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Archive of posts filed under the Geometry category.

8/15/17 — Happy Pythagorean Triple Day!

Today we celebrate a rare Pythagorean Triple day!  And especially rare, as 8-15-17 is a primitive Pythagorean triple.  We won’t see many more of those!

Here’s an animation I made to celebrate.

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Regents Recap — June, 2017: Consistency and Precision

Two prominent themes of my critical review of the New York State Regents exams in mathematics are consistency and precision in language.  Here’s a pair of problems from the June 2017 Geometry exam that illustrates both issues.

First, the phrasing of the question “What is the number of degrees in the measure of angle ABC?” is awkward and somewhat unnatural.  Second, if we are going to ask for “the number of degrees” in the measure of an angle, then the answer should be a number.  The answer choices here are not numbers: they are degree measurements.

Why not simply ask for the measure of the angle, as was done in question 10 on the exact same exam?

While the issue in question 4 is minor, we know that imprecise use of language is deeply connected to student misconceptions in mathematics.  And we know that an important part of our job as teachers is getting students to use technical language correctly.  Our exams should model the mathematical clarity and precision that we expect of students in our classes.  Far too often, the New York State Regents exams don’t meet that standard.

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Jason Merrill’s Lawnmower Puzzle

Jason Merrill recently posted a fun geometry puzzle inspired by his work on the Lawnmower Math activity for Desmos.  Here’s my paraphrase of the puzzle:

Suppose a lawnmower is tethered to a circular peg in the middle of the lawn. As the lawnmower moves along its spiral path, the rope shortens as its winds around the peg. At the moment the lawnmower contacts the peg, how much rope remains uncoiled?

When I first considered this problem it seemed hard.  After some thought, it seemed obvious.  Then, after some more thought, it seemed hard again.  That’s the sign of a compelling problem!

I enjoyed working out a solution, the heart of which I’ve included below.  Jason graciously included my solution in his post sharing his own, and he also does a wonderful job describing the journey of making simplifying assumptions, both mathematical and physical, that allow us to start moving toward a solution.  It’s the kind of work that often goes unmentioned in problem solving, especially in school mathematics, and this puzzle provides a nice opportunity to make that thinking transparent.

I highly recommend reading the puzzle and his solution at his blog.  Thanks for the fun problem, Jason!


Math Photo: Snowball Tetrahedron

I put the recent snow day, and a snowball maker, to good use!

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Ceilings of Curvature

On a visit to the Lowline, I noticed an interesting application of mathematics above us.

The ceiling is a tiling of hexagons and equilateral triangles.  But unlike a typical tiling of a flat bathroom floor, this tiling seems to create a curved surface!  Here’s a closer look:

The underlying pattern is hexagonal, but when a hexagon is replaced with six small, hinged equilateral triangles, the surface gains the potential to curve.

It’s interesting to follow the “straight” line paths as they curve over the surface.  And since this tiling is suspended from above, it’s interesting to think about what the surface would look like if it were lying on the ground.  How “flat” would it be?  Or a better question might be “How far from flat is it?”