Saturday, July 30, 2022

School Life #28: Two of the Guys

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(C)Copyright 2022, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

Sometimes it takes a minute to sink in.

I wanted to include a beat panel, but I didn't have space when I squeezed the panels. But then I messed up and couldn't "unsqueeze" them. So I squeezed a little more and then shrank the beat panel. It works. I think.

So James, new name that, is apparently on the soccer team, which we know considers Kyung "one of the guys". (According to Shaun.) More subtlely, Shaun (and possibly the team) are the only ones who call Kyung "Ky", but James chose not to do that before asking. Okay, that meant something in my head.

I did a quick look at my character page. I knew James's character didn't have a name. (I joke there about naming him in a contest -- but I wouldn't get enough entries.) Surprisingly, there is no James mentioned on the page. I figured it 50-50 that there would be.

So is this the start of something? Is it the end of something? Is it me moving on to another chapter?

Maybe to all of those.



I also write Fiction!


You can now preorder Devilish And Divine, edited by John L. French and Danielle Ackley-McPhail, which contains (among many, many others) three stories by me, Christopher J. Burke about those above us and from down below.
Preorder the softcover or ebook at Amazon.

Also, check out In A Flash 2020, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides.
Available in softcover or ebook at Amazon.

If you enjoy it, please consider leaving a rating or review on Amazon or on Good Reads.





Come back often for more funny math and geeky comics.



Friday, July 29, 2022

Signal-to-Noise 2

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(C)Copyright 2022, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

The caveat being that I strive for more signal in my own blog, but 100% math all the time would short-curcuit my brain.

This is the reason that the math jokes and questions give way to teacher and family jokes of other silliness. And why my Regents explanations try to have a conversational, rather than a purely clinical, academic tone to them. I couldn't enjoy that, and I don't think others could. Really, after the 15th math joke, anyone still here would be saying, "That's nice. What else you got?"

While I was composing this, it occured to me that I might've done something similar in the past. I did -- back in 2013 -- but the graph was different then the one I was envisioning now, so I went ahead with it. (Note the "2" in the title.) I needed a "quickie" after the last two comics.

Okay, now bring in da noise, bring in da signal!



I also write Fiction!


You can now preorder Devilish And Divine, edited by John L. French and Danielle Ackley-McPhail, which contains (among many, many others) three stories by me, Christopher J. Burke about those above us and from down below.
Preorder the softcover or ebook at Amazon.

Also, check out In A Flash 2020, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides.
Available in softcover or ebook at Amazon.

If you enjoy it, please consider leaving a rating or review on Amazon or on Good Reads.





Come back often for more funny math and geeky comics.



Thursday, July 28, 2022

Summer Friends

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(C)Copyright 2022, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

Sometimes you just need someone to listen. And sometimes you need someone to pretend to be listening.

I used the "School Life" tag, but I didn't give it a number because I couldn't fit it into the format that I've been using for those strips. I've done that before.

I spent a bit of time working on some of the other characters, and then realized that they didn't fit in, or weren't needed. So a squeezed a few down for that final panel, because everyone at school has more friends than they realize. The social network between the different groups is growing, at least for the groups that are the same age.

As for Kyung's brother, it's a story that got lost in the shuffle between the pandemic affecting the comic and my home and work life. Maybe I'll write my own fan fiction on Archive Of Our Own at some point. Or maybe someone else will beat me to it.

Like Michele's pregnancy, it's gone on longer than I anticipated. (Okay, I actually liked the idea of the pregnancy lasting for two years because almost no one ages in this strip.)

They'll be more in the days ahead. I hope. Depending upon life.



I also write Fiction!


You can now preorder Devilish And Divine, edited by John L. French and Danielle Ackley-McPhail, which contains (among many, many others) three stories by me, Christopher J. Burke about those above us and from down below.
Preorder the softcover or ebook at Amazon.

Also, check out In A Flash 2020, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides.
Available in softcover or ebook at Amazon.

If you enjoy it, please consider leaving a rating or review on Amazon or on Good Reads.





Come back often for more funny math and geeky comics.



Monday, July 25, 2022

Finished

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(C)Copyright 2022, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

That was the year that was.

This could've been a last day comic. And then it became a pool comic because I just haven't been able to keep up. A lot of teachers couldn't keep up either this year because it seemed like we were teaching at least two year's worth of material this year. Life is crazy. Let's hope it gets back to better normal than we're at now.

