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A Quick Fraction Puzzle

September 29, 2018 am30 10:30 am

This problem is not original, but pushing the solution in this way is fun. Watch, or play along. See if you can figure out what a nice ending might be.

Find a number that is 1 more than its reciprocal

Make sure the audience knows what a reciprocal is:  It’s what you multiply a number by to get 1 as the product, or colloquially, the “flip” of a number. For example the reciprocal of 9 is \frac{1}{9}. The reciprocal of \frac{7}{8} is \frac{8}{7}.

I usually allow kids to explore in any direction, but for today’s purposes I don’t want that to happen. I’m going to control the investigation.

Get the kids to make some naïve guesses: 1 is too small since the reciprocal of 1 is 1. 2 is too big since the reciprocal of 2 is \frac{1}{2} (and 2 – \frac{1}{2} = \frac{3}{2}).

So let’s start more serious guessing. \frac{3}{2}\frac{2}{3} = \frac{5}{6}, so that’s wrong. But it’s not such a terrible guess. \frac{5}{6} is only a little less than one. We need a slightly bigger number. Add 1 to \frac{2}{3}. That will give us something a little bigger than \frac{3}{2}.

\frac{2}{3} + 1 = \frac{5}{3}. Did that just give us the solution? No. \frac{5}{3}\frac{3}{5} = \frac{16}{15},  a little more than 1, but very little more than 1. But that helps us get the next guess. If we add 1 to \frac{3}{5} we will get something very slightly less than \frac{5}{3}.

\frac{3}{5} + 1 = can you imagine where I might be leading the students? What would you like the students to notice? What concepts would you like to share with them?

If you comment, mention the age of the children you imagine working with. I’ve been speaking with a few teachers, and we are trying versions of this with 10 year olds, 17 year olds, and everything in between….

I haven’t posted a mathematics puzzle or problem (for kids or adults) in quite some time. I hope there’s someone out there who still likes this.


Remind me: Who did the UFT not Endorse?

September 12, 2018 am30 6:38 am

The United Federation of Teachers (UFT) did not endorse Andrew Cuomo. For all his progressive rhetoric, and despite some progressive (or progressive-sounding) legislation, union members know that there was much more that could be done, that he’s responsible for blocking much progress, and that if it suits his needs, he can turn on us, and viciously. He is a self-serving, principle-free politician, and we do not trust him, most of us do not like him, to the point that our leaders did not endorse him. Good. Here’s the list. No Cuomo.

Today we had scheduled a Citywide Chapter Leader meeting. This is a big deal. We do it once a year. Some of us write it, in ink, in our calendars.
Seven days ago we got notice: postponed.
Has this happened before? Not in my 17 years as chapter leader.
Explanation: none. (See below).

What do these things have to do with each other? Maybe nothing. But yesterday we got summoned to a rally “to support Letitia James” (James is running for Attorney General, with NYSUT – that’s the UFT’s statewide federation – endorsement).

Is it a rally for Tish?  Nope. It’s a “Get Out the Vote” for Cuomo, Hochul, and James. Mulgrew wants us to show up at a Cuomo rally because we like #3 on the list?

Date: 9-12-18
Event: Bronx GOTV Rally for Cuomo, Hochul, Tish
Time: 6:00 p.m.
Location:  Xxxx Xxxxx Xxxxx, Brooklyn xxxxx

So much for not endorsing Cuomo.


Welcome to Another Year

September 3, 2018 pm30 10:13 pm

New York City Public Schools open tomorrow (except some PROSE, and I said public, not charter, no idea when they open) – anyway, our schools open tomorrow for teachers, with kids on Wednesday.

Welcome back.

This is our first school opening post-Janus. Will it make a difference? I hope not in a negative way, although we are sure to lose at least a few members. On the other hand, perhaps we will find the Unity leadership more responsive?

Where should we look for responsiveness? That’s easy: how much will Unity stand up for members in schools with out of control administrators? They’ve got to do better.

We don’t have a new contract yet, but I expect to have one at some point this year. I don’t expect great changes.

But welcome back. Forget that other stuff, just for a moment. You are educating kids – that’s a great and valuable thing you do, and no matter what the year brings, remember that – your work is important. It matters.


Specialized High Schools – some comments should not matter

August 26, 2018 am31 6:15 am

At the time of the proposal of the thirteenth amendment abolishing slavery*, the former senators from the southern states were not consulted.  And that was correct.

Today, as people with differing opinions discuss future admissions policies for the Specialized High Schools, those “reprentatives” of the schools who have been heretofore silent on segregation should likewise not be consulted.

The current admissions system is based on a single test, on one day. That’s the way it’s been, for a long, long time. But in 1970 or 1971, someone decided to study the admissions policy for the schools (at that time the Bronx High School of Science, Brooklyn Technical High School, and Stuyvesant High School). The New York State legislature responded by passing the Hecht-Calandra Act, enshrining the existing test in law, and limiting an already limited alternate route (Discovery).

