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 “impossible” 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
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.
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?