A Rational Approach

(Click on the cartoon to see the full image.)
(C)Copyright 2014, C. Burke.

Not only will he come between them, he'll come right in the middle of them, exactly. (2 and 3, that is. Rad 5 doesn't count.)

Suggested/predicted by William Ricker, although I will say it was kinda/sorta planned along this way in the first place.

If You Could See My Expression, You'd Know It Has No Equal

I've started reviewing for the new New York Common Core Algebra Regents Exam which will be given next week. My students are worried. They should be. No one really knows what it will look like, and the prep material is all over the place. I've given them sample tests, and tried to cover both new material and the basics.

I need to work on the basics.

I gave a quiz last week, with 10 multiple choice questions. Many of them had questions about the correlations and the correlation coefficient. Those didn't surprise me. What did surprise me what the question asking "Which of these is an example of an algebraic expression?" Most of the class picked choice 1) y = mx + b.

I had to ask them today about expressions and equations, and they seemed to know the difference. So I wrote y = mx + b on the board and asked what that was. They looked at me oddly, as if it were a trick question. Testing jitters? I don't know.

At least, no one picked the inequality, and no one picked the Pythagorean Theorem. But too few people picked the expression.

For those playing along at home, an expression doesn't have an equal sign and can't be "solved". It can only be "evaluated" if you know the values of all the variables. (Moreover, it doesn't actually have to have any variables or parentheses or operators. "6" is an expression.)

As for my students, their expressions were "puzzled" and "bewildered". Mine was more "disappointed".

We're working on it.

Venn There, Done That

I can't believe that I haven't used that title before. A quick search didn't find it -- at least, not on the first three pages of results. It's possible that I used it in a comic. However, the comic dialogue, as of this writing, is not searchable. That's on a list of things to do this summer.

Memorial Day and Weekend are now behind us, and discussion today turned to the things we did this in the past few days and things we wish we could've done. Being poolside was up there, and someone mentioned summer drinks. "Pina coladas" were offered up as a suggestion. I don't recall if the weather report came up at this point or later (more thunderstorms expected), but someone started singing the Pina Colada Song.

Naturally, with my iPad handy, I scrolled through the Venn diagram tag until I found this little gem for a Personal Ad

(I did this because the Song tag was too big, and a search on "Pina Colada" failed.

Another Venn diagram, with a comment about being unable to make a four-oval Venn diagram. Again, for the record, they are difficult to draw, but not impossible. I have updated the entry accordingly.

Memorial Day 2014

(Click on the cartoon to see the full image.)
(C)Copyright 2014, C. Burke.

Remembering Memorial Day.

Thankfully, my father, my brother, and my uncles all came home. Not all served in wartime, but that didn't make it any easier for my grandmothers.

The photo of Arlington is in the public domain.

Day 30 of 30: You Want a Piece of This Function?

This is day 30 of the 30-day blogging challenge. It's the checkered flag! Maybe I'll take tomorrow off.

I briefly thought about doing something concerned with the Indy 500, but it's almost 10pm ET as I write this and that race is long over. I'm sure the excitement isn't for those present, but it would be forced now. So, continuing from last night's discussion on functions, let's talk, by which I mean complain, about piecewise functions.

Okay, one Indy-related question:

Q: How many ways can 33 cars be arranged at the start of the race?

A: One. That's what Time Trials are for!

That could've been a comic for today had I had time to make it. Busy weekend. Busier with the cleaning and the grading. But back to functions.

Just explaining to a student what a piecewise function is is not a simple task. Explaining how to read one takes time and patience, along with repetition of the phrase, "when you see the comma, think 'when'". Oddly, I once said, "say 'when'", and it sounded like I was pouring beverages out. I had to switch that up.

Worst of all are the examples that they give. They make no sense whatsoever. They are purely abstract creations that you wonder if they might ever come up -- even just in another math class.

This isn't to say that piecewise functions aren't useful in the real world, or even in mathematics. I could even justify them in Algebra as opposed to waiting for Trigonometry/Algebra 2. But do they have to make them so confusing off the bat. (Hell, the name itself is confusing -- and might I add that it isn't even recognized by my spell checker!)

Examples of reasonable math functions that they could have brought up? First, the absolute value function, which looks like this:

"When" x is less than 0, you want to flip the sign, (i.e., take the negative of x because a negative of a negative number is positive). Otherwise, leave x alone. Note that the last condition has to cover all other possibilities in the domain. You don't want to leave a value out(*), and you definitely don't want to repeat a value, because then it won't be a function. (*) Yeah, there are times you'll leave something out, but not here, not now and not with absolute value!

Another one that they can use which makes for an interesting graph, but doesn't have any variables, is the Sign function, not to be confused with the Sine function:

Negative numbers return -1, positive numbers return positive 1 and zero returns 0. This was useful when I was programming computers, something that I'm glad I've done and something to which I'll refer often. Why not steer the kids in that direction if it's something challenging that might interest them? By the way, the Sign function would be the basis for some kind of trinary system of anything when binary gets boring.

So we have two good examples to start. So what do the books give us? Something like this:

Okay, maybe nothing that nuts, and maybe nothing with e or i in the exponent, but it might as well have been. Non-continuous functions that mean nothing even in the abstract.

On the other hand, finding relevant, relateable uses for piecewise functions was a little crazy. I could've tried my Financial Algebra textbook which is constantly trying to get the students to create some of these (and eventually did). And then there are the old standbys, which don't mean as much any more. I used to use the example of different phone plans when talking about systems of equations. This is easily adaptable into a piecewise function. There's a minimum charge for a certain number of minutes and then you have to pay, say, $.10 per minute after you used your allotment. There are two problems with this example: first, the minimum means that there will be a constant, a line with a slope of zero and no variable. Second, what kid in my class a) pays for their phone, or b) doesn't have unlimited minutes. Minute plans are already a thing of the past.

Pay phones are right behind them, but you can tell them that they're at the mall or the airports and they'll believe that they're there somewhere and just haven't noticed.

My other example suffers similar problems: if you live in an apartment and aren't allowed to have a washing machine, then you have to go a laundromat where you can do it yourself, or you can pay to have them do it for you. It's usually done by weight. I gave an example (and I haven't dropped off laundry in a long time) of a cost of $.60/pound with a $6.00 minimum, and let them figure out that you're paying for at least ten pounds whether or not you bring ten pounds. Same problem with the flat minimum and just as unrelateable.

On the other hand, if I put enough of these problems on the board, they'll see enough commas and say "When".

Like I am now because this is too much.

30 Days. When!

Day 29 of 30: Recursive Functions

This is Day 29 of my 30-day blogging challenge. I can see the checkered flag in the distance.

Common Core Algebra seems more concerned about functions and families of functions than the previous Integrated Algebra in New York State was. It used to be that it was all linear equations and if you saw f(x), you just think "y". They had their uses, but we didn't get to talk much about them -- there was already too much to talk about in that course, and we had a hard time making all of it happen.

Well, Common Core is no different. They just give you a different "all that" to cover, and they really do want you to cover it all. When it comes to functions, they not only want the standard linear, quadratic, exponential and absolute function, but also cubic, square root, piecewise and recursive. Some of them need to be graphed, some only have to be used for evaluating, for example, f(3) or f(-5).

I can get to piecewise another time. My students are getting the hang of them. Well, some of them are. Some of the time. Okay, so maybe not. And I don't blame them. I didn't have to deal with those until later on.

Recursive functions, on the other hand, I don't remember from math class at all. Seriously. I remember them from computer programming. This isn't to say that I hadn't seen them in a math class first before I programmed a recursive function, but I know what left its mark on my memory and where it took.

A recursive function is one that calls itself. It uses itself in its own definition. The two most obvious examples (and Common Core won't use obvious examples are:

Factorial: f(n) = f(n - 1) * n, f(1) = 1

Triangular Numbers: t(n) = t(n - 1) + n, t(1) = 1

To calculate f(n) when n = 5, you do the following: f(5) = f(5-1) * 5 = f(4) * 5. But you need to know what f(4) is.
Well, that's easy. f(4) = f(4-1) * 4 = f(3) * 4. Okay, so now we need to know f(3) ... and then f(2) ... and so on down to f(1), which we are given.
As I told me students: you might as well start with f(2) and work your way up. We'll have to do all that work anyway. Then find f(3), f(4) and so on. If you are programming a computer ... well, I'd probably do it the same way for simple things, but use a recursive function when the professor tells me to do so. Most of the recursive things I've ever coded worked fine with a For...Next or Do...End loop.

Likewise, t(5) would work the same way. The only difference is that I used the letter t for triangular because I already used f for factorial in the same example. Be careful! Students can actually get confused by this. (I'm not kidding.)

