Extract 4. Simeon offers an alternative way of conceptualising the length of AD
- 18 SIMEON: Well it depends on where the force is acting.
- 19 TALIA: It’s acting on this point here.
- 20 SIMEON: Uh, direction, sorry.
- 21 TALIA: Okay. So it’s not just about the point. So you said I need to measure the distance from the point, but that’s not really going to work. Because that’s fixed that distance, but you also said that it would be different if my force came from different directions. So it can’t be about the distance from point to point.
- 22 SIMEON: It’s kind of ... distance from like the axis of one force to like the
- 23 TALIA: It’s the axis of a force.
- 24 SIMEON: It’s like the direction it’s going...
- 25 TALIA: Okay.
- 26 SIMEON: ... you have to consider them in like the same direction. Like if they
were parallel and then measure the distance between them.
- 27 TALIA: What’s parallel?
- 28 SIMEON: The two ... so if you imagine like a line through the centre of the
pivot and then if...
- 29 TALIA: To this point here?
- 30 SIMEON: Yeah like a straight line. And then if the force was acting in the same
direction then you’d measure like that perpendicular distance.
- 31 TALIA: So and then the force is acting in that same direction.
- 32 SIMEON: Then you’d measure the perpendicular distance between them.
- 33 STUART: Which is this one, dot to dot isn’t it?
- 34 TALIA: So is the distance between those two parallel lines that distance?
- 35 SIMEON: It’s not the full one, no.‘Cause there’s questions like ... on a bicycle
when you have the pedals, like because ... when it’s at an angle the perpendicular distance isn’t as long so you don’t get like maximum force on it, you only get it when it’s perpendicular.
36 TALIA: Yeah so this is the thing ... we’re getting in a muddle because we’re
trying to talk about perpendicular things to points, and then you talked about parallel things which means you need lines. But that distance that you got me to mark on there is not the same as that distance between those parallel lines. Stefan?
Simeons first statement (turn 18) indicates thinking beyond the particular case of Figure 7.5, although Talia problematises his use of‘where’ by stating in turn 19 that В is a fixed point through which the force acts. Simeon’s suggestion of‘direction’is accepted by Talia in turn 21 and she offers a contrast with the reasoning that unfolded in Extract 3. In turn 22 Simeon suggests the phrase ‘axis of [a] force’ andTalia problematises this vocabulary in turn 23 and Simeon tries the informal ‘direction it’s going’ and this is accepted (turn 25). Simeon introduces ambiguity to his reasoning in turn 26 through his use ofthem’ and ‘they’ .Talia asks‘what’s parallel’and this is treated by Simeon through his next turns as problematising his understanding as he explains an alternative method to finding the required distance. On close inspection turns 28, 30 and 32, Simeon is conceptualising the distance AD in Figure 7.5 as the distance between the line BC and a parallel line passing through A. This does offer a general method and seems to be accepted by Talia in turn 34, whereas Stuart’s suggestion that the distance between the parallel lines is the length ofAB is problematised by Talia and refuted by Simeon in turn 35. In turn 36Talia is speaking to the whole class at first as she draws attention to imprecise vocabulary that she says has been used, but then she directly problematises Stuart’s claim from turn 33. She invites another student, Stefan, to speak. Extract 5 is the sequence of turns that follow.
Extract 5. Stefan explains the particular case and Shannon offers a new method
37 STEFAN: I think the distance perpendicular to the force (inaudible) if the
force was a straight line...
- 38 TALIA: So do I need this one?
- 39 STEFAN: No.
- 40 TALIA: Okay.
- 41 STEFAN: And then...
- 42 TALIA: So let me just make my force do this.
- 43 STEFAN: And then if you continue the straight line down, it would be the
point which the distance is perpendicular to the force, so...
- 44 TALIA: Like that.
- 45 STEFAN: Yeah like that.
- 46 TALIA: Okay, but if I had this here and this over here and this is doing this
47 SHANNON: You measure from the point the force is acting on to the pivot
but you resolve your force...
- 48 TALIA: This distance?
- 49 SHANNON: Um, yeah but then you resolve your force just to get the component going perpendicular to your wrench or point.
- 50 TALIA: Okay. That’s a different... can we come on to that in a minute?
Yeah that would be another way to think about it.
Stefan’s explanation (turn 37) is interrupted by turns 38 to 42 as Talia removed Figure 7.5 and made Figure 7.6 the publicly shared figure. As before, the point labels
FIGURE 7.6 A diff erent particular example of the moment of a force about a point
have only been added here for clarity. Line segments EH and FH were introduced during the course of the extract but have been included in Figure 7.6.
Stefan identifies the ‘distance perpendicular to the force’ as critical (turn 37) and explains in the particular case of Figure 7.6 how this line segment can be constructed by ‘continuing] the straight line down’ (our emphasis). In the particular case this is accepted (turn 46), but it is problematised by Talia offering a different publicly shared figure. After a silence of over five seconds, Talia invites Shannon to speak. Shannon introduces the term ‘resolve’ (turn 49) and offers a method that applies beyond the particular case of the publicly shared figure.Talia is not, however, privileging this method at this point in the lesson and so rather than being accepted or problematised, Shannon’s method is ‘parked’ (a term Talia uses herself within a minute of this interaction) and returned to later.
The lesson continued with a student applying a method similar to Stefan’s in order to identify the required distance in the figure Talia introduced in turn 46.This new figure was annotated in order to construct a line through the pivot and perpendicular to the line of action of the force. While this particular formal language was not used, presenting the two particular cases of this new figure and Figure 7.6 side by side afforded students seeing the general case through these.