My students wouldn't ask what a 'segue' is -- they know it's a thing with two wheels and a motor.
It's something I could try in class. And then see who just uses a ruler.
More Algebra 2 problems.
January 2020, Part IV
The Question in Part IV is worth 6 credits. Work must be shown. Partial credit is given.
Determine which drug is made with a greater initial amount of active ingredient. Justify your
answer.
Sarah’s doctor told her to take both drugs at the same time. Determine algebraically the number
of hours after taking the medications when both medications will have the same amount of active
ingredient remaining in her bloodstream.
37. Sarah is fighting a sinus infection. Her doctor prescribed a nasal spray and an antibiotic to fight
the infection. The active ingredients, in milligrams, remaining in the bloodstream from the nasal
spray, n(t), and the antibiotic, a(t), are modeled in the functions below, where t is the time in
hours since the medications were taken.
Answer:
To find the initial amounts of the active ingredient, calculate each function for t = 0.
The nasal spray will have 21/15 milligrams, and the antibiotic will have 3 milligrams, so there will be more active ingredient in the antibiotic. (See image below)
To find the number of hours when the two active ingredients will be at the same level, set the two equations equal to each other.
You can see that the quadratic factors into (t + 5)(t + 3), so the entire equation can be multiplied by both binomials to remove the denominators.
After that, you can solve the resulting quadratic equation, getting t = 8 and t = -3. We discard the negative because negative time makes no sense.
They will have the same amount of active ingredient after 8 hours. (See image below)
Comments and questions welcome.
More Algebra 2 problems.
Complicated integrals always Sting, don't they?
Okay, so I've used Sting jokes before (but not in a Lord of the Rings sense yet) and I don't een did this song before. But I've reused math puns over the years, too. I wouldn't force anyone to read all 1500+ comics now. (If they wanted to, sure, go ahead.)
What I can say about the final answer is that u will be raised to the 4th power, and a will be raised to the 6th. And that's a lot of constants.
To be fair, I was going to make them definite integral, but I didn't have the space. Plus I would have agonized over numbers to pick, and then checking my work, and ...
I call these Minis for a reason, gosh darn it!
Excuse the strong language, I've been rewatching Abbott & Costello movies.
Why don't you just go and find yourself already!
I needed a new character here. Henry is just to pleasant to be that cross.
This was going to be a surprise bonus comic for yesterday, but -- Surprise! -- real life and all that.
She's giving him a bit of her mind. With two bits, he gets a haircut!
Watch out if she starts carrying scissors.
Either way, if the wife says shave, she means you may be sleeping on the couch otherwise.
I used to have a coworker who was so convinced that Ken was based on him that he was shocked when I pointed out that Ken had neither a beard nor mustache.
But give him some time off, and who knows what will happen.
I have been keeping busy in my house, so I didn't get to address the issue of using the Area model (aka, the Box Method) for multiply 2- and 3-digit numbers. In the past couple of weeks, an example floated around social media of a front-line worker who couldn't do the "new math" with their child, and just did it the "old way". This, I understand. The outpouring of ridicule for the new math ... well, I guess I understand that, too, because people haven't learned it, or More Importantly, learned WHY it's being taught this way.
Side note: an issue I did bring up on Twitter is that, when first rolled out, some instructors didn't know why they were doing this either. They just did it because they were told to, and they might not have even done that much well because of resistance. Learning it that way would make the core knowledge behind it difficult to scale up.
If I ever do write an article, I'd likely reference Mr. Incredible from The Incredibles 2, when he ponders "Why did they change math?" The superhero reference is not accidental since I would be addressing an essential worker who is giving their all out there while many of us aren't leaving our homes. But, he did eventually get it, and understand it.
If a student learns rules, they may be able to recite them from memory, and maybe they can apply them through rote. On the other hand, if a student develops the Number Sense and makes the connections, the extra steps in the first exercise can be used over and over through high school into college with more advanced topics. Along the way, they can build layers upon layers on their foundation.
Odds are I won't get around to writing that blog post, because I never seem to. But at least there is this entry, so my plans aren't totally FOILed.
Or crossword puzzles, if that's your thing. Forget about finding ones to suit you're level. You just can't find any.
Mike's wearing another punny math shirt, even if you can't read it. Also, it might be time for a haircut.
My wife has made this cake a few times. Uncooked spaghetti whiskers can replace the icing, just for fun.
Given that mathematical nature of this cake, I'm surprised I hadn't used it before for a comic.
Happy Easter!
