## [Preview] Who Wore It Best: Maximizing Area

I have a recurring happy dream that I’m on Jeopardy. It’s the final round. The Trebekbot 2000 reads the final clue.

“These are the dimensions of the rectangle that has the largest area given a fixed perimeter.”

“WHAT IS A SQUARE!” I yell out while my competitors are still thinking quietly. I have disqualified myself and ruined the round, but I don’t care. I start high-kicking around the set while security tries to wrangle me away and I still don’t care because I finally found some use for this fact that takes up a significant chunk of my brain’s random access memory.

It’s a question you’ll find in every quadratics unit, every textbook, everywhere. I could have selected this week’s Who Wore It Best contestants from any print textbook, but instead I’d like to compare digital curricula. I have included links and attachments below to versions of the same task from Geogebra, Desmos, and Texas Instruments, three thoughtful companies all doing interesting work in math edtech. (Disclosure: I work for Desmos, but don’t let that fact sweeten your remarks about the Desmos version or sour your remarks about the others. Just be thoughtful.)

So: who wore it best?

Click each image for the full version.

Version #1 – Geogebra

Version #2 – Desmos

Version #3 – Texas Instruments

1. #### Lisa Bejarano

July 2, 2016 - 10:47 am -

These are hard to compare because the student doesn’t walk away learning the same skills in each activity. The Geogebra and TI tasks are basically the same except that TI requires a class set of nspires. The Desmos task goes a little deeper into multiple representations of maximizing area, but is “too helpful” in walking students through each step. Slide 5 of the Desmos task is comparable to the Geogebra & TI activities. The jump to slide 6 (the graph) may be a big leap for some students. I would instead have students use slide 5 from Desmos (or Geogebra task) to construct a table on a whiteboard or paper, and then let students discuss, debate and experiment with different ways of presenting the data, leading to the parabola presented on Desmos slide 6. If the students build this connection from their own thinking it will increase their ownership and understanding of the relationship of the graph to the rectangle.

2. #### John Golden

July 2, 2016 - 12:11 pm -

It seems like there would likely be a worksheet to go with the GeoGebra sketch. (Maybe I’m projecting.)

Lesson-wise I like the comparison of a few at the start of the Desmos lesson, and building the idea of the graph of area vs height.

But, of course, I agree with Lisa on the too helpful Desmos.

3. #### Dan Meyer

July 2, 2016 - 1:22 pm -

Dangit you two: “too helpful” is my line. You know exactly where to stick me.

John:

It seems like there would likely be a worksheet to go with the GeoGebra sketch. (Maybe I’m projecting.)

I’m just going off what’s included in the sketch. Seems only fair.

Lisa:

The jump to slide 6 (the graph) may be a big leap for some students. I would instead have students use slide 5 from Desmos (or Geogebra task) to construct a table on a whiteboard or paper, and then let students discuss, debate and experiment with different ways of presenting the data, leading to the parabola presented on Desmos slide 6.

Nice suggestion.

4. #### Vishakha

July 2, 2016 - 9:26 pm -

I actually come away liking the TI activity the most.
– Desmos activity breaks this down into too many small steps. Measuring the perimeter was painful – counting the dots!
– Geogebra leaves too much to the student’s thinking and imagination and ability to see the patterns

My favorite version of this activity is actually a Fathom version.
– Generate a bunch of random lengths and widths such that the perimeter is a chosen constant
– Create a calculated attribute for area
– Plot a scatter plot of length versus width
– Plot the areas on a dot plot
– Select the max areas and notice that in the length versus width graph the points which get highlighted are on the x=y line.
– This has the advantage that you can then vary the perimeter constant and re-run the simulation so that students can “prove” this by seeing it works for all perimeters

5. #### BryanPenfound

July 3, 2016 - 9:20 am -

I think the Desmos activity would suit what I would want in a first-year calculus class the best – I don’t mind the permieter and area counting section, I would just make the line segments a bit smaller so they are quicker to count. But it has the advantage of the students start thinking about what I want them to think about: the relationship between the perimeters and the areas, working towards an equation, and maybe even thinking about completing the square to find the vertex. I can follow up with some more examples (maybe even the same one!) where we see the ease of using the derivative. Beautiful: connecting the new information to something they (hopefully) already know.

The TI lesson is probably a bit too structured, and if I was to use their worksheet, I might as well go through the questions together with the students so I can answer common questions in real-time. Geogebra leaves a bit to be desired, and students may not be focused enough to realize the relationships that I want them to think about – and like John said, more would have to be done to get to the equation/graph. I like that the equation/graphing is built-in and a natural progression in the Desmos lesson – and isn’t an awkward hanging question “How is the length being determined?” as it is in Geogebra. That slider in Geogebra too … yikes! Heavens to Betsy give me some decimal numbers in there, not everything comes out as nice and easy whole numbers!

6. #### Steve Phelps

July 4, 2016 - 5:11 pm -

I am not sure you took advantage of all the features of the Nspire or of GeoGebra in the same way you did with Desmos (I really should take the time to do Desmos better). As is, Desmos probably wins this round, but I think the fix was in.

Modify the Nspire worksheet to take advantage of Navigator and I think the activity begins to approach the Desmos activity. Granted, it would take some expertise, but so did the construction of the Desmos things.

Similarly, do more with multiple GeoGebra applets on a page, or with differing representations (spreadsheets for example), or with the new questions/tasks features, or with a GeoGebra book, and GeoGebra now functions more like the Desmos activity.

7. #### Dan Meyer

July 4, 2016 - 5:35 pm -