about. It’s telling the person to do that. You haven't told the person to do that. That you.. You know instinctively to do it but you haven't told the person to do that

48. STUD So I should right 33% percent as one third? [writes "33%=1/3"]

49. TUTOR Well.. But are you just gonna subtract one third or are you gonna subtract.33333 Is that what you are subtracting?

50. STUD Well I am only subtracting like :::

51. TUTOR Yeah! No! - so what are you really subtracting. How did you calculate that?

52. STUD Well. :::::::::::::::::::: Well if the whole thing is nine.

53. TUTOR Ah uhm

54. STUD And making 33% plus 33% ,33%, equal 100%

55. TUTOR Equals one hundred PERCENT

56. STUD Uhm. Yeah, 33% could equal three

57. TUTOR How did you calculate that? ::: Just doing that little piece, what would you do to get that three. What did you do in your head?

58. STUD I transfer 33 to 1/3 and

59. TUTOR [interrupts ] Right- oh- you could have. And you never really needed the 33, you could have just used the third.

60. STUD [continuing] and divided nine by three which equals three,

61. TUTOR Which equals three.

62. STUD But I am not using one I am using two

63. TUTOR Right

64. STUD equals 6 [wrote 9/3=3*2=6]

65. TUTOR So, So you could.. So you did. Is that how you got your six? Oh did you get your six this[points to the expression "9-33%"] way. This is different. Here you are using subtraction and here you multiplied by two. So you need to pick one way or the other and write a complete expression

66. STUD OK.

67. TUTOR Because either way is fine

68. TUTOR We could even right both. Let’s write both. Let’s write an expression this way [points at "9/3=3*2=6"] and write an expression this way[points at "9-33%"].

69. STUD OK.

70. TUTOR But you have got to tell your reader to do this thing [points at "9/3=3*2=6"], because that's the thing you forgot to tell your reader.

71. STUD OK. ::::

72. TUTOR The person who is gonna use your formula, because you are making a formula.

73. TUTOR Because if somebody else came along, and used this they would go nine- minus 33%, or nine minus point 333. What is nine minus 1/3?

74. STUD Nine minus one third is 2/3

75. TUTOR No. Nine minus one third is eight and two thirds. Right?

76. STUD Well, one third.... So I have to mention that minus, Um, one third, of whole number

77. TUTOR Of what whole number

79. STUD Of the whole

80. TUTOR Right which is?

81. STUD [wrote "9-1/3 of whole=2/3"]

82. TUTOR Yeah, exactly, which is nine. Which actually is what in the original problem?

83. STUD Um, "D"

84. TUTOR "D", So how would we write the expression this way?

85. STUD This way?

86. TUTOR Yup, so skip a line...with using d. now write the expression

87. STUD Minus ::::::::[wrote "d-1/3=2/3="]:: equal :::2/3 :::which equals :::I don't know

88. TUTOR Nine minus one third. Does that give you the answer

89. STUD well no

90. TUTOR No, because that gives you eight and two thirds

91. STUD Yeah

92. TUTOR What do you have to do. Because if you have to do it you have to right it down for your expression?

93. STUD Say, it’s of the whole number

94. TUTOR of the whole number. You need to write that. Let’s get rid of that by rewriting d minus :::::: and of ::: and of means what

95. [writes "d-1/3 of whole number"]

96. STUD Times

97. TUTOR Times! So you can use one of your minus, plus [operators I guess]:::: So now let’s rewrite this one. This one looks great, but let’s rewrite it using just letters and times symbols instead of the words. "D minus"

98. STUD I think it.. Would it be that way?

99. [writes "d-1/3"]

100. TUTOR Ahhum, But don't forget the, That's only this part. That's one third of the whole number. And then you got to go. Don't forget this part [points at the "d-" I assume] "d" minus

101. STUD Where does that go in? [that I presume refers to the "d-"]

102. TUTOR Where do you think it goes in?

103. STUD I would just disregard that part

104. TUTOR Well what is this gonna give you[points at d*1/3 I think] Let’s say d is nine, what's this gonna give you

105. STUD Um. 6

106. TUTOR One third of nine?

107. STUD No, it’s gonna give me three

108. TUTOR It’s gonna give you three, Which you know is not the right answer. Right? What do you want to do with that three?

