Here is the original Malcolm Swan task, which I love:

Draw a shape on squared paper and plot a point to show its perimeter and area. Which points on the grid represent squares, rectangles, etc? Draw a shape that may be represented by the point (4, 12) or (12, 4). Find all the â€œimpossibleâ€ points.

We could talk about adding a context here, but a change of that magnitude would prevent a precise conversation about pedagogy. It’d be like comparing tigers to penguins. We’d learn some high-level differences between quadripeds and bipeds, but comparing tigers to lions, jaguars, and cheetahs gets us down into the details. That’s where I’d like to be with this discussion.

So look at these four representations of the task. What features of the math do they reveal and conceal? What are their advantages and disadvantages?

**Paper & Pencil**

You’ve met.

**Dan Anderson’s Processing Animation**

Hit run on this sketch and watch random rectangles graph themselves.

**Scott Farrar’s Geogebra Applet**

Students click and drag the corner of a rectangle in this applet and the corresponding point traces on the screen.

**Desmos’ Activity**

277 people on Twitter responded to my prompt:

Draw three rectangles on paper or imagine them. Choose at least one that you think that no one else will think of. Drag one point onto the graph for each rectangle so that the x-coordinate represents its perimeter and the y-coordinate represents its area.

Resulting in this activity on the overlay:

Again: what features of the math do they reveal and conceal? What are their advantages and disadvantages?

## 16 Comments

## Nathan Chow

October 6, 2015 - 1:35 pm -I really like Scott Farrarâ€™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.

## Steven Peters

October 6, 2015 - 1:40 pm -3 (Geogebra) has the advantage of showing the rectangle and the point in the space at the same time. The other three just show points. That’s a huge advantage.

1 is too distracting.

2 shows you something but you might not know why.

It’s interesting to compare 2 and 4 for pedagogical reasons, since #4 doesn’t cover the whole space and includes errors. It’s maybe less useful for teaching.

Geogebra applet for the win

## Scott Farrar

October 6, 2015 - 2:12 pm -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.

## l hodge

October 6, 2015 - 3:26 pm -Neat task.

Paper and pencil perhaps is best for provoking and allowing for a thoughtful approach to the problem. Desmos may speed up the point plotting and scaling issues, but that may not be a good thing when first thinking about the problem.

Desmos could facilitate the compilation of individual plots, which could raise some interesting questions regarding errors as well as the reasons behind more and less dense portions of the plot.

The other two could be a challenge for students to reproduce in geogebra or other software; they nicely illustrate the “answer”; and raise questions why the animation seems to be denser near the border of the region.

## Dan Anderson

October 6, 2015 - 4:41 pm -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?

## Andrew Knauft

October 6, 2015 - 4:58 pm -I love tackling the “which technology is right for this problem” question.

I would start the task with individual students working pen+paper. I see the lack of response from the medium as allowing the student to slow down with their thoughts. The student must be aware of the interpretation of the x and y coordinate as they plot their point, strengthening that awareness with each point they plot and cementing it for when they see all possible points later.

I say pen+paper, but really I do want to pool all student responses together. I think it’s really valuable that your Desmos activity uses three points that are dragged into their appropriate place: even though I, as an experienced mathematician and Desmos user went straight to the table and entered the values I wanted, the intention of dragging again offers the student time to consider the interpretation of values.

The live updating of the overlay can serve a vital role here, too. Looking at your .gif, there are some points which were transposed (or just incorrect). While students are still working it is possible to draw attention to these errors, again reinforcing the context.

Once each individual has submitted their points, and we discuss patterns, observations, etc. as a class, then I’d turn them back to individual work (when I say individual, I really mean pair/partners) on the GeoGebra sketch. I’d start it with the rectangle and point present, but give directions to plot the point corresponding to area and parameter and turn on the trace — one more opportunity to connect the variables to their context, but also an attempt to remove the “black box” aspect of the code. It’s accessible to the students, and directly relevant to the task, so the student should do it.

Watching the trace as you drag the rectangle reminds us (even if subconsciously) that we are investigating a relationship, in a way the Processing sketch hides (at least on the visual end)

## Matt E

October 7, 2015 - 4:17 am -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.

## Kyle Pearce

October 7, 2015 - 8:57 am -When I participated in the Desmos task, it had me curious. I think doing this in the classroom would get kids wondering “what the heck are we doing here?” which is a good thing. While I don’t think it really reveals what the intended goal is, I think that is actually good. At this point, I might ask students to predict what it would look like if “1000” or “10,000” or “1,000,000” people were to participate. What would it look like? Then, give them some time to tinker with this idea. Maybe some go to pencil/paper, maybe some try it in desmos. After more discussion and maybe some students sharing/showing their thinking and approaches, I’d unleash Scott Farrar’s Geogebra Applet. BAM! Pretty cool, eh, kiddies?

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.

## josh g.

October 7, 2015 - 7:02 pm -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.

October 7, 2015 - 7:06 pm -ps. I’m asking something people did give some answers to already. I just find these answers … bewildering?

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?

## Scott Farrar

October 7, 2015 - 10:49 pm -@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)

## Dan Anderson

October 8, 2015 - 7:04 am -I like the contrast between Matt E. and Josh G. You guys are on opposite sides of this dodgeball game.

@Josh I agree with Scott. I’d start with paper first (whiteboard, whatever, anything that won’t be shared) because I think the time required is a good thing. The lack of friction by going right into Desmos lead a whole bunch of math teachers and enthusiasts (Dan Meyer twitter followers) to make a mess load of mistakes. I’d anticipate the same with students.

## Matt E

October 8, 2015 - 7:30 am -@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.

## josh g.

October 8, 2015 - 8:06 am -Ok, here’s my summary plus my opinion mixed in:

**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.)

## josh g.

October 8, 2015 - 8:08 am -(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.)

## l hodge

October 9, 2015 - 1:15 pm -Why do we always plot a “dot”. Why not plot something that reflects the context.

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.