Real-World Connection: Forgetting about where factorials and triangular numbers may occur in your everyday existence, recursive functions are part of the real world. Take the function clean(n). Suppose you wish to evaluate this function for the value of n = floor. There's a problem with this because your wife may tell you that you can't clean the floor until you evaluate clean(counters). And you can't do that until you evaluate clean(cabinets).

Now keep in mind that this is just supposed to be an illustration of a recursive function. However, it isn't exactly a true parallel example. For one thing, one you can't calculate 5! (5 factorial) without first calculating 4! (4 factorial). On the other hand, you can clean the floor without cleaning the counters or the cabinets. However, in both cases, according to my wife, at least, you'd be incorrect.

Day 28 of 30: Process of Elimination

This is day 28 of the 30-day blogging challenge. Putting horse racing metaphors aside on Indy weekend, it's the final lap and we're going all out.

If you ever watched Are You Smarter Than a Fifth Grader?, then you know that Jeff Foxworthy makes a great game-show host, with his ability to crack a joke at the right time and to be topical with what the contestants have said. He's a little slow moving the game along with all the pauses, but I think that that was more the decisions of the producers or directors. It's because of Foxworthy that, two season ago, I recorded the first few episodes of Are You Smarter Than a Bible Student?, better known as The Great American Bible Challenge. It's not a show for "religious nuts". It's funny, entertaining, and educational -- true if you've read the Bible, parts of it, or are just interested in all this Bible stuff you've heard about over the years.

I don't watch it to see if I can outscore the Bible study group contestants any more than I think I could outrun Jeopardy! champions or Who Wants to Be a Millionaire? millionaires -- especially not if they're groups of nuns or rabbis. Yes, rabbis, who may have had a bit of an edge specializing in that first half of the Bible.

No, I watch it for moments that lead a confused Foxworthy to query a contestant, "Where are the pizza ovens in the Bible?"

I'm with Jeff -- my pastor never mentioned them in his sermons.

So why the long appreciation about the show now, in its season? Because of something from this week's episode. An attempt to build tension among those who know their Biblical lion-killers, but totally destroyed it among the mathematically aware. That is, me.

The penultimate round, the one that decides which two teams advance to the final round, is called The Chosen Three. One member of the team is asked a question with multiple answers. Six choices are displayed, three of which are correct and three are incorrect. There could be other correct answers which aren't one the board. To give an overly simple non-religious example:

Which of the following are days of the week?
Monday ... Tuesday ... Frog ... Jelly ... Friday ... Wisenheimer

There are more than three possible correct answers, but only three correct answers are given. The other three are incorrect. It seems to be random whether these questions are relatively easy or impossibly difficult. Or it could just be a matter of my education on the matter as I haven't attended a Bible study in years. After all, I answered one question because I remembered it from The Ten Commandments, not from Sister St. Mark's religion class.

Okay, so here's the math.

There are three teams. They go to the team with the most points first. If that team gets all three correct, they are a lock for the final round because no team will be able to catch them. If they miss one or two (or all three), they leave the door open for one or both of the other teams to pass them. (It is possible to miss all three, but there's one one combination that doesn't include any correct answers, which is difficult enough randomly, but this is also supposed to be the "smartest" player who should be able to identify at least one of the answers. That said, I think I remember seeing it happen exactly once in two years.)

So the second team gets a question about who killed a lion in the Bible. I didn't write down all the names, and I didn't even recognize all the names. The first four were Saul, David, Daniel and Samson. (And then two with more "Biblical" names.) The contestant said that he knew that David killed a lion, and that Daniel, who was known for being in the lion's den, just "palled around" with one but didn't kill it. He added that he didn't think that Saul had killed a lion. He wasn't sure about Samson, so he chose the other two.

This team needed two correct answers to be a lock for the final round. Foxworthy revealed that David did indeed kill a lion. They need one more, so tension is mounting. He says that the contestant was right that Saul did not kill one. And then jokes about Daniel. And ... waitaminute, what did you just do, Jeff?

Sure, you're going to try to increase the tension by revealing an incorrect guess, but that didn't matter anymore.

There are three choices that we don't know about. Two of them have to be correct. The contestant picked two out of the three. Ergo, he had picked one of the two correct answers! Yes! I was so annoyed that I used the word "Ergo"!

Simple process of elimination, Jeff. There weren't enough choices remaining to have had both incorrect. His other two guesses had to be Yes-No, No-Yes or Yes-Yes. There are only three incorrect choices and you revealed two of them. So while tension was mounting, and I'll grant you that at times like this a contestant might not be thinking logically, those with no cars in the race (Indy weekend!) already knew the outcome!

That was a little bit of disappointment in an otherwise fine broadcast. Well, that and the lack of pizza ovens.

Radical Approach

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(C)Copyright 2014, C. Burke.

Or maybe "The Radical Approacheth"?

Oh, c'mon! Like you didn't see the joke in the title before you even clicked!

Day 27 of 30: Moron, er, no, ... wait... 'More On Venn Diagrams'

This is Day 27 of the 30-day blogging challenge. Coming down to the wire.

A couple of days ago, in a post about Venn Diagrams, i happened to state, in an off-the-cuff manner, that you can't have a four-circle Venn diagram. Thankfully, that much is true. Had I said four-category Venn diagram, there might've been a rumble. Watch out when mathematicians rumble. They know the odds, and while they know that they may not always be ever in their favor, they know when they are in their favor, so back up, pup.

Anyway, Venn diagrams are possible with more than three categories. The trick is being careful in drawing them so that they every possibility, ever intersection, is represented. With four circles, there won't be enough overlap. If A, B, C, and D are four overlapping circles, then odds are you'll be missing a region of only AC or BD or both. Now, it is possible to create one with ovals (aka "ellipses"), but again, placement is key. You need to be exact or something will be missing, or, more likely if I'm free-drawing, there will be two non-contiguous spaces representing the same intersection. That's like someone picking up Staten Island, moving it somewhere in the middle of Pennsylvania, but still saying it's part of New York. Silly.

My other problem with it is readability. I guess you can get used to it, but it doesn't have the same "flow" as a two- or three-circle diagram has. You need to identify which ovals a given region is contained within and then what those ovals represent. Maybe it's just me.

Another method, much cooler, actually, is to go concave. (I wanted to make a "Con-Caveman" joke here, but it didn't read well. Pity.) This has the benefit of starting with the tried-and-true three-circle Venn diagram. Then a sort of rainbow is added to snake its way around the various intersections without completely covering any of them, like this:

Maybe it's me again, just that rainbow looks like a clown's sad face.

Other tricks can be used to make larger ones, and some numbers seem to be easier than others (although I don't know if it's easier to make them or just easier to read and understand them after someone else created it.)

I will identify my biggest problem with these oversized Venns: their undersized parts. The sections are just too small to be useful. If you're doing some kind of programming and you just need a bucket, well, okay. However, if you are trying to categorize things in a math class, it's just a waste of time. The exercise in creativity is making one. Using one would be an exercise in futility.

Note: Comments on twitter, particularly by William Ricker, prompted this further investigation. Comments are welcome here, there, Facebook, Google+ or anywhere else, but not everyone follows all those things. (Many days, I don't!) So comments, with no registration required, are appreciated. Just remember, if you comment as "Anonymous", you won't get any responses in email, and we'll probably not take you as seriously as someone with a face (or an avatar) who isn't linking to some spammy website.

EDIT: Corrected William's name. Not my fault! There should've been a red squiggly under it. It's not like I wear my glasses when I type on the computer any more! My fault for not just typing "Bill". Or would it be "Blil"?

Day 26 of 30: Long Day Tomorrow, But Is It Really?

This is Day 26 of the 30-day blogging challenge. Final burst of speed.

I've been sitting here at my desk by the computer for a computer of hours now writing exams (and not writing exams, when I should have been), and I'm ready to call it quits for the evening. I have to, if I want to get ready for tomorrow and retire at a decent hour, and not make an excuse for not writing an entry for today. Last thing I remember thinking is that it's going to be a loooong day tomorrow.

But is it really?

The day will be the same approximately 24 hours that every other day is.

The daytime hours will be slightly longer than today's not much more appreciable that one would easily notice. For that matter, the next day will be even longer, and the day after that longer still. I might have to contact my local representative to let him know about the global daylight problem which, if nothing is done about, I can extrapolate that we'll have 24 hours of daylight by sometime in December. How would I sleep? I'd need a mask.

It also occurs to me that the daytime hours can be stretched tomorrow more than the following day if I take a sudden trip to, say, northern Canada tomorrow, and then come back after dark. Alaska wouldn't work as well because it would require me to fly across the entire continent, whereas I can fly due North to Canada (assuming that's the route a direct flight takes).