109. STUD Times the other two

110. TUTOR Oh- OK

111. STUD I means d times 2 equals 6 [writes =d*2=6]

112. TUTOR Well what's d? D is nine.

113. STUD Oh yeah

114. TUTOR You don't want to multiple d times two, what do you want to multiple by times two

115. STUD Nine...if you are using...

116. [drops pencil in exasperation and holds head]

117. TUTOR Well let’s go back to the third. This third, When you right 33% that's just the percent, that's not the quantity. You want to subtract the quantity, not the percent, or the fraction. You don't want to subtract a third you want to subtract a third of the whole number, or a third times the whole number, SO instead of writing just 33% or just one third, you want to right THAT[ points at 1/3d], because that's the quantity not just the fraction, because the fraction just tells you what part, it doesn't tell you exactly how much. A third could be a third of two thousand, it could be a third of seven. So let try. So you want to take your number and subtract 33% but not 33%, you want to subtract 33% OF[emphasized] the whole number. So how would you write that.

118. TUTOR That's the

119. STUD Yeah

120. TUTOR You did this

121. TUTOR That's 33% percent. This is a good part of your expression. So how would we write the whole expression? there are two different ways. And you have played around with both of them. Let’s stick with this one for now. This one you take d, your whole amount, and what do you want to subtract. What exactly do you want to subtract?

122. STUD I want to subtract, Actually I want to add one third to it

123. TUTOR You want to add one third? Oh, You want to take a third and then another third. and add the two together, or multiple by two

124. STUD Add one third of the whole number

125. TUTOR OK- So write that down. So you want to take

126. STUD I want to take the 1/3 of d.

127. [writes 1*3d +1/3d "]

128. TUTOR Ahhum. Good. And you got the one third of d not just one

129. STUD Times, I mean plus, another third of d

130. TUTOR Good, and that's one way to right the expression

131. STUD which equals two-thirds

132. [ adds "2/3d=?" to the line with 1/3d+1/3d ]

133. TUTOR Right

134. STUD times d

135. TUTOR Right

136. STUD equals

137. TUTOR Good, and that is one way to write the expression and that's a perfectly good way. If you know a third is coming off, you know two thirds is left.

138. STUD Yeah

139. TUTOR You could have also used this way, which you originally started. You could have done "D" minus a third of d, because that also gives you two-thirds of d. But you can't do, You would have to say "d minus 1/3 of 3, you can't just say d minus a third because you can't just subtract a third. A third of what? So you have to have the "of what" part. Either of those expressions are great! [tutor writes "d-1/3d"]

140. TUTOR See it?

141. STUD Yeah

142. TUTOR This[points at d-1/3 presumably] is where you fell apart, because you wanted to subtract a third, but you just can't subtract a third but you can't just subtract a third of something, you have to subtract a third OF something, and that "of" triggers multiplication. See it

143. STUD Yeah, that's how I would have done it in mind, because I understand what I mean

144. TUTOR Right, Exactly, but when you are writing an expression you are writing it for any old person who comes and uses that formula. Right? Or , more specifically, but in your case, you will probably be writing this for a computer. If you are writing this for a computer your computer has to.. you have to be VERY specific for your computer, or it’s not gonna do what you want it to do. Right?

