9:49 Bill: Okay, let me get some scratch paper there. Can we use that for scratch?
Ann: Yeah. Probably.
Bill: Let me see what it is. Looks to me like thatÕs scratch paper. Okay, Let me get my stuff together here while thatÕs coming up. Okay this is kind of like aÉsimulated race, in this over and back. ItÕs a race between a turtle and a rabbit [points to them on the computer screen]. And to be able to make these things operate all we need to do is to come over here and press run turtle or run rabbit or run both [points to the on-screen Buttons]. We can do them individually like you see or we can run them both at the same time. The other things that are on the screen here, you see hereÕs the turtle, an arrow going to the right, and a turtle with an arrow going to the left, and then the just plain rabbit, and then time [points to these words in turn on the computer screen]. What this means [points to the Turtle-Over Box] is that we can set the speed that the turtle is going to run in that direction only [moves hand quickly over distance line]. And we can set the speed [points to the Turtle-Back Box] for it to come back [moves hand back along the distance line], but for the turtle we have to set both speeds. For the rabbit we just have one. Whatever weÕre setting for he goes over and back at the same speed [moves hand over and back]. Okay? ThatÕs really all it is [gestures to the computer]. Pretty simple, huh [Ann nods]? So before we get on to that, I want to ask you though, weÕre going to be talking about speeds [points to the Turtle Boxes] here in feet per second, butÉcan you tell me what it means to you if youÕre riding in the car with your folks and theyÕre driving at forty-five mph? What does that mean to you?
9:51 Ann: It means, likeÉit means, like, if youÕre going forty-five miles per hour that means that if you time yourself at an hour you would have gone forty-five miles from where you started.
Bill: Good. ThatÕs true. How about if you only went a half hour? Instead of driving for an hour you drove for thirty minutes.
Ann: For a half hour?
Bill: Yeah.
Ann: You would only go half of forty-five.
Bill: Very good. UmmÉwell, thatÕs in effect what weÕre doing here [points to the computer] except the distances we have here are in feet [points to 100 ft at the end of the distance line] and instead of miles per hour here weÕre going to have feet [points back to 100 ft and then quickly to the Time Counter] per second. Okay?
Ann: Okay.
Bill: So, as an example, if we take the turtle here [points to the Turtle-Over Box], which it currently shows that heÕs set for thirty, what does that mean?
Ann: [Short pause] Thirty feet per second?
9:52 Bill: [Nods] Uh huh.
Ann: [Pause] So, he wouldnÕt go that farÉright?
Bill: Yeah, butÉand we were talking about miles per hour before, here we have thirty feet per second for the turtle [again pointing to the Turtle-Over Box]. What does that mean to you in terms of his speed?
Ann: HeÕll go thirty feet in one second.
Bill: Good. How far will he go in two seconds?
Ann: Sixty feet.
Bill: Very good. Pretty simple isnÕt it? [Ann nods] Okay. Well, letÕs start off, I can show you how to do these things [reaches over and grasps the mouse], but you see this arrow, I can comment that youÕve been using this before, I can either use the mouse to set those or I can use the tab keys. [Taps the tab key three times] You see I can move the tab key and it moves it to the different areas? And all I have to do is go up and put in the number [types Ò20Ó into the Turtle-Over Box] and I donÕt even have to hit enter, okay. So then IÕm going to set the turtle for twenty going that way, and twentyÉcoming back [types Ò20Ó into the Turtle-Back Box]. Do you understand what IÕve done now, in effect?
Ann: [Nods] Yeah.
Bill: HeÕs going to go this way at twenty feet per second, and when he hits the end heÕs going to turn around and come back [moves finger over distance line, then back]. Okay? [Ann nods slightly] If IÕve set him for that [waves hand towards the computer], can you tell me about how long itÕs going to take him to go over and back [gestures over and back along the distance line]?
9:53 Ann: At twenty feet per second?
Bill: Uh huh.
Ann: These are in tens arenÕt they [points to random tick marks]?
Bill: Well, letÕs see. If this in a hundred out there [points to 100 ft on the distance line], how do you think that [moves hand back along distance line] might be divided up?
