Happy New Years Eve 2015

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(C)Copyright 2014, C. Burke.

At least he came to give moral support. Two to the thirteenth won't be around for a while!

Two notes: First, 2^11 - 2^5 - 1 = 2015. That is to say, if all the powers of two, up to 10, were included, the sum would be 2047, which is one less that 2 to the 11th, which is 2048. Second, let's not forget the Y2048 bug! Or maybe the Y2(11) bug, just to be creative. Yes, it's true -- some really old computers, which will be really, really old 33 years from now, will have a problem handling the year as 2 to the 11th power. It should prove as Earth-shattering as Y2K did.

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UPDATE: The Making of a Webcomic

This comic started with an 11-digit binary number, 10 ones and 1 zero. I thought it would be funnier if I used the powers of 2 instead, so I needed 11 twos. And then I decided to include the next power as an extra gag. So I needed to draw 12 twos.

Rather than use one of the two twos I usually use and instead of typing the twos, I decided to try on of the paint programs on my tablet and doodled them. I was worried that they might be too snakelike -- and then I was worried that I'd doodled a row of ducks. (I have to keep this in mind if I ever need ducks again.) Then I made the smaller numeric exponents from 12 on down to 0. Colored it in and emailed it to me PC.

Putting it together I realized that I had too many exponents and not enough twos! Oops! I made a mistake. BUT I PICKED THE WRONG MISTAKE! The problem wasn't that I didn't have enough 2s (and quickly created an extra). The problem was that if the lowest exponent was zero, then the highest exponent I needed was 11.

And somehow though all the checking and proofreading -- including all that stuff above (which I have since corrected) -- none of this popped into my head. Of course, moving to 2^11 power would be a bigger problem than moving to 2^12. Some things are stored as 10 bits (I don't know why, but they were) but nothing would be stored as 11 bits (well, maybe -- programmers are strange).

Anyway, the correction has been made. HAPPY NEW YEAR!!

Two Taylor Swifts and a Sliding Yardstick

I've come to realize that as of this month of December, I am currently exactly 2 Taylor Swifts in age. That's really not so bad when I think about it because when You Belong With Me hit the charts (in 2009), I was 2.25 Taylorswifts. So i guess it's getting better. I could continue, figure out when I'll be only 1.5 Taylorswifts, but it would start to sound like an old Abbott & Costello routine. (Not that there's anything wrong with that.)

I had the thought recently about sliding yardsticks (or sliding metrics). What if you're measuring something, and the unit that you're measuring against changes along the way. It would be a great way to obscure (or hide) data.

What brings this up, without getting political, are all the political postings I see on different social platforms by people who should be smart enough to know better. (Scary thought: they are smart enough, but they assume we're all dumb enough.)

You don't judge a book by its cover and you don't evaluate major chunks of the economy based on a graphic containing a pithy quote and carefully selected factoids(*) underlined with a "witty" applause line, wrapped up in 140 characters, meant to be repeated and reported by people using the phrase "Alls I knows is...".

(*) factoid - Not a "little fact", but something that "looks like a fact", but isn't. This could be extended to the presentation of facts in a way that they seem related but are really not.

If that's all you know, you need to be more educated. Look beyond the numbers. Check those that confirm your biases and those you dismiss out of hand as made-up lies (What other kind are there?). However, when the fact-checking doesn't happen, a low-information individual is created -- someone who doesn't know all the facts, who knows that they don't have all the facts, but believes that every fact and factoid that they do have is enough, and no other piece could possibly make a difference. Any other data is obviously wrong, and if not, needs to be ignored, you Fill-In-The-Blank-ist Bad-Noun!

Oddly, though no piece of information can sway them, they are shocked that their one piece of information hasn't yet swayed you.

What spurs this on?

Comparisons of budgets and deficits and debts and prices and any number than can (and did) fluctuate wildly, so that specific highs and lows can be picked out and averaged or compared or manipulated alongside the assignment of blame or credit, regardless of the merit of that assignment. Is the latest increase big or tiny? Is it bigger or tinier than that other guy's? And what percent increase in the load was the straw that broke the camel's back?

Are these things appropriate fodder for debate? Absolutely, as long as that debate is longer than a sentence and the rebuttal is more than "You're a butthead. Neener. Neener!"

Shouting down your opponent doesn't mean you've won the argument. It means you're rude and willing to stay ignorant of facts you don't have.

My entire class can ignore me when I say that 0! = 1. They can all insist that it should be 0, and maybe even argue why. but they'll be wrong in class and wrong on every test after.

As for me, I've given way too much thought to it. I'll be exactly 1.5 Taylorswifts old when (T + 25)/T = 1.5

Yes, I can calculate it. No, I don't want to. Unlike Costello, I know that she isn't going to catch me. The answer is that the ratio is getting smaller, but I'm still getting old.

Another list of random websites I jotted down years ago

While at a professional development session years back, the facilitator was talking about all the different sites on the Internet that could prove useful other than just having a blog. Some were placing to search and get ideas. Others were places to go and send your students (possibly). I thought of it as places to market my comic. I think that this is how I learned about the old digg and reddit.

At some point, I have to check if these sites exist and add links. Also, I have to check which can be viewed in public schools.
In no particular order, here is what I scribbled down:

• Wheel of Lunch
• Hype Machine
• Census Dashboard.
• listpic.com
• twitter
• delicio.us
• doggdot.us
• Slashdot
• imgur

Other notes in the margins:

• feeds
• podcasting
• host ip.info
• GeoMicro
• 1337 jargon
• Digg
• Kazaa
• Joost

I recognize some of these, and have used some. Digg has completely changed and is useless now. Imgur has recently been blocked by the DOE, most likely for porn, gore and death -- if you type in a random five-letter combination, you will likely find a picture, but only God knows of what!

Disposition of Gift Cards

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(C)Copyright 2014, C. Burke.

There could even now be a gift card in my wallet waiting to be used ... or expired.

Sadly, many do get stolen with handbags, or hacked online. One of the reasons I have old iTunes cards unused is that the very first one I cashed in (for my child) was immediately hacked and spent and an additional $50 was run-up on my credit card. The iTunes card was a total loss because we just couldn't find a human being at Apple to talk to. The credit card was easier to deal with.

That being said, when I can buy anything, I never know what it is I want to buy. If it's just a store card where I'll eventually shop anyway, that isn't so bad. But, as noted in the comic, I'm just as likely to be shopping for a present for someone else, or for run-of-the-mill, day-to-day junk that I need, and not anything special or gift-like in nature.

Caffeine, on the other hand, while being day-to-day, every day, is always a special gift! I'm set for the next month.

A Christmas Extra

From a suggestion from a Facebook friend, Paul Wesley Adams, our Xmas Xtra:

Merry Christmas 2014: Yule Logs

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(C)Copyright 2014, C. Burke.

Merry Christmas to all my readers. To those not celebrating, enjoy your day.