No, tomorrow will just seem longer because I have to give a couple of tests, but I still have to prepare work for a couple of different classes. Loaded day. To make matters worse, I always have to make multiple versions of tests to discourage cheating because when I don't, the temptation is to great to try to put one over on me. It's too late in the year for me to fail someone for this, so best to remove the temptation. Trouble is that only one version is written at the moment, so some tweaking will occur in the morning. Lunch will be spent finishing (and proof-reading) the other test for the other class. What can you do?

So it's just a day where things might weigh a little heavy on me, and there will be times when I busier than usual because the deadline just can't be pushed back any further. However, the workday will be just as long as any other.

Oh, wait a minute. Tomorrow's Thursday? I have after-school tutoring on Thursday until the end of the school year.

It's going to be a looooong day tomorrow.

Day 25 of 30: Ven Diasel in The Fast & The Refreshing

This is Day 25 of the 30-day blogging challenge. Almost done and nearly out of horse-racing metaphors as we pull away in the homestretch!

There's a shop in school where some of the students run things (supervised by staff, of course). I want to be clear that I'm not talking about the school store that sells notebooks, gym clothes and hoodies, and general school spirit items. No, this one has coffee, snacks, bagels, fruit, pizza and cold beverages.

I stopped in during my lunch period for a cup of coffee (half-decaf, it was late in the day and I didn't want to finish off the pot of regular, nor ask them to make another pot, which would've been mostly wasted). As I'm prepping my coffee, another staff member comes in and tasks the women by the counter with this challenge: I need something cold to drink that's sugar-free, no caffeine and not carbonated. (And not water.) She had to stop and think about it and look over to the refrigerator.

And I thought to myself: Wait! I can do this! I'm a MATH TEACHER!

Stand back! This looks like a job for Venn Diagrams!

If we make three circles, one for caffeine, one for sugar and one for water, we can place items in these circles and see what doesn't have a home! What is the complement to the union in the Universal Set that is the Refrigerator?!

Now, there is a slight problem in that you can't have a four-circle Venn diagram. However, the fourth set was "Water" and the only element contained in that set would be "Water", so we'll leave that aside and deal with it later. Basically, the top ten choices fall into the Venn diagram as follows:

As you can see, I came to the realization that the most suitable beverage was "Sugar-free Lemonade". However, as I was on the other side of the room by the coffee maker, I had no clue if that was an option. However, given the varieties of flavored iced tea drinks from a company that also has fruity drinks, it seemed like a logical bet. If not, I'm sure that they found a suitable sugar-free fruit-flavored drink in among all the bottles.

What did I find? Another problem to torture my students with!

I wonder if Venn diagrams are part of the Common Core?

People

(Click on the cartoon to see the full image.)
(C)Copyright 2014, C. Burke.

I'm sure Pi sees things many times.

Day 24 of 30: Rubik Had a Cube Long Before the Borg Did

This is Day 24 of the 30-day blogging challenge. The finish line is in sight.

Rubik had a cube long before the Borg did. Oh, sure, the Borg Collective might be an ancient race, or perhaps just an extremely old one, but we didn't sight any of their cube until sometime in the 90s, and that was glancing three or four centuries into the future even with that. Rubik had a cube while I was in high school. According to what I've read today, Rubik's Cube is 40 years old. This is a little surprising, because that puts its invention in the early 70s (1974, obviously ... I know, I'm a math teacher!), but the cube as a phenomenon didn't occur until the early 80s, at least seven years later. I remember. I had them in high school. My brother even set me up in business once at a block party, solving them for a quarter a pop. Actually, I only made fifty cents because two of the cubes were unsolvable. And I had to prove it to them.

Unsolvable cube Number One was completely finished except for two corners which were rotated out of place. They couldn't be rotated back into place. Rotating them involved a minimum of three corners. That's just the math. To move one thing, something else had to be moved. There was no way that those could be moved. Why? "This cube has been taken apart and put back together." I didn't say that they had done it or that it had been done on purpose. In fact, knockoff cubes fell apart fairly easily because they didn't spin well. If you didn't completely twist, say, the front face and then tried to turn the top face, a corner or two might fly off. Should this happen, someone would scoop up the loose piece and snap it back in. The problem is that you had a two of of three chance of snapping it in incorrectly. At this point, it would almost have been better just to have taken the entire cube apart and resemble it in the original state of six solid faces. But that was a daunting task. Snapping in one or two loose pieces didn't pose a big problem. Reassembling the entire cube could be problematic. I don't know anyone who ever tried it.

It took a while to explain, but they finally accepted my explanation of Unsolvability. In retrospect, I should've charged her the quarter anyway. I did solve it as far as it could be solved.

Unsolvable cube Number Two: another knockoff. It didn't have the "Rubik's" logo and it had the wrong color scheme. On a real cube, white was opposite blue, yellow opposite green, red opposite orange. (I wrote that from memory without checking ... hope I'm right!) Knockoff could have colors anywhere. This one did. To make matters worse, a couple of the center pieces had fallen off. The center of a face didn't really go anywhere. Yes, you could move them around, but they were essentially the plus and minus ends of the x-, y-, and z-axes. They determined what the other eight square around them should be.

Now this didn't pose a problem for me because I was a high school teenager and, therefore, a Super Genius. She had one side completely done, all yellow. However, they weren't in the correct places. You see, it's not enough that all the yellows are on one face, but the corners have to be in the correct positions as well as the edges so that they line up with the center pieces on the adjacent sides. I looked to see which colors yellow was touching so I could figure out which color was on the opposite side from yellow, by process of elimination. The opposite side wouldn't touch yellow. Except that they all did. He had peeled off the stickers and put the yellows on one side. Going around the top ring, yellow touched all five other colors, which is impossible. Thankfully, that didn't take much time or effort to figure out, so I didn't feel like I was cheated out of a quarter (not that he was going to pay me in advance -- he didn't believe I could do it even after I'd done two.)

My method wasn't one of the quick ones, and I never solved a cube in under two minutes. Forget about the 25-second methods. They weren't going to happen. I'd have to relearn. At that point, it was all about color recognition and fast reflexes. That wasn't going to happen. I was happy enough to know how to solve one. (And thanks to Bro. Steve for handing out the instructions in our Number Theory class.

One final note: I like to play a game called Fluxx produced by Looney Labs, founded by Andrew and Kristin Looney (no, really). I discovered online from one of her friends, that Kristin (not yet Looney) was a Regional Rubik's Cube champion on TV's That's Incredible years ago. Here's the link. Enjoy.

Day 23 of 30: 23 Skidoo

This is Day 23 of the 30-day Blogging challenge. Final burst of speed.

This will be a short one. I've been out all day celebrating a graduation. Graduation is not an ending, but a beginning. A time to prepare for what comes next. And what came next was paella at our favorite Spanish restaurant. We don't go there often, so when we do, it's a bit of a treat.

On to the today's topic: 23 skidoo. I could go on about the number 23 and the Law of Fives, but that would be silly because when it come to the best information, You're Not Cleared For That. But to keep in on a math topic, 23 skiddo is a phrase that has at its origin the Flatiron Building in Manhattan, appropriately in the Flatiron District. It an oddly triangular building sitting on a just-as-oddly triangular block formed the intersections of Fifth Avenue and Broadway near 23 Street. Broadway runs diagonally through most of Midtown, and crossing the major avenues creates these triangles, which wind up with fancy names: Herald Square, Times Square, Columbus Circle. There are no squares in Herald Square (but there is Macys). At Times Square, they have tried to refer to the "bow tie" made by the triangular blocks; I don't think this had caught on, and I will refuse to acknowledge it if it does. Columbus Circle does, indeed, have a circle.

As for 23rd Street, it has the Flatiron Building and the famous phrase. Why is it so famous? Well, the story goes, the police with shoo away the loiterers and vagrants who hung around there ("Skidoo!") because the usual design of the building would cause the wind to shift as you passed by. It was a little uplifting, if you catch my meaning, and those guys couldn't get enough ankle. (The skirts were long then. I'd say "unusually" long, but that was usual for then. Wonder what they'd think of today's skirts.)

Okay, back to math tomorrow. And Congratulations to my daughter on her graduation with Honors.

Day 22 of 30: Too Good for Answers; Too Good for Wikipedia

This is Day 22 of the 30-day blogging challenge. Pushing for the last extra

It's Toot Your Own Horn Day at Mr.BurkeMath because if you don't toot it, no one else will. Sitting around waiting to get yourself discovered is a great way to keep a private journal that no one knows anything about. Tooting your own horn gets you some attention, although sometimes it just gets you the horn slapped out of your hands.

Two "f'rinstances": first, years ago, I decided that I needed to get my webcomic listed with anyplace that lists webcomics. I thought that (x, why?) was too special to keep it a secret among a handful of friends and relatives. (Even my students didn't know too much about it.) I got listed in a few of them, but I couldn't crack the biggest one: wikipedia. Note: I'm deliberately neither highlighting the name nor adding a link. You know where it is if you want to go there.)