Ann: Into tens.
Bill: Pretty good to me.
Ann: [Pauses, staring at the computer screen] Okay, so h-how long would it take him?
Bill: You can use paper by the way, too and the calculator [picks up and puts back the calculator]. DonÕt hesitate to use either one.
Ann: [While Bill was saying the last two sentences, Ann was gesturing over and back on the table and asking:] Go back ----? Go back and forth?
Bill: Yeah, how long will it take him to go over and back?
Ann: [Stares at the computer. Then writes 100Ö20 in long division form on her scratch paper. Then she writes Ò5Ó as her answer] Five seconds.
Bill: To go both directions, or just one way?
Ann: Just one way.
Bill: Very good, how did you get that?
Ann: I divided twenty into a hundred.
9:54 Bill: [Nods] Super. Can you tell me then how long it wouldÉsee he goesÉ[gestures with hand over and back] the race is like a relay race except each person runs by themself. He runs all the way over, and he turns around and runs right back, too [moves finger over and back].
Ann: So, it would be ten seconds if he wentÉ[waves hand over and back].
Bill: 10 seconds. And you got that by dividing twenty intoÉ
Ann: A hundred.
Bill: In this case you got that [glances down at AnnÕs paper] by dividing twenty into one hundred, right? Very good. Should we try it and see?
Ann: Okay.
Bill: LetÕs run the turtle and see what happens. [Ann activates the turtle] There he goes. You see that little counter down here [pointing to Time Counter] is keeping track of our time. Bingo. Right on the money. Good job. Let me ask you this, if we set the rabbit for twenty-five meters per second [types in Ò25Ó into the Rabbit-speed Box], how long will it take him to go over and back?
Ann: [Writes 100Ö25 on scratch paper] HmmmÉ [grabs calculator and types 25Ö100] Twenty-five hundredths of a second [looking to Bill]?
9:55 Bill: I donÕt think you divided right.
Ann: I didnÕt think so.
Bill: What you have on paper is probably correct. Twenty-five into one hundred. So here youÕd put in one hundred divided byÉ[Ann tries but fails to use the calculator for 100Ö25] there you goÉ you hit the wrong button.
Ann: One hundred divided byÉ[successfully uses the calculator for 100Ö25] equals four? [Looks quizzically at Bill, then at computer, then writes Ò4Ó above the long division].
Bill: [Pause] So what does that tell you there? Twenty-five into one hundred is four.
Ann: That it would be four secondsÉforÉone way across
9:56 Bill: Okay. [draws a distance line below her long division work]. That seems to puzzle you. Why is that?
Ann: I donÕt know. [Pause].
Bill: Okay.
Ann: [Counts the tick intervals by tapping pencil first three, then four times] Yeah [nods].
Bill: Wha-what seems strange to you?
Ann: [Shrugs shoulders] I donÕt know. I just didnÕt understand.
Bill: You mean the twenty-five didnÕt sound right or what?
Ann: The four seconds.
Bill: Oh, four seconds.
Ann: Yeah.
Bill: Is it too long or too short, do you think?
Ann: [Shrugs slightly, then lowers gaze] I thought it was too short. Sounds too short.
Bill: Well, we are looking [Ann says something unintelligible] for the time it takes him to go both over and back, so how long is that going to take him?
Ann: That would take him eight seconds.
Bill: Eight seconds, do you want to give him a try? [Ann activates rabbit, who finishes in eight seconds]. YouÕre right on the money. Very good. You had a frown on your face, what were you puzzled about there?
9:57 Ann: [Shrugs, then pauses] Just wondering [moves the mouse back and forth while watching the computer screen] how they can make him move back and forth so quickly.
Bill: Oh, how does the program do it?
Ann: Yeah [laughing].
Bill: YouÕll have to ask Dr. Thompson about that. I donÕt know. I didnÕt write the program. LetÕs take another step now. Hey, weÕre buzzing right along here. LetÕs say we leave the turtle going over at twenty meters per second, err, IÕm sorry feet per second and you want to change that one [note: Turtle-Back Box] to fifty. Plug in fifty there [Ann types Ò50Ó into the Turtle-Back Box]. And then you know what IÕm going to ask you. How long is it going to take the turtle to go over and back [waves finger over and back]?