The Hecht Calandra Act was passed to stop a study, to preclude discussion.

And today we have de Blasio’s proposal to radically change the admissions process. Now, lots of people want to talk about THAT.

There are people with a direct personal interest in the school: faculty, administration, current students and parents, aspiring students and families.

For the past fifteen years, and more so in the past five-ten, the numbers of Black and Hispanic students at these schools (now eight schools: the three named in the law and five newer schools) has plummeted. In some of these schools, there are faculty and administrators who worked to reverse or slow this process of segregation. But most did nothing.

We have people who no longer have a personal stake in the school, who are advocating that the law be respected. These alumni are not stakeholders. They are advocating that we bow down to a law passed to prevent discussion. They have forfeited their right to be heard. We are correct to ignore them. (Even if in the 13th hour they are advocating systemic change – while insisting the law is unchallengeable.)

I have one guy, always silent on the issue, who now twists every conversation about specialized high schools to advocate that the mayor drop the five new schools, and respect the law by leaving the original three (including his alma mater) alone. I ignore him.  Good people should too.

Some specialized high school administrators have been silent in the face of this obvious accelerating segregation. Their right to be heard should be contingent on their not defending the “SHSAT-must-never-be-examined” Hecht-Calandra Act.

That still leaves many of us. Their are current students and parents – both for and against the changes the mayor proposed. They should be heard. There are communities where young students have been preparing for the test. There are alumni who HAVE been speaking about the horrible segregation and have been working for change. They all have a right – perhaps even an obligation – to engage in the conversation.

And there are faculty members who have been trying to address this issue. I served on a UFT task force with several of them, from all eight schools. And there are school communities, including administrators, who have been looking to modify admissions. In my school, the entire faculty and principal agreed to a bold proposal that we presented to the DoE a couple of years ago, only to have the DoE and their lawyers shoot us down. The principal and I were in the process of renewing our proposal (since we have a new chancellor) when de Blasio’s plan was announced.

But for those who ignored a law that told people that they couldn’t even talk about specialized high school admissions, for those who ignored obvious and deep segregation, and especially for those who are no longer connected to these schools – why should your opinion matter?





August 25, 2018 am31 4:27 am

Sometimes I write my own math problems:



But I think this is the first time I over a decade that I’ve written a multiple choice question.

Specialized High Schools – I support public schools

June 12, 2018 am30 7:38 am

As I start rambling on about specialized high schools, I need a starting point. And here it is:  I am a strong supporter of public education. I support equity. I am anti-racist. I have a sense of fairness that is always on. But none of this is simple.

Public education in the United States – well, it doesn’t begin at the beginning. I mean publicly funded, publicly run, secular, unsegregated schools, roughly K-12 (though I’m not sure how we got to exactly those 13 years), with compulsory attendance. I don’t think we’ve ever had exactly that in this country. There’s not some “golden age” we can point to…  we think of the one-room schoolhouse, but that was protestant, not secular. And, AFAIK, the New England schoolhouse was spread to the midwest and then partially imposed on the south in the early, mid, and then late 19th century, unevenly and incompletely. (There’s a book called “Pillars of the Republic” by a guy named Kaestle that does a good job on this).  There were also little private tutoring schools (“Dame Schools”?) going back to the 18th century. And then with the rise of the big cities, charity schools, which give rise to our large urban schools. And these are sort of merged into an almost universal system. But golden age?  With segregation? With the vestiges of their religious origins?

And then there’s been a parallel system of religious (used to be mostly Catholic, but there’s quite a range now) and private (or “independent”) schools. So this has never been universal. And over the last two decades there has been the growth of the privately run “charter” schools.

All of this is to say that my support of “public education” is of an ideal, not of the way schooling takes place in the US today, certainly not in NYC today, and there’s not some golden model I can point to at some date in the past.

I think I’ll talk about equity and equality next.



New York City’s Specialized High Schools and Admissions

June 11, 2018 am30 7:11 am

New York City’s Specialized high schools are in the news;  Chancellor Carranza, Mayor de Blasio, and State Assemblyman  Barron have proposed changing how students gain admission to these schools. That proposal is stalled until January, which gives us time to discuss and reflect.

Most students who get placement in these schools do so by scoring high on the Specialized High School Admissions Test (SHSAT). In recent years the number of Black and Hispanic students admitted has dwindled . The “diversity” issue has generated occasional arguments and proposals over the last few years. But this proposal comes from the Mayor, and has been headline news…

There are eight schools that would be affected. I teach at one of them. In fact, 16 years ago, I was the founding math teacher at one of them. And the chapter leader for these 16 years…

I certainly have some thoughts on specialized high schools and admissions. And over the next few days I will be sharing them.