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Math Photo: Slope Field in the Sky

Slope Field in the Sky

I finally made it down to the new Fulton St. subway station.  Such a beautiful view from below, like the slope field of a differential equation.

Each of the 88 glass blades of the Sky Reflector-Net is positioned to channel sunlight down throughout the station at different times of day and days of the year.  Maybe next year I’ll stop by on the summer solstice.

MOVES 2015

MOVES 2015I’m excited to once again be participating in the MOVES conference at the Museum of Mathematics!

MOVES, the Mathematics of Various Entertaining Subjects, is a biennial event run by MoMath that celebrates recreational mathematics.  This year, the conference will be headlined by John Conway, Elwyn Berlekamp, and Richard Guy, co-authors of Winning Ways for Your Mathematical Plays, a classic book on mathematical games.

I’ll be running a session on the Activity Track called Games on Graphs”, where we will explore some elementary graph theory through a few simple graph-based games.   Most importantly, we’ll talk about how to create new games that can further our mathematical investigations!

You can learn more about the conference here, and see the full program here.

Regents Recap — June 2015: Common Core Geometry and Transformations

Here is another installment in my series reviewing the NY State Regents exams in mathematics.

June, 2015, saw the administration of the first Common Core Geometry Regents exam in New York.  I led a teacher team that worked to adapt our curriculum to the Common Core standards.  One of our primary concerns was how the new transformation-based approach to Geometry inherent in the Common Core standards would be represented on this Regents exam.  In particular, we had no idea how establishing congruence and similarity via transformations, an apparent emphasis of the standards, would be assessed.

A total of 18 of the 86 points on the exam (around 21%) were associated with transformations:  six multiple choice questions and two free response questions.  Here are a few examples, with associated Common Core standards in parentheses.

Question 10 is about rotations that map regular polygons onto themselves (CCSS.MATH.CONTENT.HSG.CO.A.3).

2015 CC GEO 10

Question 16 assesses the concept that dilation can alter length but must preserve angle measure (CCSS.MATH.CONTENT.HSG.SRT.B.5).

2015 CC GEO 16

Question 24 addresses establishing congruence by rigid motions (CCSS.MATH.CONTENT.HSG.CO.B.7).

2015 CC GEO 24

Question 30 asks the student to explicitly connect transformations to congruence by recalling the fundamental principles that underlie rigid motions (CCSS.MATH.CONTENT.HSG.CO.B.6).

2015 CC GEO 30

And Question 33 asks to student to first produce a “traditional” proof of congruence, and then interpret the congruence through a rigid motion (CCSS.MATH.CONTENT.HSG.CO.C.11).  This is a simple way to connect the two concepts.

2015 CC GEO 33

Overall, the manner in which transformations were tested aligned with our expectations, both in scope and in content.  Our Geometry worked closely together throughout the year, integrating a variety of resources from New York state and elsewhere, but mostly felt in the dark about what the test would look like.  In the end, we were pleasantly surprised.  But we also noticed how much different the Common Core Algebra exam was in year two, so we know that we may be surprised again.

Math Photo: Hexagonal Rabbits

Hexagonal Rabbits

The tilling station is one of my favorite exhibits at the Museum of Mathematics.  These rabbit tiles create a hexagonal tiling of the plane.  Pick any rabbit, and you’ll notice six rabbits all around it; this is exactly how hexagons fit together to tile the plane.

What I really like about this tiling is the the various levels of triangles that emerge.  Triangles of rabbits, one of each color, mutually intersect at ears and paws.  And I can’t help but seeing the monochromatic rabbit triangles!


Regents Recap — June 2015: Trouble with 3D Geometry

Here is another installment in my series reviewing the NY State Regents exams in mathematics.

The Common Core standards have brought a slight increase in three dimensional reasoning into high school Geometry.  I think this is generally a good thing:  3D geometry is typically given short shrift in this course, but is a beautiful and intriguing topic.

It can also be a confusing topic, as these problems from the inaugural Common Core Geometry Regents exam demonstrate.

2015 CC GEO 1

According to the scoring guide, the correct answer is (4) a cone.  Technically, however, the correct answer is (3) a right triangle.

Rotation is a rigid motion:  it does not change a figure’s size or shape.  If a right triangle is rotated about an axis, it will remain a right triangle.  Presumably, the intent of this question is for the student to identify the solid of revolution formed by revolving the triangle about an axis.  But that is a different question than the one posed.  Ironically, the notion that rigid motions preserve size and shape is one of the fundamental principles in the transformation-based approach to geometry embodied by the Common Core standards.

Here’s another problematic 3D geometry question.2015 CC GEO 6

According to the answer key, the correct answer is (2).  But the actual correct answer is all of these.  While most cross-sections of spheres are circles, some cross-sections of spheres are single points (when the cross-sectional plane is tangent to the sphere).  All the given objects have single point cross-sections as well, thus, could all have the same cross section as a sphere.

This is certainly not the first time we’ve seen problematic three dimensional geometry questions on these Regents exams (here’s a particularly embarrassing example), and I’ve been chronicling mathematically erroneous questions on these tests for years.  Errors like this are often dismissed as insignificant, or “typos”, but because of the high-stakes nature of these exams, these errors have real consequences for students and teachers.

If these exams don’t model exemplary mathematics and mathematical practice, their credibility in evaluating the mathematical practice of students and teachers must be questioned.