Ann: Four seconds.
Bill: [Looks to the computer screen, then to Ann] Okay. How did you come up with that?
Ann: Because half of a hundred is fifty, so it would be two fifties to equal one hundred.
Bill: Okay.
Ann: And then two plus two is four.
Bill: Okay, but remember heÕs going over [points to the Turtle-Over Box] at twenty feet per seconds.
9:58 Ann: Yeah [laughs]. Okay, so itÕll take him twenty to get backÉor fifty. [Writes Ò2 sec. get backÓ then looks at the computer screen. She then writes Ò+5Ó under the 2.] Eight [writes Ò8Ó under the addition column]. It would take him eight seconds because it would take him five seconds to take him across [moves pencil over in the air] through twenty.
Bill: Okay [nods]. And how did you get the fiveÉthe five seconds?
Ann: Because I already did it up here [circles the 100Ö20=5 long division problem on the scratch paper].
Bill: Ah, from earlier work. Okay, so our total time was going to be what?
Ann: Eight seconds.
Bill: And thatÕs these two added together [reaches over and points to 2+5]?
Ann: Yeah. [Note: The 2 could possibly be misinterpreted as a 3 due to the way Ann wrote the number.]
Bill: Five plus two?
Ann: Seven [Bill winks]Éeight same thing.
9:59 Bill: [Looks surprised] Oh, okay, the same thing.
Ann: Close [scribbles a Ò7Ó over the 8].
Bill: [Chuckles] All righty. Do you want to try it? LetÕs see [Ann activates turtle]. You said five seconds to get there right? Bingo, Looks like youÕre right on. Look at him go right back. Wee! How about that [Ann nods slightly]? Good, letÕs do this now. LetÕs skip one down and set theÉturtle [types Ò30Ó for the Turtle-Back Box] for thirtyÉand have you do the same thing.
Ann: How long would it take him to get back?
Bill: Over and back.
Ann: Over and back. We already know it will take him five seconds with the twenty.
Bill: Okay.
Ann: So thirty into a hundred instead. [Writes 100Ö30 in long division form. Then uses the calculator. Looks at the calculator display, then to Bill] I donÕt think I did that right [giggles].
Bill: Why? What did you come up with? What did you do and then what did you come up with?
10:00 Ann: [Reading off of the calculator display] 333333333.
Bill: [Reaches over and looks at the calculator display] Is there a decimal in there somewhere?
Ann: Yeah.
Bill: Yeah. So whatÉisnÕt thisÉwhat is this, seconds weÕre dealing with?
Ann: Yeah.
Bill: So how long would that be?
Ann: That would beÉthree seconds [looks at the calculator display, counting the threes after the decimal point] and ten million. Three thousand and ten millionth.
Bill: Oh, I see. No. LetÕs forget about all those other threes back there [points to the calculator display].
Ann: Three seconds.
Bill: Why donÕt we just say itÕs going to beÉbecause here [points to the Time Counter] weÕre only measuring to the first decimal place.
Ann: Yeah.
Bill: The tenths place. So why donÕt we look only at the tenths place here [again pointing to the calculator display]. WeÕll round to that and call it 3.3.
Ann: 3.3 [writes Ò3.3Ó over long division 100Ö30].
Bill: Okay. And then your total time over and back?
Ann: [Writes 5+3.3 in column form. Rewrites the 5 so that the decimal points align. Then adds them and writes Ò8.3Ó] 8.3 seconds.
10:01 Bill: LetÕs see [Ann activates turtle]. And again you got that by dividing thirty into the hundred, right?
Ann: Yeah.
