Comments on: [3ACTS] Best Squares

By: Two Schools of Math Teachers | Matt Vaudrey

Two Schools of Math Teachers | Matt Vaudrey — Mon, 07 Aug 2017 22:51:43 +0000

[…] We had just gotten our issue of CoMmuniCator (the monthly publication of CMC) which featured two-page descriptions of lessons, like visual patternsÂ and drawing the ideal polygon. […]

By: Greg Schwanbeck

Greg Schwanbeck — Thu, 13 Dec 2012 18:23:11 +0000

@Dan After extensive testing I have concluded that the error with the postscript not coming through results from a problem existing between keyboard and chair. How embarrassing.

By: Patrick Honner

Patrick Honner — Thu, 13 Dec 2012 00:33:09 +0000

I’m proud to be cited as a minor inspiration in this task design, which I think is great.

To Bowen’s point, we definitely want students to develop the tools to analyze, probe, and ultimately refine mathematical ideas. As Kevin responded, one of our roles as teachers is to model this behavior so that students can learn it.

In addition, content knowledge plays a big role here in helping the teacher ask good questions and shape the conversation. This is partly what makes the teacher better suited for this role than students (in some, or most, cases). And this role is one that I don’t see technology readily replacing.

On a final note, did anyone suggest scoring larger squares higher than smaller squares? I bet it’s easier to draw a good small square than a good big square.

By: l hodge

l hodge — Wed, 12 Dec 2012 22:27:16 +0000

@DanKarl, Sorry, I lost track of the fact that they didn’t draw a square. Consider the circles instead. Take a metal circle and bang on the outside with a hammer so it dips inward a bit. Take a copy of the steal circle and bang on the inside with a hammer so that it bubbles outward a bit. Both shapes have the same perimeter, and were arguably distorted by the same amount, but the first one would score higher on the area to perimeter metric. Somewhat similar sorts of issues apply with plotting four points for a square.

The area measures are not very sensitive. Even with a really lousy job of drawing a square, like one side or angle being 20% more than another, you would still be within a couple of percent of the area of a square with the same perimeter.

By: Dan Meyer

Dan Meyer — Wed, 12 Dec 2012 21:59:54 +0000

@Greg, the post-script on this post must not've come through on your end.

By: Greg Schwanbeck

Greg Schwanbeck — Wed, 12 Dec 2012 21:50:47 +0000

Have you seen the “eyeballing game” before: http://woodgears.ca/eyeball/

It’s similar to some of the tasks you’ve blogged about recently.

By: Dan Meyer

Dan Meyer — Wed, 12 Dec 2012 19:25:19 +0000

Really useful commentary and criticism from James Key and Bowen Kerins. I tossed them up into the post itself. @Dan:

How did you find the area and points for your circles? Geogebra?

No, I vectorized the paths in Adobe Illustrator and used a plugin that tells you the perimeter and area of an closed shape.

By: Dan Karl

Dan Karl — Wed, 12 Dec 2012 15:16:01 +0000

Dan,

I love these problems.

How did you find the area and points for your circles? Geogebra?

I really like that definition of a perfect square or circle or equilateral triangle as it not only applies to geometry, but into Algebra 2 and calculus as we discuss maximizing equations (in this case the area.)

@Hodge – What in the world are you talking about? If

one “side” is actually two segments that meet at a point inside the “square”

then that wouldn’t be a quadrilateral anymore. And it wouldn’t happen because they are only drawing four points, not five….

By: James Key

James Key — Tue, 11 Dec 2012 16:12:45 +0000

“Draw two points and then the point exactly between them.”

Sorry to be nit-picky, but while the above task certainly meets the requirement for “minimal language demand,” I think ONE MORE WORD is required for the sake of precision: “halfway.” There is not a unique point that is “exactly between” two given points; there are many. But there is one point that is “exactly halfway between them.”

I know that “conciseness” and “precision” sometimes compete with one another, and I confess that I often strike the balance poorly.

By: Kevin Hall

Kevin Hall — Tue, 11 Dec 2012 14:18:33 +0000

It would be our task because students may need a teacher to model this kind of thinking for them. I’ve always found that it’s surprisingly difficult for students to search for counterexamples, but when it comes to proof, it’s an early rung in the ladder of abstraction.