Classroom narrative: Meeting Lulu

submitted by: Nathalie Sinclair
on Tue Jun 10 16:08:47 2003

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Course: Math 7
Topic: Co-ordinate geometry
Resource type: Tool
Catalogue entry:
http://mathforum.org/mathtools/tool.html?id=951
Resource location:
http://hydra.educ.queensu.ca/cgi-bin/Maths/maths.cgi?do=activity&activity=lulu
Story:
  In the first class, I introduce the students to the first part of the Lulu microworld and let them experiment independently for a good five minutes. As I hear conjectures about whether it is possible to meet Lulu (who moves twice as far in the opposite direction as 'me'), I initiate a whole class discussion to talk about this question. The students explain how Lulu is moving and why it is impossible to meet Lulu at the park. Since the students don't always wonder whether it would always be possible to meet with Lulu, I suggest that they move to the second part of the Lulu microworld where the initial positions are different, such that it is impossible for "me" and lulu to meet. Again, I let the students explore on their own. Since they are able to randomly position the players, they are faced with the problem of determining when it would be possible to meet with Lulu.

In attacking this question, the students begin by calling out positions that they have tried that either do or do not work. We make a list of these on the blackboard. After about ten minutes, a few students conjecture at the same time that the starting positions have to be 3 units apart. We check this conjecture against the list on the blackboard and the students quickly realized that any multiple of 3 would also work. The students write their findings in words and experiment with a few possibilities just to make sure.

This allows us to move on to the following part where Lulu uses different movement rules. It is not long until the students are able to describe this movement rule as being at right angles. We spend the remainder of the class trying to come up with a generalisation for when it would be possible to meet Lulu. We make long lists of successful and unsuccessful trials but are unable for a long time to come up with a better description. One student then proposes a relationship that provides successful meetings positions but that does not cover all the ones that we have on the board. In fact, this problem is quite difficult to solve without recourse to algebra. Some of the students lose interest in the problem and continue on to other parts of the microworld. Some decide to experiment with different movement rules. Others try the "plan ahead" page where they try to meet Lulu in one move by specifying a vector.

On the second day, the students and I discuss what we have done the previous day, what things the Lulu microworld allows you to do, and what questions they have come up with. Now that they are familiar with the microworld, I ask them to play around with the functionalities they have seen and to see whether they can discern any other interesting relationships or patterns. Most of the students start playing with the traces left as the two players move on the screen. A few investigate the shape that Lulu traces if "me" traces a square. Others use the different movement rules in order to trace out attractive designs. Still others experiment with trying to cross all the streets with Lulu, without going on one street more than once. A few of the students spend quite a bit of time playing with the "plan ahead" page, trying to come up with a systematic way of being able to meet Lulu in one move. On student decides to experiment with creating his own movement rules and identifies rules for which "me" and Lulu will never meet, for example.

After about 1/2 hour, when most students have played with one aspect or the other of the microworld, I encourage them to experiment with creating their own movement rules. They spend the latter portion of the class doing this and become quickly adept at manipulating the equations themselves instead of always using the equation builder. The students end the class by writing down some of their discoveries, and some of the difficulties they had.

On the third day, I have written up some of their discoveries and I share them with the rest of the class, inviting them to investigate them further. These include the dilation effect of the shape traced of Lulu moving twice, three, or four times as far as "me", and the rotation effect on the shape traced of Lulu moving at right angles. These investigations allow students the work with creating their own rules and to use their understanding of the way Lulu moves in order to investigate some geometric relationships.

Send comments to Nathalie Sinclair at nathsinc@educ.queensu.ca

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Tue Jun 10 16:08:47 2003