What’s The Difference?

Judith Kysh:

An integer-sided rectangle with area A is called a rectangular personality of A. Which integer from 1 through 100 has the most rectangular personalities?

Scott Farrar:

Which area value has the most possible perimeters?

This is the sort of thing that keeps me up at night.

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


  1. What I wonder more is what makes this question (regardless of how you phrase it) so much less exciting to me than the original question?

    For me, I think it’s partly that this one has one right answer rather than a flavor of open-endedness. There’s much less investigation here, much less opportunity for curiosity to come into it. And it’s partly that, thinking about solving this problem, it feels tedious and unenlightening. Sure, we’ll learn something about prime factorizations and divisor counting, which can be fun, but hunting through prime factorization patterns to find the one that maximizes the divisor count just doesn’t sound all that great to me.

    A question like “Is there an area less than 100 with 4 possible perimeters? How about 6? More?” starts sounding more like fun. Maybe that’s because I know that I can find several points of satisfaction along the way by finding one, or breaking a record, or getting curious about what numbers are possible. Or maybe I’ll just end up wondering why they didn’t ask about 5, and go off hunting for that.

  2. The second one sounds catchier, to me, but it also implies some stuff that probably shouldn’t be. The first one would have my kids asleep half way through the question. (Yes, i know. My kids can fall asleep fast)

    I’d probably start off with the second one, and refine it as the analysis goes on.

    I struggle with this too. I’ve totally thrown out worrying about the difference between “negative” and “minus” because worrying about it doesn’t make the kids do better – it just causes them more stress and distraction.

    I’m sure some mathwars type will happily skewer me for that.

  3. The first version is confusing because it introduces irrelevant terminology. The reader is led to wonder why it’s called “personality”, and this distracts from the mathematical question.

    The second question is clear and concise. It is also more open-ended, since the area is not restricted to the interval [1,100].

  4. @Jim Hardy That’s another reason I like the previous problem’s open-endedness better — questions like that might naturally come up when you’re graphing things. On the other hand … units! So in what sense do we really mean “same value”? It gets a little difficult to talk about even if it is a natural question.

    @Mr. K I happily volunteer for the skewering! Specifically, I think you ought to use correct terminology carefully, because it might help some students, although if you’re not comfortable making the students do it, I might think that’s OK. Still, I wonder how much of the confusion about, for example, absolute values has to do with students not seeing any difference between “negative” (the adjective describing a number) and “opposite” (the unary operation).

  5. I think the name “rectangular personality” was needed in the original prompt since we didn’t *already* have the graph of Perimeter v. Area.

    If we are already working on drawing rectangles and finding perimeters, areas, and plotting points. Then we can communicate what we mean to the student more easily. My question “Which area has the most perimeters?” relies on the work the students have already been doing.

    The weakness of Dr. Kysh’s question (when standing alone) is the extra definition. We had to say “learn this vocab” before you begin doing actual work. A drawback of setting, not of the content, of course. (Note: Dr. Kysh was posing this to a group of adults)

    CPM Algebra Connections (2006) introduces the first part of Dr. Kysh’s question with question 1-5 on page 6:

    “The area of the rectangle at right is 24 square units. On graph paper, draw and label all possible rectangles with an area of 24 square units. Use only whole numbers for the dimensions.

    b. Find the perimeter of each of these rectangles.

    c. Of these rectangles, which has the largest perimeter? Which has the smallest? Describe these shapes. Remember to use complete sentences.”


    I think you’re right. Having a definite answer can make the problem less open ended.

    However, I don’t see this as a new question. I see it as a subset of the first exploration. You can come up to a kid who is working on plotting rectangles and ask, almost in passing, “how many rectangles have area 24?”

    Or better yet, you can put them in a situation where they ask that question. Imagine a student who created Dan’s response: https://blog.mrmeyer.com/wp-content/uploads/110927_1hi.png I could say “really? that zone is all rectangles? … What if I only want integer sides?” And that opens up a whole new thing to explore.

  6. I misread the original problem. The difference is clearly that one is being presented as a whole problem the other comes out of something already under discussion. There is a lot of hidden context in #2.

    So, the question becomes – how do we create contexts that lead to interesting questions?

    I think Dan’s been attacking that for years now.

  7. I’ll just jump in arbitrarily here after Mr. K because I think he’s right about the original problem. It’s fully specified, which is the logic of paper, of words preserved in ink on the page, not the logic of the classroom where this sort of problem has room to open up and breathe.

    Scott’s version is subject to some negotiation between the teacher and the class. Like, “Wait, the area value of what? What are we talking about here?” The student gets experience problematizing a space. I value that in applied math problems which is why, in what I call the second act, you have the moment where the student has to decide what information, tools, and resources would be nice to have on hand in order to solve the problem. “What do I need to know?” is a question we’re constantly asking ourselves in the world Out There, but rarely in the classroom In Here. Scott’s pure math problem captures that interaction nicely.

  8. I also like the fact that Scott’s problem does not state anything about integers. This leaves room for a more interesting discussion, and an easy out if you don’t want to take on question #1.

    I like question 1 as well, but only in an extra credit or math club type setting.

  9. @scott. so with regular polygons and circles, you get these (apparent) quadratic loci. When you remove the restriction of regularity, you get this region bounded by the curve generated by the square.

    For an arbitrary triangle, how should you drag just one vertex so that the (perimeter,area) point will lie on a line? Or, in a isosceles triangle, leave the base fixed and drag the vertex angle. What is the equation of the locus generated?


  10. If I gave these questions in parallel (student choice) I bet the strongest students would pick question #1 because of the extra vocab and complexity of language. However, the second question is by far the richer.

    I’m wondering how to adapt this investigation for my grade 8 class. We are about to begin studying squares and roots moving towards Pythagorean Theorem and I want them to first find context to consider the factors of numbers (factor lists and prime factorization).

    I traditionally use “the locker problem” as an opener but find that many kids don’t have the experience and language to make the most of it.

    I was thinking that after some time looking at areas on geoboards I could ask, “What area between 1 and 100 has the most rectangles?” Then move to a discussion about factors — What number has the most factors? What numbers have an even/odd amount of factors? What do these factor lists have to do with the prime factorization of the numbers?…

    Thoughts? Suggestions?

  11. The first questipn and second seem to the casual observer to be very different. The wording of the first does not make clear what the intergers 1 through100 refer to. Since the only measurments explicitly mentioned in the problem are “sides” and ” area” i assumed it was talking about the area. But every number has only one rectagular personality, by definition. So, it’s confusing. To work, youd have ask the final question with more words. And it’s already overly wordy.