Two things to note. First, I hadn't expected Michele to take the entire year off. It just sort of happened that way because, again, no time to work on any kind of story line. Second, I'm sure somewhere in my notes, if not in a comic, I gave little Miss Wayne a first name. I couldn't find it. I believe that I was going with Mary (Something), like her mother. I'll have to research this.



I also write Fiction!


You can now preorder Devilish And Divine, edited by John L. French and Danielle Ackley-McPhail, which contains (among many, many others) three stories by me, Christopher J. Burke about those above us and from down below.
Preorder the softcover or ebook at Amazon.

Also, check out In A Flash 2020, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides.
Available in softcover or ebook at Amazon.

If you enjoy it, please consider leaving a rating or review on Amazon or on Good Reads.





Come back often for more funny math and geeky comics.



Thursday, July 21, 2022

The Most Common Mistakes I Saw on the Algebra 2 Regents Exams

Last month, June 2022, I participated as a scorer for the New York State Algebra 2 Regents Exam. I was assigned problems 31 through 34, which accounts for one-third of the available points for the open-ended responses. I can safely say that I read and scored hundreds of papers, as a conservative estimate. And over the course of nearly a week, several common errors and misconceptions emerged more than I would have imagined.

What follows are the most common mistakes we encountered in our team for these four questions. First, let me start off by saying that no particular student or school is being singled out. These occurred from school to school, district to district.

Second, I will stick to the avoidable mistakes, the responses where the student knew the proper procedure or at least showed some idea of how to solve the problem but show careless mistake or misunderstanding cost them one or more points. These are the problems with students may have done 95% of the work but only receive 50% (or less) of the credit.

Question 31

Graph y = 2 cos(1/2 x) + 5 on the interval [0,2π], using the axes below.

Among students who knew how to create the graph, most knew that the midline was 5, the maximum was 7 and the minimum was 3. However, there were three common errors, any two of which would result in zero credit.

First, the frequency was 1/2. Many responses had other frequencies, either 2 or some other value.

Second, the interval was stated. If they drew a line that went out of those boundaries without clearly marking off [0,2π], then a point was deducted. Similar to this, and possibly more aggravating, were the graphs that fit into the correct interval, but then included arrows at the endpoints. Arrows indicate that the graph continues when, in this instance, it should not.

The third of the common mistakes was not labeling the x-axis. The scorer could not assume the scale or where, say, 1/4 π was based on the curve above it.

For this question, I'll also point out that a large number of the papers I scored included students who relied on the table of values in their calculators. This was apparent from the fact that the x-axis was labeled 1,2,3,..., etc. In almost every instance, the student lost at least one credit for going beyond 2π. Many of these papers had a second error, making the question worth zero credits.

Question 32

A cup of coffee is left out on a countertop to cool. The table below represents the temparture, F(t), in degrees Fahrenheit, of the coffee after it is left out for t minutes.

[TABLE OMITTED]

Based on these data, write an exponential regression equation, F(t), to model the temperature of the coffee. Round all values to the nearest thousandth.

Of the students who knew the procedure, among the paper I scored, rounding errors were almost non-existant. This made me happy as I expected more of them.

The number one error that caused a one-credit deduction was using x instead of t, as in either of the following:

y = 169.136(0.971)x

or

F(t) = 169.136(0.971)x

This was probably the most careless and costliest error that I encountered. A lot of work is involved putting the data into those tables to get this response, and a single letter reduces the credit by half. It's maddening to me as an educator as well as a scorer.

There were few who did Linear regressions by mistake, which would be worth 1 credit if done "correctly". However, many of those also used x, which cost the other point.

For those who didn't know what to do, many wrote a function or equation for exponential decay.

Question 33

On the set of axes below, graph y = f(x) and y = g(x) for the given functions:

f(x) = x3 - 3x2

g(x) = 2x - 5

State the number of solutions to the equation f(x) = g(x).

It was quite clear from the response that many, MANY students had no idea what that last sentence meant or how to find solutions to f(x) = g(x), despite the graph that they had just created.

For the record, if they had answered "3" without a graph, they would have received one credit. And if they drew an incorrect graph and then responded with the number of intersections that appeared on their graph, they would have received one credit. (Consistent error, they already lost the points on the graph.) The student did not have to explain nor justify nor provide coordinates. They only had to provide a number. Many times that number did not match the graph or was omitted completely.

