I really like Scott Farrar’s Geogebra Applet, as the feedback is instant. The relationship between the points plotted and the rectangle being created is visceral. It hits you in your gut.

It’s hard to feel that connection in any of the other representations.

I’d love a combination of all of the above.

It appears that the Desmos activity had a drawback of no error checking. (Or, to call back your talk at CMC Asilomar, if this were a video game it would ask you to try again if your perim/area were impossible) In a customized Desmos applet, could you ask for width and height during the brainstorm rectangle phase, and then the second screen asks for perim and area? (couple more options here… you could still accept perim and area that do not match the given width, height: just flag them separately and then you can filter them out of the overlay or include them depending on where you want to take the lesson next)

It also appears that someone was investigating rectangles of a certain type, thus creating a rather straight line. Nice conversation starter there…

I think I am partial to the ability of my geogebra one to have the non-numerical entry point. On the other hand, they aren’t directly improving their skills in that arena either. A student is not hindered by calculation, but neither are they supported in it. But… I have liked giving a task to students where their interaction is manual and analog– meaning they are digging into the constraints and boundaries of a mathematical concept by physical motion and visual feedback. Does a student become immediately aware of and curious about this region that they cannot seem to get the dots into?

The processing one I like as a contrast to the hand-implemented geogebra. How would we systematically experiment around this idea? And when we devise a systematic approach, computers can be helpful in executing it. We can see the density of the dots is not uniform — what is the interaction between this perim v. area relationship and the way we have implemented our experiment? (I would guess the rectangle dimensions are chosen at random, thus creating a would-be-uniform display if not for the fundamental perimeter-area idea.)

The hand graph is limited in iteration, prone to calculation error, and slow. However, its requirements are all internal (save paper and pencil) rather than computer-based. Graphing by hand is perhaps the handle onto all of these other ideas. Is there research around how the concept of cartesian graphing changes if we grow up experiencing it in the computerized sense without the pencil and paper sense? I wonder!

For now I’d love to mix all of these in the classroom. When I spoke about this problem at Asilomar 2013, I used Malcom Swan’s initial prompts and provided grid paper (http://www.scottfarrar.com/asilomar2013/Handout%20Farrar2013.pdf) for the attendees to grapple with the problem. Only afterwards did I move to the geogebra version (2013 version that has a slightly different scope: http://www.scottfarrar.com/asilomar2013/htmlversions/01%20area_v_perim_v20131109.html )

And I would start the same way with students now. Here’s a rough timeline for a 1-2 day lesson:

1. paper rectangle in groups. Make a Perim v. Area graph on paper in your group that has your four rectangles. (this may correct some errors right away, like if kids put width and height instead of perim/area, their group may correct them)
2. Groups share on Desmos, add more rectangles. They are released from having to graph them on paper now. We are letting that go because Desmos will pick it up. Thus allowing students to step up their conceptual / abstracted approach.
3. class discussion, noticings, wonderings, address errors… blank region may become clearly curious here. Ok, now we’ve used the Desmos tool, but if we want to dig in with more rectangles, lets switch tools.
4. Geogebra sketch and/or programmed iteration. Groups come up with things to try and experiment with on the geogebra sketch, or devise a script that would generate perims and areas and plot them like Dan Anderson’s.
5. Return of the single case. Which points are on the edge? Can investigate with Geogebra sketch by hand. Or: can hand the class a mission on Desmos: “come up with rectangles that you think are on the edge”.
6. depending on the level of the class you can lead this towards some kind of proof or justification, entice students to make reasoned conjectures about what is going on.

Now we’ve taken the stepping stones from drawing a single rectangle to an aggregate set of many rectangles and their (p,a), to a set larger than what we could human-ly create. When we’re on step 4 there, we’re getting a handle on the abstraction of perimeters and areas of rectangles, thus allowing us to experiment with the abstraction just as easily as we worked with a single rectangle on paper at the beginning of class. But then we can go back and use each tool for what it helps with.

That, I believe, is the power of education technology: giving concrete handholds to abstract concepts. But tech is at its most powerful when working in concert with non-tech methods. And the possibilities of tech are so wide that no single tool was the magic bullet that enables this lesson– each one exposed a new vector to take.

Finally, remember Swan’s prompts (that did not aim to use technology) originally got at the existence of this concept by suggesting an impossible point and then suggesting that there are many others to find. His prompt defines these two regions (that we might call white and red for our purposes) with two examples: one red and one white. I wonder what the difference is about investigating the existence of white and red by imagining and reasoning on paper vs. bouncing up against their boundary on the geogebra applet.

———

Great juxtaposition, Dan.

Whoa, I love Scott’s approach. Much better than mine, you can actually link a rectangle with a point. If I were working this with a class, we’d start out on paper/whiteboards gatherings some individual data. Then go to the activity builder – Dan Meyer – approach to gather the class set of data. Then go to the Scott Farrar’s geogebra applet to explore what’s happening at the edges. Why are there edges? What are the shape of those edges, why?
I suppose the only reason for my (more removed) approach would be to investigate why the curvy bit is darker (hence more likely to occur) than the rest of shaded area, as shown in this sketch: http://www.openprocessing.org/sketch/218787
Thoughts?

I only have time to concur with those who said that we should start with pencil & paper, and put off technology for… well, longer than we’re comfortable with. They don’t need tools made by other people in order to have revelatory experiences, and we often inadvertently send the opposite message.

Then again, that will be my response 90% of the time.

I’m seriously confused by people saying the pencil and paper version offers something exploratory / revelatory that isn’t represented in the other ones.

What does graphing via pen and paper give you that, eg. the Desmos activity doesn’t? The Desmos version still had people placing their own points on the graph, and (I believe) it only showed results from other people after you’d submitted your own.

The only thing pen and paper seems to give you here is that you draw the axes yourself and have less data to find patterns in.

@josh g.

It may be from an (irrational?) fear that without creating the graph themselves, the students may not have the mental grapple on the particular graph they see pre-made.

I think its not a paper v. digital thing though– think about how kids engage with pre-printed axes on paper vs. creating their own: they have to take time to read and parse what the graph is set up for. And I think we see on the Desmos app many people who did not read the axes as perim and area.

So, I personally used doing it on paper first as a structured ramp up to quickly parsing the properties of the pre-made graphs.

(Also if everyone is a graphing whiz, then it doesn’t cause significant delay… but probably more common that there are students who could use some extra practice setting up the graphs. I think these kinds of activities are usually great for supporting some of the tangential skills because they get to *apply* the skill and if their graphing is wrong you just tune it up and move on without dwelling on it in a pure-review setting)

@josh g.

It’s okay, you can call me out by name, I deserve it. :-)

I suppose I was being a bit hyperbolic. It’s true that, in this case, since the point was to get students to understand how perimeter and area are related, then that should be the primary focus of the activity. The question is what technology (using the looser definition that includes pencil & paper) best serves that purpose. And I suspect that depends largely on your kids and their backgrounds and needs.

I think I am reacting to what I’m seeing as an attitude that, in general, pencil and paper < (digital) technology. I believe that p&p has its own advantages that are not immediately apparent, and should not be hurried through to get to the cool, shiny stuff.

(And, yeah, doing work on paper first and then putting that data into Desmos is a decent mix, especially if your students will have trouble with correctly plotting points.)