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Math Photo: Spiky Symmetry

These cacti caught my. I can see both a dodecagon and a star in the 12-fold symmetry of the cactus in front. And to my surprise, the cactus behind it has thirteen sections!

I wonder about the range, and deviation, of the number of sections of these cacti. And what are the biological principles that govern these mathematical characteristics?

Regents Recap — August 2017: Yes, You Can Work on Both Sides of an Identity

In a controversial post last year, I argued that it’s perfectly acceptable to work on both sides of an equation in proving an algebraic identity. While it’s common to tell students “You can’t cross the equal sign” in this situation, doing so is mathematically legitimate as long as the new equation is true under exactly the same circumstances as the original.

For example, when proving an algebraic identity, multiplying both sides of an equation by 2 is permissible, because = y and 2x 2y are true under exactly the same conditions on x and y. Squaring both sides of an equation however, is not, since

x^2 = y^2

can be true under conditions that make y false, say, when x and y-2.

The post in question, “Algebra is Hard”, was a response to a June 2016 Regents scoring guide that deducted a point from a student who, in proving an algebraic identity, multiplied both sides of the equation by a non-zero quantity. The student was penalized for “not manipulating expressions independently in an algebraic proof“, a vague and meaningless criticism.

“Algebra is Hard” received quite a bit of attention, and while many agreed with me, I was genuinely surprised at how many readers disagreed. Which was terrific! Of course my argument makes perfect sense to me, but it was great to have so many constructive conversations with teachers and mathematicians who saw things differently.

But my argument recently received support from the most unlikely of sources: another Regents exam.

Take a look at this exemplar full-credit student response to an algebraic identity on the August 2017 Algebra 2 exam.

Notice that the student works on both sides of the equation and subtracts the same quantity from both sides. Even though the student did not manipulate expressions independently in an algebraic proof, full credit was awarded.

The note here about domain restrictions is an amusing touch, given that it was the explicit domain restriction in the problem from 2016 that ensured the student wasn’t doing something impermissible (namely, multiplying both sides of an equation by 0).

So in 2016 this work gets half credit, and in 2017 this work gets full credit.

While it’s nice to see mathematically valid work finally receiving full credit on this type of problem, it’s no consolation to the many students who lost points for doing the same thing the year before. What’s especially frustrating is that, as usual, those responsible for creating these exams will admit no error nor accept any responsibility for it.

Be sure to read “Algebra is Hard” (and some of the 40+ comments!) for more of the backstory on this problem.

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The Math Behind Gerrymandering and Wasted Votes — Quanta Magazine

The U.S. Supreme Court is currently considering a case about partisan gerrymandering in Wisconsin and Texas. One of the keys to the case is the “efficiency gap”, an attempt quantify the partisan bias in a given electoral map. For my latest article in Quanta Magazine, I explain and explore the efficiency gap using simple examples, and talk about some of the implications of this particular measurement.

Imagine fighting a war on 10 battlefields. You and your opponent each have 200 soldiers, and your aim is to win as many battles as possible. How would you deploy your troops? If you spread them out evenly, sending 20 to each battlefield, your opponent could concentrate their own troops and easily win a majority of the fights. You could try to overwhelm several locations yourself, but there’s no guarantee you’ll win, and you’ll leave the remaining battlefields poorly defended. Devising a winning strategy isn’t easy, but as long as neither side knows the other’s plan in advance, it’s a fair fight.

Now imagine your opponent has the power to deploy your troops as well as their own. Even if you get more troops, you can’t win.

The full article is freely available here.

Global Math Week Symposium

James Tanton is on a mission to bring joyous mathematics to the world.  His Global Math Project is about to launch Global Math Week:  during the week of October 9th, over 600,000 students from around the world will enjoy a shared mathematical experience based on Tanton’s Exploding Dots, a wonderful, surprising, and awe-inspiring take on place value.

James has been traveling the world for the past year spreading the good word about mathematics and his exploding dots.  If you haven’t yet signed up, I encourage you to do so.  The mathematics is wonderful, relevant, and inspired, and the Global Math Project has lots of resources at their homepage.

To kick off Global Math Week, the Global Math Project together with the Museum of Mathematics will be hosting a symposium at NYU’s Courant Institute.  Mathematical luminaries like Po-Shen Loh, Henry Segerman, and many others will be on hand to celebrate.  And I’m honored to be participating in a panel discussion on Uplifting Mathematics for All, where we will discuss how to make mathematics meaningful, fun, and coherent in and out of the classroom.

So get ready for Global Math Week!  Hopefully this is the first of many to come.

Regents Recap, August 2017: How Do You Explain that Two Things are Equal?

Sue believes these two cylinders from the August, 2017 New York Regents Geometry exam have equal volumes. Is Sue correct? Explain why.

Yes, Sue, you are correct: the two cylinders have equal volumes. I computed both volumes and clearly indicated that they are the same. Take a look!

Wait. Why did I only get half-credit? What’s the problem, Sue? You don’t think this is an “explanation”? The two volumes are equal. The explanation for why they are equal is that I computed both volumes and got the same number. I don’t know of any better explanation for two things being equal than that.

What’s that? You wanted me to say “Cavalieri’s Principle”? But if I compute the two volumes and show that they are equal, why would I need to say they are equal because of some other reason?  Oh, never mind, Sue. See you in Algebra 2.

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