It's been a busy time, and truth be told, this was a joke that I wrote down a month before last Christmas and then forgot to do. I planned to do it right this year, but just didn't have time to illustrate it the way I would have liked (and I know my limits). That's not to say that I won't try. So if you come back, maybe not today, maybe not tomorrow, but soon, it'll get updated. If you don't come back, you might regret it, but you'll always have the rough draft.

Christmas Song Color Wheel

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(C)Copyright 2014, C. Burke.

Given that it's the color of the season, you'd think that there'd be a song called Red Christmas.
Yeah, there's a parenthetical EP by The Killers, but that's not what we're talking about here.

Also, I'm aware that the song is just called "Heat Miser", but that wouldn't have been as amusing.

The color wheel above is, more or less, from the Magic: the Gathering game, where each colors have two ally color and two opposing colors. Over the years since I stopped playing it, they've elaborated on that, or so I hear.

Likewise, I stopped regularly reading comics before there was an entire color spectrum for Green Lantern, but I couldn't find a good way to illustrate R-O-Y-G-B-I-V along with Black and White (and white was needed for Bing's song). Moreover, there'd be a lot more blanks to fill in.

(blog:) What's Your Favorite (x, why?) Christmas Comic?

This is the eighth year that I'm doing Christmas comics, and in that times I've amassed nearly two dozens posts with the Christmas tag. (And there are random posts missing the tag that should have it.)

This begs the question: Which is your favorite (x, why?) Christmas comic?

An older one? A newer one? A song or carol? A puzzle?

I probably should have made an online survey ... but past experience tells me that I'll get fewer than 10 responses (likely much fewer).

The Twelve Days of Christmas (Reprise)

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(C)Copyright 2014, C. Burke.

Expressing my True Love mathematically and recursively.

This comic calls to mind my Twelve Triangle Numbers of Christmas from way back in 2007.

Once again, I used pseudo-code here, and not any particular programming language. I wouldn't program it exactly like this, but I wanted it to be readable.

As for the summations, those actually caused me some problems -- or at least a bit of thought about how they work, which is something I'll have to refresh my memory on. However, the one thing that occured to me is that I wanted the gifts in order, not just an actual sum of the number of gifts, and I wanted it to match the code. That meant that I had to count backward (or count down). Unfortunately, summations (to my knowledge at this point) count up. That meant minus signs and +1s, which I eliminated by starting j at 0 instead of 1.

More on this tomorrow, probably.

(blog): Still More About That Jeopardy Non-Common-Core Math Category

Spotted on the Web: The Political Hat had a post about the Common Core in Jeopardy category. What makes this interesting is that there was an "Update" to the common with a link to my column from two weeks ago. (Yeah, me!) Maybe this is the reason that Michelle Malkin of Twitchy.com suddenly started following me out of the blue. (Double Yeah, me!)

The fame is going full-circle. Or inside-out. Or something. Moving up from the H-list to the G-list. I'll be on the D-list before you know it!

The Political Hat bills itself as "Politics, News, and Drollarity with a Tip of the Hat".

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And that's all for tonight. Too late to post a fresh comic, which won't be seen, or to type my intended blog entry, which is longer than usual, meaning twice the time to type and three times the time to edit and correct.

Until tomorrow, this is Day 19 of 31.

(blog): Another Jeopardy Math Category

Just a couple weeks after Jeopardy had its Non-Common Core Math category during Kids Week, they ventured again into that academic field, but this time with adults. They fared a little better. For one thing, they made it through all five clues, including a Daily Double, and they didn't wait until everything else was exhausted.

Unlike the kids' clues, these didn't all have to do with the numbers 1, 2, 3, 4, 5 being added and multiplied.

The first question involved calculating a 20% tip on a $16.00 fare. The contestant managed to calculate the tip correctly ($3.20), but got the answer wrong. Neither of his opponents picked up that his calculation was correct, but that the clue wanted the total, which was $19.20, as Alex pointed out. (He will always point things like this out. I think he delights in that sort of thing. But to continue . . . )

The second clue asked for the greatest common factor of 84 and 105. Oddly (to me), the first contestant said 7, which was incorrect. I had already checked 3 (yes) and 9 (no) using simple Rules for Divisibility when the second contestant said, 3, which is smaller than 7. Had he or the third contestant realized that 7 was, in fact, a common factor as well as 3, then they should have realized that the GCF was 21. I have to say that if you don't see 7 right away, you won't get 21 in the allotted time. On the other hand, 7 was called and it should have been a quick check.

The third clue checked your knowledge of time. If your friend meets you 130 minutes after 11:30 am, when would you meet him? Two hours and ten minutes later would be 1:40 pm.

The fourth clue, the Daily Double, dealt with the surface area of a cube 10 cm wide. (I don't remember the actual unit, so I'll say it was centimeters, but it was definitely 10.) The contestant started to answer, paused and then answered correctly that it was 600 square centimeters.

Finally, the $1000 clue was a simple two-step equation: If 3x - 11 = 43, then was it x? Alex was surprised at how fast the answer was given. I was still dividing 54 by 3 -- there was $1,000 on the line, and I didn't want to rush and get it wrong.

Overall, the topics were spread out a bit, and with the exception of the clock math, they were good questions for an Algebra Regents review sheet -- especially, the one asking for the total and not just the tip.

The dangerous part for me in all this? The only thing worse than being a math teacher who didn't make it a true Daily Double would be being the math teacher who made it a true Daily Double and got it wrong. Luckily, that wouldn't have been a problem.

On the other hand, I had the same fear with the million-dollar question on Are You Smarter Than a Fifth Grader?, so I guess it's a good thing that I don't go on these shows!

Update: Typo fixed in question 3.

Order of Operations Mnemonic

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(C)Copyright 2014, C. Burke.

Remember to place all your holiday orders of operations early so they'll arrive in time ... and in the correct order.

(blog): 360, 180, 90, 2 and 1/2: I'm Talking Arcs and Inscribed Angles

Assisting today in a Geometry class. It was Day 2 of Arcs of a Circle. Yesterday, the teacher covered central angles with the class, which meant that today's lesson moved to the inscribed angles. She carefully and diligently explained what makes an inscribed angle, and how the line segments intercept an arc in a similar way that they saw in the previous lesson. And then we got into the relationship between the arcs and the two types of angles.

The measure of an arc of a circle is equal to the size of the central angle which intercepts it. The measure of the inscribed angle is half the size of the arc it intercepts. Pause. What does this make the relationship between the central and the inscribed angle. Pause. Wait. Rephrase? Response? Good -- but try again in a full sentence...

"It's half the size." sounds good. No, really, it does -- it means someone's paying attention and either getting it, or somewhat getting it. Following up: "What is half of what?" The inscribed angle is half of the central angle.

Okay. So if the inscribed angle is 60 degrees, how big is the central angle? Let them think about it. Did they come up with 120 degrees? Or 30 degrees? If the smaller angle is half the bigger angle, then the bigger angle is ... ? (Okay, it's a leading question, and I hate leading questions, but sometimes you do need to just pull that one number out of them so you can move on.)