145. STUD Yeah

146. TUTOR OK, let’s try number 6

147. STUD [reads problem] __Sue made 72 dollars by washing 6 cars to buy holiday presents. She decided to spend 32 dollars on a present for her Mom and then use the remainder to buy presents for each of her s sisters. She will spend the same amount on each sister. How much can she spend on each sister?__

148. STUD Now uhm ::::::: times thirty two equals :::::: 40 writes "72-32=40 "]

149. TUTOR Remember that we are trying to write an expression, we are not trying to do any work.

150. STUD Yeah, divided by "s" equals money spent on my sisters [continues on writing a single line by adding the "/s" onto the line above to get "72-32=40/s=money spent on sisters"]

151. TUTOR Now write one big expression that shows everything, without having , without having done any work

152. STUD Can I just use like numbers or letters,

153. TUTOR Yeah, WELL use the numbers, but here. You actually.. Remember how I talk about all the time, this[?] is not really equal to this. Are these two things really equal? [points to the "72-32" and the "40/s"]

154. STUD Ah NO

155. TUTOR [repeats his no] Kind of yucky, so let’s down here right one expression, without doing any math. Pretend you forgot how to subtract. Can you right an expression with doing any of the subtraction, division, or multiplying?

156. STUD Yeah, I think so

157. TUTOR That shows the whole thing?

158. STUD :::::::[writes 72-32=a/Stud] ::::::

159. TUTOR Right, instead of a, let’s just use this. Pretend

160. STUD Oh I get it :::: it will be 72 minus 32 equals something minus thirty two divided by s.

[writes 72-32=72-32/s = money spent"]

161. TUTOR so really you don't want to put the equal sign, you just want to write that Now!, what is being divided by s? because order of operation says you divide before you subtract. So what is really being divided by s?

162. STUD Ah [writes parenthesis]

163. TUTOR right, now you wanted to write this equals, because you want to go one step at a time. Bad habit, because then you get equal signs between things that aren't equal. So here

164. STUD So here I really don't need this [??]

165. TUTOR Right. You really want this [??], and then we do...then we work down, to show that everything above it is equal to [inaudible]

166. STUD It’s just because I can't write on this [student crosses out "72-32=" that was part of the "72-32=72-32/4"

167. TUTOR So you see what I mean?

168. STUD Yeah

169. TUTOR [talks over the student] Yeah, that's like forgetting to capitalize at the beginning of the sentence. It’s just yucky. OK Laughs

170. TUTOR What if someone said it wasn't thirty-two dollars, this doesn't give me the right answer. I want to spend 30 dollars on my mother. Now you can change it easily. Here, it not as easy to change, because you don't know where the 40 came from. OK, number 7what does number 7 says

171. STUD [reads problem] __John and his wife Beth have been saving to give the 5 children presents for the holidays. John has saved 972 dollars for presents and Beth has saved "b" dollars. They give each child the same amount. Write an expression for how much each child gets.__

172. STUD The same amount between them, or the each amount between just one person.

173. TUTOR It doesn’t matter which way.. If each person gives them the same amount, there gonna get the same amount at the end, right?

174. STUD Ah, yes. So it would be 972 divided by 5 equals = a [wrote "972/5"]

175. TUTOR Just leave, because well make one big expression.:::::

176. STUD "b divided by 5" wrote onto 972/5 = b/5"

177. TUTOR So how much does each kid get?

178. STUD Ah. :::::::They get :::::::They get h+c? I don't understand this one part

179. TUTOR Hoe much are they gonna take in?

180. STUD They are gonna take in this divided by 5 and this divided by 5

181. TUTOR AND

182. STUD Ah Plus, So it would be :::::::::writes "(972/5)+(b/5):::

183. TUTOR Next one. Do you need those parenthesis there?

184. STUD Not really

185. TUTOR Why not?

186. STUD Ah, because division get done first left to right

187. TUTOR Right so those aren't necessary [(b/5)+(972/5)]this one here, those were. You need to tell your person you got