One of the panels on Friday started off slowly for a few reasons: first, two of the panelists hadn't shown; second, no moderator had been chosen in advance and it didn't appear that either panelist was particularly interested in leading the discussion; and, third, it didn't seem like they had much to say. Basically, they seemed unprepared. Now I know this con was put together quickly, but people were only put on panels that they had shown interest in a month earlier when preparing for the full-scale, on-site convention.
However, this post isn't meant to gripe about any particular panel. What happened, happened, and it told me one thing. Specifically, I needed to be prepared for my panel.
To this end, on Saturday, I sat down with a notebook and mind dumped whatever came into my head about the topic of Serial Vs. Episodic.
I managed to fill an entire page. As luck would have it, I barely had reason to look at it. A few points were fresh in my mind, and the rest didn't matter because of the directions the hour moved.
On the other hand, I still had some thoughts and opinions on the matter. So I'm going to jot them down here. If it's not a topic of interest to you, you might want to back out now. There is no seconfary topics afterward.
Okay, then.
First and foremost, I miss episodic TV. Everything is going serial, and that's not necessarily a good thing. Don't get me wrong: there are good serials out there, and I even enjoy a few of them. But there are so many on cable. And if you don't know about them early enough, you're likely going to just skip it and wait for it on a streaming service.
Episodic TV will never go away, as long as we have sitcoms, and independent stations have schedules that need filling with syndicated programming. If it's evening and you want to relax, but you don't want to start something, what are you going to do? Scroll through the programming guide until you hit Friends or Frasier or Big Bang Theory or Golden Girls. Maybe your tastes from more toward drama and some variation of Law & Order or C.S.I.. You have a story with a start and a finish, and you can switch it off. (Or watch another if you're up for it!)
With serials, it's all or nothing. You're not going to tune into whichever episode is on tonight -- not even if it's a rerun of an episode you've already seen. Out of context, the show loses something. Generally speaking, for me, most of these shows lose something in the retelling. For me, it's one and done. I rarely rewatch them.
In fact, when some of these 10-12 episode series come back after 9 months off the air, my viewing schedule is probably to busy to binge the previous season, to refresh my memory of the characters and the plotlines. Especially if it's going to remind me that nothing of substance happened for nine episodes, so that everything could come to a head in the final episode -- AND THEN there was a cliffhanger anyway! And the "entire season is one story" went right out the window.
This happens in longer, network series as well, when they hire one villain for the season. You know that they will fail to catch them, week after week, until the finale. Some of the show needs to be about something else.
This isn't to say that a show can't have seasonal themes, multi-episode arcs, and character growth. It's not the 60s anymore, so the reset button doesn't need to be hit at the end of every single episode. But have some part of the story start on one show and finish on the same show. And don't undo it the following week.
I had a note in the margin about Marvel's Agents of SHIELD, which had a good season where they went through three story arcs, even doing a callback to the first (Ghost Rider) in the third. There was some good storytelling that year.
By contrast, you have shows like Marvel's Jessica Jones and The Runaways. I thought Jessica Jones was a great show, but the first season got a little formulaic in the middle episodes. Jessica wants to catch Kilgrave, but she won't kill him. Because of this, he continually manipulates her, putting her into situations where she has to let him go so she can save someone else. (Now, okay, a hero won't let the innocent person die in order to catch the villain, but she seems to put herself into that kind of situation, repeatedly.) This problem was compoundd by the person she saved being killed anyway in a later episode, so nothing she did actually mattered. (Then again, that might've been the point that they were trying to hit home. In which case, message received.)
The Runaways is a show I only recently encountered. I'd heard about it, and knew it was somehow comic-related, which knowing anything about the comic. Had it been on the CW or Fox, running for 20 episodes, the first season would've been better. For one thing, so much of the first season could have (and should have) been dealt with in THREE episodes, not ten. And without trying to spoil things, everything again builds up all-eggs-in-one-basket for the finale ... and it ends on a cliffhanger anyways, with the story unresolved.
(Side note: cf Arrow, season 1.)
One of the "probelms" people sight about 22-24 episode shows is that they run out of material, and there's always *that one* episode, every year. Funny thing is, "THAT" episode will be the one that everyone remembers, either because they love it, or they love to hate it. That said, I'll acknowledge that stretching The Prisoner from six to 17 episodes caused a clunker or two.
I'm rambling now, but I covered most of the points I was ready to make, should they come up.
