Tuesday, January 16, 2018


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(C)Copyright 2018, C. Burke.

Maybe it should be point PU?

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Friday, January 12, 2018


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(C)Copyright 2018, C. Burke.

At least there was nothing negative.

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August 2017 Common Core Algebra 1 Regents, Part 2

I recently realized that students are reviewing the August 2017 Algebra 1 Regents, and I never wrote up the open-ended problems.
Here are the Part I questions, answers, and explanations.

August 2017, Algebra 1 (Common Core), Part I

25. A teacher wrote the following set of numbers on the board:

Explain why a + b is irrational, but b + c is rational.

Answer: The sum a + b is irrational because the sum of an irrational number and a rational number is always irrational.
The sum of b + c is rational because the square root of 225 is 15, which is a rational number. The sum of two rational numbers is always rational.

26. Determine and state whether the sequence 1, 3, 9, 27,… displays exponential behavior. Explain. how you arrived at your decision.

Answer: The sequence displays exponential behavior because it has a common ratio.
3 / 1 = 3
9 / 3 = 3
27 / 3 = 3

27. Using the formula for the volume of a cone, express r in terms of V, h, and π. how you arrived at your decision.


V = 1/3 π r 2 h
3 V = π r 2 h
3 V / (πh) = r 2
square root of (3 V / (πh)) = r

28. The graph below models the cost of renting video games with a membership in Plan A and Plan B.

Explain why Plan B is the better choice for Dylan if he only has $50 to spend on video games, including a membership fee.

Bobby wants to spend $65 on video games, including a membership fee. Which plan should he choose? Explain your answer.

Answer: According to the graph, Bobby can get 14 games under Plan B but he can only get 12 games under Plan A.
According to the graph, if Bobby plans to spend $65, then both plans will give him the same number of games because that is the point where the two plans are the same.

29. Samantha purchases a package of sugar cookies. The nutrition label states that each serving size of 3 cookies contains 160 Calories. Samantha creates the graph below showing the number of cookies eaten and the number of Calories consumed.

Explain why it is appropriate for Samantha to draw a line through the points on the graph.

Answer: Samantha should draw a line through the points on the graph because she could eat only 1 or 2 cookies, or even a part of a cookie.

30. A two-inch-long grasshopper can jump a horizontal distance of 40 inches. An athlete, who is five feet nine, wants to cover a distance of one mile by jumping. If this person could jump at the same ratio of body-length to jump-length as the grasshopper, determine, to the nearest jump, how many jumps it would take this athlete to jump one mile.

Commentary: I'm surprised that they didn't include the unit conversions necessary to complete this problem. However, they are in the back of the test book.

Answer: The grasshopper is 2 inches and can jump 40 inches, which is a ratio of 40:2, or 20 times his body length.
The person is 5'9", which converts to 5.75 feet (because 9/12 of a foot is .75 feet). Multiply this by 20: 5.75 * 20 = 115 feet per jump.
One mile is 5280 feet, so 5280 / 115 = 45.9, which rounds to 46 jumps would be needed.

If you got as far as finding 115 feet (or 1380 inches), you would have gotten one point. You need to get to 46 jumps, rounded correctly, to get the other.

31. Write the expression 5x + 4x2(2x + 7) - 6x2 - 9x as a polynomial in standard form.

Answer: Use the Distributive Property, and then Combine Like Terms
Standard form means the term with the highest exponent goes first and then in decreasing order.
Remember that the sign goes with the term after it, and the first term here is positive (+).

5x + 4x2(2x + 7) - 6x2 - 9x
5x + 8x3 + 28x2 - 6x2 - 9x
8x3 + 28x2 - 6x2 + 5x - 9x
8x3 + 22x2 - 4x

32. Solve the equation x2 - 6x = 15 by completing the square

Answer: b = -6, so b / 2 = -3, which means that (x - 3) will be in the answer.
Also, (b / 2)2 = 9. Add 9 to both sides to begin.

x2 - 6x + 9 = 15 + 9
(x - 3)2 = 24
x - 3 = +(24)(1/2)
x = 3 +(24)(1/2)

It isn't necessary to reduce (radical (24)) into (2 X radical (6)).

End of Part II.
How did you do?
Corrections, comments, questions are welcome.