They moved along with the notes and did a couple of practice problems before moving on to the next step. What if two inscribed angles intercepted the same arc? What could we conclude about the two angles? The teacher waited to see if they could reach the statement before she gave it. It took a moment to realize that A was half the arc and B was half the same arc, so angles A and B had to be congruent even if we didn't know how big the arc was. We didn't need to know. But if we did, we could work things out.

Then things started getting complicated because when you start putting in two many line segments and too many inscribed angles, triangles start forming. Wait! What are we supposed to do with those?! Treat them like three inscribed angles, of course, but don't forgot those properties of triangles, either. Particularly, the one about the sum of the angles!

So if we had a problem that looked like this:

... we have enough information to fill in both angles BAC and BCA as well as arcs AC and BC. We just might not know that we know yet. Not unless we remember some other facts about circles and triangles. The total measure of the central angles in a circle is 360 degrees, so the total of the arcs of the circle is also 360 degrees. The sum of the angles of a triangle is 180 degrees. Notice that if each angle of the triangle is inscribed that make each part of the circle twice the size of inscribed angle -- and 360 degrees is twice as big as 180!

Excellent discovery, if they can make it on there own. One student was hovering about it while he was talking. If he made the connection, he didn't share it with me, but he was close to it.

Finally, why is 90 so important that I included it in the title?

Because many of these problems use diameters as one of the line segments. A diameter cuts the circle in half, into two semicircles, each 180 degrees. The central angle formed by the two radii joining into a diameter is a straight angle, measuring 180 degrees. Any inscribed triangle using the diameter as one of its sides would, by necessity, have an angle that measures half of 180 degrees, which is 90 degrees.

Wait a minute!

So any inscribed triangle using the diameter of a circle is a right triangle? And any inscribed right triangle has to include the diameter?

It's almost as if someone planned it that way. Maybe not, but that's how we planned the lesson.

(blog): Last Friday's #NYCMathTweetUp Meet-Up

As promised yesterday, more about my weekend meet-ups.

On Friday Night, a gathering of #mathednyc took place at the offices of Offices of New Visions for Public Schools on West 13 Street in New York City, dubbed the #NYCMathTweetUp. I'm not sure how the capitalization worked with hashtags. Prior to the event, I read that n > 30 had RSVPed, where n is "the number of people interested". However, upon arrival, I discovered fewer than that made it there on a cold night, but this didn't discourage anyone.

The icebreaker game started as soon as we stepped off the elevator and filled in our name tag. We had to pick another tag from the tree with the name of another guest. I thought to jokingly call these hashtags, but they actually contained our Twitter handles instead. The goal was to locate that person and start a conversation. I joked that I might find my partner, but she might excuse herself because she was still looking for her partner. And I say "she" because I did, in fact, locate @khatrimath before the evening ended. I didn't have many comments to make at that point, except that I thought that the two of us were the only ones with math in our Twitter handles.

Guests were asked to bring a math activity that we could try out. I promptly forgot all about that. Not a problem, however, because (1) David Wees @daidwees supplied us with a page of New Visions Math Tweetup Puzzles, and (2) we didn't actually do any. But that was fine, because the conversations were flowing nicely. (I even stopped talking long enough to allow other to speak... a few times, any way ...)

The one major activity of the evening (after eating too many sliders) was an activity entitled Things That Suck. As educators, we divided ourselves into three camps -- Totally Sucks, Does Not Suck, and Need More Info/Might (Not) Suck -- on each of several burning topics, including Homework, Regents exams and Professional Development. This last one being the reason I tweeted that the evening was "like Professional Development, but, you know, fun", and why the first response to that was "the kind that doesn't suck", by the aforementioned @khatrimath.

The evening wound down and the conversation continued at a local establishment a few blocks away, where the picture I posted yesterday was taken.

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Update: What the heck, I'll add a little about Saturday night, too. I was debating whether or not to add a little more about the Celtic Cross concert because the readers here are usually looking for mathy-geeky comics or math-education discussion. But I'm Irish, and I had a good time, so why not? It's not blog, right?

I uploaded a couple of videos to youtube. I'll link you to their cover of Little Talks, which I referenced in a comic last summer. Search on user cjburke23 to find a bunch more. It was a fun show, and I had a sneak peek at their playlist so that I'd know when to film songs I hadn't gotten before. Unfortunately, they strayed from the playlist a few times. After the end of the evening, the crowd was screaming for "One. More. Song. One! More! Song!", and they obliged (and even played two).

It was at this point, the lead singer, Kathleen Vessey Fee called me over (I was close to the front) and said, "Chris, give someone your phone!". Then she handed me the Cowbell and the stick to play it. Unfortunately, I didn't know anyone well enough to hand over my videocamera (it's not a phone) who wasn't also dancing. The thing is: I knew that this would likely happen, given that it was a birthday night out for me, and this was secretly why I invited a bunch of people. So there isn't any video of it (not that I'm aware of), but I played along on the Black-Eyed Peas Tonight's Gonna Be A Good Night, which then led into Taylor Swift's You Belong With Me.

It was at this point that the jaw of the young lady pictured below dropped halfway to the dance floor as did those of several of her friends, all of whom were surprised that I appeared to know the song and could sing along with it. So I got a picture with my first "groupie", Jill. The gentleman with her took it, so I'm safe. And it was about time we were introduced because she and her friends are regulars at these shows, and we've bumped into each other on the (very small) dance floor before.

So no math discussions that night, not even about the Guinness t-test, unless that was a taste test.

(blog): It Was a Meet-Up Weekend

Until recently, I had a lot of make-up weekends, lost Saturdays and Sundays were I made-up the backlog of work that I let snowball from Monday to Friday (or the week or weeks before). However, this weekend was a Meet-up Weekend, where I got to meet up with people I didn't know personally beforehand, but now I can say I have.

Warning: These meetups keep me out late -- not complaining. On the contrary, glad that I'm out of a rut. On the other hand, I haven't made up the sleep yet, so this is a short intro post, and the actual events will get described tomorrow. (No comic tomorrow, either. I haven't done one yet, and don't intend to in the next 20 minutes or so.)

Friday night, if you are a math educator in or near New York City and you blog, tweet or post anything about education, you should have joined us at the NYC Math Education Tweetup at New Visions for Public Schools. According to my email, about 30 people were scheduled to attend; however, I think the weather might've scared a few off. Likewise, parking in Manhattan (I took a train) can be a nightmare and finding the correct address was a minor adventure in oddly-labeled buildings. Again, I'll go into more detail tomorrow. I didn't have a decent camera on me, but pictures were taken, and I'm waiting for more to show up online or in email. One picture did show up at the "after-party" -- that's me on the right.

