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

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Version #2 – Desmos

160702_2

Version #3 – Texas Instruments

160702_3

Steve Phelps suspects I stacked the deck in favor of Desmos here, taking full advantage of our platform while taking only partial advantage of GeoGebra and the Nspire. John Golden concurs, hypothesizing that “there would be a worksheet to go with the GeoGebra sketch.”

So a note on sampling: the GeoGebra example is the most viewed lesson on the subject I could find at their Materials site. The Texas Instruments lesson is the only lesson on the subject I could find at their Activities site. I told Steve, and I’ll tell you, that if anybody can come up with a better lesson on either platform. I’ll be happy to feature it. This isn’t much fun for me (or useful to Desmos) if I stack the deck.

Both Lisa Bejarano and John Golden call out the Desmos lesson as “too helpful” – they know how to make it sting – in the transition from screen 5 (“Collecting data!”) to screen 6 (“Here! We’ll represent the data as a graph for you.”).

160705_1

I’l grant that it seems abrupt. I don’t think this kind of help is necessarily counterproductive, but it doesn’t seem as though we’ve developed the question well enough that the answer – “graph the data!” – is sensible. The Texas Instruments version has a solution to that problem I’ll attend to in a moment.

My concern with the GeoGebra applet is that the person who made the applet has done the most interesting mathematical thinking. I love creating Geogebra applets. I generally don’t have a good story for what students do with those applets, though. In this example, I suspect the student will drag the slider backwards and forwards, watching for when the numbers go from small to big and then small again, and then notice that the rectangle at that point is a square. The person who made the applet did much more interesting work.

Let me close with one item I prefer about the Desmos treatment and one item I prefer about the Texas Instruments treatment.

First, my understanding of Lisa Kasmer’s research into estimation and Paul Silvia’s research into interest led me to create this screen where I ask students “Which of these three fields has the biggest perimeter?” knowing full well they all have the same perimeter:

160705_2

Still later, I ask students to estimate a rectangle they think will have the greatest area. That kind of informal cognitive work is largely absent from the TI version, which starts much more formally by comparison.

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TI does have a technological advantage when they allow students to sample lots of rectangles and quickly capture data about those rectangles in a table.

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Desmos is working on its own solution there, but for now, we punt and include prefabricated data, which I think both companies would agree is less interesting, less useful, and more abrupt, as I mentioned above.
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That’s my analysis of these three computer-based approaches to the same problem. What’s your analysis? And it’s also worth asking, “Would a non-computer-based approach be even better?” Is the technology just getting in the way of student learning?

You can also pitch your thoughts in on next week’s installment: Pool Table Math.

2016 Jul 8. Steve Phelps has created a different GeoGebra applet, as has Scott Farrar.

2016 Jul 9. Harry O’Malley uploads another GeoGebra interpretation, one that strikes a very interesting balance between print and digital media.

About 

I’m Dan and this is my blog. I’m a former high school math teacher and current head of teaching at Desmos. More here.

34 Comments

  1. Following your analogy: the wear is the least important. The body it is. Geogebra is open source software. Other don’t. So with Geogebra you can make your “tailored suite”. With others, you can just buy suite.

    See [1]. Perhaps you could consider release desmos as open source product. You would benefit from the community and it from you (Apache webser for example did that)

    Regards,

    [1] https://opensource.org/osd

  2. Following my post: “reproducibility” is one key of the science (Other people can reproduce your results) But the only way to reproduce the results is to know what “the black box software” I use really does. So open source software is important from scientific viewpoint.

    This complements the previous response.

  3. I agree with your assessment of these activities, Dan. My questions come from what the goals of the activity are.
    Isn’t the basic concept, what shape rectangle provides the maximum area for a fixed perimeter, really something that can be investigated by upper elementary or middle school students? I can imagine an investigation where 5th grade students measure the side lengths of different rectangles and calculate the areas to create a table of data. I can imagine a classroom of 8th graders using a digital activity to link the geometric and graphical representations discussed in this post.
    So, if there’s a revisit of this concept at the high school level, shouldn’t it focus on *why* a square maximizes the area (under certain conditions) and why a rectangle with specific dimensions maximizes the area (under other conditions)? None of these activities really addresses that question, as far as I can tell. And why restrict the maximizing shape to rectangles? Why not just ask which shape maximizes area for a given perimeter (and why does that shape produce the maximum area)?
    If the fundamental question in a high school classroom is “Why?” then there are more interesting questions to ask and the context of a dog pen or a garden or a field is pretty fake.