Thursday, January 11, 2018

The Roman One

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(C)Copyright 2018, C. Burke.

100 what 1 did there?

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Monday, January 08, 2018

New Mersenne Prime

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(C)Copyright 2018, C. Burke.

Joke's on Ken. You can buy a vowel!

First, here's a previous comic from 2008 about Mersenne primes. They've found a few more since then.

For those who don't know, Mersenne primes, they are a special subset of primes that could be written in the form of 2prime# - 1.

I'd say I'm waiting for the speculation about 2Mersenneprime# - 1, but I wouldn't be surprised if that's already been done.

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Thursday, January 04, 2018

Problem of the Day: Quadratic and Absolute Value Functions

During the break, an online friend, who knows I'm a math teacher (the user handle @mrburkemath is generally a tip-off), sent me the following math review problem:

"Given the functions h(x) = |x - 4| + 1 and k(x) = x2 + 3, which intervals contain a value of x for which h(x) = k(x)?
This was followed by a list of intervals in the form number < x < number. I've left these out as I have to assume the entire problem is copyrighted from a review book. I'm hoping my excerpt is covered by "fair use".

First thing I said what, in a purely multiple-choice format, just plug the two functions into a calculator (or use an online app, if you're at home without one) and look for the answers. Then select the intervals that include those values.

Otherwise, we can work it out. First thing to realize in that dealing with quadratics and absolute values, there can be, at most, two real number answers. (The problem didn't involve imaginary roots.)

The friend wasn't sure if she was supposed to set each equal to zero and solve, or set them equal to each other. It's in the question: h(x) = k(x).

So here is what we have:

h(x) = k(x)
|x - 4| + 1 = x2 + 3

First thing is to isolate the absolute value, by subtracting 1 from each side:

|x - 4| = x2 + 2

Many of you can skip the mini-review I'm going to do right now.
When solving an absolute value equation like |x - 4| = 7, you have to split the equation into two possibilities, one positive, one negative, and solve each.

x - 4 = 7x - 4 = -7

Applying that rule to this equation, you get
When solving an absolute value equation like |x - 4| = 7, you have to split the equation into two possibilities, one positive, one negative, and solve each.

x - 4 = x2 + 2x - 4 = -(x2 + 2)
x - 4 = x2 + 2x - 4 = -x2 - 2
0 = x2 - x + 6x2 + x - 2 = 0
No real roots Factor (x + 2)(x - 1) = 0
x = -2 or x = 1

If you check the discriminant (b2 - 4ac) for the first equation, you will get a number less than 0, meaning that there are no real roots.

Going back to the original problem, they wanted you to select any of the intervals that contained either -2 or 1 or both.

Had they asked for the points of intersection, the values of the functions for those x values, then you have to plug them in:
h(-2) = |-2 - 4| + 1 = |-6| + 1 = 6 + 1 = 7, (-2, 7)
h(1) = |1 - 4| + 1 = |-3| + 1 = 3 + 1 = 4, (1, 4)

Simple, right?

Monday, January 01, 2018

2017: The Return of the Itinerant Teacher ... To Being Itinerant

2017: A Teaching Year in Review

There's an adage that annoys teachers: Those who can, do; those who can't, teach. And yet we will also ponder the fate of those who "can't teach" -- often they become "consultants" after a year and a half in the classroom, becoming self-appointed experts at telling the rest of us how to do it.

In my case, it's not a matter of "can't teach" as much as "won't let me". Now, I'm not assigning blame, for a couple of reasons: first, I don't wish to burn bridges, nor kill my own career; second, I'm not entirely blameless in the situation. Stuff happens.

Looking back, 2017 started as a year of promises, and in five and a half months, many planted seeds seemed to be taking root and flowering. Bearing fruit, if you will. And then someone razed the garden.

In December 2017, after a few months in the ATR (Absentee Teacher Reserve) Pool, subbing for a few different schools, I received an inquiry about a math position in Park Slope. My previous school year had been dreadful, resulting in a "Developing" rating and the loss of nearly 20 pounds. I had been happy to be in the pool for those months, subbing for other teachers, occasionally pushing into and assisting in math classes, and generally covering whatever needed covering. As restful as this was, however, I couldn't see this being a career path for me, so I looked into the school that had contacted me.