They have a big, long list of comics. Many older than mine, which makes sense because if not for them, I wouldn't have thought about it in the first place. But look at the list and you'll see that many of the old-timers, the classics have completed their runs. I've been about this for going one seven years now, which is a long time to invest in a hobby using technology that's ever-changing, ever-advancing. There are only a relative handful listed after my debut, but the key thing is that they are after my debut. What the heck?

Granted, some of them are by industry people who moved onto the web. Some by artists who just started something new, but have been around. But some were just by regular guys who just struck a nerve with the right people, I'm guessing. Anyway, if they could get into wikipedia's list, surely, I could. (Yes, I called you "surely".) So I edited the page and added myself. Nope, that didn't work. I was deleted with the reason, "Only comics with their own wiki page could be added to the list" (paraphrasing).

Easy fix. I created a page and imported my wiki page from comicgnesis.com (the good one, not the one with the typo in the address). Not so fast. The page was nominated for rapid deletion (avoiding the usual deletion process with, presumably, takes more time. I'm not famous enough? Maybe I'm just too good to be listed in wikipedia after seven years.

Go ahead. Just try to add me, wikipedia. Someone try adding me to your webcomic list. I'm still too good for you!

The crazy part of all this? I'm actually listing in wikipedia on the GURPS Autoduel page, which was actually deleted years ago and replaced with either a stub or just a mention on the list of GURPS books. The page is actually out of date because its details only the first edition of the book and makes no mention in the article of the second edition. However, the thumbnail is the cover of the Second Edition book, which I was the co-author of. I'm listed as Designer, which is a little overblown, but I'll take it. My name is in red, meaning that I don't have a personal page. I've been tempted to add one ... but I don't want it marked for rapid deletion.

Second, Answers.com was looking for math contributors. At least, it said it was looking for math contributors. They made me jump through a couple of hoops, submit a sample article, sign in and answer a bunch of questions, ask a bunch of questions for others to answer ... That last one was no easy task. Just about any math-related question and even non-math-related questions that I could think of had already been asked in one form or another, so they didn't count.

In the end, they decided to go another way. They didn't just pick different contributors; they just changed what they were looking for. And I wasn't it. I didn't have enough exposure for them. Or maybe, just maybe ... I was just too good for them and would have made the rest of their stuff look weak in comparison. Yeah, that's the ticket.

I decided to show them a think or two. (Yes, I typed "think" on purpose.) I stuck around and answered even more questions, raising my profile, showing off my knowledge and letting them know what they passed up on. And then I realized, I was giving it away for free! That's fine for me to do here, but not there.

So I keep it here. That helps me fill up 30 posts in 30 days after all.

A third site I'd like to mention, and let me state that I have no problem with them whatsoever: TV Tropes. Warning: I will not be held responsible for the hours you lose reading that site. It's a fun page. It was used, abused, avoided, averted, inverted and reverted by the master of comics at Irregular Webcomic, David Morgan-Mar. His comics would reference various tropes whenever he took one and stretched it out beyond recognition, and then one of his regular readers with add a notation to the trope linking his comic to that page! That was dedication, and Mr. Morgan-Mar deserved it for what he inspired. Envy much? Yeah, I have. I would love to be included in a TV Tropes listing, any of them. I know I've used some tropes -- I've been called on it in the past. However, I don't want to become a member just to add my own stuff. That's a little wrong.

And I'm afraid of any rapid deletions.

Day 21 of 30: Picking the Songs

Day 21 of the 30-day blogging challenge. Making a run for it.

One of my students has been following me on twitter and checking out the recent comics. He wondered when I was going to do Frank Sinatra, because I said last Sunday, "Next up, something with Sinatra."

Two things here: first, he misunderstood the tag line joke about what comes next when watching movies with my mother as a "coming soon" ad; second, the young man enjoys Sinatra in a way that few people younger than me, who aren't Michael Buble, do.

And then instead of Sinatra, I ran a One Dimension parody today (which mistakenly loaded last night on the blog). So I started thinking about something: How is it that I choose the songs that I parody?

Not an easy answer.

There are basically two types of songs to choose from: the standards that everyone knows (or at least you hope they know, recognize or are otherwise familiar with), such as Bing Crosby, the Beatles, and Broadway showtunes. And then there are the songs with constant airplay on the radio, generally crossing over into the Top 40 so there's more exposure to the audience, and to me. Yes, I listen mostly to Top 40 than to edgy stations. I also listen to country music sometimes, but unless it crosses over, then there's less of a chance that people will recognize the tune without me spelling it out. If I have to link to a video so that the reader can say, "oh, that song, then it doesn't work.

A couple of times I've picked a song because it was catchy -- and played often -- so I starting putting my own words to the song. It helps if there's a rhyme or a near-rhyme that suggests some topic in Algebra or Geometry that I can work with. It also helps if that topic isn't slope because I seem to hit that one more than others. A couple of times I've picked a song that was overplayed on the radio and I just wanted to do something -- anything! -- with it. See: File It Under Wood, for example.

Sometimes I can only squeeze out one verse and I wait a while for a breakthrough. Other times entire songs pour out of me that I have to trim down. (I had some extra couplets for the 700th comic.) And sometimes the songs go through my head so much that I can't (or don't) remember if I've used them already or not, or if I only meant to use them.

In particular, check out Slope-Intercept Form and This Comic Brought to You By the Letter B.
I could try to make the case that I was expanding my original idea, but I'd forgotten it entirely. And, truth be told, those weren't the first parodies I'd written to Let It Be, but that was a long time ago, and a little political in nature, and I shy away from that here. It's also one of the first songs I ever learned -- I heard it a lot when I was a kid -- and it's a simple tune. It's perfect for inclusion in a comic like this.

Have I used obscure songs? I'm guessing I have. A Canadian friend was unfamiliar with Brandy, which became a recent song about Tangent. And, of course, what might have been a magnum opus, my dedication to the end of summer and to all hard-working teachers returning to school, Star Teachers, to the tune of the 80s anime TV show, Star Blazers. If you listen to its opening, its quite catchy -- there's a video of a high school orchestra performing it (without words) -- and I had it in mind before I started writing. And then my problem was that I had this long, long song and I had to illustrate it! So crafty reuse of material occurred, and I think I introduced (or elevated) a new character or two in that one.

Did I ever get too obscure? Almost. I have a favorite Irish band, Celtic Cross -- Warning: music will start playing if you click that link., but they're a local band, playing around the Northeast U.S. mainly, although the lead singer did perform the National Anthem before a Steelers game two seasons ago. I had a great idea for St. Patrick's Day to make a mathy version of one of the songs and mathy versions of the band and . . . what the hell was I thinking? I would have loved it, and everyone else would've scratched their collective heads, however many that might've been at the time. So I scrapped that idea -- I don't even remember what the idea was anymore.

That's about it. If there's anything else you'd like to know, just comment and ask. Or just tell me your favorite (or least favorite). If you just want to see more of the parodies, you can check out my Music and Song tags -- they almost entirely overlap -- to find all the songs I've used. I may have to do the same, just to refresh my memory. And then maybe check out Sinatra's catalogue of work.

Today's Comic: Scroll Down a Bit

I entered the incorrect date last night when I scheduled today's comic and it published last night by mistake. Click the link or scroll down the main page to see it. I was going to delete it and do it again, but it already had a bunch of views, so decisions, decisions . . .

Oddly, the timestamp is 1 am even though I actually published it last night, probably around 9 pm.

Curious.

January 2014 Algebra Regents, Part IV

Day 20 of the 30-day blogging challenge. In the homestretch!

Finishing the January 2014 Integrated Algebra Regents, which started with the multiple-choice, the rest of the multiple choice, and Parts II and III.

This is Part IV, the last three questions. (Two and a half, really, as you'll see.)

37. On the set of axes below, solve the following system of equations graphically for all values of x and y. State the coordinates of all the solutions. y = x² + 4x - 5, y = 2x + 3

As I've stated in the past, the graph is the easiest four points you'll ever get, especially because you are given a graphing calculator. Type in the equations, hit Graph, check the Table and you're good.

Points get lost when people don't label the lines and state the co-ordinates of the points of intersection. Note: Even if you graph a line incorrectly, you still need to state an "appropriate" solution, even if you have to estimate a fraction.

Graph the lines, you'll get the correct answer.

38. Solve algebraically for all values of x:

To solve this proportion, cross multiply. Use the Distributive Property. Here's a hint: if they saw to find all solutions to an equation, there's a good chance that this will be a quadratic equation with two solutions. You need to find both of them.

After distributing, get all the terms on one side of the equation, leaving only a zero on the other side. Factor the polynomial into two binomials. Using the Zero Product Property, we know that one of the two terms must equal zero, so we can solve for x, as illustrated below:

39. Doug has four baseball caps: one tan, one blue, one red, and one green. He also has three jackets: one blue, one red, and one white. Draw a tree diagram or list a sample space to show all possible outfits consisting of one baseball cap and one jacket.