  4. I like that the Desmos treatment has an element of prediction to it. That makes it more memorable in the long run, and I suspect will lead to better integration.

  5. The platforms have different advantages in my view. Geogebra is much more flexible for designing an applet to either prompt a question or demonstrate an idea. Desmos is much easier to use as a sort of interactive text. Whether the activity is overly helpful or missing a worksheet isn’t necessarily a reflection of the platform.

    “How does the area change as the width changes (constant perimeter)” or “How is the space re-organized as the width changes” are interesting questions. Maybe investigate this a bit geometrically, as roughly sketched out below, before bringing out the sledge hammer (plotting points & getting an equation):

    Desmos activity with a geometric approach

    Geogebra demonstration with a geometric approach

    A variation on the graphical demonstration would be to request observations on this plot:
    Area vs Perimeter plotting rectangles instead of points.Perhaps focus on something other than “max area”.

  6. Wendell Luckow

    July 6, 2016 - 6:04 pm -

    Is the point of the activity to find the answer via a graphed parabola, or is the point of the activity to gain a deeper understanding of the structure of a parabola?

    There are some neat things I really like from the TI activity. In section 3 (The symbolic proof), students see the formalization of parameter written as p=2l+2w. I think it would be neat to utilize technology somehow to come up with that equation. If my fixed parameter (or fence, or whatever) is 20 feet, then I might think that one possible length and width is 1 and 19. But if I drew it out, I’d quickly realize that my intuition was incorrect.

    After that, I think it would be good to represent both length and width as a single variable. For example, now we know that 2l+2w=20, so I can split that into two different cases, both in terms of w:

    l=-w+20 and
    w=w

    Using Desmos, I could graph these two equations (y=-2x+20 and y=x). At any given x value, the sum of the two outputs should be 20. Earlier, you wrote:

    “…but it doesn’t seem as though we’ve developed the question well enough that the answer – “graph the data!” – is sensible”. I argue that by graphing the two lines this way, the graph is sensible, not as an answer, but as a visualization of what parameter is.

    Now, hopefully I know that area is just length times width and I can actually multiply these two together to get A=l*w=(-w+20)(w)

    Now, using Desmos, I could graph the product y=(-w+20)(w) and actually see that a parabola is the product of two lines! Not only that, but the line of symmetry runs right through the intersection between the two lines. That’s why the solution is a square!

    https://www.desmos.com/calculator/zwyoutz96s

    The big conceptual difference I see is that in the above activities, we discover the fact about the square first, and then graph it. In the method I just explained, students graph it and then discover it’s a square. I understand it’s more abstract this way (which in many cases isn’t a good thing), but I also think it gives more purpose to the graphs, provides a stronger answer, and gives students a solid understanding of the structure of parabolas.

    One neat thing about presenting it this way is that it creates a context where technology is fully utilized. I’m pretty anti-digitized worksheets. My favorite example of technology use (which seems like a big theme in this particular blog post) is from a high school classroom where students used the audio program Audacity to calculate the distance to the moon! They found an audio clip of ground control talking to the astronauts on the moon, and whenever something was said, there was a slight echo moments later (the sound going to the moon and coming back down to earth). Students measured this distance, divided by two, and then multiplied in the speed of light to calculate the distance to the moon! Not only that, but it was accurate enough to see that the moon does not fly in a perfectly circular orbit. That’s a great example where technology is necessary to have a meaningful lesson, not just a neat tool to increase engagement over regular paper and pencil.

  7. Great stuff… Can I just make a plea for all Maths (yes we say Maths in Australia and New Zealand!) education?

    That sounded grand (and it is)…

    As blogs like this globalise our education communities, turning school-level conversations into global conversations, can I make a plea for a formalisation of units. I understand that in the USA inches and feet are your thing (you are in with Burma and Liberia only!) but the rest of the world has shifted gears to metric long ago. I would just argue that metres, kilometres, centimetres etc just feel so much more related using the prefixes. Having these great problems couched in inches and feet (and yards even!) provides a cognitive block before students even get started here. Sure you could argue that a quick Google will translate the measurements, but often these problems (like this) are about the underlying structure and it makes it easier to get to the heart of the problem if we say the rectangle has a perimeter of 60m (not 18.288m).