Additionally, some students tried to solve f(x) = g(x) algebraically, with various degrees of success. Again, this just reinforces my belief that they didn't know what solutions the questions was asking for.

Speaking of solutions, there was a substantial subset that assumed that "solutions" meant the "zeroes" for each of the graphs, which were listed separately.

While looking for graphing errors, most of the scorers I met worked with for this question focused on the domain -2 < x < 3, unless the lines were significantly incorrect beyond those points. It was usually clear when a point was misplaced or omitted (maybe they misread their table) and when a student was just sketching something that looked like what he saw on the screen. For example, many graphs of g(x) not only weren't slope of 2 and/or didn't have a y-intercept of -5.

I found the graphing errors a little disturbing because the functions were calculator-ready -- not manipulation or calculations had to be done before entering them into the calculators.

There were some "forced" errors. That is, a number of students tried to force one line or the other to intersect at integers, particularly (1,-2) or (1,-3).

This questions was worth 4 credits, which were essentially 2 credits for a correct graph of f(x), 1 credit for the correct graph of g(x), and 1 credit for "3" or a number consistent with the graph provided.

Question 34

A Foucault pendulum can be used to deomnstrate that the Earth rotates. The time, t, in seconds, that it takes for one swing or period of the pendumulm can be modeled by the equation t = 2π√(L/g), where L is the length of the pendulum in meters and g is a constant of 9.81 m/s2.

The first Foucault pendulum was constructed in 1851 and has a pendulum length of 67 m. Determine, to the nearest tenth of a second, the time it takes this pendulum to complete one swing.

Another Foucault pendulum at the United Nations building takes 9.6 seconds to complete one swing. Determine, to the nearest tenth of a meter, the length of this pendulum.

The most common mistake, by far, was A CALCULATOR ISSUE, so it only cost one credit instead of two. Many students, somewhere in their calculations, divided by 2π instead of dividing by (2π).

To state this more explicitly, many wrote (9.6/2π)2 correctly on their papers and then entered it exactly that way into their calculators, instead of entering (9.6/(2π))2. As a result, 9.6 was divided by 2 but then MULTIPLIED by π. THis usually resulted in a response of 2230.8, unless another error was made.

The second most common computational error was rounding in the middle of the problem. This meant that they lost accuracy and their responses would be off by as much as a meter.

A concepual error that I encountered quite a few times involved this equation:

9.6 = 2π√(L/9.81)

Some students attempted to square both sides of the equation to get rid of the radical, but they did not square the 2π. A second conceptual error involved dividing by 9.81 instead of multiplying.

Final Comments

Arguing with the format of the Regents exams won't get me anywhere -- particularly in this space and on this blog. (Please, just don't.) However, as long as students are stuck with them, educators need to stress the pitfalls of cutting any corners or simply not checking the work thoroughly. Mistakes happen not just through laziness or carelessness, but also through stress or just misreading or misinterpreting a few words.

The number of papers where students had absolutely no idea what to do tells me as an educator that I need to review a topic. But the ones who knew what they were doing but were tripped up on silly things, I really feel for them. I don't want them to be discouraged by a grade that doesn't affect their ability. It's even more frustrating because they'll never know why they received the score they did. (They might get a breakdown in points, but not any explanations.)

And honestly, I want to make sure that next year's students don't get tripped up by these things.

Tuesday, July 19, 2022

Bip

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(C)Copyright 2022, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

Once more to be sure. Bip.

This comic has occurred to me probably as many times as I've worried if I "bipped" the car or not. And when I bip it again just to be sure. It could also happen at a parking lot or on a residential street down the block from the store I need to go to. The alternative would be to walk back and check the door. But that would be silly.

I thought about doing this as six panels where there's a question mark at the door before he bips it again. The last panel could've been walking in front of the garden. But then I thought the idea came across with just four boxes. I hope I'm right.

Bip.



I also write Fiction!


You can now preorder Devilish And Divine, edited by John L. French and Danielle Ackley-McPhail, which contains (among many, many others) three stories by me, Christopher J. Burke about those above us and from down below.
Preorder the softcover or ebook at Amazon.

Also, check out In A Flash 2020, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides.
Available in softcover or ebook at Amazon.

If you enjoy it, please consider leaving a rating or review on Amazon or on Good Reads.





Come back often for more funny math and geeky comics.



Thursday, July 14, 2022

Breath

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(C)Copyright 2022, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

Just breathe.