I got a little turned around in the village and thought I was heading in the correct direction for the train. I should've taken the hint when I was standing at the corner of West 4th Street and West 12th Street that something was amiss. I did figure it out, and eventually trains did get me home much later than I expected.

Unfortunately, my body won't sleep late once the sun's up unless it's extremely cloudy, so I lost a couple of hours. However, that didn't prevent me from joining Celtic Cross at a concert. (By joining, I mean that we were all at the same place at the same time.) The music didn't start until after 10pm. Funny thing, back in college, I never went to a late-night show like this. Now, I seem to do it every couple of months. It was an extra-special outing for me in that some family and friends were going to join me for a change to celebrate my birthday last week. Unfortunately, by showtime, every one of them had a legitimate reason for not being there. But I didn't care. I met other people, fans of the band, some of whom I've seen before, even if we hadn't spoken. I also found out others were having birthdays. At one point, three of us met and discovered that we were 30, 40 and 50. Unfortunately, one of us walked away before we could get a picture. I won't mention which it was, but here are the other two (me and Elizabeth).

I wore a red shirt to be festive for Christmas and all that, but by that part of the evening, it was too hot, so I wore it open. I got some compliments on the Celtic Cross design on my shirt, which has no connection to the band other than their name. (That is, their logo is different.)

Again, I can go into detail about the fun I had in a post tomorrow, but if you need to know why I like this band so much and follow them around, here's a little bit of a hint:

(me and Kathleen Vessey Fee)

(blog): Happy Consecutive Number Day!

It's seems like every day, like every number, has something special about it, but as George Orwell might've said, but wisely chose not to, "Some dates are more special than others". (Had he said something about some numbers being more equal than others, that could've been a problem, but would've fit well into his theme. But that's a subject for another blog post.)

Today is Consecutive Number Day because it is 12/13/14. We've had a Consecutive Number Day for each of the past 11 years -- TWO, if you use the Commutative Property of Dates (also known as the DD-MM-YY format). This last fact makes this year's date even more special because there won't be a 12-13-14 is London or Madrid or pretty much everywhere else in the world where they use that format because there isn't a 13th month.

Finally, it's special because after 12 years in a row, this one will be the last one for a long time. Now, don't fret yourself into a worry. You don't have to wait until 2103 for another one ... unless you're some kind of crazed purist. Me? I like to run the numbers, have fun with the numbers.

The first Consecutive Number Day that I took note of happened when I was in ... kindergarten, no, wait, Pre-K! Yeah! That's the ticket! ... in 1978. It was 5/6/78 and we even had to pause at 12:34 pm (I was asleep for 12:34 am) to note that it was 12:34 5/6/78.

Eleven years later, digital watches were common and we knew the correct time to the second, if we ever bothered to set it correctly, so we knew when it was 1:23:45pm 6/7/89. We could do the same the following year, although 7/8/90 wasn't quite as much fun.

Two thoughts come out of this. First, given the accuracy of today's computing, I had to wonder if the atomic clock could tell us when it was exactly 1:23:34567891011 12/13/14 (and, if so, did some geek take a selfie of it?). Second, it gives us something closer to look forward to.

Hold the date now in your datebook app for 1/2/34. Sadly, there isn't a 5:67 am or pm. Not even in London. Or even in Pasadena, where they'll be hosting the Rose Bowl because New Year's Day is on a Sunday.

(x, why?) Mini: Catching a Cab in the Rain

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(C)Copyright 2014, C. Burke.

It's all about the green, about the green, no yellow...

If you think about it, the Flash's powers wouldn't help him catch a cab, although he could beat it to where he wanted to go.

And if Green Arrow shot one with an arrow, he'd need to use another trick arrow to jack up the car while he changed the tire.

Of course, the probability of Batman catching a cab in the rain is 100%, metaphysical certitude, because, you know, he's Batman.

What's It All About, Algie?

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(C)Copyright 2014, C. Burke.

It's all about the pun.
And it's about a month later than I planned it.

And once I'd decided on going with an allusion to What's It All About, Alfie, I had to lose the examples that didn't go with Algebra. There were a number of variations that I had written down (some better than others) and some I forgot because I wasn't near that piece of paper. Going to happen.

But if you're going to round the bases, make sure you know who's on first. I don't know.

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Update: So I discovered that Get Smart used the title What's It All About, Algie decades ago in Season 5, Episode 23, which is very illuminating according to the Law of Fives.
"The Law of Fives" was also the name of a (x, why?) comic published 5/23/10.

(Blog): MATHice in Scienceland

Only three days into my new substitute assignment, and I have a few observations. Now, i don't want to get political or start anything -- that's not me. I try to roll with it, whatever it is, and not stir the pot. Especially, considering that I have to revisit this pot for the next month. But, here goes ...

One of the school's Living Environment teachers has been out sick. (I don't know how many the school has.) I've covered her classes every day this week, so far, and likely will continue to do so. I don't mind, but the kids do. They're objecting to the work posted on the board with no one to teach them. They're frustrated and confused and don't like to be told to read the text book. Some of them do what they have to do and try their best. Some of them, I think they just want a teacher to tell them the answers to the questions. And some don't seem to care at all. And none are happy that there's a math teacher in their science class.

Today, I tried to talk to a few of them. It was a good day for this because many of the freshmen were gone on a trip and some of my classes had fewer than a dozen students present. Unfortunately, they just wanted to complain (which is fine, so far as that goes), but they didn't want to hear any solutions -- not the ones that are possible or likely. They need to step up and take matters into their own hands, but they don't have the maturity for it.

Have you heard me complain lately that today's freshmen are less mature than the junior high school students I needed to get away from a dozen years ago?

I don't have all the facts, just some anecdotals. At my last school, while they were losing teachers to budget cuts, I knew that the Living Environment (aka Biology) teachers were safe. (This is my impression, mind you, not a statement of facts.) They were needed more than any of the other science teachers because they were the freshmen class, and everyone has to pass that class and Regents exam. If I extrapolate this, Living Environment teachers are likely in high demand, meaning that they have options as to what school they teach in. Which means that they'll leave a poor school in a heartbeat if something better becomes available.

Without passing judgment, I tried to explain to the kids that if someone was looking for a position, would they pick a school that's been around for 50 years or one that was reopened in the past five? That's not the perfect measure for adults, but that's something the kids could understand without feeling insulted.

Not all 50-year-old schools are great, but they've "reorganized" the poorer performing ones in recent years. But if you're looking for a position (which I've done too much of in recent years) and you see a school where, say, 80% of the kids graduate in 4 years and 70% go on to college, and a second one where, say, 40% of the kids graduate in four years and who wants to contemplate the low number going to college, which do you think the teacher is going to lean toward? If one school has staff who have been there for more than a decade, and the other is only five years old and no one's been there for more than three of those five years, which do you think the teacher is going to lean toward?

If a teacher were to walk into my classroom while I was subbing and saw what little respect the students had not only for me (the sub), but for the property of their actual teacher and (let's face it) for themselves, would they want to come to this school?