    Think of it like airline travel – wouldn’t it be confusing if all the European airlines only spoke German when flying into the USA? Long ago the whole industry decided that English was the standard language so everyone can communicate clearly.

    We speak the same language of ideas and ways to motivate and inspire our students around the world – this is just one small thing that would make that language clearer for me and the rest of the metric world.

  8. A very interesting discussion, though it possibly misses a crucial point. All three pieces of software are excellent and offer different affordances. GeoGebra has many more functions available and geometric functionality; Desmos offers very interesting inter-student and teacher-class interactivity (as demonstrated in your post); TI will have its own strengths but I am too unfamiliar with it to be able to comment more.

    What these examples demonstrate well, though, is that software on its own is wholly inadequate for teaching. One needs to think about pedagogy: what is one aiming to achieve in the lesson (or lesson sequence)? How can we go about achieving this? Perhaps software offers a route, and maybe one tool is more appropriate or better than another for a particular purpose. And even if we are going to use software, should we invite students to decide what to do with the software during the lesson or should we give them pre-prepared materials? Starting with the software and asking “how can we use this to teach X?” is much more restrictive than than asking “how can we teach X? We have software A, B, C … at our disposal IF we wish to use it to help.” If I ask this question, then it may be that I will choose to use software B to teach this topic, but maybe I will use a card-sort, or some other stimulus instead. The examples given in the blog post all seem to be of the first variety (“How can I use software X to teach the max area problem?”, though some of the prompts and questions in both the Desmos and TI activities are excellent).

    This reminds me, too, of the use of PowerPoint presentations in some classrooms to teach topics: they are restrictive in that they presume that the lesson will follow a pre-planned route, whereas I wish to have the flexibility to go down different avenues should the need or opportunity present itself. (There are times when PowerPoints are useful; again, this is a tool and should be used appropriately.)

    In my ideal classroom, students will have built up experience with a variety of general-purpose mathematical software tools (calculators, maybe graphical calculators, Desmos, GeoGebra, CAS systems), and then when investigating a new problem, such as the max area problem, would choose to use an appropriate one if it seems helpful to them. But as the teacher, I would then continue to push them to develop their understanding, and not just use the software to give a numerical answer. Having pre-prepared software tools available might be useful to me in some contexts, but only when they fit my learning objectives.

  9. Here’s a couple GeoGebra applets I had that somewhat relate. Instead of going for the “maximize area” question directly, I wanted to just show there are times when areas are larger and smaller.

    https://www.geogebra.org/m/VebYwHxq

    Start with a somewhat direct question asking if a rectangle is “bigger” than the triangle: http://i.imgur.com/kBDUpcI.png but allow you to change the rectangle’s shape while keeping perim constant.

    Then I move to two rectangles, same perimeter. But the conceit is we’ll pour one into the other. If you spill game over. http://i.imgur.com/u1NMRZq.png But if you get 3 pours in, you can play more with the rectangles and try to keep going.

    Still a work in progress to get to the maximal area idea directly. But the idea here is to not go right for it. This pouring might create some frustration– but hopefully not too much– at this idea that the same perimeter can enclose many different areas. We don’t say there’s a max, we just show sometimes you spill.

    This itch can be scratched by showing there IS a maximum. This maximum solves the pour-spilling problem. Or: its the aspirin to the headache of overflowing rectangles.

    I do think you have a great point about the design of many geogebra applets. The author frequently does a lot of the thinking. No itch, no headache. Like you say, the kid drags back and forth and sighs.

    I’m not prepared to defend my approach as better than Desmos or TI activities, but I do say its better than the GeoGebra thing you found. Or… if not better, mine should come before that one in any sequencing.

    Final side note:

    Interesting the different ways we’ve all approached the constant perimeter communication. The first Geogebra applet and TI give a number. Desmos tells directly that its constant — but after letting the user question that assumption. My strategy was to tell it with the visual aid of the triangle’s segments wrapped around the rectangle.

