Comments on: What’s The Difference?

By: Jinjer

Jinjer — Sat, 24 Dec 2011 21:50:15 +0000

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.

By: The Multiplication Table – Part 1 « Uncover A Few

The Multiplication Table – Part 1 « Uncover A Few — Mon, 07 Nov 2011 18:07:40 +0000

[…] questions for future posts. Â In the meantime, how does the multiplication table relate to the area and perimeter questions from Dan Meyer’s blog? Â (Does it relate to this more interesting question, too? Â Maybe not as strongly.) Â For some of […]

By: Den Rattee

Den Rattee — Tue, 01 Nov 2011 16:49:19 +0000

ps. pseudo context alert on the locker problem. I hope Dan isn’t too offended.

By: Den Rattee

Den Rattee — Tue, 01 Nov 2011 16:43:07 +0000

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?

By: Steve Phelps

Steve Phelps — Sat, 29 Oct 2011 08:53:46 +0000

@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?

http://geogebraithaca.wikispaces.com/file/view/z10_os2-1-1.pdf

By: luke hodge

luke hodge — Sat, 29 Oct 2011 02:05:47 +0000

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.

By: Scott

Scott — Sat, 29 Oct 2011 01:06:23 +0000

An updated version of the geogebra graph. I wouldn’t show this to kids until they’ve explored it themselves for a long time.

http://scottfarrar.com/blah/perimeter_v_area2.html

It creates some pretty nice imagery: http://scottfarrar.com/blah/area-v-perim-snip.PNG

By: Joshua Zucker

Joshua Zucker — Fri, 28 Oct 2011 22:49:56 +0000

@Steve, brilliant question! And, in the other problem, what does the graph of a constant-length diagonal look like on the area vs perimeter graph?

By: Steve Phelps

Steve Phelps — Fri, 28 Oct 2011 22:39:50 +0000

Which have diagonals of integer lengths?

By: Dan Meyer

Dan Meyer — Fri, 28 Oct 2011 22:02:03 +0000

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.