End result: if the school is poorly-performing, then you aren't going to be able to attract a qualified teacher whose future depends on the performance of immature individuals who have been taught to respect themselves above all else to the exclusion of having any respect for others, without actually recognizing that they aren't respecting their own future as they find "disrespect" in everything anyone tells them that is contrary to their own preconceived (and immature) notions of reality.

Is that the school's fault? Is it the kids' fault? The parents' fault? The teacher's fault?

I don't know. I'll play it safe. I'll blame Bloomberg. He didn't create the mess. He just made it worse by trying to remake the schools in his own image. And he's not my boss anymore, so I can get away with that now.

(Blog): 8 Lies Math Students Love to Tell!

In honor of the marking period ending and grade submissions coming due, we present to you ...

The Top 8 Lies Math Students Love to Tell

8. I'm doing my work!

7. Why am I failing? I handed in all my homeworks and passed all my testses (sic).

6. I was just about to get started.

5. Hey, why'd you erase it? I was copying that.

4. I said that!

3. I'm finished.

2. Mr. Burke, you the greatest math teacher evah! (week before grades are due and/or parents coming to school)

And the Number One Lie Math Students Love to Tell. . .

1. I did my best. I tried really, really hard.

Blog: Jeopardy and Non-Common Core Math

Last week, Jeopardy had a Kids Week and on Friday night, one of the categories was Non-Common Core Math. As a math teacher and just someone who likes numbers, I was curious what the category would be. The kids, on the other hand, well they were curious, too, at first, but then ran away.

It started innocently enough, with the $200 answer being: "1 + 2 + 3 + 4 + 5". Quick mental math gave the question, "What is 15?" A simple exercise in triangle numbers, which are formed by summing consecutive numbers. It's one of those things which most kids will see and do even before they hear the phrase "triangle numbers", and long before they know they memorize the formula. Besides, a small sequence like this is quicker to add (if you don't have it already memorized) than computing a formula.

Things got trickier with the $400 answer, "1 - 2 + 3 - 4 + 5". There was some hesitation as the contestants (I almost typed "students") worked that one out before one of them arrived at "What is 3?" (I didn't tape it, so I can't review it to see if someone got it incorrect first. I don't remember.) There are two short cuts for this problem, and both have to do with pairing. If you noticed that each pair "1 - 2" and "3 - 4" yield a result of "-1", you have -1 + -1 + 5, which is 3. If you noticed that "-2 + 3" and "-4 + 5" yield a result of "+1", then you had 1 + 1 + 1, which is still 3. If you just oscillated your numbers, you took more time and you probably didn't buzz in in time.

The kids gave up on the third answer: "1 * 2 * 3 * 4 * 5". Given the ages of the kids, I would have thought that at least one of them had seen factorial before, and this was the definition of 5!, although the numeric equivalent was needed. Perhaps they got stuck on 24 * 5, not thinking to reverse the order (20, 60, 120, 120). Whatever the reason, no one got the answer, and they bailed on the category.

It proved so unpopular that when Trebek cautioned "less than a minute to go" (a.k.a < 1 min), the two Math clues remained, and were the final ones of the game.

The $800 answer: "-1 * 2 * -3 * 4 * -5". This actually bothered me that none of the kids gt it. First, for the reason Alex gave. Second, because I had to listen to Alex give it. The previous clue had a result of 120. The numbers multiplied are the same, only some of the signs have changed. Multiply a negative times a negative times a negative and the product will be negative (times two more positives, which won't affect it). The question should have been "What is -120?", which should have been easy considering the previous question gave them the number, and they only had to add on the sign.

The final reason to be annoyed? Alex took so long to explain what should have been obvious that we didn't get to see the last clue. Would it have had division? Exponentiation? Mathematical minds want to know!

But that clue won't be revealed, and it's likely that they avoid such mathematical categories during future Kids Weeks.

---

UPDATE: I wrote a little More about that Jeopardy category after the blog The Political Hat referenced this entry.

And Jeopardy had Another Math category, with adults, shortly after this.

(x, why?) Mini: Eights

(Click on the comic if you can't see the full image.)
(C)Copyright 2014, C. Burke.

But four and four are eight, so whatever the problem, it's not a math problem.

But here's a bonus problem: when was the last time we saw these Eights together?

Blog: How Do We Make a Scatterplot?

I was going through a shoebox of stuff in my basement. I found a handful of paperbacks I'd started, some old stubs and statements and, underneath all that, some pages ripped from both spiral and composition notebooks. Most of them appeared to be random questions and answers to homework assignments from April, 2012 (not too bad), but with them was a lesson plan dated December 8, 2004, almost exactly 10 years ago. It was my first year teaching high school, and only my second full year teaching. I remember the students enjoyed the activity, but in my morning class and my afternoon class. However, I don't think I ever did it again. A few reasons I can think of -- I spent the next few years teaching Geometry, teaching Special Ed, and switching gears from Math A to Integrated Algebra. Every year, it seemed, the ninth grade curriculum was being rewritten or reordered. And, frankly, some groups of students were better than other when it came to activities (try to close an activity and bring the class together for a summary without half of them packing their bags!), and some times of the year lend themselves to this sort of thing more than others.

So, basically, this is the sheet of paper I kept. I'm sharing it with you and, thus, keeping it electronically, so I can throw this scrap away.

December 8, 2004

Aim: How do we make a scatterplot?

SWBAT make a graph comparing numbers of words spoken per second/minute.

Activity: Students will break into groups. Each will have a sports article. Their task is to take turns "auditioning" for a sports announcer job (like "Ron" on "G-Unit Radio")

Note: Ron was a school aide who did the morning announcements with a little bit of a flair, and he referred to it as "G-Unit Radio", the "G" for the school name and "Unit" for "Family" ... and having nothing at all to do with a similarly-named hip-hop group from Jamaica, Queens.)

Each article is marked off at 10-word intervals (for easier counting). One group member will be the timer, another the recorder. Students will take turns talking for 10, 15 20 and 30 seconds and then counting the words. After the data is recorded, data will be graphed.

Q: Is there a relationship between time and # of words?

HW: Finish the graph based on the data.
Work on December HW packet.

---

I remember when making groups, one of the keys decisions in grouping was how many of them had a watch. This was before everyone has a phone as well.

I also remember being told to come up with a Christmas vacation assignment. I hate those, and some of my students would be traveling (some outside the country) and did I really want them to bring books along? So I make a packet of HW for the month of December and told them it was due the day after the day we got back because everyone always forgets it the first day back. (Some of them were doing the entire packet the day we got back, which might explain we I haven't done that since.)

I hope you enjoyed my trip down memory lane. Try the activity yourself, and let me know how it works. FYI: I was in the middle of a free six-week trial subscription of Sports Illustrated, courtesy of whatever store I had started my Christmas shopping in, and I pulled an article from there. The sports pages work, too, especially if your school is listed in them!