Bill: Very good. [Race ends] Looks to me like youÕre pretty accurate. See what it says here [pointing to the Time Counter]? Let me show you also what we can do up here in the options menu, just toÉmake it a little bit interesting [uses the mouse to add digits onto the Time Counter]. See thereÕs the additional 333333s. If I had more places I would continue to get them all the way across. Okay. So youÕre right on the money. ThatÕs very good Ann. LetÕs go back to that position [uses the mouse to have the Time Counter display only to the tenths place]. I want to set them now for theÉturtle to go forty [types Ò40Ó in the Turtle-Over Box]. And thatÕs forty over and heÕs going to go twenty back [Bill types Ò20Ó in the Turtle-Back Box. Ann writes 100Ö40 in long division form]. And the rabbit now is going to race him, and weÕre going to set the rabbit for twenty-fiveÉfeet per second [rabbit stays at 25].
Ann: So how long would it take each of them to go over and back?
10:02 Bill: WhoÕs going to win the race? [Pause] Why donÕt you go ahead [points to extra scratch paper which Ann ignores] and use a separate sheet if you want so we can kind of keep this stuff organized if you want.
Ann: [Ann draws a box around the 100Ö40. Then she uses the calculator for 100Ö40.] Aaagh [makes a mistake and recalculates. She then writes Ò1st .025Ó and below that Ò2st 2.5Ó]
Bill: Make sure your decimal pointsÉyouÕre reading them correctly off of the uhÉcalculator.
10:03 Ann: [Moves the decimal to .25 on Ò1st.Ó The she uses the calculator for 2.5+5.0 and writes, in an addition column labeled ÒTÓ, 2.5+5.0=7.5 in column form. Then she writes ÒRÓ and under that Ò25Ó which she circles and boxes. Next to that, Ann writes 4+4=8 in column form, circles the problem, and draws and arrow to the 25.] The turtle will win.
Bill: HeÕll win, by how much?
Ann: ByÉfive tenths of a second [she says it half answer, half question].
Bill: Okay. Do you want to run them [gestures to the mouse]? LetÕs see [Ann activates race]. While theyÕre running can you explain to me how you got your answer? How did you come about deciding that they would win by that much?
10:04 Ann: OkayÉ[looking down at her work] I tried to do a hundred into forty [points with pencil to her 100Ö40 in long division form] forÉfor the first time for the travelÉthat the turtle [gestures to the computer screen] would travel.
Bill: Uh huh [nods].
Ann: And then, when I gotÉit took me two tries, but when I got 2.5 [points to the column labeled T], I added that plus the five that I knew I would already have, the same thing going back up [draws a squiggly line from the T column to the first problem she worked on, 100Ö20=5 in long division form] to here again.
Bill: Yeah, okay.
Ann: And then for twenty-fiveÉ
Bill: So, you figured out the turtleÕs total time was seven and a half seconds?
Ann: Yeah.
Bill: I see that [glancing over at AnnÕs paper]. Yeah, okay.
Ann: And then I took the twenty-five [draws a line from the 25 under the R to the second problem, 100Ö25=4, and circles it] and I went back to whatÉto the information I got up here.
Bill: Ah, good for you.
Ann: [mumbles something].
Bill: So it would take the rabbit eight. Unfortunately, I started you explaining that when they were running the race, but uhÉlooks like something came up to eight seconds [gestures to the Time Counter], but I'm not sure which one it was. Should we run them again just to see?
Ann: Okay.
Bill: I think it continues to run even after the turtle wins or the rabbit wins, but letÕs see [Ann activates race again and they watch. The turtle wins by half a second]. Yep, looks like you were right.
Ann: Yeah, ---- turtle.
10:05 Bill: Good for you. Okay, do you want to try one more of those?
Ann: Okay.
Bill: LetÕs try one with thirty for the turtleÕs speed going over and forty coming back. And this time weÕre gonna setÉ oops, I donÕt want thirtyÉforty, forty coming back and the rabbit weÕre going to set at thirty-five [sets the Turtle-Over Box to 30, the Turtle-Back Box to 40, and the Rabbit-speed Box to 35. Ann writes, in the upper right portion of the scratch paper, 100Ö30=3.4 and 100Ö40=2.5 in long division form, and then writes 3.4+2.5=5.9 in column form. Her 100Ö30=3.4 is only two inches from a previous problem, 100Ö30=3.3].
Ann: Forty, thirty-five?