This was a throwaway joke that occurred to me when doing something else. I wanted to go more elaborate and then realized that it had to be kept simple. It was also supposed to be three horizontal panels, but then the dimensions would've been crazy.



I also write Fiction!


You can now preorder Devilish And Divine, edited by John L. French and Danielle Ackley-McPhail, which contains (among many, many others) three stories by me, Christopher J. Burke about those above us and from down below.
Preorder the softcover or ebook at Amazon.

Also, check out In A Flash 2020, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides.
Available in softcover or ebook at Amazon.

If you enjoy it, please consider leaving a rating or review on Amazon or on Good Reads.





Come back often for more funny math and geeky comics.



Thursday, July 07, 2022

Nibbles and Bits

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(C)Copyright 2022, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).
Barely half a byte!

The Nibbles and Bits line popped into my head the other day when I thought about the 8-bit characters and how I haven't used them in a while. It wasn't until I started typing the dialogue that I suddenly started channeling Sheldon Leonard from It's a Wonderful Life. It wasn't on purpose, but I played into it. To the point that I nearly forgot to make a "nibble" reference.

For those who don't know (FTWDK), a nibble is a four bits, or a half byte of data. I thought my Comp Sci teacher was making a joke way back when. Nope. Someone else made that joke long before, and it stuck.

The witch seemed to be the appropriate character to mention nibbling on the little ones and the little zero.

Now I have to think of some new 8-bit characters, and then find the tools I used to create the ones I have.

And maybe name them.



I also write Fiction!


You can now preorder Devilish And Divine, edited by John L. French and Danielle Ackley-McPhail, which contains (among many, many others) three stories by me, Christopher J. Burke about those above us and from down below.
Preorder the softcover or ebook at Amazon.

Also, check out In A Flash 2020, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides.
Available in softcover or ebook at Amazon.

If you enjoy it, please consider leaving a rating or review on Amazon or on Good Reads.





Come back often for more funny math and geeky comics.



Wednesday, July 06, 2022

June 2022 Algebra 2 Regents, Part IV



This exam was adminstered in June 2022. These answers were not posted until they were unlocked on the NY Regents website or were posted elsewhere on the web.

More Regents problems.

Algebra 2 June 2022

Part IV: A correct answer will receive 6 credits. Partial credit can be earned.


37. The population, in millions of people, of the United States can be represented by the recursive formula below, where a0 represents the population in 1910 and n represents the number of years since 1910.

a0 = 92.2
an = l.015an-1

Identify the percentage of the annual rate of growth from the equation an = l.015an-1.

Write an exponential function, P, where P(t) represents the United States population in millions of people, and tis the number of years since 1910.

According to this model, determine algebraically the number of years it takes for the population of the United States to be approximately 300 million people. Round your answer to the nearest year.


Answer:


The growth rate is 0.015, which is 1.5%.

The exponential function would be P(t) = 92.2(1.015)t

To find the amount time until the population is approximately 300 million, set the function equal to 300 and solve for t.

300 = 92.2(1.015)t

300/92.2 = (1.015)t

300/92.2 = (1.015)t

log(300/92.2) = t log(1.015)

log(300/92.2)/log(1.015) = t

t = 79.243...

Approximately 79 years.

Alternatively, you could have answered the last two parts as follows:

P(t) = 92.2e.015t

300 = 92.2e.015t

300/92.2 = e.015t

ln(300/92.2) = .015t ln e

ln(300/92.2)/0.015 = t (1)

t = 78.6548...

Approximately 79 years.

Note that the decimal portions vary a little, but they both round to the same number of years.




End of Exam

How did you do?








More to come. Comments and questions welcome.

More Regents problems.

I also write Fiction!


You can now preorder Devilish And Divine, edited by John L. French and Danielle Ackley-McPhail, which contains (among many, many others) three stories by me, Christopher J. Burke about those above us and from down below.
Preorder the softcover or ebook at Amazon.

Also, check out In A Flash 2020, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides.
Available in softcover or ebook at Amazon.

If you enjoy it, please consider leaving a rating or review on Amazon or on Good Reads.



Tuesday, July 05, 2022

June 2022 Algebra 2 Regents, Part III



This exam was adminstered in June 2022. These answers were not posted until they were unlocked on the NY Regents website or were posted elsewhere on the web.

More Regents problems.