Blog: Knew It Was Going to Happen, And Yet . . .

Taking me cue from the discussion the other day about randomness and probability, we come to my last day at my current school. I stepped in to a temporary position while another colleague was on maternity leave, and now she is returning. As luck would have it, today was also the last day of the marking period. Normally, I'd be bringing a massive amount of work home to grade, but since I won't be going back to return anything, I had to leave a pile for me to check. I left answers keys and hints that she might just want to "check" the work and score it all. At this point, most of the kids are "good" about leaving two-thirds of the papers blank if they don't know what they're doing. Anyone finishes the page in what looks to be a neat and unrushed manner probably has a handle on things.

Today's events: I brought in doughnuts and put them out in the Teacher Center a dozen at a time during my preps. Some of the folks who hang out there (many stay in their rooms or elsewhere where there are conference tables) knew that today was my last day and that I'm having a birthday next week, when I'll obviously no longer be there. It was a little busy today. Friday's are little crazy because the periods are very short and there's 90 minutes of PD (professional development) at the end of the day for the teaching staff. (This was something that was voted on my the teachers last year.) I was doing double duty making sure that I packed what I wanted and tied up loose ends while creating work for the kids who I'd kept in the dark about the switchover. There can be some troublesome elements in the class, and they didn't need encouragement to misbehave, which they'd do despite the fact that I'm handling the second marking period grades this weekend.

Now, before I forget, when I arrived at work this morning, there was a bag with a bottle in it on my desk. I believe it's essentially an Italian version of Bailey's, which was confusing because it was a gift from a lovely Russian woman. So by the time that PD came around at the end of the day, after a lot of well-wishing (and an incident I'll mention at the end), I hadn't thought more about it. I had to go to the library for yearbook photos and I was going to take the remainder of the doughnuts, and I was supposed to go out for a drink with one or two coworkers.

That's when I ran into Pamela. (I'll call her "Dr. Pamela" this once, just to give her her due on the title, but I'm leaving off the last names right now. I don't know why.) She apologizes that it's Friday and the end of the day, but it's my last day, and could I help her set up that document camera she found in her desk drawer. (True story: I put it in that desk drawer a year and a half ago because it was the only piece of the computer not tied down. No one noticed since. I wasn't in the school all last year.) Well, sure thing. We'd talked about it once, and I hadn't done it, so it was the least I cou---

SURPRISE!

Seriously? The document camera? You lured me into a surprise party with a document camera??? Points for originality. So I celebrated my last day and my landmark birthday with my Math Department colleagues. Unfortunately, my boss couldn't make it, being pulled in "six directions at once". (I'm guessing: each way on the x, y, and z axes.) And others from the Teacher Center were likely corralled with their own departments at the moment, so they missed out.

And the thing is, I knew it was coming. I thought I knew. And yet ... Still a random moment and I expected none of what was there. And just to show you that I'm not the only person who think mathematically all the time, take a look at the equations and expressions on this word wall:

I'm leaving that school in capable hands. And, who knows, I might find myself returning.

As for the unfortunate incident, the less said the better. In literally the last minute of teaching at this school, a fight broke out. I'd hope it was just a flare-up, some hurt feeling that a slap on the arm would resolve and diffuse and not go anywhere else. Sadly, it did go somewhere -- out into the hall. A crowd did form, but I have to give a shout-out to students who tried to separate, restrain(*) and calm the participants. (*: when it comes to "restraining" a student, many are looking to be held back just to save face. However, that was true only for the instigator.) The kids came to each others' aids before it got worse. But like the inevitable car wreck you witness, you know what's about to happen. I wish I could say more, but I've probably said too much already. I was bummed out for the prep period before my PD, which in itself was a bit silly to go to, so I was even happier about the little diversion. Maybe it someone wants to join me for that new job/landmark birthday drink, we can talk!

Blog: What is Interval Notation?

Another question for my students: What is Interval Notation?

Short Answer: It's a topic that's worth, at most, two points on the state exams, and that most of you aren't likely to see again. On the other hand, given the number of students I've known who have scored 63 or 64 on that same state exam, ignore this at your own peril.

Another Flippant Short Response: It's actually a very simple topic to grasp if you give your instructor, your facilitator, five minutes to explain and really use that time to understand, not just copy down a few rules in your notebook. Or you can try to figure it out for yourself from the worksheet and then read the textbook when you get home.

Seriously, it's a simple thing.

A couple of days ago, we were talking about Compound Inequalities. Interval notation is another way to express the range of data in the solution to an inequality. Why do we need another way to express this?

This is Algebra. We don't need no steenkin' reasons! However, I like to say that it's shorthand, even if it uses the same number of characters, with the exception of not requiring any underscores. For that matter, as I'm using a simplified form of a mark-up language, the less-than symbol has another meaning, and it actually takes me three keystrokes to ensure that you see this: <.

With Interval notation, there are no < symbols or variables, either. Just the boundaries.

For example, if we needed to represent the compound inequality -3 < x < 5, we could take the two endpoints and stick them in a pair of brackets, like this: [-3, 5].
This means the same thing and seems quicker to write, and doesn't require as much space (or, at least, doesn't seem to, depending on where you're writing it).

Well, that's all fine, but what if you have "less than" instead of "less than or equal to"? Simple, we don't use square brackets; we use parentheses, instead. That means -3 < x < 5 could be written as (-3,5).

Here is where you need to pay attention. I've said in the past that notion is important, and different symbols mean different things. This is one instance where notation can mean two different things. (-3,5) can be a inequality for one variable, such as x, or it can be a point on the co-ordinate plane, using two variables, (x,y). How do you know which? Context!

It's also important to note that you can (and the state will!) mix and match these: -1 < x < 4 would be written as [-1,4).

Finally (for now), what if it isn't a compound inequality. What is you only have, for example, x > 5? In this case, five is the lower boundary, but what is the upper boundary? The graph has an arrow going up to the right, continuing forever until it hits, as Buzz Lightyear might intone, Infinity, and Beyond! ... or infinity, at any rate. This would be written as (5, ∞).

Note that because infinity is not an actual number (that is, x cannot actually equal infinity), it will always have a parenthesis next to it, not a square bracket.

Finally, a proud moment because one of my students put themselves out there and took a chance. I asked where the answer goes to with x < 5. I got blank stares. "Well, think about where it goes off to the right." The student very cautiously hazarded the guess, "negative infinity?".

Absolutely. He had never heard of such a thing before and didn't know it could exist. Then he realized that he didn't know that it couldn't exist. I liked the way he thinks. I hope there's more of that because if you think of it, this isn't any more complicated than the compound inequalities they're representing.

Okay, maybe that's not saying as much as I'd like.

Postscript: For future reference, the infinity symbol (∞) is the ampersand (&), followed by "#8734" (no quotes).

Blog: Is It Still a Random Event If You Know It's Coming?