  10. Christine Lenghaus

    July 8, 2016 - 4:39 am -

    Yes, I take in a loop of string and in front of their eyes, show the students that even though the perimeter is fixed the area is not – it is not intuitive and they collect their own data which they then graph.

  11. Thanks for the comments, everybody.

    Pam Rawson:

    So, if there’s a revisit of this concept at the high school level, shouldn’t it focus on why a square maximizes the area (under certain conditions) and why a rectangle with specific dimensions maximizes the area (under other conditions)? None of these activities really addresses that question, as far as I can tell.

    Thanks for your comment, Pam. The question of “Well what are the learning goals?” seems to recur throughout this series, making an apples-to-apples comparison a little tricky. To your specific question, I think the Texas Instruments version comes closest, as it includes extensions for calculus students involving the derivative.

    l hodge:

    Geogebra is much more flexible for designing an applet to either prompt a question or demonstrate an idea.

    Then what is the best platform for embedding provocative GeoGebra applets in an instructional framework? I ask because nine times out of ten when I see an interesting applet (including the ones you posted) I’m not really sure how to make the best use of them in a classroom.

    Julian Gilbey:

    Having pre-prepared software tools available might be useful to me in some contexts, but only when they fit my learning objectives.

    No argument here. That’s why my closing question asks if anybody can find a better print treatment of this same objective.

    @Scott, really interesting alternate. It seems so different from any of the other treatments it’s hard for me to compare.

    It’s clear that these microworlds are hard to design in any platform. As we’ve already discussed, they risk the designer doing much more interesting work than the student. But they also risk burying so much plumbing and wiring below the surface that it becomes a black box, with the student only able to wonder non-mathematically, why pushing this button makes that one turn a different color. This is all very challenging.

    @Greg, all of the examples I cited are either metric or disregard specific units entirely. What am I missing?

    @Wendell, thanks for offering your alternate treatment.

    @Xavier, thanks for your comment. At this point, while the Desmos Activity Builder software isn’t open source, individual activities (like the one I posted) can at least be duplicated and modified without restriction.

  12. Steve Phelps has also created an alternate GeoGebra applet, which I’ve added to the post above.

    Two questions for Steve:

    One, what is the point of capturing individual data when the trace is on, effectively capturing all the data?

    Two, is this applet for the teacher to use? The student? I’m not clear what either should do with it.

    With his GeoGebra applet, Steve has demonstrated that he understands the math. I’m just not clear how the applet helps students understand the math.

  13. Harry O'Malley

    July 8, 2016 - 4:09 pm -

    Here’s a much better GeoGebra version:

    https://www.geogebra.org/o/FxQhsZeY

    It is for student use, with teacher facilitation. Looseleaf paper or graph paper are required to record observations and think with.

    Here are some of the key features of my version here:

    1. It focuses on the algebraic expressions from the start, but does so in a way that still allows students to think about them intuitively and visualize them. I have done this task (Fixed Perimeter Rectangle Area Maximization) with students enough times and find that waiting until the final stages of the task to introduce the symbolic version of the problem does not give them enough time to internalize their meaning. In this version, they are a natural part of the reasoning process from the start.

    2. It is scaffolded enough to drive the task but forces students to make the connections necessary to glue the pieces together.

    3. It does not specify a method for maximizing the area expression in the final step. This allows different student perspectives and representations to drive the conversation at the end. If a graphic or more formal symbolic method is desired, the teacher can decide to ask for it verbally once students have exhausted their methods.

    A nice extension is to have students recreate the animated rectangle (design a new slider and new parametric points) for a rectangle with a different fixed perimeter.

  14. Really, really provocative, Harry. I added it to the main post.

    I think the part about that applet that fascinates me most is how it puts so much of the work on paper, treating the applet as an object of analysis, but not the medium of analysis. It may just be a matter of taste, but the balance of powers between computers and paper feels just right in yours, and perhaps tilted too far towards computers in the three I originally highlighted.

    If I had my preference, GeoGebra would allow us to obscure more of the wiring. My first look at the applet is pretty dizzying. Ideally, I think we’d invite students to pick a value for m as their first task (which they’d perform without seeing later tasks) and then move onto the point plotting. The fact that GeoGebra requires you to write down all the steps in advance feels like we’re playing by paper’s rules in a digital medium.