Algebra 2 June 2022

Part III: Each correct answer will receive 4 credits. Partial credit can be earned. One computational mistake will lose 1 point. A conceptual error will generally lose 2 points (unless the rubric states otherwise). It is sometimes possible to get 1 point for a correct answer with no correct work shown.


33. On the set of axes below, graph y = f(x) and y = g(x) for the given functions.

f(x) = x3 - 3x2
g(x) = 2x - 5

State the number of solutions to the equation f(x) = g(x).


Answer:


Use your calculator and the table of values for the cubic function. You should be able to graph the linear function from the slope and y-intercept.

YOU MUST GRAPH THE FUNCTIONS. NOT SOMETHING THAT ONLY **LOOKS LIKE** THE FUNCTIONS YOU SAW ON YOUR CALCULATOR.

You'd be surprised (or maybe you wouldn't) how lazy some students were when it came to graphing this problem. The most charitable thing I could say was that they were "careless" about it.

There is no reason to have a mistake in g(x). It's a simple linear function, starting at (0,-5) with a slope of 2 (up 2, over 1). And down 2, to the left 1.

For the cubic function, the five points on the graph below should be on your graph. And the extremes, just have lines with arrows. Do NOT make them vertical.

There are 3 solutions to the equation f(x) = g(x) because there are three places where the graphs intersect. You didn't have to explain, you just had to say "three". If your graph was incorrect and you only had two, you could respond "two" and get credit for that -- you would've lost the point on your graph.

According to the rubric, if you said "3" and your graph didn't show three, you still got a point because you might have solved it algebraically. Or it might've been a wild guess.





34. A Foucault pendulum can be used to demonstrate that the Earth rotates. The time, t, in seconds, that it takes for one swing or period of the pendulum can be modeled by the equation t = 2π √(L/g) where L is the length of the pendulum in meters and g is a constant of 9.81 m/s2. The first Foucault pendulum was constructed in 1851 and has a pendulum length of 67 m.

Determine, to the nearest tenth of a second, the time it takes this pendulum to complete one swing.

Another Foucault pendulum at the United Nations building takes 9.6 seconds to complete one swing. Determine, to the nearest tenth of a meter, the length of this pendulum.

Answer:


First of all, the year is not needed for any calculations. (Yes, that needs to be said.)

The first part is straight calculation.

t = 2π √(L/g) = 2π √(67/9.81) = 16.420363... = 16.4

This was worth one point. If you rounded incorrectly, you lost that point.

For part two, you had to work backward to solve:

t = 2π √(L/g)

9.6 = 2π √(L/9.81)

9.6 / (2π) = √(L/9.81)

(9.6 / (2π))2 = L/9.81

(9.6 / (2π))2 * 9.81 = L

L = 22.900... = 22.9

This portion was worth three credits. One computational or rounding error cost 1 point. One conceptual error costs 2 points. Both would lose all three credits.

The biggest mistakes I saw had to do with 9.6/2π instead of 9.6/(2π). This was a calculator issue, so it should have been scored as a computational error, especially if the equation was written on the paper. Many students rounded their numbers in the middle of the solution. DON'T DO THAT.

A conceptual error would be, for example, squaring the equation after the second line above, but not squaring the 2π to 4π2.





35. In order to decrease the percentage of its residents who drive to work, a large city launches a campaign to encourage people to use public transportation instead. Before starting the campaign, the city's Department of Transportation uses census data to estimate that 65% of its residents drive to work. The Department of Transportation conducts a simulation, shown below, run 400 times based on this estimate. Each dot represents the proportion of 200 randomly selected residents who drive to work.

Use the simulation results to construct a plausible interval containing the middle 95% of the data. Round your answer to the nearest hundredth.

One year after launching the campaign, the Department of Transportation conducts a survey of 200 randomly selected city residents and fmds that 122 of them drive to work. Should the department conclude that the city's campaign was effective? Use statistical evidence from the simulation to explain your answer.

Answer:


95% of the data falls within 2 standard deviations from the mean. The Mean is 0.651 and the Std Deviation is 0.034.

0.651 - 2 * 0.034 = 0.583, or about 0.58

0.651 + 2 * 0.034 = 0.719 or about 0.72

The interval would be from 0.58 to 0.72.

For the second part, note that 122/200 = 0.61. Since that number is in the 95% confidence interval, then the decrease was not statistically significant. So the campaign was not effective.