It's only the Third of December, and already there's a digression. That's why I love blogs: the very randomness about them. Randomness, probability, math. The circle of circles.

I have been bouncing around the New York City educational system for a few years now, and that's okay. I know that I'm wanted and there will be a soft landing somewhere. I figured that if I didn't get back to where I wanted to be in a reasonable amount of time, I'd find another suitable position. In the meantime, I'd be in the Absentee Teacher Reserve (ATR) pool. The downside to being in the pool is that you never know where you're next assignment will be: good neighborhood or bad, conveniently located or not. The upside? Well, no lesson plans, no mindless nights grading mindless papers writing meaningful comments, which likely wouldn't be read anyway. And I can most likely reset my alarm for a later time. Currently, I have a tendency to get to work an hour early to prep for a first period class because I don't wish to do all that work the night before when I could be sitting in front of the television, watching the insides of my eyelids.

A quick summary of the past few years: my school was shrinking because Bloomberg was trying to close it down despite its high grades on his own report card system. It stayed open but lost staff, including half the Math Department (which was three people, not counting the Support Services teachers). That event inspired this comic: Kiss Today Good-bye. After nearly a decade, I was leaving what felt like home.

That fall, I spent a month in Williamsburg at a small school that really didn't need any substitutes, and I mostly circulated the rooms, trying to help, which was easier to do when the subject was math. That was followed by a week at a school ten minutes from my home. At that point, a teacher at my old school retired, and I was on my way back. Things had changed a little. I had mostly freshmen, unlike the previous year, and the schedule was a little ... challenging. One could theorize that the classes were particularly rough because they wanted to encourage the former teacher to retire sooner. I think he had his plans set from the first day of school. It was a bit of a rough year, but I was promised a better horizons the following year ... and then I was excessed again.

My personal Year from Hell started in August. (I actually asked an A.P., who called to request an interview, if he had time the following day as I would be busy on the day he proposed, attending a funeral.) Professionally, it wasn't a bad year at all. I was called into Bed-Stuy, which sounded worse than it was, and spent a month there, watching them build the sets for The Knick, literally, right outside my window. Note: I am NOT misusing the word "literally". I have pictures. And Clive Owen even retweeted one of them! (Actually, it later turned out it was a FAN account, but I didn't know that at the time.)

This was followed by a few week in Park Slope, at my old zone school, which had been broken up years ago. Park Slope went through Yuppie-ville into Hipster Haven, but many of the kids traveled there from other areas. It wasn't an impressive place, and said to say that the teachers I worked with weren't fully versed in the topic they were teaching. I was able to nonchalantly correct a couple of things as if something was simply misspoken, but one thing in particularly was just so wrong, I kept my mouth shut. I figured that it was a tough topic and I could help individually with the students, but I couldn't undermine the teachers for the rest of the year. They weren't idiots, and I didn't want them to be thought of that way. (And I did explain their mistake later, during a prep period.)

Sometime in October, I got a call from a school in Staten Island. It would be a bit of a hike, and the Verrazano Narrows bridge has an incredibly expensive toll. (Not kidding -- even with a discount, it was costing me more than $10/day to get to work over one bridge.) It was supposed to be temporary, which worked out for me, because I knew that there was going to be an opening for me in Brooklyn at any given (i.e., random) moment. Except that there wasn't. For whatever reason that the teacher was on leave (I wasn't told; I never asked), she didn't come back. I was there until June, and I had settled in and had started to make friends. I attended a graduation dinner for a colleague, a summer barbecue. I even met a parent who I hadn't seen since we were both 13, some mumblemumble years ago.

At the end of the year, the AP asked me what was next. She didn't ask if I wanted to stay. maybe she assumed I wanted back in my old school ... or just my old borough on the other side of the bridge.

Which brings us back to randomness and inevitability. I went back to my old school during the summer for Regents Review classes, and subbing for a little spending money. One of my former colleagues was 8 months pregnant. She would be out on the first day of school. The AP of school organization started the paperwork to get me back on a temporary basis, and I was happy to go back. The only problem was that I hadn't a clue how long I would be back for. No one seemed to know how much time she would be taking. It was going to be six weeks plus whatever sick leave she had saved up (as far as I could guess) but I hadn't a clue how much that would be. Three months? A semester? Two marking periods, it turns out.

And here I am. Ready to leave again. Probably for the last time. I don't seem them -- at this moment -- turning things around. I don't see them expanding the student population, and with it, the budget. How long do I stay in the ATR pool? Sooner or later, I will have to find a position. Sooner or later, I will be subbing in a good school and I'll seek a position. (And maybe even regret not seeking last year's position again.) But in the meantime, I can count on two things:

A high probability of randomness, and an absolute certainly of zero lesson plans.

Blog: What is a Compound Inequality?

This one is for my students ... should they ever come by here.

Everything in Arithmetic seemed to be about finding the answer. Let me add emphasis to that: THE Answer. Then along came Algebra, and, suddenly, THE Answer wasn't as important anymore. Don't get me wrong -- it was still important. However, how we got the answer and why it was correct seemed to matter more. Just writing, for example, "5" on the paper in that big, empty space wasn't good enough. Even if it was "obvious" that it was 5. Why? Because the next problem might not be so "obvious", so we still needed to know the rules so we could attack the next one, and the one after that, and so on, as they got more complicated.

But even as we showed our work and checked our answer, we knew one thing for sure: There are AN answer. One. Singular. The value that makes the equation True.

Until inequalities came along. Why did we even start that chapter? How could there be problems with not only more than one answer, but an infinite number of them, an entire range of values, shooting off into infinity. The answer isn't seven, it's greater than seven. Does that mean it's eight? Well, yes, but it's also nine, ten, eleven, twenty-seven, thirty-one and a half, the square root of 92, a googol (not a search engine). It's all those real numbers. The ones bigger than seven.

Okay, so equations have one answer (or maybe two?? -- what do you mean, "we'll talk about that?"), and inequalities have arrows that point to the left or the right and go on forever. That's it, right?

Weeeeeellllllll . . . .

Do you know how in English class (or ELA or whatever), you can have compound sentences, which are two sentences joined together by a conjunction. (Cue: Schoolhouse Rock's "Conjunction Junction".) Those conjunctions are "AND", "BUT", and "OR". In math, we can have compound inequalities, and they can be joined by "AND" or "OR". What about "BUT"? Here's a secret for you: "AND" and "BUT" mean the same thing:

I went to the store, and it was closed.

I went to the store, but it was closed.

So we don't use "BUT" in Algebra. ("What about when talking about someone's face?" "That's just wrong, Gordo." -- and now because it's rude, so much as that would be misspelled.) Likewise, "OR" isn't a conjunction in math. It's something called a disjunction, and we'll address that later. One topic per day, please.

If you wanted to get a "B" on your report card, you would need to score AT LEAST 80 and LESS THAN 90. (Exactly 90 or higher would be an "A", and while that would be great, let's be realistic here: A's are tough to get. But aim high.)