Sample space: tan/blue, tan/red, tan/white, blue/blue, blue/red, blue/white, red/blue, red/red, red/white, green/blue, green/red, green/white.

A tree diagram would have four branches with the cap colors and each of those branches would have three branches repeating the jacket colors.

Find the number of Doug’s outfits that consist of a cap and a jacket that are different colors.

There are 12 outfits, only two are the same color (blue/blue, red/red). 12 - 2 = 10. If you have a tree diagram or sample space, you are not required to show work for this problem. It will be assumed that you counted from the work you did above. If you do show work, it has to be relevant and correct. If you're tree or sample is incorrect, this answer must be consistent.

On Spirit Day, Doug wants to wear either green or white, his school’s colors. Find the number of his outfits from which he can choose.

Again, they will assume that you counted: there are 6 combinations. If you calculated without a tree diagram, remember not to count the green/white outfit twice.

* * *

And that's it. Sorry it took so long to cover this, but for those of you who are currently reviewing for June, I'm hoping that it worked into your schedule.

One Direction, Two Direction: The Slope of My Line

(Click on the cartoon to see the full image.)
(C)Copyright 2014, C. Burke.

... Red Direction, Blue Direction . . .
Rising up the chart.

And you know what they say when your slope is undefined, right?

I was working on another verse, but decided to just cut it here.
Suggestions? Please them in the comments.

EDIT: I entered the wrong date, and this published last night with a timestamp of 1am. Go figure.

Pausing for Station Identification?

This is Day 19 of a 30-day challenge, and I'm not giving up this easily.

Tonight may be the first night that I get to bed at a decent time, so I'm not spending a lot of time on this entry. And when I went to Twitter, trolling for ideas for a quickie topic, I discovered that the #WeirdEd chat tonight was going to be about Pixar! Seriously! Can I miss that? How long can I participate?

And can I tie it into math, other than counting the sequels: 1, 2, 3 . . .

There's plenty of education opportunities there, but finding the numbers isn't quite as easy. Unless I count back the years to the time when I had many of those toys from Toy Story, but, uh, no.

First education question was this: "In NEMO Marlin has to learn to let go and trust Nemo. How do you learn to trust your students to do right on their own?"

How do I? Other than to put the work in front of them and stop hovering. Oddly, they need to trust themselves more, take that first step and discover that they don't need to call me over so much. I try to get them to work in groups and rely on each other. They can relate to each other better than I can sometimes. This is true even if their initial conversations are as far apart as Buzz and Woody's were. (see what I did there?)

Okay, that's all I have until tomorrow. I could talk more about the stock market, and how the lessons went.

Or about the AP visiting my Common Core Algebra class for about 10-15 minutes today, and my students informing me that she took "like six pages of notes!". Or the fact that when they asked why she was there, they seemed to believe me when I said, "You have an important test coming up next month, and she wants to know that you're ready."

The best part about today was that they were thrown off just enough by the A.P.'s presence that they settled down a little, just a little unnerved. I was, too, but I ran with it, and asked 99 questions, at least, and just kept asking. Using names and getting volunteers, and holding my clipboard. In hindsight, I wish I'd checked off random boxes on my clipboard just for the show.

Okay, I think I have enough for today. If this page looks like this tomorrow, that will be cheating.

Good night.

Wonder Twin Powers Activate! -- er, I mean: DiagnosticOn!

This is Day 18 of the 30-day blogging challenge, for anyone still keeping track. Like me. Or my students who want to see if I make it.

Once upon a time, I used to teach a subject called Math A. It was something that the state decided should be taught to freshmen and sophomores in high school instead of the inferior (I'm just inferring from the switch) Sequential I. Sequential I, which would be followed by II and III, just as Math A would be followed by Math B, was something that came long after I got out of high school -- I had Math 9, in eighth grade, for that matter -- and disappeared by the time I became a teacher. Math A eventually gave way to the far superior Integrated Algebra, which I say lovingly, and yet mockingly, because it's all essentially the same math! Little tweaks here and there (e.g., relative error instead of percent of error) and switching the order of the topics and the maker of the textbooks, but essentially the same.

Through it all, when it came to scatter plots, we showed them how to make a line of best fit. All we ever seemed to care about was if the line that they drew was "reasonable". There wasn't a "right" answer which we were expecting because that was something that was a little beyond them. Sometimes, we had them come up with the equation of the line of best fit, but this confused many because they had different lines, so they had different answers. Still, it was a good exercise because it reminded them how to write the equation of a line from a pair of points. It also got them used to dealing with fractions, ant that is a whole different topic altogether! ("That is a whole different topic!" they repeated in unison)

Along comes Common Core and someone realized that these kids are already using graphing calculators for their tests (which weren't allowed just a decade ago, but are now required), let's teach them how to use them. I'm all for that, except that the students never seem to have them. We can't require them to buy them (except, perhaps in specialized schools) and class sets have a habit of breaking and disappearing. When this happens, not everyone is learning how to use them and then no one can help them on test days. Now, this isn't a problem for most things -- after all, a lot of it is intuitive, and a lot of it is repeated often throughout the term, like graphing.

But then there's linear regression. That's one of the last things we teach. It doesn't get used again. I hope it's stuck in their brains instead of them stuck in the mud and left behind.

Today was the first time that I had to teach it to freshmen. I'm familiar with the process of using Lists (behind the STAT key) and calculating y=ax+b (with the use of a instead of m). That's all good. The forty-six bazillion decimals, instead of a more familiar 7/5, for example, get a discussion on proper rounding. But today was the first time that I ever had to teach about the correlation coefficient, which before now, I've really only seen in an Excel spreadsheet -- in my final grad school class. (Okay, I probably saw it sometime during a Statistics class in college, but I know that I didn't own a graphing calculator in those days. I was lucky to have a Commodore 64 with a modem at home.

Nothing on the CALC menu gave me any hints as to where I could find it, but a quick look at the Internet told me that I had the power to find it all the time. It was there in front of me. I should have found it with my brain or felt it with my heart, but I had the courage to look it up online instead.

The answer I needed will NOT show up unless you go to the CATALOG, scroll down to the middle of the letter "D" and find DiagnosticOff and DiagnosticOn. There aren't any check marks, but the thing defaults to Off. I had to instruct my students, many of whom knew little about the CATALOG in general, how to turn this on and why. They wondered why it wasn't on my default. (Okay, they didn't actually say "default", but that's what they meant!) I can understand why it defaults to radians instead of degrees, but this makes no sense to me.

The take-away from all of this is when we collect all the calculators and reset their memory, we not only have to reset each and every single calculator to Degree mode, but we'll have to go to the Catalog to turn DiagnosticOn. Twice the pain for little gain because we don't even know if they'll answer the questions, or if they'll even remember today's rather abstract lesson.

Bonus Post: (x, why?) Comic Spotted in the Wild

Maybe it's not the "wild", but it is online. And I didn't really "spot", she gave me the link (after asking permission a while ago).

Stacey Roshan is a teacher who does "Flipclass" videos online. The opening/exit comic in this lesson looks awfully familiar. And the lesson is well-presented, too.

This lesson is on rational exponents (aka "fraction as exponents") and I won't spoil which comic was used. Check it out for yourself.

Attention, all educators: If you would like to use one of my comics in a lesson or video or anything school-related that is not "for-profit", please contact me. The answer will most likely be Yes (unless you're extremely rude or planning something extremely bizarre). In fact, you may proceed as if the answer were "yes", but please contact me anyway. I'd like to know about it. Just don't alter the comics, please.

If my comic will be archived on a website, please have a link back here to the blog, or to the (x, why?) comics page.

Day 17 of 30: Taking Stock

This is Day 17 of a 30-day blogging challenge.

More than halfway through the thirty-day challenge, and most of the way through the semester. In fact, the third marking period started a week ago and final exams are less than a month away. The students only have a couple of full five-day weeks remaining, and one of those is an exam week, so there won't be much learning going on then, just demonstrations of knowledge acquired. So it's time for a project.

There's always time for a project. I need something to boost grades, particularly in my Financial Algebra classes where some of the students seemed determined to do just enough to squeeze by ... and then miss the mark by a few tenths of a point, where "few" is defined as "thirty to eighty, possibly one hundred twenty".

I took over this class in late October, when they were in the middle of Chapter 3. It hadn't occurred to me at the time that they hadn't covered Chapters 1 and 2. (Actually, they had started with Chapter 6, which I recently discovered when I tried to cover it and it seemed oddly familiar to them, like a dream, or a dream within a dream...)