  15. Chester Draws

    July 10, 2016 - 1:59 pm -

    I’d like someone to please answer Pam’s questions:

    So, if there’s a revisit of this concept at the high school level, shouldn’t it focus on *why* a square maximizes the area (under certain conditions) and why a rectangle with specific dimensions maximizes the area (under other conditions)? None of these activities really addresses that question, as far as I can tell. And why restrict the maximizing shape to rectangles?

    Because I don’t see the value is any of this for high school students. They already know area of rectangles and this is a very awkward and backhand method of introducing quadratics.

    What learning there is will be further diluted by the introduction of technology, which is adding one more layer between them and the concepts. Yes, for some of the examples it is a thin layer and some a fat layer, but it’s still one more thing that distracts them from the algebra.

    (Initially I thought your statement Dan “It’s a question you’ll find in every quadratics unit, every textbook, everywhere. was wrong, because I could not recall ever seeing it. But examination of my textbooks showed you were right. I must have looked at them and thought “I see no useful learning in that” and moved on.)

  16. Michelle Rinehart

    July 12, 2016 - 6:01 am -

    Great post! I found the comments/discussion about the learning goals/intentions of each activity to be particularly interesting. Just because these three activities all relate to the same “content,” are they all trying to do the same thing?

    Two quick things:

    1) Here is another approach related to this content from TI:
    https://education.ti.com/en/timathnspired/us/detail?id=2492E2C37DE24D6DACCAC05AE0E4FBF1&t=81A3FC9ACCC24ABAA3BF97F1A038AFA9

    2) Also, re: your comment of “Would a non-computer-based approach be even better?”, here’s a take on this activity developed by Pam Harris as part of her Focus on Algebra work.
    https://drive.google.com/open?id=0Bz9gwqNrxjI8Nm9IdUNIR2M0cWc

  17. I feel like I am a person of reasonable intelligence (citation needed). I am not a math teacher and I read these to try transfer some of Dan’s ideas to science education. This idea was probably covered a few times in the math classes I took, but I didn’t “learn” it until I tried to set up my garden this summer (at 30). I had 64′ of landscape timbers in 8′ lengths. It took me 2 calculations to figure out square was best and a 3rd to see the pattern.
    To Pam Rawson’s point, I was left wondering why? It was pretty easy to see the pattern, but understanding why was the more thought provoking.
    How about a Nana’s Garden 3 Act?

  18. Dan – your work in this one is metric (thank you!) but most of all the other 3 acts and a load of stuff on 101Qs is oldskool. The student handout for the TI task has feet. I guess my post was a plea to everyone for future conformity to a single standard that would ease one cognitive barrier for our students (and millions of others outside the USA, Burma and Liberia). Great work!

  19. Julio Santos

    July 12, 2016 - 7:00 pm -

    It is very interesting how three different companies take their approach on the same concept and idea. Each one has its pros and cons. Thanks for sharing.

  20. @Michelle, thanks for passing along the other TI activity. I saw it but mentally categorized it as an extension to the task we’re chatting over in this post. Pam’s worksheet is certainly good food for thought.

    Erik, Chester, and Pam, what is your answer for “why the square maximizes the area of a rectangle with a fixed perimeter?” My answer is something like, “Because when you maximize the area function, that’s the shape you get. The function is an inverted parabola so we know that’s the global maximum.” Or, more intuitively, something like, “imagine the shape you and a bunch of friends would get from trying to push outwards on the rectangle at the same time.”

    I can’t design the experience for helping students understand “why” if I don’t understand “why” myself. And my best explanations seem pretty well tied up in Desmos activity already.

  21. Being a long-time proponent of GeoGebra, my natural urge was to adjust Mr. Meyer’s GeoGebra to make it more helpful in the classroom. Just as an example, I have added a graphing component to his file ( see http://ggbm.at/NMnEykCm), but that is nowhere near the ultimate of what could be done.

    Also, the best presentation has to be adaptable “on the fly” and GeoGebra enables you to show or hide things at a click of a button. (Mr. Meyer’s was created on a Cartesian grid that he hid prior to posting), You can make items visible conditionally, hide them altogether, etc. The options are plentiful.

    This could be adjusted to be meaningful at any grade level from elementary school to calculus. It all depends on how you adjust it. GeoGebra gives the opportunities, but the teacher must take advantage of them.