188. STUD Because you have to minus the two numbers first

189. TUTOR right, you got to do the subtraction first

190. STUD [reads problem]__Bob left at 3 P.M. and drove 550 miles from Boston to Pittsburgh to visit his grandmother. Normally this trip takes him "h" hours, but on Tuesday there was little traffic and he saved 2 hours. What was his average driving speed?__

191. STUD ::::Well ::: Ah :::: so he save two hours :::ahum:::::::::

192. TUTOR Do you know how to calculate average driving speed?

193. STUD I think, but I forget

194. TUTOR Well average speed, as your mom drove here did she drive the same speed the whole time.

195. STUD No

196. TUTOR But she did have an average speed. How do you think you calculate he average speed?

197. STUD It would be h hours divided by 550 miles an hour.

198. TUTOR So which way is it? It’s miles PER hour. So which way do you divide?

199. STUD It would be 550 divided by h

200. [write 550/h=mph"] OK so now, that's how you calculate miles per hour. So now how about for this problem? Read the problem again. Because you got the right idea. You know how to calculate average speed. But what exactly do you have to do for this trip

201. STUD Um. Well he save two hours, but I don't know how that is important

202. TUTOR Well how do you calculate... Not for Bob but for your mom, how did you calculate what her average speed was driving to CMU this morning?

203. STUD Ahm, I guess you would I would have done it 550 divided by h

204. TUTOR yeah [even though the 550 is not for his mom?] That's how you calculate average speed but what exactly is it? 550 represent what?

205. STUD Miles per hour

206. TUTOR No.

207. STUD Oh 550 miles

208. TUTOR Right

209. STUD Divided by h

210. TUTOR Which represents?

211. STUD Miles per hour

212. TUTOR No what does h represent?

213. STUD Hours

214. TUTOR Hours! So what are you getting? What are you dividing by what?

215. STUD Oh miles divided by hours.

216. TUTOR Right TOTAL miles divided by

217. STUD Total hours

218. TUTOR So let’s calculate it for this guy, That's exactly the concept, TOTAL miles divided by TOTAL hours [writes "550/h"]

219. TUTOR Is that what it is?

220. STUD Yeah

221. TUTOR Is 550 the total miles? [neat!]

222. STUD Yes

223. TUTOR Is h his total hours?

224. STUD Yes

225. TUTOR Is it??

226. STUD Oh no h-2

227. TUTOR OK- right this again and write it correctly so that order of operations and stuff works

228. STUD [Writes "550/(h-2)"]

229. TUTOR Exactly, so where did the 2 go in?

230. STUD The two hours he saved on traffic

231. TUTOR To calculate the total hours, so good.

232. TUTOR How we doing, we got lots of time. All right thinking harder. These are pretty good. Let’s try number nine. Okay

233. STUD Okay

234. TUTOR [Laughs and mentions hard work]

235. STUD [read problem]__Julie was trying to raise money to help fight Cancer. She got 7students to each donate "s" dollars and "t" teachers to each donate 10dollars. Write an expression for how much she collected?__

236. TUTOR Number 10, or no number 9

237. STUD [writes 7*s +t*10=money to fight cancer"]

239. TUTOR Good. Next problem

240. STUD [reads problem] __Cathy took a "m" mile bike ride. She rode at a speed of "s" miles per hour. She stopped for a "b" hour break. Write an expression for how long the trip took?__

241. STUD uhm :::::::::::::::::::::: writes "s/m+b"::::::::::::::::::::::::::::::::::

242. TUTOR How do you calculate the amount of time it takes you? If your, if your, if your riding at, let’s make it simple. If you are riding at 20 miles per hour, OK and you go 100 miles, how many hours did that take you?