I could understand not wanting to start right out of the gate with the Stock Market, and its somewhat complex ideas. Going for the chapter with checking accounts seems a better place to go, or possibly the one dealing with getting a job so that you have something to put into that checking account. After that, you can think about writing checks to your stock broker. Oddly, I just finished talking about retirement and pensions and social security. We've mentioned investing, but having gone there yet. Now is the time -- because there really isn't anything else and we have about 20 days of classes to kill.

Soooooo... stock market project. Not a game. No buying low, selling high, trading between friends and making a million dollars. Just one project, tracking one stock. Something that won't drop a lot of bookkeeping on me. Just checking their work.

The big question will be if I decide to pre-approve their stock choices, just so no one else picks the same stock ... and hands in the same exact paper. Last semester, students did a project about financing a used car. They all picked different cars and had cover sheets showing their choices, and yet every car seemed to cost exactly the same amount of money. Interesting.

I could write the stock symbols on index cards and hand them out. They could trade if they wanted to, but they couldn't both have the same card. However, that would actually take some of the fun out of it: being forced to follow a stock I gave them instead of finding one of their own in a sector that like: technology, fashion, entertainment, shopping, food/beverage, ... or boring stuff like big oil or steel production. I could tell them to try their initials as a stock symbol to see if it's taken. (There's no CJB, but CB is "The Chubb Corporation".)

I should add those descriptions, give them a little more incentive. Or and I have to forbid penny stocks. Nothing under $5.00. I don't need the wild swings and students "gaming" the system, which could be easily gamed. Why not? It's not like there's real money involved with the SEC looking over their shoulders.

Well, unless someone misunderstands the instructions to "invest" $1,000 and uses real money. In which case, if they make a lot of money, I want my cut.

Day 16 of 30: Thinking of Mom

This is the 16th day of a 30-day blogging challenge. My comics count toward the daily blogging requirement, but I'm trying to write every day anyway.

The day is almost over, but Happy Mothers Day. At least, I hope it was happy. I spent it with my in-laws and my little nephew (and two fully-grown nephews, as well as my kids). I just want to talk about Moms: no math, no probability, no graphing, nothing.

In total, there were five Moms there. They had a good day. The cooking had been divided up among all the men, for a change. (My brother-in-law is now in charge of pulled pork forever. Good thing he enjoyed making it!)

For my regular readers, you know that it hasn't been the greatest of years for me and my family (and my extended family). I've lost six people since last August. Of those, five were Moms. For once, I was happy for social media because I followed how everyone else was doing. Some seemed to be doing better than me, and they had it rougher than I do.

I lost my Mom, my niece, my aunt and two cousins. Follow the family tree and you'll realize that I have a sibling who lost her mother and her daughter. One cousin passed away not long after her daughter became a first-time mother (and that little boy is credited with my cousin's survival until then), and my other cousin's daughter lost both her mother and grandmother. There are sad feelings all around.

But looking online, I see the notes to their mothers. I see the old pictures and fairly recent ones. There are poems and sayings and feelings. They're sad, but they're not miserable. They aren't in a funk, so why should I be. She's not forgotten (as today's comic will attest to).

I read a lot of remembrances online. Many of my "friends", both real-life and online-only, have lost parents, some when they were teens or younger. They're reminded every year. But they're happy memories, even if they aren't happy.

This "rough patch" the family went through is over, God willing. And we'll move on. Mom would've wanted us to.

And she also would've wanted us to post a picture of her when she was young and vibrant and dancing. But don't mention the year unless you want to be haunted.

Mothers Day 2014

(Click on the cartoon to see the full image.)
(C)Copyright 2014, C. Burke.

Next up: something with Frank Sinatra. Doesn't really matter what.

Oh, and Cool Runnings. That was Mom's feel-good movie whenever she was down.

January 2014 Algebra Regents, Part II and III

Note: Part I of the test was covered in two earlier posts: 1 - 10 and 11-30.
You can also check out the Regents tag.

Part II

31. Express (84)^.5 / (2 * (3) ^.5) in simplest radical form.
In other words: Express the square root of 84 divided by the product of 2 times the square root of 3, in simplest radical form.

Divide 84 by 3. Do NOT divide 84 by 2. There are Rules for Radicals, because square roots are numbers with the same exponent (i.e., one-half).
84 / 3 = 28, so this becomes the (square root of 28) / 2. Radical 28 can be factored into radical 4 times radical 7. However, the square root of 4 is 2. So you have 2 times the square root of 7 over 2. That simplifies into the square root of 7. Here's a picture:

32. The cumulative frequency table below shows the number of minutes 31 students spent text messaging on a weekend.

Determine which 10-minute interval contains the median. Justify your choice.

There are 31 pieces of data (in this case, students). The median is the amount of time spent by the 16th student because he or she is in the middle. (Think if they were lined up in order.) 41-70 is less than 15, but 41-80 is over 15, so the 15th student is in the 71-80 group. Note: If you said 41-80, you would have lost a point because it specifically says that it must be in the ten-point interval.

33. Kirsten invested $1000 in an account at an annual interest rate of 3%. She made no deposits or withdrawals on the account for 5 years. The interest was compounded annually. Find the balance in the account, to the nearest cent, at the end of 5 years.

Use the Compound interest formula. If you don't remember the compound interest formula, use I=PRT to figure out the interest for the first year. Add the interest to the $1000 principal, and use the formula on the new value for the second year. Repeat until you have all five years. This is acceptable. It is a lot of work for two points, but it is acceptable.

The actual formula you should be using is B = P(1 + (r/n))^nt, where n is the number of times per year that you are compounding interest. Since it's only one time per year (annual rate), n = 1 and it falls out of the formula.
Substitute and you get B = 1000(1 + .03)⁵ = 1159.274, which rounds down to $1,159.27.

If you think you might be wrong with the formula, use the first method.

Part III

34. Graph and label the functions y = |x| and y = |2x| on the set of axes below. Explain how increasing the coefficient of x affects the graph of y = |x|.

The absolute value means that y is equal to the value of x or the value of 2x without the sign (i.e., positive). You will have two V-shaped graphs with their bottoms on the origin (0, 0). (I'm not really sure how much more I can say about this without just posting a graph.)

Explain: Increasing the co-efficient of x will make the graph more narrow.

35. Terry estimated the length of the edge of a cube to be 5 cm. The actual length of the side is 5.2 cm. Find the relative error of the surface area of the cube, to the nearest thousandth.

Relative error -- one of my least favorite topics because a) it doesn't lead into another topic in any meaningful way, and b) if you express the answer as a percent, which you were required to do just a few years ago, you will lose a point. Seriously.

The Surface Area of a Cube is six times the area of one face, which equals six times an edge squared: 6s².
Relative error is the difference in the two surface areas, divided by the actual surface area. So ((6)*(5.2)^2-(6)*(5)^2)/((6)*(5.2)^2)=0.075443787, which is 0.075 to the nearest thousandth. Again, rounding errors will cost you a point. Don't make them.

36. From the top of an apartment building, the angle of depression to a car parked on the street below is 38 degrees, as shown in the diagram below. The car is parked 80 feet from the base of the building. Find the height of the building, to the nearest tenth of a foot.

The angle of depression is the same as the angle of the triangle where the car is. The wall is opposite the car. The ground is adjacent to the car. This a trigonometry question involving the tangent. That is, Tan 38 degrees = x / 80.
So x = 80 * Tan 38 degrees, which is 80 * 0.781285627 = 62.50285012, if I've calculated correctly, which is 62.5 feet.

to be continued . . .

Day 14: Does Primetime Need Writers? My Students Are Hatching Plots!

For anyone keeping track, this is Day 14 of a 30-Day Challenge. Thirty days of updates. Counting today's comic, this makes two for today.

Yesterday's "Life Insurance" lesson in Financial Algebra class didn't go -- to use the vernacular -- too, too badly. There were diminishing returns as the day wore on. Basically, period 3 with the co-worker worked better than period 4 solo, when I didn't have another adult to bounce thoughts back and forth. I did my best in period 4 to cover the most crucial topics. But period 7, late in the day, wasn't the time to bore the students with such a topic, especially when it became evident that whatever worksheet I might have had wouldn't be given out. (And it wasn't.)

But the kids did have some interesting questions, many of which I expected, although perhaps not in the way they were presented. There were questions about policies and suicides, which were quickly replaced by hiring someone to kill you or beneficiaries killing the policyholder. Not that they had dollar signs in their eyes -- as I informed them, first they'd have to find someone with a sizable life insurance policy to name them as beneficiary. Then they'd have to make it look like an accident.

Well, I didn't have to tell them that part. Most of them were already planning ways to do just that. And, of course! they'd get away with it! They're teenagers, after all! They know how to do everything! (Except, perhaps, some of the problem in Chapter 8 ... and Chapter 6 ... but I digress.) I think the big unanswered question was, "How would they know?"