This is an example of a compound inequality. If we were to graph all the averages that result in a grade of B, there wouldn't be an arrow. Sure, x > 80 would have a closed circle above 80 and an arrow pointing to the right. And x < 90 would have an open circle and an arrow pointing to the left. However, the "AND" in the condition tells us that we only want the points that make both true, the area that they overlap. So we wind up with a line segment with one closed endpoint and one open endpoint, representing a range of numbers -- still infinite! -- that are solutions to the inequality x > 80 AND x < 90.

One last thing. Compound inequalities with AND can be written without the AND. Look at the endpoints. The minimum number (80) has to be less than or equal to the actual average (the variable) and that has to be less than the maximum score (90), so we could write it as 80 < x < 90.

Confused yet? Don't worry, you'll get the hang of it.

Blog: 31 Posts of December? There's Proportionality in There Somewhere

I didn't write a novel in November, like many set out to do each year, and I didn't successfully tackle my goal of 100,000 words in the past year or so (before hitting a landmark birthday), but I did think that I could challenge myself to author another 31 blog posts this month.

And that almost fell apart on Day 1. It's late in the evening, and I'm exhausted. So I've punted and given you an introduction post announcing my intent without actually getting started. I've fulfilled the obligation with more noise and less content.

Noise and content, as we know, are inversely proportional in blog posts, as well as message boards and other forums. The more of one, the less of the other. Bob McAllister, host of the old -- old -- kid show Wonderama, used to sing "The more you Fingel, the less you Heimer, The less you Heimer, the more you Fingel" when he introduced his character, Professor Fingelheimer. That was probably my first real exposure to the concept of inverse proportion even though it didn't mean a fingel-thing!

With noise and content, you really can't have one without the other. Well, you can, theoretically, but 100% content is boring and bland while 100% noise would be ... should I name names? You know which blogs those are: lists of 20 random things required 72 clicks to get through.

So this is mostly noise and nostalgia, and tomorrow will be some wonderfully thought out piece about inequalities that might even merit a second draft.

Unless a comic comes to me in the next hour or so that just can't wait. Or my work schedule demands a little bit more of me.

Which would leave you with a little bit less of me.

Thanksgiving Inequality question comic

I needed an activity for the last day before Thanksgiving with snow on the way -- and I gave a test yesterday. New topics are useless. Most approaches to working today won't work.

So I adapted my first Thanksgiving comic (below) to make it an inequality, which is the topic we most recently covered, and lightened the colors to make it more copier-friendly.

I asked them to create their own Thanksgiving-themed math comic on the bottom of the page (which didn't have to be about inequalities), and to solve their own problem. (There doesn't have to be a problem -- if they come up with a good joke or explanation of a problem, that's fine, too. Most likely, they'll copy mine -- that's usually what happens.)

Inequalities

(Click on the comic if you can't see the full image.)
(C)Copyright 2014, C. Burke.

Is that an inequality? Confound it!

(x, why?) Mini: Too Close

(Click on the comic if you can't see the full image.)
(C)Copyright 2014, C. Burke.

Can't get much closer. Maybe that's why the Police sang Don't Stand So Close to Me.

(x, why?) Mini: Paragon

(Click on the comic if you can't see the full image.)
(C)Copyright 2014, C. Burke.

How many sides does a paragon have?

Falling Leaves, Part 2

(Click on the comic if you can't see the full image.)
(C)Copyright 2014, C. Burke.

Reality doesn't work that way. Ninety percent of the leaves accumulate in the gaps between cans. And that's assuming someone move the birdbath in the middle of the garden, which doesn't jibe with a plan born out of laziness.

By the way, the mat is actually written in English. It has not been translated from, say, Klingon for the benefit of the viewer.

Falling Leaves, Part 1

(Click on the comic if you can't see the full image.)
(C)Copyright 2014, C. Burke.

It's like sinking a free-throw without even putting down the broom. Which is impressive if you've ever tried to throw a leaf.

Perimeter, Right Triangles and Radicals

In my Algebra class today, we were reviewing the process for adding and subtracting radical numbers. Previously, we simplified irrational numbers, such as the square of 80, which becomes 4 times the square root of 5.

To give them a more thoughtful question than just what is the sum of SQRT(75) + SQRT(48), where the only "thought" is to get past thinking that it's SQRT(123), I decided to pull out Right Triangles and that Old Favorite, the Pythagorean Theorem. Now, I didn't want them to just simplify the irrational number, I wanted some kind of addition in the problem. That brings us to Perimeter of a Right Triangle.

There are basically two types of problems you can offer up for consideration: problems with one radical number, and problems with two radical numbers. Three is just being mean, and overly complicates things -- on the other hand, it could make it interesting. Give the students the length of the legs (the base and the height), establish that there is a right angle between them, and let them go to work.

In the first kind of problem, you can pick any two numbers and you'll most likely have an irrational hypotenuse. Okay, but boring. It's more interesting if you make one of the legs irrational in such a way to make the hypotenuse rational. Surprise them.

It keeps it interesting when you consider these two problems, which look very similar, but are very different.

Consider the square root of 13 triangle first. If we square 6, we get 36. If we take the square of the square root of 13, we get 13. 36 + 13 = 49, which is the square of the hypotenuse. Therefore, the hypotenuse is 7.

Finding the perimeter is as simple as adding the three sides, which in this case means combining the like terms, which would be the integers 6 and 7. The perimeter is 13 + root 13.

In the square root 12 problem, the numbers had to be carefully selected. In this case, there will be two irrational numbers. If anything is to be combined, then the radicals have to simplify to the same radicand. Otherwise, the exercise is pointless.

If we square 6, we get 36 again. If we take the square of the square root of 12, we get 12, of course. 36 + 12 = 48, which is the square of the hypotenuse. Therefore, the hypotenuse is root 48.

We can't add any of the numbers as they are written, but we can simplify both of the radicals, as shown:

Both of the radicals have the square root of 3 in simplest form, and their coefficients can be added. The perimeter is 6 + 6 square root 3.

Interestingly, in both examples I wrote for lessons today, a double number appeared in the answer. This is actually something else to be careful about. Some students might see that as a co-incidence. Others might see it as a pattern and come to expect it.

There is an easy solution to that: give them some more problems to work on!

Brains and Muscle

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(C)Copyright 2014, C. Burke.

The Formula being Marriage = Brains X Muscle X Height. Or something like that.

And packing a garage for the winter isn't just a 3-D jigsaw puzzle. You still need to access to the winter things.
Or you could toss some stuff to the curb on bulk pick-up day, but that never happens.

One day, maybe we'll have a garage big enough so that there's also room for the car.

Binge

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(C)Copyright 2014, C. Burke.

If everyone waits until the end of the season to watch, will the show make it to the end of a season?

Decomposing Numbers

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(C)Copyright 2014, C. Burke.

They're decomposing numbers ... there's less of them every year ...

Happy Halloween!