    As a retired teacher I have found the GeoGebra people to be extremely cooperative supportive despite the fact they have not made a penny off of me.

  22. Graphing is a very convincing argument that the maximum area is a square. But, it doesn’t give a sense of why.

    Imagine a “tall” rectangle. Cut a thin slice off the top and then tape a slice of the same “thinness” to the side to create a newrectangle. The perimeters of the original and the new rectangle are the same. The second rectangle clearly has more area because we gained more area from the slice taped to the side than we lost from the slice removed from the top.

    Do it with numbers if you like. Make the slice 0.1 units wide and the original rectangle 3×8. We lost “3 x 0.1” from the top and gained “(7.9) x 0.1” on the side. Also, the width increased by 0.1 & the length decreased by 0.1, so the perimeter is the same.

    If the width increases, the area increases. What kind of rectangle has the maximum width? My intent with the geogebra and desmos activities in a previous comment was to sketch out a possible way to get at the above idea for why it is the square.

    Harry, that is a really nice idea for providing support in getting an equation for the area.

    Also agree with Dave that geogebra is better for adapting on the fly (when the applet is the object of analysis).

    If geogebra is to be the” medium of analysis”, perhaps just make a handout that helps the student build their own applet. An extension might be to build an applet for a pen where a river serves as one side of the fence, or a divider separates the region, or a different shape, etc.

  23. l hodge:

    Imagine a “tall” rectangle. Cut a thin slice off the top and then tape a slice of the same “thinness” to the side to create a newrectangle. The perimeters of the original and the new rectangle are the same. The second rectangle clearly has more area because we gained more area from the slice taped to the side than we lost from the slice removed from the top.

    I found this visual really effective.

  24. Erik Ellingboe

    July 13, 2016 - 7:41 pm -

    What went on in my head was much like the quote you pulled from l hodge, but I visualized using graph paper. This gave me a tangible “unit” of area to use.

  25. The two dynamic programs are the way to go. As the updated posts mentions, the geogebra sketch has been under sold. Its my my experience that geogebra does have a longer learning curve but there is more room for customization. However, in this desmos sketch, the problem is presented in a more “classroom friendly” way.

  26. I agree with Pam Rawson’s initial question about what is the purpose of the activity. I am reading this for course I am taking this summer and I teach high school English, but I know that purpose is important. Throughout the post, there are comments about some being more difficult than others and the con of having preset data. But depending on the ability level of the class and the purpose behind the activity, I can see an argument for preset data being a helpful scaffolding tool.
    On a more general sense, I think that this kind of group comparison is interesting. I also think the commentary and discourse is very cool. Way to go math folks!

  27. Alison Childers

    July 22, 2016 - 3:03 pm -

    My two cents: This year I want to let students design ‘dining tables’ that have a desired number of people around it (all of the kids in the class), and let them find the ‘table’ that would have the most area for food. Then after they have come up with lots of designs, they can actually cut out a model of their tables and pin them to a graph.

    https://drive.google.com/open?id=0B-5elry4t9OOOEMtLVRqWVQ0WEk

    I did this with purely rectangles when I did this graph, but would love to extend it to something like a table that they can actually create. As a fun product of the lesson I then want students to stand up and actually make a couple of the tables to see that it really is the one with the biggest area, like this teacher did with 6th graders here at around 9min if you don’t wanna sift through the whole video.

    https://www.teachingchannel.org/videos/real-world-geometry-lesson

    The purpose of using this problem for me is to introduce the idea of quadratics giving you the ability to maximize a quantity, and also to introduce why there is symmetry in a parabola, because you can have a 3×4 table or a 4×3 table and they give you the same area.

  28. Alison:

    This year I want to let students design ‘dining tables’ that have a desired number of people around it (all of the kids in the class), and let them find the ‘table’ that would have the most area for food. Then after they have come up with lots of designs, they can actually cut out a model of their tables and pin them to a graph.

    Love that image.

  29. This is one of my favorite questions, and I (recently) used it with 8th graders while teaching a demo class for the school that (now) employs me. Two comments:

    1) If phrased in its (contrived-ish) standard form of, given a fixed amount of fencing — create a rectangular enclosure that maximizes interior area, then there are some very (I think) interesting variations.

