Going through the NY Regents Algebra exam questions in the NYT

Saw this article about the NY Regents Algebra exam in the NYT today:

NYT Article on the NY Regents Algebra Exam

The article made me sad on a lot of levels, but it did include 5 sample problem and I thought it would be fun to go through those problems with my older son.

Here are the problems and my son working through them:

Problem 1 – a problem about lines and slopes:

Problem 2 – a problem about algebraic inequalities:

Problem 3 – a problem about systems of equations and arithmetic (this problem makes me cringe, but my son has a clever idea about how to simplify the calculations):

Problem 4 – a problem about the graph (and the roots) of a polynomial equation:

Problem 5 – A problem about arithmetic and geometric series

So, I thought all but the 3rd one were good questions. I honestly have no idea why the exam writers felt it was important to write that question using the amounts of dollars and cents that they did. It seems to me that this choice made the problem needlessly complicated.

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An Abstract Algebra question from John Golden

Saw this question from John Golden on Twitter a few minutes ago:

My immediate idea was two prior Zometool projects that we’ve done that touch on rotation groups, but they require a Zometool set.

My two next thoughts were a bit more technical – Galois Theory and Elliptic curves. On reflection, though, I feel like both are pretty tough tasks for one class.

So my next idea related to three things I’ve seen on Twitter recently.

(1) Start by watching the first 10 minutes or so of this wonderful public lecture by Jacob Lurie from last year’s Breakthrough Prize:

In the first part of the talk he discusses rings and touches on Emmy Noether’s work on the subject in the early 1900s.

Here’s how I used this video with my kids last week (we did not explicitly dive into abstract algebra, but we did talk about clock arithmetic):

Using Jacob Lurie’s Breakthrough Prize Lecture to inspire kids

(2) Next check out this video linked by Steven Strogatz last week:

In this video you learn about a few incredible ideas related to abstract algebra. For example, when you adjoin i to the integers, you get new primes, but you still have unique factorization. However, when you adjoin the square root of 5, you lose unique factorization. These ideas are just one step removed from what Lurie touched on in his lecture.

Oh, and the punchline of this video about the square root of 163 is pretty amazing!

(sorry not TeX-ing this, I’m writing in a hurry)

So, even just stopping with the ideas in this video you’ve got some neat facts that are pretty accessible (and cool!).

(3) Finally, if you have time, take a look at this “new to me” proof that e is irrational that Dave Radcliffe tweeted about last week:

https://twitter.com/daveinstpaul/status/669205374034034688%5Bembed%5D

Essentially this proof looks at numbers of the form A + B*e where A and B are integers. This set of numbers isn’t a ring, but it is at least another example of expanding a number system. For a one day lecture it seems close enough to what’s going on in part (2) above to keep the class flowing. Plus, it is sort of fun to see this proof that e is irrational.

It is also easy to skip of the first two parts take longer than expected.

Anyway, that’s my “pondering this Twitter question for 20 minutes” idea.