There was several pluses that stuck out: it was in my old neighborhood, which was easy enough to get to; it was a small school inside a big building, shared with other small schools; and I had previously worked in the building, so I knew that there wasn't a history of "trouble" in the halls and stairwells.

The principal and AP were nice people, answered my questions as well as they could. Note that my first question, considering previous experiences, was "Is this position provisional or permanent?" Was I being hired to teach math at the school or to fill a void until the end of the year?

Honestly, I couldn't tell you at the time which answer I expected or even which I would've preferred, so long as I had an answer and knew what to expect. I was told that he had been told by "downtown" that whomever he hired that's who he hired. That was it.

So in January 2017, I was making progress getting to know my new Algebra and Geometry students, just as they were making progress. I found strengths in some low-performing students. A few others were suddenly "turned on" to math, which they hadn't been in middle school. And I made important connections with a couple of students who seemed troubled in their own ways. I found ways to reach them, to be able to talk to them (somewhat). They knew they could talk to me if anything was troubling them, disrupting their class time, distracting them from learning.

I made enough progress that I was once able to say to a reluctant talker, "I'm sorry but I only have a minute, so can we jump to the point where you start speaking to me?"

It was abrupt, but she opened up right away and asked what she had to, and I was able to answer her. Next time, I let her go back to her natural process before speaking, but I noted she spoke up a little sooner.

By spring, we could practically have a conversation without prompting. By June, she told me that she saw me as a kind of "mentor" figure. That same day I was called into the office for a meeting. I thought it was a post-observation conference or the end-of-year review (a little early). When I saw the ELL teacher in the room, I wondered if there had been an issue with one of my students.

No, the ELL teacher was also the union rep. She was there because I was being excessed. Let go. From a "permanent" position. I chose my words carefully because I didn't want to end my career. And I might have. I felt betrayed. And I felt I was betraying my students who had made connections with me and whom I thought I'd see again the following school year.

In the principal's mind, the position hadn't been filled yet. He was still searching, not that he'd ever mentioned this in the prior six months. Any protest or argument from me was cut off with a simple question: would I like to still be considered for the position?

As much as I wanted to tell him to shove it -- OF COURSE, I WANTED TO BE CONSIDERED FOR THE POSITION. I thought it was MY position. I was already making plans for next year, what to change in the curriculum, how to approach Geometry with my current Algebra students, how I wanted to redo the classroom. That ended it. Except I knew I wasn't going to be considered or we wouldn't be having the conversation. I'm too expensive, and new teachers are easy on the budget.

So I started making my good-byes to some of the students. I hope they do well. A couple of them were following me on social media and still contacted me early in the fall semester, but the contact has fallen off. I hope they found new mentors, ones that aren't going anywhere.

This would be a good point for a musical interlude, in place of my summer break, so I can stop rambling and recollect my thoughts.

The end of the summer brought another plot twist: the city and the union agreed to change the rules about placement. If I didn't find a position, if a school did not hire me, then there was a good chance that the city would select an open position somewhere in Brooklyn (possibly beyond) and place me there anyway, whether or not I wished to go there and without regard to the school's desires or ability to fit me in their budget. And that placement would be permanent unless -- here's the catch -- the teacher received a developing or unsatisfactory rating!

So what did this mean for me as a teacher: I could find myself in a horrible school, in a horrible neighborhood, in a horrible situation like the one that caused me to drop to a weight beginning with the number 1 (which I hadn't seen since the year started with the number 1). I might not be able to get out of it, if I couldn't find an open position. Moreover, if the school didn't want me there, then they could make my life miserable so I'd get an unsatisfactory rating!

Granted, it was no picnic for the schools, either, and it was no surprise that I got a few inquiries after this announcement, while at the same time, open positions started disappearing. I applied to quite a few places, but, sorry, I wasn't going back to middle school, and I wasn't traveling to the Bronx. (Note: it's a minimum two-hour travel, one-way, by subway to get to the Bronx from my house.)

Making my search more desperate, an email arrived stating that my first temporary assignment was at Cobble Hill High School for American Studies. Nice neighborhood, not a lot of teenagers in it. I spent one single week at that school and I rated it as probably the worst week of my teaching career. That week was an absolute disaster, and I didn't want to repeat it. I only name the school so I can give credit where it's due. I had maybe 1 or 2 difficult assignments during my eight weeks there in September and October. While the school still has its share of problems, I didn't encounter anything like the last time. And I'll give a shout-out to Stephanie (I hope I spelled it right), for taking good care of me and the other ATRs assigned there. On my last day there, I told her that I wish I could take her with me.