But here's a final note: I grew up watching cartoons, rerun endlessly, and we had no problem watching them again and again. Speed Racer could be repeated viewing forever. On the other hand, when Star Blazers came out, it was something totally new. And then, the second season came on with the the Comet Empire. And then ... it started showing the first season again, and my friends were like, "we already saw this." Not that the handful of us affect the ratings much, but the show was taken off the air withing weeks of starting reruns. Fans didn't need to watch the entire serial again. But maybe another episode of Gilligan's Island ...
I'm not saying that he already owned that hat. He could've constructed it from paper. But I'm not saying he didn't either.
What can we say about triangle PST?
Look at line ST. It's a tangent line, so it intersects the circle at one point. Call that point U. We don't know the length of SU or TU. We don't know if they are congruent to each other or not. But we do know one thing.
SU and SQ are two tangents to the circle from the same point, and they have the same length. Likewise, TU and TR have the same length. This is enough information to tell us something about the perimeter of the triangle:
PS + SU + UT + TP = PS + SQ + RT + TP = (PS + SQ) + (RT + TP) = PQ + PR = 2 PQ = 2 PR
The perimeter of the triangle is equal to the sum of the lengths of the two larger tangent lines. And because those tangents are equal to each other, we know that the perimeter equals twice the length of one of the tangents.
Also, the triangle could like a clown hat.
Long-time readers and followers may know that I enjoy going to science fiction conventions, and they may also know that I tend to go to one of them per year as my weekend away. I'm not talking about Comic-Con, which is a bit "too much" for me -- all hype and promotion. No, I mean a smaller con, located a little (but not much) farther away, where I spend the weekend.
In the past, that was Lunacon, which has fallen into the annals (or "annuals"?) of sci-fi history. Now, there is it's pseudo-replacement, HeliosphereNY, held in the spring in Tarrytown, NY (a stone's throw from Sleepy Hollow). This past weekend was supposed to be my weekend away, but, like everything else in the country right now, it was cancelled.
However, you can't keep sci-fi fans down for long. They were like, Let's Go! It was On with the Con!
HeliosphereNY held a virtual convention online. I was impressed that it came together so quickly, and that it ran as smoothly as it did considering they didn't even know if they would have panelists.
A couple of things led to its success: first, much of the programming work had already been done. It was just a matter of finding panelists who were available and scheduling some of talks which they had expressed interest in.
Second, and this is the biggie, I have to send kudos to the Filk community. I'm not a filker, and I don't spend much time in the filk room, listening to their concerts or open singing. But they are a very organized group, a close-knit one, a society unto their own. And they had an entire slate of performers ready to fill one track of programming. I don't think there was a point during the weekend where there wasn't some concert or other going on, even when there were no discussion panels.
Side note, here: A number of years ago, at one of the last Lunacons, the filkers wanted to invite Leslie Fish, a well-known filker on the West Coast, to the con. However, they needed to raise money to bring her to NY. So they started an indiegogo campaign. I donated to that fundraiser, without asking for any reward, for the simple reason that I wanted it to succeed. Not because I wanted to hear the concert, but because if it did succeed (which it did), that meant it could be used as a viable route for other guests (or events) in the future. (Which didn't happen, alas.)
Back to the weekend: Friday night was a little touch and go. The participants had to accustom themselves to using zoom, and the moderators and administrators had to deal with "zoom boomers" who came in to disrupt. The convention was free, and word was put out online in places where people ho might attend generally congregate. This also brought out the wackos. Despite the waiting room feature, people slipped in and had their little fun. As a results, for many of the panels, the microphones had to be turned off for all participates but the panelists. There was a chat window for questions. Thankfully, I didn't encounter anything rude or lewd on video.
For me, Friday was a mixed bag. One panel had no moderator and it seemed to lack direction, like the panelists weren't sure what they would talk about even after it started. The Horror panel, which featured readings, was going well, but was getting "bombed". Unfortunately, I was a casualty of that, as the incorrect person who removed from the room. (I got sincere apologies from the admin and the panelist after that one. No hard feelings.)
The "big" panel for me on Saturday was "I Wrote a Thing, Now What?", with advice from writers/editors on what to do, and what not to do. And a reminder that I need to write another story for another shot at another anthology, even though it may not get funded. Sadly, given this downturn, people are watching more TV, but reading less. Book sales are down. Even audio books, but they mostly sold to people driving to work. Notable about this (for me) was that Alex Shvartsman and Ian Randal Strock were both panelists. I was on two panels at HeliosphereNY 2019, and I sat next to both of them, one at each panel.
