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Regents Recap — June 2014: What is an “Absolute Value Equation”?

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

The following question appeared on the 2014 Integrated Algebra exam.

2014 ia regents 23My question is this:  what, exactly, is an “absolute value equation”?  According to the scoring key, the correct answer to this question is (2).  This suggests that the exam writers believe an “absolute value equation” to be some transformation of y = |x|.

But “absolute value equation” is not a precise description of what the exam writers seem to be looking for.  It would be hard to argue that y = |2b^{x}| is not an “absolute value equation”, but that appears to be the graph depicted in (1).  With some work, all the given graphs could be represented as equations involving absolute values (an exercise left to the reader).

I doubt this imprecision caused any student to get this question wrong, but as I have argued again and again, these exams should stand as exemplars of mathematical precision.  These exams should not model imprecise language, poor notation, and improper terminology.  We do our students a great disservice by constantly asking them to guess what the exam writers were trying to say.

Regents Recap — June 2014: When Good Math Becomes Bad Tests

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

It is a true geometric wonder that a triangle’s medians always intersect at a single point.  It is a remarkable and beautiful result, and the fact that the point of intersection is the centroid of the triangle makes it even more compelling.

This result should absolutely be a part of the standard Geometry curriculum.  It important and beautiful mathematics, it extends a fundamental notion of mathematics (symmetry) in new ways, and it is readily accessible through folding, balancing, compass construction, and coordinate geometry.

But here’s what happens when high-stakes testing meets meaningful mathematics.

2014 regents geom -- medians

This wonderful result has been reduced to an easy-to-test trick:  the centroid divides a median in a 2:1 ratio.

It’s not hard to see how such a fact can quickly become an instructional focus when it comes to centroids:  if that’s how it’s going to be tested, that’s how it’s going to be taught.  Of course, teachers should do more than just teach to a test, but there’s a lot riding on test results these days, and it’s hard to blame teachers for focusing on test scores when politicians, policy makers, and administrators tell them their jobs depend on it.

This is just one example of many, from one test and one state.  This is an inseparable component of standardized testing, and it can be found in all content areas and at all levels.  And for those who argue that the solution is simply to make better tests, keep this in mind:  New York has been math Regents exams for over eighty years.  Why haven’t we produced those better tests yet?

Exploring Correlation and Regression in Desmos

exploring correlationI’ve created an interactive worksheet in Desmos for exploring some basic ideas in correlation and regression.

In the demonstration, four points and their regression line are given.  A fifth point, in red, can be moved around, and changes in the regression line and correlation coefficient can be observed.

The shaded region indicates where the fifth point can be located in order to make (or keep) the correlation among the five points positive.  The boundary of that region was a bit of a surprise to me!

You can access the worksheet here.  Many interesting questions came to mind as I built and played around with this, so perhaps this may be of value to others.  Feel free to use and share!

Math Photo: Grey Geometry

Grey Geometry

I like the many shades of grey, and the many shades of conic section, present in this spire.


Math and Dinner — NYMC

NYMC LogoI’ll be giving a Math and Dinner talk for the New York Math Circle on Monday, August 4.  Math and Dinner starts with some interesting mathematics at NYU’s Courant Institute, and after the talk the conversation continues over dinner at a nearby restaurant.  The series provides a fun and casual way for NYMC participants to chat about mathematics, teaching, and learning.

My talk is titled An Array of Matrix Explorations.  Here is the description.

The world of matrices is rich and diverse, connecting ideas across disparate disciplines and extending familiar mathematical notions into unfamiliar territory. In this talk, participants will explore some common concepts in the high school curriculum–algebra, geometry, and trigonometry–through the mathematics of matrices, where their depth and connections can be seen in exciting ways.

The talk is free, but participants pay their own way for dinner.  You can find out more about the series, and register, here.