This is a tricky question. So tricky, in fact, that it tripped up those responsible for creating this exam.

Dilation is a similarity mapping (assuming, as we do, that the scale factor is non-zero), and translation is a congruence mapping. Thus, any composition of the two will be a similarity mapping, but not necessarily a congruence mapping. So in the above question, statement II will always be true, and statements I and IV are not always true.

Statement III requires closer attention. Under most circumstances, translations and dilations map lines to parallel lines, and so the same would be true of their compositions. However, if the center of dilation lies on a given line, or the translation is parallel to the given line, then that line will be mapped onto itself under the transformation.

This means that the answer to this test question hinges on the question, “Is a line parallel to itself?”

If the answer is yes, then statement III will always be true, and so (3)* II and III* will be the correct answer. If the answer is no, then statement III won’t always be true. and so (1) *II only* will be the correct answer.

So which is the correct answer? Well, that’s tricky, too. The answer key provided by New York state originally gave (3) as the correct answer. But several days later, the NYS Department of Education issued a memo instructing graders to accept both (1) and (3) as correct. Apparently, the state isn’t prepared to take a stance on this issue.

Their final decision is amusing, as these two answer choices are mutually exclusive: either statement III is always true or it isn’t always true. It can’t be both. Those responsible for this exam are trying to get away with quietly asserting that (*P* and *not P)* can be true!

Oddly enough, this wasn’t the only place on this very exam where this issue arose. Here’s question 6:Notice that this question directly acknowledges that the location of the center of dilation impacts whether or not a line is mapped to a parallel line. It’s not entirely correct (a center’s location on the line, not the segment, is what matters) but it demonstrates some of the knowledge that was lacking in question 14. How, then, did the problem with question 14 slip through?

As is typical, the state provided a meaningless and generic explanation for the error: this problem was a result of *discrepancies in wording*. But there are no discrepancies in wording here. This is simply a careless error, one that should have been caught early in the test production process, and one that would have been caught if production of these exams were taken more seriously.

**Related Posts**

- Regents Recaps
- More Mathematical Misunderstanding
- The Worst Regents Question of All Time
- Another Embarrassingly Bad Math Exam Question
- Algebra is Hard

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Consider this multiple choice item from the June, 2017 Common Core Geometry exam.

Instead of testing a student’s understanding of triangle congruence, this question exposes a serious lack of mathematical understanding among the exam creators.

A superficial reading of the problem suggests that (3) is the correct answer. In (1), the two triangles share only three pairs of congruent angles; in (2), two sides and a non-included angle are congruent in each triangle; and in (4), the triangles share only one pair of congruent sides and one pair of congruent angles. None of these scenarios (*AAA*, *SSA*, *SA) *seems sufficient to guarantee that the triangles are congruent. And in (3), the triangles have one pair of congruent sides and two pairs of congruent angles; this (*ASA** or SAA*) is sufficient to conclude the triangles are congruent, so (3) is apparently the correct answer.

But closer inspection shows that, in fact, (1) and (2) are correct as well.

Consider choice (1). While it’s not exactly clear what it means to map angle *A* onto angle *D*, it must require that point *A* gets mapped to point *D*. Similarly, point *C* must be mapped to point *F*. If a rigid motion maps *A *to *D* and *C* to *F*, then segment *AC *must be congruent to segment *DF*. We now have one pair of congruent sides and three pairs of congruent angles: the triangles are congruent (by *ASA* or *SAA*), and choice (1) is a correct answer.

In (2), we are given that segment *AC* is mapped onto segment *DF. *This means that point *A* gets mapped to point *D* and point *C* gets mapped to point *F*. And since segment *BC *is mapped onto segment *EF,* we know that *B* is mapped onto *E*. Therefore, the vertices of triangle *ABC* are mapped via rigid motion onto the vertices of triangle *DEF. *This is sufficient to conclude that the triangles are congruent, and choice (2) is also a correct answer. (It’s also worth noting that, since the triangles are given as acute, *SSA *is actually sufficient to guarantee that the triangles are congruent. This mathematical error turned up in a separate question on this exam.)

As it stands, the only option that is not a correct answer to this question is (4).