They weren't satisfied with, "Trust me, they'd know. They have investigators who seen this sort of thing before." When that didn't convince them, I added, "And you haven't even done this once." (That I know of...) "It's not like you'll get it right the first time. It'll probably take four or five tries to get it to really look like an accident. The problem is dealing with the fallout from the first three or four tries."

I think that convinced them to give up their plans for fraud and murder. One can only hope. My final thoughts on the topic (which I actually repeated when the topic came up the next period and the one after that), check out an episode of Law & Order or Bones or any other criminal show. They just know.

That being said, I think my students could hatch a plot worthy of prime-time, even if it unravels, as all such criminal plots tend to do when the investigators take the case.

But one show they can't write for: when I showed the final slide of my presentation with the Mortality Tables, I introduced it with "Valar Morghulis". Not a comment. Not even from the geekier kids. I had to fall back on another punch line: talking about using life insurance to pay for funeral expenses because they are quite costly, a student asked why cemeteries are so expensive. I couldn't stop myself from telling her: "Cemeteries are crowded and they're very popular.... People are dying to get in."

Oh, yeah. I did.

Problem and Solution

(Click on the cartoon to see the full image.)
(C)Copyright 2014, C. Burke.

An open circle is like a boundary. Kind of like a fence sitter. Those fence sitters are always trouble, you know.

Day 13: What If They Gave Tutoring And Nobody Came?

What if they gave tutoring, and nobody came? I almost found out today.

It's been an interesting few weeks. I signed up for Tuesday and Thursday afternoon Regents prep, mainly because I could use the hours. Don't get me wrong: I'm happy to help a handful of kids in the morning, during period 1 (which was "period 0" in my old school). There's just something about having a large group of students to work with. Too often, it just feels like I'm teaching another class, and like students in those classes, too many of them just wait for the knowledge to appear instead of pursuing it, running down whatever avenue they have to until they find it. But I gave it a shot. The AP sounded desperate for bodies inthe classroom, and I thought it might be helpful for some "artifact" or "rubric" or "agenda item" or whatever those 22 points or 39 steps we're being rated on actually are.

Then I found out that she had so many people, that I'm only doing it on Thursday. That actually seemed like a waste of my time -- I'll be getting half the hours that I thought I would.

Then -- we're talking next morning -- one of my students approached me and asked if I'd be interested in moderating (or assisting in moderating) the new Anime Club. Of course, I would! Unfortunately, it meets on Thursdays. I've wanted to poke my head in over there, knowing a little bit about Anime (although very little about Manga). I even had the ulterior motive of trying to lure a group up to Lunacon next year. It worked for old Dom and his kids from the Bronx High School of Science so many years ago. (Many of that group still turns out year after year. Sadly, we lost Dom Corrado in the past year.) But that's a topic for another column.

Last week was actually the first Thursday of tutoring. I had six kids. Not bad, except four of them were mine, and they were more interested in reviewing for the test we were having the following day than for the Regents exam next month. (They do want to review for that. They just didn't at that moment.) I had two others retaking the Integrated Algebra Regents. We reviewed an old exam, and I think some learning occurred, but it's always hard to tell with those exams. Are they checking their answers or just filling them in? Are they copying the work or just circling a letter. I wasn't collecting or grading the sheet, so writing down the correct choice isn't a helpful strategy, but it does get employed often.

I have no test tomorrow, so I had none of my own students today. And I lost half of the other students. Were you paying attention? How many did I have left?

That's right: one.

I had printed out ten copies of the open-ended questions from the same test. I gave him about ten minutes to work on the problems to see where he had trouble and to allow time for other students to appear. I knew they wouldn't, but I hoped.

And then he leveled with me, that he wasn't sure how to do any of the problems. Not a good sign. I wound up "teaching" each of the questions to some extent or other. There were a couple of "a-ha!" moments where he couldn't believe the question was as easy as it turned out to be (a good reason to get started on each question and try something). In others, he went through the paces and got the answer, but it seemed like an experiment he didn't think he could duplicate.

With only one student to ask questions, we finished a little early. It wasn't my room, and I didn't have any extra materials with me.

We'll see how much interest there is next week. Will he bring a friend or two? Will he skip out, too?

What will I do if no one shows up?

Probably punch out and go watch Anime.

Day 12: I'm the Sugarless Gum of Math Teachers

I'm the sugarless gum of math teachers. Four out of five classes have high scholarship. This is a new experience for me. I've had good marking periods before, but never this good. Did I suddenly become a highly-effective facilitator of learning? Or did I just happen to take over a bunch of classes at a school where more kids seemed determined to go to college (or at least get that last math credit)?

Smart money (and empirical data) would bank on the latter.

Granted, I was a little surprised when I calculated the grades. I know that there were some poor showings here and there throughout the marking period (roughly six weeks and some vacation times squeezed in), but most of the people having difficulties only had it in one spot or the other. It wasn't continuous. And the homework and assignments were being handed in (even if there wasn't enough "originality" to that pile on the corner of the desk). It seemed worse that it was mostly because of the noise level in the classroom. Now, I've had APs who like noisy classrooms. If they aren't noisy, then discussions aren't going on. No discussions means learning isn't taking place. However, the inverse of that is most certainly Not logically equivalent. By this I mean that Discussions DO NOT mean learning IS taking place.

Sometimes it's just noise. However, that is on me. That's a classroom management issue, and I have to adjust my approach any way I can. The main way to do this is to establish rules on Day One. However, my Day One and theirs were NOT the same day. I started in the end of October, and I was their third teacher. I was in a new school and I tried to ease my way into my "temporary" role (I wasn't supposed to still be there at this point). That was a mistake, but it's one I'm dealing with. They get a participation grade, and for some it's lower than others, but that's only a small part of the grade. The testing is the largest part, and they are, for the most part, doing better. For this, I'm thankful. And they aren't just squeezing by, they're getting high numbers.

A little true time: the classes with the highest grades are "Financial Algebra", and anything involving logarithms have been dropped. It's math that they can use (although I'm not looking forward to tomorrow's lesson on Life Insurance, which will be even less exciting than Pensions has been), but it's still a lot of compound interest formulas and other computations. And it's planning and thinking about the future. Good stuff. The best part: I don't have to be crazed about the amount of material in each assignment or on the tests.

My Common Core Algebra class is an entirely different situation. I have to push and push and push, and still I'm behind. However, I'm making sure that they know the material I cover. I just keep reminding them that knowing 90% of 70% of the material is 63%, which isn't a good result. So while I'm happy to review to answer everyone's questions, we still have to move along. Today, they were asking me when we were going to start reviewing for the exam, which is less than a month away. I told them the truth: when I finish covering the material.

Sadly, this brings me to the one class that's totally disengaged. I've called parents. I've written anecdotals. I've failed students. "Whatever" is the response. "You're just going to fail me anyway, so why bother?", which is only slightly more annoying than, "When am I ever going to use this?"

They might be lower performing students, but they can do the work when challenged to do so (or mildly coerced by a parent). I've been told that if you raise the bar, the students will rise up to meet that challenge. Not all of them. Some will look at that bar and walk away. Some will want to do a limbo contest no matter how much you lower that bar. There are carrots to dangle in front of them: no summer school, seniors get to go home earlier, hey wasn't there a college you were interested in?

Still working on it for next year. Highly effective? Pipe dream. I hate to say that I'm wondering what next year will bring.

In the meantime, I'm still thrilled by four out of five, and how high they were.

And I'm starting to understand that fifth dentist a little more.

Day 11: I'm Still Here... I'm Still Blogging ...

It's been a brutal couple of days, getting everything out of the way, cleaning off the desk, and getting grades ready for submission. I usually have my own calculations, done through a spreadsheet, which can look at the best scores that students have handed in. Instead, this time around, I'm relying more on the automated online process. Partly because parents and students at this school check them out more frequently. Partly because they will challenge me if I differ by even a point from the way the computer grades it. Yes, some of them, a select few, are absolutely nuts while simultaneously being in own their own, or their little precious's, failings. I massaged the numbers online, putting in zeroes where they were needed, and slightly higher numbers when I didn't want to crush or cripple a decent grade.

But it's not an automated system. They grade shown is not the one that gets put into the system unless I put in there. In general, I put in a figure that matched or exceeded what the computer showed. The trick is to make the computer show what I want it to show (or less than I want it to show). Then I come off looking pretty good. Score one for Number Sense.

Back tomorrow. Hopefully, with the new marking period under way, there will be time now. Time enough at last.

Day 10: It's 5 - 5 (Revenge of the Fifth?)

The date is 5/5, and you know what that means! That's right! It's Day 10 of my 30-day writing challenge.

Oh, and it's something called cinco de mayo, literally, the Fifth of May, the day after May the Fourth.

I know it's late and the day is almost over. On the other hand, for those who are celebrating, it's probably only beginning!

With this limited space and shortness of time, allow me to briefly split an infini-- er, I mean, to briefly explain the story behind the story of this old cinco de mayo classic from 2008.