36. Solve the system of equations algebraically:

x2 + y2 = 25
y + 5 = 2x

Answer:


Student 2x - 5 for y in the first equation (the circle) and solve for x.

x2 + y2 = 25

x2 + (2x - 5)2 = 25

x2 + 4x2 - 20x + 25 = 25

5x2 - 20x = 0

5x(x - 4) = 0

5x = 0 or x - 4 = 0

x = 0 or x = 4

y + 5 = 2(0)
y = -5
(0, -5)

y + 5 = 2(4)
y = 3
(3, 4)




End of Part III

How did you do?








More to come. Comments and questions welcome.

More Regents problems.

I also write Fiction!


You can now preorder Devilish And Divine, edited by John L. French and Danielle Ackley-McPhail, which contains (among many, many others) three stories by me, Christopher J. Burke about those above us and from down below.
Preorder the softcover or ebook at Amazon.

Also, check out In A Flash 2020, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides.
Available in softcover or ebook at Amazon.

If you enjoy it, please consider leaving a rating or review on Amazon or on Good Reads.



Monday, July 04, 2022

Enjoy the Fourth!

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Update:

(C)Copyright 2022, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).
Happy Fourth of July!

The same thing happens every year at every holiday. How can I celebrate it that I haven't before that fits into the format of this comic. Given the odd schedule I keep now, it's better to just skip a holiday than to put a mediocre or placeholder comic for any particular day. I did finally realize that I never did a "fourth dimension" comic -- which is good because I almost went Star Wars. But this comic, as simple as it looks, was a bit time-consuming and this past weekend was busy with family and extended family. Then wasn't a lot of time to just sit at a PC and work on a comic, and when I did have time, I didn't have the inclination to do a good job. (I was tired, but a good tired. Mostly. But tired.)

Anyway, as I type this, Independence Day is over and I hope everyone was able to have a good and safe one.

Saved for posterity:






Sorry for the placeholder comic, but family demands this past weekend trump other concerns. This will be updated by tomorrow.

(C)Copyright 2022, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

I hope everyone had a pleasant and safe Fourth of July/Independence Day! As you can see, I did not have time to complete today's comic, but you can get the gist of it.

This is that odd time of year when summer vacation starts and so many things that had to wait until summer vacation suddenly have to get done. On top of that, things have sort of snowballed for the last couple of weeks with new things popping up. That's life. I'm hoping for a boring July.



I also write Fiction!


You can now preorder Devilish And Divine, edited by John L. French and Danielle Ackley-McPhail, which contains (among many, many others) three stories by me, Christopher J. Burke about those above us and from down below.
Preorder the softcover or ebook at Amazon.

Also, check out In A Flash 2020, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides.
Available in softcover or ebook at Amazon.

If you enjoy it, please consider leaving a rating or review on Amazon or on Good Reads.





Come back often for more funny math and geeky comics.



Sunday, July 03, 2022

June 2022 Algebra 2 Regents, Part II



This exam was adminstered in June 2022. These answers were not posted until they were unlocked on the NY Regents website or were posted elsewhere on the web.

More Regents problems.

Algebra 2 June 2022

Part II: Each correct answer will receive 2 credits. Partial credit can be earned. One mistake (computational or conceptual) will lose 1 point. A second mistake will lose the other point. It is sometimes possible to get 1 point for a correct answer with no correct work shown.


25. Does the equation x2 - 4x + 13 = 0 have imaginary solutions? Justify your answer.

Answer:


You can determine the number of solutions by using the discriminant, or by graphing and stating the number of x-intercepts. A negative discriminant means imaginary solutions. No x-intercepts means imaginary solutions.

The discriminant is b2 - 4ac = (-4)2 - 4(1)(13) = (-4)2 - 4(1)(13) = 16 - 52, which is < 0. So the discriminant is negative and the solutions are imaginary.

If graphed, the range of the function would be entirely above the x-axis, so there would only be imaginary solutions.





26. The initial push of a child on a swing causes the swing to travel a total of 6 feet. Each successive swing travels 80% of the distance of the previous swing. Determine the total distance, to the nearest hundredth of a foot, a child travels in the first five swings.

Answer:


You can use the formula for the sum of a finite series, or you can manually calculate 80% for the second through the fifth swings. Either is acceptable but the latter leaves more room for errors to creep in. Two errors is 0 credits, regardless of the work done.

Using the formula from the back of the booklet:

Sn = a1(1 - (r)n) / (1 - r)

S5 = (6)(1 - (0.80)5)) / (1 - 0.80) = 20.1696

The total amount is about 20.17 feet.