Within a few days, the NYS Education Department issued a directive to count all answers to this question as correct. As is typical, no admission of an error was made: the problem was blamed on *discrepancies in wording*. Of course, there are no *discrepancies in wording* here: this problem as written, reviewed, edited, and ultimately published is simply mathematically incorrect. Its existence demonstrates a fundamental misunderstanding of the underlying concepts.

This isn’t the first time an erroneous question has made it onto one of these high-stakes Regents exams. In fact, there were at least three mathematically invalid questions on this exam alone! Over the past five years I’ve documented many others, and each time it happens, it raises serious questions: Questions about the validity of these exams, how they are experienced by students, how they are scored, and the lack of accountability for those in charge.

**Related Posts**

- Regents Recaps
- Trouble with Dilations (and Logic)
- The Worst Regents Question of All Time
- Another Embarrassingly Bad Math Exam Question
- Regents Recap — June 2016: Algebra is Hard

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Founded in 2011, the mission of 100kin10 is to recruit, train, and retain 100,000 excellent STEM teachers in 10 years. To that end, they lead a coalition of nearly 300 public and private-sector organizations committed to supporting STEM education in the US.

Over the past two years, 100kin10 has been working to identify the key systemic problems in STEM education. Today’s STEM Grand Challenges launch maps out the interconnected landscape of those problems and issues a series of challenges that address the problems identified as most pivotal among the network.

At the event, I’ll be speaking about my personal and professional experiences with some of the grand challenges, and I’m looking forward to hearing from a variety of different perspectives on how we can best improve STEM education.

You can learn more about 100kin10 here.

]]>Celebrate Permutation Day by mixing things up! Try doing things in a different order today. Just remember, for some operations, order definitely matters!

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The theme of the conference is *Big Ideas In and Out of the Classroom*. I’ll be delivering the opening keynote, *Connecting Big Ideas In and Out of the Classroom*, and will also be running a related workshop later in the conference. I’m honored to be providing the opening remarks for this event, which is undoubtedly the first of many.

The Summer Think has been organized and orchestrated by teachers from Math for America’s various fellowship programs, and features two dozen sessions proposed and facilitated by teachers. In addition to the underlying support offered through their fellowship programs, MfA has provided organizational support for the conference.

Teacher-led, teacher-driven professional development lies at the heart of Math for America’s programs, and the fact that over 50 teachers are participating in a conference less than two weeks after the end of the school year speaks to the impact it has on MfA’s teachers.

You can learn more about the conference, including a list of sessions and presenters, here.

]]>The World Science Festival is a week-long celebration of science and the arts in New York City. Now in its tenth year, the WSF has drawn over a million and a half visitors to its science-themed activities, which are designed for the public and hosted all over the city.

I will be participating in the WSF’s *Ultimate Science Sunday*, a day full of interactive exhibits, demonstrations, and games. I’ll be constructing a large aperiodic tiling with visitors, as well as sharing some cool 3D-printed mathematics.

The 2017 World Science Festival runs from May 30th to June 4th. You can see the entire *Ultimate Science Sunday* program here, and find out more about the World Science Festival here.

*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!

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

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The article,* Making Math with Scratch, *highlights a Math for America workshop I ran for teachers that centered on approaching mathematical concepts through the lens of coding and computer science. Several projects I use in my classroom are featured, and I also discuss why I like teaching with Scratch and how it’s become a valuable part of my approach to teaching math.

The purpose of the workshop and Patrick’s classroom activities are to demonstrate the power of bringing mathematics and computer science together. “Ultimately the goal is to show how math and computer science are great partners in problem solving. And Scratch provides a terrific platform for that.”

I’m excited to share the work I’ve been doing with math and Scratch over the past few years–including talks and workshops at conferences like Scratch@MIT, SIAM ED, and the upcoming NCTM Annual meeting–and I really appreciate this nice profile from Scratch Ed.

You can read the full article, *Making Math with Scratch, *at the Scratch Ed website.

What catches my attention in this photo, after the blue and white squares, is how the beams slowly bend away from center. I suppose knowing the size of those beams, and some trigonometry, would allow you estimate the location of the camera.

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