Bill: Yeah, heÕs going to come over and back at thirty-five. [Note: referring to the rabbit.]
Ann: [Writes 100Ö35 in long division form. She then uses the calculator and writes Ò2.8Ó over the new division problem.] This time the rabbit would win.
10:06 Bill: And his time is going to be?
Ann: His time would be 2.8.
Bill: Hmm. OkayÉ. You want to run them and see? [Ann activates race] You were right, the rabbit won.ÉBut his time was a little bit different. How was that?
Ann: I just took the first two here [points to the calculator display].
Bill: [A pause, then looks over to the display] The other ones were just decimal places, right?
Ann: Yeah, they were justÉ
Bill: Yeah, so we just round those off, but how far did he run in [points toward AnnÕs paper] 2.8 seconds?
10:07 Ann: HeÉgot all the way across [moves finger over the distance line] and he was coming back. He was about thereÉ[points to middle of the distance line] when he stopped.
Bill: Ah.
Ann: When he got to 2.8.
Bill: [Slowly] Okay. [Pause] Why donÕt you run the rabbit just by himself, so that way we can kind of test, right? And see what goes on [Ann activates rabbit]. What did that tell you?
Ann: Oh, I get it. I didnÕt add it up.
Bill: Yeah.
Ann: I should have [scratches out the 100Ö35 long division].
Bill: Why would that be?
Ann: Huh?
Bill: What was wrong with your calculations?
Ann: I needed to double it.
Bill: Why?
Ann: To get over and back.
Bill: Ah, you just calculated it over.
Ann: I went one way.
Bill: Okay, good. IÕll tell you what, it looks like youÕve got this thing wired, so I want you to see if you can work these out [hands Ann a problem set] and youÕre welcome to write on this paper or on your own scratch paper if you want. Now what here I would like you to do is letÕs just start off at the top, you can probably already figure out some of these anyway. WhatÕs the rabbitÕs speed have to be to give me these times [points to left and right columns on problem set with pencil]? Over and back, now remember. And we can test him on there [gestures toward computer], or you can test it on there if you like.
10:08 Ann: [Pause] Okay. [Pauses. Writes what looks like 100Ö5=2.8 in long division form [Note: perhaps in confusion over 100Ö35?]. Draws a line over and back above a half-distance line she used previously, circles 100Ö5, then scratches it all out. Writes 100Ö5 in long division form again. Types on calculator 100Ö, stops and looks to the paper, then back to the calculator.] Ohhh. [Scratches out the long division. Writes Ò100Ó and below that Ò5Ó. Runs out of room on the scratch paper, so turns it over.]
10:09 Bill: Here [points to pile of scratch paper], just grab another piece. There you go.
10:10 Ann: [Spends much of her time staring at paper or tapping the desk with her pencil. Then she draws a number line, dividing it up into three tick intervals]
Bill: Gotta give you some more room there, youÕre almost off the table [moves the calculator, mouse and keyboard so that there is more desk space. Ann draws a new number line under the first, this time eleven tick intervals long] You want to change one?
Ann: Yeah.
Bill: The rabbit? [Ann nods] Okay. All youÕve got to do is put in your number.
Ann: IÕm almost sure IÕm wrong, though.
Bill: WhatÕs an easy way to find out?
Ann: [Chuckling] I donÕt know [Note: about the answer, not the Òeasy way to find outÓ].
Bill: What speed do you think he should go at?
Ann: FiftyÉprobably [types in 50 for the Rabbit-speed Box].
Bill: Okay, and before you push the go line, why fifty? I mean, how did you come up with fifty?
Ann: It seemed logical? I donÕt know [shrugs shoulders]ÉItÕs a guess.
Bill: Did that [note: the 50] have something to do with your drawings [points to AnnÕs scratch paper] there?
10:11 Ann: No. I messed them up, I did it wrong.
Bill: Okay. Well, letÕs take a run at it. LetÕs see what it is. He really zips doesnÕt he?
Ann: Forty.
Bill; Why forty?
Ann: Because if it took four seconds at fifty, then if you subtract ten, it might take him a whole second longer to get back.
10:12 Bill: Okay, he hung up, weÕll see if he goes, there he you go.