Doing this in steps:

S1 = 6
S2 = 6(.8) = 4.8
S3 = 4.8(.8) = 3.84
S4 = 3.84(.8) = 3.072
S5 = 3.072(.8) = 2.4576

S1 + S2 + S3 + S4 + S5 = 6 + 4.8 + 3.84 + 3.072 + 2.4576 = 20.1696 = 20.17





27. Solve algebraically for n: 2/n2 + 3/n = 4/n2

Answer:


Excuse me for saying that this is ridiculously easy, but this question could appear on an Algebra 1 exam. You don't even need to worry about n = 0 in the problem unless you somehow introduce 0 as a possible solution.

2/n2 + 3/n = 4/n2

(n2(2/n2 + 3/n) = (4/n2)n2

2 + 3n = 4>

3n = 2

n = 2/3

That's it. Two credits.





28. Factor completely over the set of integers:

-2x4 + x3 + 18x2 - 9x


Answer:


Remove the GCF, which is x (or -x), then factor by grouping. Then check if anything else can be factored.

Personally, I prefer for the leading coefficient to be positive when I'm factoring, so I would factor out (-1) along with the GCF of x.

-2x4 + x3 + 18x2 - 9x

(-x) (2x3 - x2 - 18x + 9)

(-x) ((x2)(2x - 1) - 9(2x - 1))

(-x) (x2 - 9)(2x - 1)

(-x)(x + 3)(x - 3)(2x - 1)

Anything equivalent is acceptable. For example, (x)(x + 3)(x - 3)(-2x + 1).





29. The relative frequency table shows the proportion of a population who have a given eye color and the proportion of the same population who wear glasses.

Wear
Glasses
Don't Wear
Glasses
Blue Eyes 0.140.26
Brown Eyes 0.110.24
Green Eyes 0.100.15

Given the data, are the events of having blue eyes and wearing glasses independent? Justify your answer.


Answer:


Two events are independent if P(A and B) = P(A)P(B).

Let A be the probability of Wearing Glasses. Let B be the probability of Blue Eyes.

P(A) = 0.14 + 0.11 + 0.10 = 0.35.

P(B) = 0.14 + 0.26 = 0.40

P(A)P(B) = (0.35)(0.40) = 0.14

P(A and B) = 0.14 from the table.

Since the amounts are the same, the events are independent.





30. For x ≠ 0 and y ≠ 0, ∛(8lx15y9) = 3ax5y33. Determine the value of a.

Answer:


You can simplify the cube root or you can focus on the 81 portion of it.

∛(8l) = ∛(3 * 3 * 3 * 3) = 3 4/3 = 3 a

Therefore, a = 4/3.





31. Graph y = 2cos(1/2 x) + 5 on the interval [0,2π], using the axes below.


Answer:


The midline is 5 and the amplitude is 2, so the range is from 3 to 7. The period of cos(nx) is 2π/n, which is 2π/(1/2) = 4π. So over the range of [0,2π], only half of the wave will be seen.

Your graph should look something like this:

Do NOT continue past 2π outside of the interval you were given.

Do NOT put Arrows on either end of the line because that indicates going beyond the interval.

DO number the axes, especially the X-axis.

DO make it look like a curve and NOT a linear function.

Remember that this is only a TWO credit problem, and you lose one point for each graphing error. That means that 2 (or more) "little" mistakes means you will get zero credit, regardless of the amount of work you point into it.





32. A cup of coffee is left out on a countertop to cool. The table below represents the temperature, F(t), in degrees Fahrenheit, of the coffee after it is left out fort minutes.

Based on these data, write an exponential regression equation, F(t), to model the temperature of the coffee. Round all values to the nearest thousandth.

Answer:


Put the information into L1 and L2 in your calculator. Run an Exponential Regression. Write the answer using F(t) and t. Do NOT use y or x in your final answer.

If you entered the data correctly, you should have gotten a = 169.136 and b = 0.971. This means that your equation should have been:

F(t) = 169.136(0.971)t

I scored this problem on literally hundreds of exams. The biggest mistake among those who did the work correctly was to use y and x, or F(t) and x, either of which cost a point. This mistake was far more common than rounding to the wrong number of decimals, or even misentering the data and getting a and b values which were slightly off.




End of Part II

How did you do?








More to come. Comments and questions welcome.