243. STUD Um 5

244. TUTOR 5 and how did you get that 5? How did you use the numbers 100 and

245. STUD 100 miles divided by miles per hour

246. TUTOR So you took the miles and divided it by the [garbeled, probably "speed"]

247. STUD Miles divided by s plus b equals time [writes m/b+t ]

248. TUTOR Right, OK, whenever I get these.. did you see how I had to stop and think? I have stop and think for these to? so I always remember to stop and think, which way do I have to divide, because I know I have to divide, which way? OK? So you have to figure out which that is? OK number 11

249. STUD [reads problem] __Debbie has two jobs over the summer. At one job she bags groceries at Giant Eagle and gets paid 5 dollars an hour. At the other job she delivers newspapers and gets paid 7 dollars an hour. She works a total of 30 hours a week. She works "b" hours bagging groceries. Write an expression for the total amount she earns a week.__

250. STUD ::::::::::::::::::::::::::::::::::writes b*5+(30-b)*7::::::::::::::::::::::::::::::::::::::

251. TUTOR Beautiful, excellent, good work That one was tough

252. STUD [reads problem]__Michael starts a business selling lemonade. He buys 35 dollars worth of supplies including lemons, pitchers, cups and advertising. He sells a 16 ounce glass of lemonade for 2 dollars. If he sells "g" glass of lemonade, how much profit will he end up making__

253. STUD ::::: 2 dollars for each cup:::35::: minus 2 * g [writes "35-2*g"]

254. TUTOR Which number do you want to be bigger? Which number is gonna be bigger? 35 or 2*g?

255. STUD Ah.. ::: well let’s see what it’s gonna be Else he loses money and [writes "2g-35"]

256. TUTOR So this would calculate how much money he lost? Now if you do this, will this calculate how much money he lost?

257. STUD No

258. TUTOR What if he sold 5 glasses what's the answer gonna be.

259. STUD Ten

260. TUTOR That would be ten. What's ten minus 35?

261. STUD Oh I got it., money made, then 35

262. TUTOR No let look at this I think you can. This should work for both, because what's the answer? If he sold 5 glass what would the answer be?

263. STUD Um - minus

264. TUTOR What does that represent?

265. STUD Well you could just take off the minus and it would be how much money he lost

266. TUTOR Exactly. So that minus just represents lost. So this works for both of them

267. [writes 10-35= -25"]

268. TUTOR So the math even at negative number, those negative numbers allow it to tell us all the answer, even with one expression, that's why those negative numbers are so cool.

269. STUD [reads next problem] __A jacket that normally cost d dollar goes on sale for 2/3 of its original price. How much does the jacket cost on sale?__

270. STUD We already did this, but :: two thirds time d

271. [writes 2/3*d]

272. TUTOR And the only thing I am gonna tell you is that when you write it like that you have to put parenthesis, because otherwise it looks like 3 times d or you could write it like this

273. STUD Actually I think...

274. TUTOR which is the same thing

275. TUTOR point 66666 which is 66%, but this way is always better.

276. STUD [reads problem] __Rebecca makes a "h" hour car trip. For 3 of those hours it was raining and Rebecca drives at 40 miles per hour. The rest of the time it was sunny and she drove 55 miles per hour. Write an expression for the total distance Rebecca drives.__

277. STUD h-3 times 55 :::: would be ::::: plus ::::::::: 3 times 40 [writes "(h-3)*55mph + 3*40mph"]

278. TUTOR Right and usually with the expression we don't put the units in

279. STUD [reads problem] __John drove 300 miles to grandmother's at 30 miles per hour. He drove back at 20 miles per hour. He drove a total of 600 miles. What was his average speed?__

280. STUD 600-30+20 divided by :::::::::::::: two :::::::: no this parts wrong ::: writes 600-[(30+20)/2] and then scratches out the 600-"

281. TUTOR Right

282. TUTOR That [points at (30+20)/2"] looks great but it doesn't work. OK You would think it would, you are just averaging, but it doesn't work. What did we define average speed as earlier?

