Riffing off of Scott’s geogebra applet, here is what you getplotting a scaled version of the corresponding rectangle instead of a dot. Visually it is clear that the rectangles near the “border” are square-like and those far from the border are “skinny”.

I think one could also argue that scaling provides an opportunity to use number sense and proportional reasoning when done with pencil and paper. Just depends on your group and what you are trying to accomplish I suppose.

**P&P**
Pros:
– slows the student down
– requires thinking about axes, scale
– gives practice on setting up a graph from scratch
Cons:
– slows the student down
– no immediate feedback re: mistakes
– harder to pull together everyone’s data

**Desmos thing**
Pros:
– quicker to plot
– much easier to pull together everyone’s data
– focuses on the main idea, less time spent thinking about how to turn a blank page into a graph
Cons:
– didn’t error-check (although this could be fixed, which would be a huge Pro point of immediate feedback on errors)
– might be too easy for kids to click randomly without thinking hard?

For whatever it’s worth, while I’m very interested in pushing the why of using pencils-vs-tech, I cannot survive in a classroom where I don’t have a big whiteboard to randomly add doodles and things to, even when my main notes are on a projector. I scrawl things on paper while I’m helping students. We need both, and it isn’t easy right now to quickly jot down mathematical thoughts on a computer (although Desmos and MS Office have improved on that greatly vs where we were ten years ago). But I get really suspicious when I smell either romantic nostalgia for the past *or* tech-hype for what’s shiny. Neither is a good basis for decision-making.

(Except for why I carry around this cool Mongol 482 pencil my wife found in our basement. This thing is totally awesome.)

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.

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)

If I’m teaching graphing, sure I want people to understand how to choose appropriate axis scales, make sure that they get the (x,y) concept, etc. But the point here was to dig into how surface area and perimeter relate, wasn’t it?

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.

While I’m sure this could be done in a number of ways to inspire that curiosity, what I do know is that it would be much more difficult to generate the same enthusiasm if we were to start with the paper/pencil task and move on from there.

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