First of all, let me state for the record that this comic is NOT based on a true story.

Oh, no. It most certainly IS the True Story.

It was early in 2008, not May, when I taught a double-period class of Integrated Algebra students. Actually, given that it was 2008, it was still called Math A back then, which combined Algebra and Geometry in a two-year cycle. Given that it was a double-period class, I could take my time with the explanations and allow plenty of time for student activity. The strange thing about this class was that it was a team-teaching class. The stranger thing was that the other teacher was not a Special Ed teacher, but someone else from the general ed Math department. The strangest part was that I didn't actually have one co-teaching, but, in fact, had two co-teachers: one in each of the two periods.

This last thing becomes less strange when you realize that one of the two was the Assistant Principal who was obligated to teach one claas. So he picked half a class. And then most of the time he was a no-show because of "meetings" and "observations" and "bears, oh my!". If you knew him, this wasn't strange at all. Uh-uh. At times, he could have been the A.P. from the comic.

The other co-teacher, Mr. Prato, loved the class, and it was a shame that he had to leave halfway through it. He could relate to the students and work with the individuals while I worked with the entire class. (There are at least six ways to co-teach, and this one generally worked for us, although on some days/topics, we'd switch roles and he'd take the lead.)

Anyway, this one day, Mr. Prato came in, apologized to me about being swamped with work and asked if I minded if he sat in the back and caught up with paperwork. Well, everyone needs a break, and it wasn't like he was ditching (like some A.P.s I could name!). So, sure, go ahead.

Back to the class. I went a long introduction to parabolas without much technology to speak of in the room. We plotted these things out. I showed them the standard form for the equation and went through what happens to the parabola when a is positive, what happens to the parabola when a is negative, what happens to the parabola when a is a big number, what happens to the parabola when a is a fraction, what happens to the parabola when b isn't there, what happens to the parabola when b is negative, what happens to the parabola, parabola, parabola, parabola.

After so many iterations and over a half hour, I asked some question (I can't recall exactly what) of the students about how changing something or other would affect the parabola.

A student sitting near me, two seats from the front, cautiously raised his hand with a slightly puzzled look, and asked, "What's a parabola?"

Seriously, I am not kidding. His head hadn't been down. He hadn't been talking or doodling or been otherwise engaged. He'd been staring at the board -- or at least in the general area of the front board. All I could figure after the fact was that he'd been sleeping with his eyes open. But none of that had occurred to me then.

I responded EXACTLY as I did in the comic. I even flubbed it: I meant to say it was a "drink", not a "dessert", but it's just as well. I wouldn't want to promote alcohol use among minors anyway, right?

Mr. Prato stopped dead and looked up at me. "Mr. Burke, I almost fell out of my chair."

The funny thing was I don't think anyone laughed. The student was puzzled. I was annoyed. And I don't think anyone else knew what to make of what I'd said.

I realized then, yeah, maybe I'd have to work on the sarcasm.

Day 9: A Quick Note About the End of the Marking Period

I have a comic for today, but I still wanted to have a blog update, if only to comment for the 30-day challenge.

Crunch time arrives again as the marking period ended Friday and grades are due. I have a pile of stuff that needs to be graded (I've plowed through the homework and a lot of the classwork) and I spent a good part of the weekend grading tests. I hate tests on Friday and I hate last day of the marking period tests. For one thing, one about the absent students and the make-up tests? When can I grade those? Probably never. Okay, not never. But this stuff takes time. And I didn't have a choice in the matter: if I wanted to have a test, the Math department had to have their tests on Friday this time around. So that's what I did. I don't want to be the teacher who slices through with a red pen and says, WRONG! WRONG! WRONG! -- despite what my students might think. As it is, I hate multiple-choice tests with their all-or-nothing component.

On the other hand, if I spend two minutes per test, that's well over 200 minutes, close to four hours, just going over the exams. And that's not even recording them online.

But the fact is that it takes more than two minutes if I look at their work to find their mistake. It's too minutes in itself just to write the comments down to guide them to get it right the next time. Except that they don't read the comments because they don't believe that there is a next time. After all, we just did that already. It's not like there's another test on the stuff where we'll use it again. Or a final exam.

So a lot of time was spent, but it had to be done.

Oddly, what shouldn't have to be done (but does) is grading the homework and all the little assignments that will count for perhaps one-fifteenth of ten percent. At the valuation, does it make much difference grading it as Satisfactory or Good? Does that extra 10% of 1/15 of 10% really matter much? As long as it isn't a zero, it won't have a big effect on their grade. And the zero doesn't have much of a negative either.

But the system is online and there's no faking it. So back to grading.

And I'll save the blogging -- and the new comics -- for another day.

May the Fourth

(Click on the cartoon to see the full image.)
(C)Copyright 2014, C. Burke.

Princess Threea and Luke Fivewalker, in case you were wondering.

This seems to be a bigger deal every year, so i figured I couldn't ignore it and keep my geek cred.
Frankly, before maybe five years ago, I hadn't really heard the pun. Don't know where it started or when it really caught on.
It was a clue on Jeopardy! a few years ago, so that's where I'd start looking.

Also check out Lucas Numbers, Movie Night and, of course, Hamlet II: Claudius Strikes Back.

Day 8: One for the Roses

This is Day 8, for those keeping track, and if you are keeping track, is that track Churchill Downs? This afternoon will be the 140th running of the Kentucky Derby, the first leg of horse racing's Triple Crown.

The "Sport of Kings", as it once was known, is accessible to all, young and old. And to math teachers. Seriously, there are nineteen horses in the race, numbered 1 to 20 (skipping number 11, which was scratched), and people place bets not only on which horse will come in first, but on which will come in second and third as well!

Sounds like a permutation problem to me!

Disclaimer: Gambling has its risks. There are no sure bets. Winners of horse races are not random numbers of equal probability. Bet with your head, not over it. Four score and seven years ago, yabba dabba doo, m-o-u-s-e.

When you place a bet on a horse to win, you put down your $2.00. Let's say that you want to "Dance With Fate" and you bet on the horse of the same name. (This is not unusual. People bet on horses because of names without any regard to the background and track record of the horse in question. Fools and their money?) As of this writing, Dance With Fate has odds of 14-1. That means that should he win, you'll be paid $28 plus your original $2.00, give or take. (Betting is parimutuel, meaning it depends on all the money bet in the race, and the odds shown are rounded.)

So your $2 bet gets you $30, which is nice, but it will barely buy you dinner, or, for that matter, pay for the snacks and beverages you probably consumed while waiting for the race. (Granted, you would've had to pay for them anyway.) If you want a bigger payout, you have to risk a larger amount, or take a riskier bet.

The next bet is to pick the exacta where you pick the horses which will finish first and second, in that order. You want to have it both ways? Sure, you can: just make two bets! That's allowed. But, first, let's look at this one bet. How many possible exacta bets are there? There are 19 horses which could theoretically win the race (despite what the experts will say about some of them). That means that there are 18 remaining who can place (i.e., come in second). Applying the Counting Principle, we find that there are 19 X 18 = 342 possible outcomes for first and second.

As previously stated, you can make multiple bets, put at $2 per shot, it'll get a little pricey. No one's planning on $2 x 342 = $684 on betting slips (I hope). However, there is one thing you can do to improve your payout: "wheel" your bet. If you are planning on putting money on one particular horse to win, you can play the exacta with each of the other 18 horses. At $2 apiece, you'll bet $36, but the combination of payouts (and I won't even pretend to be able to explain how its calculated) will likely (but not definitely) be higher than if you bet the entire $36 on your horse to win. But, again, if your "winning" horse fails to win, you get nothing.

Trifecta, Anyone?

Finally, there's the trifecta: picking horses to win, place and show (i.e., third place, obviously). How many possible permutations are there? Glad you asked!

It's 19 X 18 X 17 = 5,814 for all of them, or 1 X 18 X 17 = 306, of you're sure that your horse is a winner. Hint: that's not a good bet to make, even if you're playing for buttons or jellybeans with your friends.

Let's narrow it down. Suppose you pick any four of the nineteen horses. How many bets do you have to make to cover every permutation of horses coming in first, second and third?

It's 4! = 4 X 3 X 2 X 1, which the odd horse finishing out of the money. That's $48 for a $2 bet. Remember: I'm not commenting about the viability of the bets; I'm just talking about the math.

Handicapping Today's Derby Race

Okay, so you read the sports pages, saw what the experts had to say, checked all the stats for the past year, and looked into the parentage of each of the participants. (No, you didn't, but let's say you did.)

Which horses do we bet on? That's easy!

Pick your kids' ages or birthdays and your lucky number. Wheel them and make a bet. Enjoy the race and a cool drink as you kiss your money good-bye.

That's what I'd do, but I won't because they closed the OTBs around here.