283. STUD Um

284. TUTOR Had the words total in it. Had to do with totals.

285. STUD Ah total : Ah total miles plus 50, plus total miles per hour

286. TUTOR No, no. To calculate miles per hour what do we need?

287. STUD Miles and hours

288. TUTOR Miles and hours. We need TOTAL miles and what else?

289. STUD Total hours

290. TUTOR Exactly

291. STUD So:::

292. TUTOR So for his complete trip

293. STUD So seeing it’s 600, it has to be half, 300 miles could be the first half

294. [writes "300"]

295. TUTOR You have to deal with this idea of total

296. STUD I am gonna figure out the hours for each half of the trip and then add them together.

297. TUTOR Exactly

298. STUD Uhm ::::::::::::::::::::

299. TUTOR You can, pick an easy problem and figure it out

300. STUD :::::::::"writes divided by 30=10 hours 300/20=A"::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

301. TUTOR Remember that you don't need to do the calculations because I am gonna ask you at the very end to actually just write the expression without doing the math. OK.

302. STUD So it would be 10 plus A? mumbles

303. [writes "10+A" and then adds in front of it "600*"]C; is that how you calculate miles per hour

304. STUD Ah no

305. STUD Mumbles [writes "(10+A) * 600"]

306. TUTOR That just the exact same thing you had before, expect you are that you are just multiplying "A" times 600 but I suspect you want to do the whole thing? How did we calculate. Once again what is the definition of average speed? Average miles per hour?

307. STUD Um

308. TUTOR Tell me out loud. Definition of average speed

309. STUD :::

310. TUTOR It has to do with total. The word total has to be there.

311. STUD :::

312. TUTOR How did we do it over here?[points to previous problem where speed was calculated]

313. STUD 550 total miles per hour total?

314. STUD Ah, total miles

315. TUTOR OK

316. STUD Then h-2 total hours

317. TUTOR Right, so we took, total miles and did what?

318. STUD Divided Ah

319. TUTOR Divided it by total hours writes ["600/(10+A)"]OK beautiful. You weren’t given ten and given "a". What were you given? \So the person that is reading this doesn’t have any arithmetic already done.

320. STUD 600 divided by 30, no it’s 300 divided by 30..write 300/30

321. TUTOR Good that's the tens; plus three hundred divided by 20 adds "+300/20"

322. TUTOR OK

323. STUD ::::600 divided by that then divides by putting the 600/ in front of it. for a final expression of "600/(300/30+300/20)"

324. TUTOR Yup, and after next year you won't right it with that division

326. TUTOR OK the next one is a good one.

327. STUD [reads problem] __A car salesperson is paid a base salary of $200 per month plus the additional amount of money in commissions for each car she sells. She sold four cars last month and received "x" dollars that month. If she sells "h" cars this month, write an expression showing how much she earns this month.
What's her commission?__

328. TUTOR Good question.

329. STUD Oh I have to find this out don't I

330. TUTOR I think so

331. STUD Um x [minus] that

332. STUD It doesn't say how many car. Ah 4 cars last month so it would be x-200 divided by 4

333. [he puts in the parenthesis only after writing "x-200/4"]

334. TUTOR OK

335. [he has written (x-200)/4 = commissions" and then adds a "c" before the commission apparently to indicate that commission’s will have the variable "c" stand for it]

336. STUD Equals commissions or "c". So this month would be 200+c*4 = salary

337. [writes 200+c*4=salary]

338. TUTOR This month?

339. STUD Yeah

340. TUTOR Read it again.

341. STUD This month, or she sold h cars

342. [scratches out the 4 and replace it by "h"]

343. TUTOR You want to skip a line so you aren’t messy

344. [starts writing "200+c*h" and tutor interrupts after the "c" to say

345. TUTOR OK now is C something that was given to you?

346. STUD No

347. TUTOR Where did it come from? What numbers

348. STUD So have to do this [scratches out the "200+c*h" he just wrote]

349. TUTOR You have to rewrite the WHOLE expression.

350. [writes 200+(x-200)/4*h=salary]