    Fixing numbers for concreteness: Suppose you have 60m of fencing. What are the dimensions of the rectangular enclosure that maximizes interior area … if you also have access to a very long wall? (So, now you only need to worry about building 3 sides of the enclosure.) Once you find the answer, does it jibe well with your intuition? If so, why? If not, how can you update your thinking? (You can ask other questions, e.g., about what happens if you have access to a 10m wall. And for something like Desmos, what if you have access to an N meter wall, where you can vary N? Etc.)

    2) With regard to, “I finally found some use for this fact that takes up a significant chunk of my brain’s random access memory” … here is another way to think about the problem, which I didn’t see (may have missed?) in the post and comments above. (I *bet* it could be implemented in Desmos in some interesting way…)

    When you have a rectangle with (say) perimeter 60m, any pair of adjacent sides add up to 30m. (Why?) So, omitting units, let’s just *start* by using the square, i.e., all four sides have length 15, and the area is 15×15 = 225.

    I think that this maximizes the enclosed area. If you think otherwise, then you will need to change some side; but if you change 15 to, say, 15+n, then maintaining the perimeter requires changing its adjacent side(s) to 15-n.

    Then the area becomes (15+n)(15-n) = 225 – n^2 …

    Since n^2 is non-negative, subtracting it off will either lower the total area (uh oh) or leave it alone (iff n=0). The latter case means we oughtn’t vary the 15 at all. That is, the square yields maximal area.

    The above constitutes the underlying mathematics, and *not* a “lesson plan” … but, I think it is an interesting way of thinking about the problem!

    (Going beyond: The alternative *idea* in my latter comment is ess’ly a geometric interpretation of the AM-GM Inequality. This needn’t be broached explicitly when presenting it to students, but knowing this piece of background info can/could be helpful in situating the “fact” within a more general mathematical body of knowledge related to inequalities and means, which have applications in finance and elsewhere.)

  30. Benjamin:

    Since n^2 is non-negative, subtracting it off will either lower the total area (uh oh) or leave it alone (iff n=0). The latter case means we oughtn’t vary the 15 at all. That is, the square yields maximal area.

    OMG. All of the light bulbs in my head.

  31. A variation about Alison comment [http://blog.mrmeyer.com/2016/who-wore-it-best-maximizing-area/#comment-2424942]: I have 10 tables of 2m width and 1m length. How can arrange them in order to have a big table with maximum area? What about if the tables are 1×1 meter?

  32. >OMG. All of the light bulbs in my head.

    (Continuing with the perimeter of 60 and units omitted.)

    As I have returned to this question many times over the years, it seems to me remarkable how the way of denoting the sides is presented as if it is “canonical” — that is, call one side x and the other 30-x [because it is the Right Thing To Do].

    Now you have a function: Area(x) = x(30-x) = 30x – x^2.

    Since x > 0, this is a downward facing parabola; i.e., its maximum occurs at the vertex.

    Calculus? d/dx 30x – x^2 = 30 – 2x, which is 0 when x=15.

    Algebra II? Vertex is at x = -b/2a = -30/2(-1) = 15.

    Etc.

    The suggestion from earlier is *not* to pick the same way of labeling sides always used, i.e., *not* to pick x and 30-x. If you want, you can think of the suggested approach as a substitution: let n=15-x, which means x=15-n.

    Now adjacent sides are 15-n and 15+n.

    So, this time, Area(n) = 225 – n^2.

    Only “sense-making” (if forced to classify by topic, I would call this Real Analysis) is needed to realize that the area is now maximized when n=0.

    Thus, adjacent sides are 15-0 and 15+0, i.e., 15×15 square.

    From the standpoint of integrated algebra and geometry, this “substitution” is shifting the parabolic function that represents the area by 15, so that it is now symmetric about the origin.

    I’m not sure why the problem’s history involves this marriage between x and 30-x; I find parabolas generally *much* easier to deal with when they are symmetric about the origin, i.e., when the coefficient of x is 0.

    In fact, this idea of shifting — which is a frequent if confusing topic at the (graphical) intersection of algebra and geometry — is how I consider questions about quadratics more generally, e.g., how to find the x coordinate of the vertex and how to uncover the quadratic equation; cf. http://matheducators.stackexchange.com/a/9709