Ann: Smiley face, I do it a lot.
Bill: You got it, right?
Ann: Yeah.
Bill: Okay, weÕll write that in here and go on to the next one.
Ann: [Reading off of problem set, whispering:] 10 seconds.
Bill: The next one is ten seconds, right?
Ann: Oh, um,ÉI know how you get the answer, but [chuckling] I donÕt know how fast [presses ÒclearÓ on calculator repeatedly].
Bill: How would you get it?
Ann: You would, ummÉsee if these [points to the tick marks] were split up in ten sections it would take a second to get over each [moves hand over and back].
Bill: Okay, remember now weÕre talking over and back [moves hand over and back].
10:13 Ann: So, it would take five seconds. So, itÕs [writes 5+5 in column form. Draws a number line with five tick marks, and the tick intervals labeled one through five and the last tick has 100 over it. Writes three squiggly lines along number line. Then writes Ò10Ó and circles it. Types 10 into Rabbit-speed Box and activates it. Counts the seconds it takes the rabbit softly to itself.] Yep, thatÕs too slow. [Writes Ò+10Ó next to the circled 10. Writes Ò20Ó and puts scratch marks around it.]
Bill: How long is it going to take?
10:14 Ann: Twenty, twenty seconds. [Writes Ò20Ó again, this time boldly, and circles it.] ÉOkay.
Bill: [Pause] YouÕre exactly right. Is there some relationship here [Ann yawns] Éthat you, say, remember the first ones that we were doing when we had a set speed, and you were calculating the time? All weÕre doing is the reverse of that now [flips hand around and shows its backside]. Do you have any ideas as to how you might approach getting these, figuring out what speed it has to be?
Ann: [Quickly] No [shakes head slightly and chuckles].
Bill: Okay. ThatÕs all right. [Ann types in 20 and activates rabbit] So twenty is your bet today, right? HeÕs off and running. [They watch it go. Before itÕs done, Ann writes 20 on the answer sheet]. You knew that one for sure, right?
10:15 Ann: Yeah. As soon as it went half wayÉ
Bill: [Interrupting] Let me ask you before you do this next one, letÕs say heÕs running it at uhÉ40 feet per second. How long is it going to take him to go over and back?
Ann: [Looking down at the problem sheet] Forty feet per second?
Bill: Yeah. [Pause] ItÕs not one of them listed there [gestures to problem sheet], I donÕt think. From the others we were doing earlier, you know, we were setting the speed and seeing how long it took him. How long should it take if you set it for forty feet per second?
10:16 Ann: Forty feet per secondÉ[Searches through scratch paper. Pauses. Uses calculator for 100Ö40] 2.5 seconds to get there and 2.5 seconds to get back.
Bill: Yeah [nods] You see, that was this one right here [points to one of the previous problems Ann did]. LetÕs go on with the next one, I wanted to go back and make sure that you understood about what we were doing before and you do so thatÕs fine. How about eight seconds? How long is it going to take him, or what speed, I mean, shall he have to race at to get there and back in eight seconds.
10:17 Ann: [Long pause while searching through scratch paper] Twenty-five.
Bill: Okay, letÕs set him and see [Ann puts 25 into the Rabbit-speed Box]. Now how did you come up with that?
Ann: I-I looked back here [holds up old scratch paper] and this is, weÕre talking about eight seconds, four plus four is eight, and I got that when I was trying to figure out how long it would take the speed of twenty-five.
Bill: [Looking at AnnÕs paper] Ah, okay [nods], all righty. [Pause] Okay, you want to run the rabbit [Ann activates rabbit]. I wonder what it would do if we just ran it with the turtleÕs [points to Turtle-Over Box] saying zero speed over and forty back [chuckles]. Maybe thatÕs why itÕs giving us that smiley face. [They watch the race end] What are we aiming for?
Ann: Eight seconds.
Bill: Eight seconds, right on the button. Good for you. [Ann writes Ò8Ó on the answer sheet]
Six is next [meaning give Rabbit a speed that would make him go over and back in 6 seconds]
Ann: We didnÕt do that one.