Okay, so what about that forced assignment? It might still be coming for all I know. We keep hearing that math is a shortage area. And there are openings, but schools are still playing games.

In mid-September, a former colleague, now an assistant principal reached out to me through the DOE email and through Facebook (we're not "friends" on Facebook) to let me know about an opening in Queens. I wasn't thrilled with the idea of Queens, but with the future uncertain, it paid to check it out. Basically, it was a temporary position for someone who might be coming back soon, might be filing an extension, or might be retiring (after exhausting extensions). I met with the principal, saw the classrooms, spoke with some students, and, honestly, couldn't think of a reason not to be there, except the location and the travel time wasn't that bad. I was "basically" hired right there and they were going to put the paperwork through. By the middle of the following week, I got an email from Datacation, saying that an account had been set up for me with the online grading system at the school. I checked online and saw the rosters for four of my five classes. I just waited for the call to report.

And I waited.

And I waited.

And I checked the online grading system. I was still in there, but I only had one class, and it didn't have any students in it.

I can only assume that the teacher returned to work. You might think that someone might've informed me of this, especially after they pursued me and framed it as doing them a big favor. I was more disappointed about the snub then about the loss of the position.

Speaking of snubs. When the summer was drawing to a close and prospects were dwindling, I reached out to my old AP at the school I taught at for a decade (most of that before the current AP was there). The school that had excessed me -- twice. Why would I do such a thing? Because I still go back to their end-of-year parties to say good-bye to retiring colleagues. Last June, that school lost two-thirds of (non-ISS) math teachers. That is, they lost two out of three. Now considering that the AP of Operations told the Summer School principal that I am the "go to" guy for math, you might think that they would give me a call about coming back there.

You would be wrong. The call never came. I am Facebook friends with my former AP (although I have reason to believe that she's "muted" me), so I knew that she and her husband were on a European vacation this summer. When she returned, I contacted her through DOE email (not through Facebook -- that would be tacky). No response. Not even, "we have somebody." And here's the thing, she was at the end-of-year party. We'd spoken. She heard about what had happened to me, so she knew that I was available. We didn't discuss it -- again, it was a party and that would be tacky.

One last one: One school I contacted in the summer didn't return my messages, and then it no longer listed an opening. Early October they send me an email telling me that they've scheduled me for an interview AND a 20-minute demo lesson with such-and-such parameters and expectations, and it would be the day after I read the email. EXCUSE ME? I emailed back, "Sorry, but tomorrow is my annual check-up, and I need to be in the doctor's office. Can we reschedule?" They never replied.

So that's the way my year was going. I'm currently at a nice school close to my home. The UFT representative even approached me about staying there on my first day. If he asks me on my last day, I might ask him to see what he can do.

Life can be easy for someone in my situation if you can roll with it, but I can't see keeping myself afloat like this for another ten to fifteen years.

So while I'm happy for a relaxing end to 2017, I'm hoping to a more satisfying 2018.

(x, why?) Mini: Happy New Year 2018!

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(C)Copyright 2018, C. Burke.

Happened 24 years ago, and will happen 24 years from now, but it's not a 24-year cycle.

Here's one list I found on the Internet.

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Sunday, December 31, 2017

Another New Years Eve

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(C)Copyright 2017, C. Burke.

Sometimes you just have to surprise people

Or get replaced with a pod person.

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Monday, December 25, 2017

Christmas Star

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(C)Copyright 2017, C. Burke.

Westward leading, still proceeding.

A Merry Christmas to all my followers and a Happy Holiday Season.

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Saturday, December 23, 2017

Long Lay the World

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(C)Copyright 2017, C. Burke.

They were a long way from the fjords.

Technical problems kept this from being posted on Friday. Oopsie. At least it made it before Christmas!

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Wednesday, December 20, 2017

(x, why?) Mini: Bauxite

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(C)Copyright 2017, C. Burke.

Happy Aluminum Hydroxide Day!

No, I couldn't wait for December 26.

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