I was on two panels this year as well, both on Sunday. The first was Episodic or Serial, and I seemed to be the lone voice of dissent. It was the opinion of the panel that serial is the way TV is going now. Actually, I didn't disagree with that, but I wasn't enthused that this is where we're heading. I miss episodic TV, and many people at home in the evening, will flip on a half-hour sitcom and just enjoy. I think the other panelists agreed with me here. I'll have more to say about this panel in Part II of this report.
The second panel was "Privacy in the Information Age". I volunteered for this one because I thought it an interesting topic. I used to work programming computers, and now I use them in education. I signed up before the teaching world was turned upside down. This panel came along at the right time. Speaking of time, it was probably the best-attended Sunday 3pm panel that I've ever been to. Shoutout to Joseph R. Kennedy for stepping up and leading the discussion with lots of information and stories at his disposal. Most of my contributions were anecdotal, but the three panelists had a good hour.
Not much to gripe about during the Gripe session, and I even stopped in for a while during the Dead Dog filk.
Overwhelming approval by the organizers, presenters, and participants. We thought it great, and would love to do it again. However ... we'd all rather just be there in person next year. So here's hoping for 2021!
More Algebra 2 problems.
January 2020, Part III
All Questions in Part III are worth 4 credits. Work must be shown. Partial credit is given.
35. Algebraically solve the following system of equations.
x + y - 1 = 0
Answer:
The first equation is for a circle, and the second is a line. There will be either 1 solution, if it's tangent, or two if it's a secant line. (Zero solutions is unlikely, but not out the question with Regents exams. However, they'd usually be some indication if it was a possibility.)
Solve the linear equation for y:
Substitute into the first equation:
If x = -2, -2 + y - 1 = 0, and y = 3
If x = 2, 2 + y - 1 = 0, and y = -1
Solutions: (-2, 3) and (2, -1)
Write an exponential regression equation that models these data rounding all values to the nearest
thousandth.
36. The table below gives air pressures in kPa at selected altitudes above sea level measured in
kilometers.
x Altitude (km) 0 1 2 3 4 5
y Air Pressure (kPa) (km) 101 90 79 70 62 54
Use this equation to algebraically determine the altitude, to the nearest hundredth of a kilometer,
when the air pressure is 29 kPa.
Answer:
First of all "kPa" stands for "kilopascals". It's a unit of pressure, and nothing that you need to concern yourself about, or freak out about if you've never seen it before.
Put the data into the list on the calculator, and run an exponential regression.
y = a(b)x
a = 101.5228...
b = .8826...
y = 101.523(.883)x
Use the formula to solve when y = 29.
29 = 101.523(.883)x
29/101.523 = (.883)x
0.28564 = (.883)x
log(0.28564) = x log(.883)
log(0.28564) / log(.883) = x
x = 10.07
Comments and questions welcome.
More Algebra 2 problems.
More Algebra 2 problems.
January 2020, Part III
All Questions in Part III are worth 4 credits. Work must be shown. Partial credit is given.
Sonja only has 40 feet of wire to use for the project and wants to cut 20 pieces total for the mobile
using her pattern. Will she have enough wire? Justify your answer.
33. Sonja is cutting wire to construct a mobile. She cuts 100 inches for the first piece, 80 inches for the
second piece, and 64 inches for the third piece. Assuming this pattern continues, write an explicit
equation for an, the length in inches of the nth piece.
Answer:
80/100 = .8 or 4/5. 64/80 = .8 or 4/5. r = .8
The formula is an = 100(.8)n - 1
Remember that the exponent is n - 1 because the first number has to be 100, so the exponent needs to be zero.
To find out whether she has enough for 20 pieces of wire, for the sum of the first 20 lengths using the formula:
Sn = (a1 - a1(r)n) / (1 - r)
S20 = (100 - 100(.8)20) / (1 - .8) = 494.235... inches
40 feet of wire = 40 * 12 = 480 inches.
She will not have enough wire. She will need approximately another 14.24 inches of wire.
34. Graph the following function on the axes below.
State the domain of f.
State the equation of the asymptote.
Answer:
Plug the function into your graphing calculator. You will see the "easy" points are (-7, 2), (-1, 1) and (1, 0). Also, f(0) is approximately 0.6. The asymptote is x = 2.
Graph it as follows:
The domain of f is x < 2. (Not x < 2.)
The equation of the asymptote is x = 2. (It says equation, so don't just says that "it's 2".)
Comments and questions welcome.
More Algebra 2 problems.
More Algebra 2 problems.
January 2020, Part II
All Questions in Part II are worth 2 credits. Work must be shown. Partial credit is given.