Bill: Huh?
Ann: We havenÕt done that one [meaning that six seconds was not a previous result from Activity 1].
Bill: Uh uh [no].
Ann: Notes donÕt help [chuckles, rearranging her scratch paper.]
Bill: But thatÕs all right. ThereÕs plenty of paper there. WeÕve got plenty of space to write.
Ann: Okay. Six É [writes 100Ö15 in long division form, then uses the calculator to find the result][1]
Bill: So, what did you do, divided fifteen into a hundred?
Ann: [Looks quizzically at the result]. ThatÕs wrong. [Responding to BillÕs question:] Yeah.
Bill: That was for a guess-and-test kind of thing?
Ann: Yeah. [Writes Ò6.6Ó].
Bill: Okay. And what did you come up with for that?
Ann: Six point six, thatÕs over already, and thatÕs for just one way.
Bill: Ah.
Ann: [Pause. Then she mumbles:] 100 divided by 10 secondsÉ[writes 100Ö10 in long division form in the upper left corner of the scratch paper. Then she looks like she might solve it by hand, but then she uses the calculator for the result. Looks at the display for a moment, then hits the ÒclearÓ button on the calculator.]
10:19 Bill: That hundred divided by ten gave you ten? [Pause] Is that right?
Ann: [Uses calculator again for 100Ö10] Yeah.
Bill: ThatÕs right. Ten times ten is a hundred. Can you think of a more efficient way, rather that having to go down and guess and check and guess and checkÉ This would work. ThereÕs nothing wrong with it [shrugs shoulders]. But is there a quicker or more efficient way of doing it do you think?
Ann: Yeah. [Begins to scribble on page aimlessly] There is, but I canÕt do it that way.
Bill: Oh, what way is that? Oh, Ôcause you donÕt know what it is yet?
Ann: Look at the answer book.
Bill: Oh [chuckling]
Ann: We donÕt have an answer book [puts pencil down].
10:20 Bill: No, there isnÕt an answer book for this stuff.
Ann: SoÉ
Bill: ThatÕs true. [Ann uses the mouse and keyboard to enter 5 in the Rabbit-speed Box and activates it.] Now if youÉFrom before, if you set him for five miles, err five feet per second, how long would it take him to get over there [gestures with hand over]?
Ann: [To self :] Five feet per second. [To Bill :] Over six seconds.
Bill: Yeah.
Ann: Well over six seconds [nervously plays with hands].
Bill: But the way you were doing it before [gestures to scratch paper], you know, the first, the first things we were working on here. How did you figure out how long it would take him?
Ann: We didnÕt do it É last time. [meaning 6 seconds]
Bill: I know we didnÕt do it with five seconds, but how did you figure out how long it would take him if he has a set speed [points to Rabbit-speed Box], in this case of five feet per second?
Ann: I É added. I think. I donÕt know.
Bill: On all the first problems you were doing, all these [points to first piece of scratch paper having AnnÕs calculations]?
Ann: No, I divided.
Bill: Yeah.
Ann: For the [inaudible], for the time ones, to find out the time.
Bill: Okay. So if I divide the É What is this time [points to the Rabbit-speed Box] going to come up with? Can you tell?
Ann: Umm [looking to the computer screen] About forty seconds, probably.
Bill: Yep [nods] Does that sound right to you?
Ann: No [chuckles]. ThatÕs way too far.
Bill: True. [Pause] So, what is it É [Ann plays with the mouse] LetÕs go back and review again. What is the target time weÕre aiming for here?
Ann: ÉUmmÉSix seconds.
Bill: Okay [nods slightly]. And we know five is moving way too slow, right?
Ann: [Nods] Yeah.
Bill: Was ten moving too slow?
Ann: ÉYeah.
Bill: Yeah. [Pause] And fifteen was too fast. Or was that too slow?
Ann: [Looks down to the scratch paper] I donÕt know. No, it was too slow.
Bill: WeÕre aiming for six seconds now, [points it on the scratch paper] right?
Ann: Uh huh [looks back to scratch paper then to the problem set three times]
Bill: And how long did you calculate that it would take him to go overÉrunning at fifteen feet per second? Your calculationÕs right down there [points to it on the scratch paper].