351. TUTOR Good

352. TUTOR Let’s see if that last problem is any harder

353. STUD [read problem] __A candle that has been burning for 2 minutes is 8 inches long. Three minutes later the candle is "x" inches long. Assume that the candle will burn at this same rate. Write an expression for the height of the candle after the candle has been burning for "m" minutes.__

354. STUD So assuming that for each minute it’s x inches long it’s a little tricky

355. TUTOR Ahum

356. STUD :::::::: x inches divided by 8inches equals

357. [writes "xinches/8 inches=rate of burning " {does this quantity name the rate or the amount burnt?}]

358. STUD rate no it would be minus, [changes the minus 8 to divided by 8"] inches is rate of burning

359. TUTOR Inches minus inches is gonna give you answer in what units?

360. STUD Inches

361. TUTOR Inches, is that a rate unit?

362. STUD It would be one inch per minute

363. TUTOR Yes, that’s what you want. so the per stands for what operation?

364. STUD divided by. so I was right!

365. [starts to change the minus back to a dividing sign]

366. TUTOR No, no, no, but wait a second, you want inches divided by inches?

367. STUD No

368. TUTOR That's gonna give you

369. STUD

370. TUTOR You are on the right track because you want inches per [[left a sentence completion]

371. STUD Yes, I was think If I divided by minus x inches by minus 8 inches it would be the inches per minute

372. TUTOR No what does that give you?

373. STUD That would be x-8

374. TUTOR Which is longer, x or 8?

375. STUD x

376. TUTOR [Read the problem again]

377. STUD "assuming the candle has been burning for 2 minutes is eight inches long. Three minutes later the candle is x inches long."

378. STUD Oh three minutes later. so it would be five minutes "assume that the candle will burn at the same rate. Write an expression for the height of the candle after the candle has been burning for "m" minutes. :::::

379. TUTOR Now you got a good start. I like what you did but what does it give you?

380. STUD It give me uh um the number of inches the candle has burned in the time of three minutes

381. TUTOR Right! So will that help you find a rate?

382. [To the line that now reads "8 inches-x inches=rate of burning" he adds "in 3 minutes" which is improper]

383. STUD Yeah

384. TUTOR Yeah because rate is what? How is rate defined?

385. STUD Its’ ::: I don't know

386. TUTOR You just told me

387. STUD It’s the time or something like that

388. TUTOR It’s got a "per" in it

389. STUD Yeah

390. TUTOR So what is it for this situation?

391. STUD The amount of candle that has burned in so many minutes

392. TUTOR Right. So and candle burned is measured in what in this problem

393. STUD Inches

394. TUTOR Inches. it could be measured in grams of wax or something. But in this problem it is measured in inches. So we want our rate to be what? Inches [left time for completion]

395. STUD um. wait, if it’s going less it will go x

397. TUTOR Yeah-- that you can go ahead and fix, so that's gonna give you inches. How are we gonna have to fix that to give us the rate?

398. STUD And we can divide eight minus 8 by three to give us one minute

399. TUTOR That’s gonna give us the rate. Exactly!

400. STUD Eight

401. TUTOR The unit rate!

402. STUD Eight divided by oh eight. [on a new line he writes 8inches/" and stops ] I mean to say x times 8.[ changes the line to say "x*8inches/"] Oh that's wrong.

403. TUTOR I can see you are getting tired, this is our last problem.

404. STUD Eight minus x inches :: divided by [writes (8-x inches)/3=rate in 1 minute of burning"]

405. TUTOR Right

406. STUD Three equals rate for one minute mumbles

407. TUTOR OK so what's the actual question?

408. STUD m times (8-x) inches divided by three [writes "m*(8-x)/3"]

409. [Note: this is the amount that has burned not the end height of the candle but the tutor accepted the student’s answer.]

410. TUTOR Because rate times time gives you inches,

411. TUTOR Beautiful! Those are hard. Those are good ones.

End of transcript