31. Biologists are studying a new bacterium. They create a culture with 100 of the bacteria and
anticipate that the number of bacteria will double every 30 hours. Write an equation for the
number of bacteria, B, in terms of the number of hours, t, since the experiment began
Answer:
B(t) = 100(2)t/30
Start with 100 bacteria. It's doubling, so the base is 2. The time is measured in hours, but it doubles every 30 hours, so the exponent is t/30.
32. Graph y = x3 - 4x2 + 2x + 7 on the set of axes below.
Answer:
Put the equation in the calculator and check the table of values.
You'll see (-1, 0), (0, 7), (1, 6), (2, 3), (3, 4), (4, 15)
Also, you'll see (-2, 21), way below the screen, and (5, 42), which is way above.
Use this data to make the graph.
Also note that (2, 3) is not a minimum point, so when you sketch the curve, don't make it a turning point. Go through the point and curve after it.
Given the values, I used a scale of 1 on the x-axis and a scale of 2 on the y-axis. Label your scales.
Comments and questions welcome.
More Algebra 2 problems.
Odd is my choosing to switch back to black & white for no particular reason. It didn't make anything easier, which is why I went back to color before this.
These past episodes not based on any specific incident, but sometimes I hear the drama of middle school. Sometimes high school, too, but those students are usually aware that I'm still in the room. Although, sometimes, they just don't care that I'm in the room.
I also planned to have Daisy standing off the side, but there just wasn't any room for that. (x, why?) is mostly free format, but there are a few things that I've seen to lock in specific dimensions and a "look".
More Algebra 2 problems.
January 2020, Part II
All Questions in Part II are worth 2 credits. Work must be shown. Partial credit is given.
29. Given the geometric series 300 + 360 + 432 + 518.4 + ..., write a geometric series formula, Sn,
for the sum of the first n terms. Use the formula to find the sum of the first 10 terms, to the nearest
tenth.
Answer:
Find the formula for Geometric Series in the back of the booklet:
Sn = (a1 - a1rn) / (1 - r), where r is the Common Ratio of the series.
360 / 300 = 1.2
432 / 360 = 1.2, etc.
r = 1.2
Sn = (300 - 300(1.2)10) / (1 - 1.2) = 7787.6...
S10 = 7787.6
30. Visible light can be represented by sinusoidal waves. Three visible light waves are shown in the
graph below. The midline of each wave is labeled L.
Based on the graph, which light wave has the longest period? Justify your answer.
Answer:
Check the length between maximum and minimum.
A has a max at 60 and a min at 340. 340 - 60 = 280
B has a max at 180 and a min at 400. 400 - 180 = 220
C has a max at 380 and a min at 60. 380 - 60 = 320.
Light wave C has the longest period.
Comments and questions welcome.
More Algebra 2 problems.
On the bright side, everything is awkward at a certain awkward age.
What my old story lines were and where they were going? I was still deciding and working on plotting. Circumstances change, and, as usual, something that starts as an easy little side project starts to become more involved and complicated.
On the other hand, if I wanted to move forward with the new circumstances of the day, I thought I should wrap up old stuff before moving forward.
Like everything else, we'll see what tomorrow brings. Or doesn't.
More Algebra 2 problems.
January 2020, Part II
All Questions in Part II are worth 2 credits. Work must be shown. Partial credit is given.
27. Describe the transformation applied to the graph of p(x) = 2x
that form the new function q(x) = 2x - 3 + 4
Answer:
The graph would shift 3 units to the right and 4 units up.
More info:
The + 4 is a vertical move. Whatever calculations are done involving x, they have been completed by the time the 4 is added, so the shift is vertical.
Changing 2x to 2x - 3 means that the new function will have lower exponent values. It won't rise to the same level until the input is 3 higher. That is, 3 to the right, not the left.
You can graph the two functions in your calculator to see the difference. You can sketch them on the paper, if you want, but it isn't necessary. You just need to describe the transformation.
28. The parabola y = - 1/20(x - 3)2 + 6 has its focus at (3,1). Determine and state the equation of the directrix.
(The use of the grid below is optional.)
Answer:
The parabola has its vertex at (3, 6), which we know because it's written in vertex mode.
The vertex is the midpoint between of a line between focus and directrix.
6 - 1 = 5
6 + 5 = 11
The equation for the directrix is y = 11.
Comments and questions welcome.
More Algebra 2 problems.
I've never done that thing were webcomics switch. So I just left the characters do it. Sort of.
In the early years, I really didn't know too many other comic writers. And the ones I knew, I couldn't really match their style. Plus, I would always forget until the end of March was practically here, and it would be too late.
Now, I'm not sure that comics still do this.