10:22 Ann: ItÕs 6.6 just to go one way.
Bill: So,ÉthatÕs still too slow, isnÕt it?
Ann: Yeah.
Bill: [Nods] Okay. So which direction do we have to go, above or below?
Ann: Above [pointing up].
Bill: [Nods] Okay. And if he ran twenty feet per secondÉ[Ann immediately types 20 in the Rabbit-speed Box.] You already figured that one out what heÕs going to get, right [Ann activates the rabbit, and waits for it to give her the result]?
Ann: He would getÉfive seconds, no.
Bill: Five overÉ, [waiting for a response].
Ann: Éfive back [rabbit finishes].
Bill: And thatÕs still too slow, isnÕt it?
Ann: [Pause, then tries 30 for the rabbit, watching it go] ItÕs too fast. Maybe. Maybe not. [Rabbit finishes] Too slow. [Tries 31 for the rabbit. It still comes in too slow]
10:23 Bill: YouÕre getting closer though. [Ann pauses while thinking. Then she types 32 and tries to change it] You have to go to backspace, I think, to erase the two [referring to the second digit of 32].
10:24 Ann: [Changes the Rabbit-speed to 33. Watches the rabbit go.] That would be right. [Rabbit finishes] How can you minus this one-tenth of a second?
Bill: Does he have to run right at thirty-three? Or twenty? Or twenty-five? Can he run a part of a feet per second in terms of speed? Like 33.1?
Ann: Probably.
Bill: Yeah [nods], we can go in there and see. [Both move forward to type on the keyboard] Well, go ahead [Ann types 33.1 and activates the rabbit. They watch it go.] Good shot. Now, weÕre almost out of time, weÕve only got about a minute left, it looks like [Ann writes Ò33.1Ó in the ft/sec. column next to 6 sec. on the problem set page]. Ann, what IÕd like you to think aboutÉthereÕs no assigned homework on this tonight. UmmÉbut you know the way we were going [moves hand over the distance line] first of all, we had a set speed and you were trying to figure out what the time was. And now weÕre going the other way around, and what IÕd just like you to think about over the evening: Is there a more efficient way to do this [points to the problem sheet], this reverse process, instead of having to guess and check? And weÕll pick up where we left off tomorrow [Ann nods]. Anything else you want to try on the thing [Note: computer] while weÕve got a few minutes or seconds left?
10:25 Ann: Yeah.
Bill: What do you want to do? [Ann types Ò.1Ó for the Rabbit-speed Box and activates the Rabbit.]. Oh, you want to be here for a long time.
Ann: I want to see how long this will take.
Bill: Well, you should be able to calculate it out canÕt you do that the way you were doing the first ones?
Ann: Yeah, but I just want to see. I likeÉ HeÕs not going to get back very quick.
Bill: No, I think the bellÕs going to ring before he even gets down there [Note: to the 100 ft tick mark] to be honest with you. From that [gestures toward the computer screen] can you calculate how long it should take him? YouÕve got the calculator there. Feel free to use it.
Ann: Over forty seconds.
Bill: Forty?
Ann: Over forty.
Bill: Over, I would believe that. How far has it gone so far?
Ann: Not even a foot and itÕs at twenty seconds.
Bill: Each one of these [points to a tick mark] are how far?
10:26 Ann: ThatÕs ten feet.
Bill: Okay, so he hasnÕt even gone ten feet yet and itÕs twenty-five seconds so far. So can you figure out ho-how long itÕs going to take him.
Ann: Maybe.
Bill: Give it a try, Ôcause weÕre out of time anyway.
Ann: Fifty seconds. No. Sixty, sixty seconds.
Bill: No. ItÕs going to be a lot more that that. He hasnÕt even gone ten feet and itÕs already [points to the Time Counter] up to thirty-five. Thirty, sixty, ninety, itÕs going to be whole bunches.
Ann: I know! Um, two hundred seconds. Two hundred.
Bill: Well, maybe weÕll find out when we come back next time. But right now youÕve got to go to class.