Love the idea! I’ll volunteer now to help with coding/creating/trialing this future endeavour :)

Q1: The teacher should formalize the discussion. They should have the ability to pull up the completed drawings of each student and then discuss the meaning with the class. Afterwards, they’ll want to do some more concrete examples. Finally, I’d suggest that the textbook then allow students to fiddle with the Maps one – provide them with a ruler and a scale and see if that changes things. If they try the same approach they’ll get practice finding a circumcenter, but they can also notice that measuring the actual walking distances may change the answer slightly.

Q2: The first student through needs to have a method of reviewing his/her work as other students finish. In a class of 25 I don’t foresee there being a huge time difference between the first and second students. Once that hits, they both get immediate feedback with regards to how right they were – the map will be pretty obviously the same, or different.

Perhaps the students who finish can move some things around and upload a new “question” to a stored database for the class. The first couple of students can try and create good questions (and answers!) and other students, as they finish, can either review their work compared to the classes or try one of the challenge problems.

Love the thinking!

The burning question I have after looking at this is, why is the average line a bit wrong? (Especially the blue/green line.) Is it just this particular random sample, or is there a cognitive bias in effect here? Direct overhead views are unusual in the real world, where we tend to see things in perspective. What angle do our brains think we’re looking at when we look at maps? Can we use this information to create maps that correct for our cognitive bias? (Assuming that’s what’s going on.) How would we have to transform the map to do so? Would the bias have been the same if we didn’t start with a map (of things we would have seen in perspective if we were standing at a spot at the bottom of the map) but we instead started with an abstract diagram? What do we do for people who ask questions that are interesting but derailing?

In college we’d go play Lasertag and I hated it. I’m red/green colorblind, and the vests for Lasertag were red for one team and green for the other. I’d end up shooting my own team half the time. My only suggestion is to think of us minorities and change the ice cream stands to blue, red, and yellow.

While the computerized implementation is cleaner, this activity can and has been done on paper … well, sort of: on transparency paper. Give each kid or small group a transparency to dot many dots, then combine the transparencies on an overhead projector.

It works really well, you still get overlapping regions. (The same sort of work can help students visualize the solution to an equation or inequality in the plane.)

For the computerized version I would love to see this sort of “overlapping” instead of the all-kids-combine page just flying in, because I might not believe that image came from everyone. With transparencies, I have to believe it, it’s right in front of me.

Sweet stuff. Now make a similar activity for the incenter ;)

If you allow the first student through to see the picture as it gets revised (via a reload button or some auto-update), I don’t see a terrible problem (except for the usual classroom dilemma of what you do with any student that finishes fast).

Also: agree with Alex on the colorblind issue.

@Dave: I’ve been messing around with using nonlinear lesson design on a electronic lesson. I would recommend a Choose-Your-Own-Adventure style text prototype in Twine or something of the like.

I like the activity of guessing, collecting all the guesses, and painting regions. However I feel it is more about regions (Voronoi diagrams) rather than circumcenter. This activity leads effortlessly to asking “What about four points?” Which has nothing to do with points of concurrency but definitely back to the key idea of a perpendicular bisector dividing a region into two regions so that any point in one region is always closer to the one endpoint of the segment in that region. Which is nicely extended to 3, 4, 5, … n points. Ten years ago I probably would have asked, given two points A and B (two post offices?), how would you divide the region into two regions so that any point in the region containing point A is closer to A than B and the same for point B. Then months later toss out the problems of three points (fire stations?) and then four points (Starbucks?) in the plane. (See Discovering Geometry 4th edition page 152 exercise 11, followed by page 165 exercise 13). Rather than scaffolding, now I see the greater value of jumping right to the more perplexing problem of three or four points and let them struggle.
Was the direct investigation of looking for the point in the park that was equally distant from all three (take your pick: entrances, water fountains, bathrooms, …) points, too boring? You’d have your Act 1 with students voting, compiling all votes and see who was closest and how close the aggregate voting gets to actual circumcenter. A nice follow up for the circumcenter would be to ask is the circumscribed circle the smallest disk that can cover any triangle?
Another direct investigation that would directly get at the incenter would be to ask “What is the largest disk that you can get that would fit in a triangular region and where would you place the center of the disk?
Again, the important teacher part after the class voting is to make sure the idea is gotten across that the “answer” is not correct because that is what the majority voted for.

@Michael: “I like the activity…however I feel it is more about regions (Voronoi diagrams) rather than circumcenter.”

I was thinking the same thing. At no time did anyone ask, “Which point in the park is the same distance from all 3 stands?”

@Dan: “But if you’d like to use Kate’s tweet to motivate the need for the circumcenter, to give students a reason to care about the circumcenter, we’ll need to start much lower on the ladder of abstraction.”

As Michael points, out this activity did not make me “need” the circumcenter. I found the activity interesting and fun, and I’m excited about what you and Dave are working on. But this just goes to show that we can have cool digital textbooks, with engaging prompts, compiled responses, etc etc — and *still* end up falling short *if we don’t ask the right questions.* The questions should feel logical and natural for the student.

Here is an easy remedy for your task. Ask students:

#1. Which points in the park are the same distance from stands A and B? Paint ’em.

#2. Which points…from stands B and C? Paint ’em.

#3. Which points…from all 3 stands? Paint.

Now the teacher follows up with a lesson on the perpendicular bisector, answering the question “How do we know the exact right answer?”

I would say that the abstraction needs to be formalized. Give students some (virtual) tools to make a geometric construction. Keep the interaction, and what others did in there. Check the geometric construction (which can be done by linking to a proof checker). I think the line that is ‘off’ shows that a formalization is necessary. So I would add some visual misconceptions.

BTW, I would say there is something pseudo-contextual about this task. Of course the circumcenter is of no use for someone standing in a park, but I doubt that geometric distance to a point is the main consideration for someone any way. Things like ‘where is my car?’, ‘where do I want to go after my ice cream’, ‘are people playing football over there or not’ are probably more important in determining what stand to go to.

From a school lacking technology– You could do this with a class set of transparency sheets overlapping. No computer required.