More Regents problems.

I also write Fiction!


You can now preorder Devilish And Divine, edited by John L. French and Danielle Ackley-McPhail, which contains (among many, many others) three stories by me, Christopher J. Burke about those above us and from down below.
Preorder the softcover or ebook at Amazon.

Also, check out In A Flash 2020, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides.
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Saturday, July 02, 2022

June 2022 Geometry Regents, Part 4

The following are some of the multiple questions from the recent January 2020 New York State Common Core Geometry Regents exam.

June 2022 Geometry, Part IV

A correct answer is worth up to 6 credits. Partial credit can be earned. Work must be shown or explained.


32. The coordinates of the vertices of D.ABC are A(-2,4), B(-7, -1), and C(-3, -3). Prove that D.ABC is isosceles.
[The use of the set of axes on the next page is optional.]

State the coordinates of D.A 'B 'C', the image of △ABC, after a translation 5 units to the right and 5 units down.

Prove that quadrilateral AA'C'C is a rhombus.
[The use of the set of axes below is optional.]

Answer:
The breakdown was 2 points from proving the triangle is isosceles, with one point for finding the lengths and one point for a statement saying it was isosceles and why. One point was given for finding all three set of coordinates of the image. The final three points were for proving that the object was a rhombus, including the work and a statement. If the student only showed that it was a parallelogram but not enough for a rhombus, that was one credit.

Points could be earned on the last portion proving that it was NOT a rhombus if incorrect coordinates for the image were found. In this case, supporting work still had to have been shown and a proper concluding statement needed to be given.

In the first part, you could use the distance formula, or you could use the graph and show it through the Pythagorean Theorem.

AB = √((5)2 + (5)2) = √(50)
AC = √((1)2 + (7)2) = √(50)

Triangle ABC is isosceles because AB ≅ AC.

To translate the pre-image to the image, add 5 to each x coordinate and subtract 5 from each y coordinate. You didn't have to show this, just the response.

A(-2,4) --> A'(3,-1)
B(-7,-1) --> B'(-2,-6)
C(-3,-3) --> C'(2, -8)

You didn't need to show your work on the graph. However, the visual is a great aid for solving the problem.

To prove that the shape was a rhombus, you had to show that four sides were congruent, OR you could show that the diagonals bisected each other. If you showed that the diagonals bisected each other, that only showed that it was a parallelogram. In that case, you still needed to show two consecutive sides were congruent.

Conversely, if you only showed two consecutive sides, then you still needed to show that it was a parallelogram.

The four sides of the rhomus are AA', A'C', C'C, and CA.

Length of AC = √(50). Shown above, you don't need to do it again..

Length of A'C' = √(50) because a translation is a rigid motion that preserves distance. This must be stated, or you could use the distance formula again.

Length of AA' = √( (5)2 + (5)2 ) = √(50)

Length of C'C = √( (5)2 + (5)2 ) = √(50)

All four sides are congruent, therefore AA'C'C is a rhombus.

Alternatively,

The slope of AC' is (-8 - 4)/(2 - -2) = -12/4 = -3

The slope of A'C is (-1 - -3)/(3 - -3) = 2/6 = 1/3

The slopes of the two diagonals are inverse reciprocals, so the diagonals are perpendicular. Therefore, the quadrilateral is a rhombus.

Other methods were possible.

End of Part Exam

How did you do?

Questions, comments and corrections welcome.

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Friday, July 01, 2022

June 2022 Geometry Regents, Part 3

The following are some of the multiple questions from the recent January 2020 New York State Common Core Geometry Regents exam.

June 2022 Geometry, Part III

Each correct answer is worth up to 4 credits. Partial credit can be given. Work must be shown or explained.


32. As modeled below, a projector mounted on a ceiling is 3.74 m from a wall, where a whiteboard is displayed. The vertical distance from the ceiling to the top of the whiteboard is 0.41 m, and the height of the whiteboard is 1.17 m.

Determine and state the projection angle, θ, to the nearest tenth of a degree.

Answer:
Find the angle in the large right triangle, and the angle in the skinny right triangle. Subtract those and you have θ. In both cases, you have the opposite and adjacent sides, so you need to use tangent.

tan x = 0.41/3.74
x = tan-1(0.41/3.74) = 6.2561...

tan x = (0.41+1.17)/3.74
x = tan-1 ((0.41+1.17)/3.74) = 22.9021...

θ = 22.9021 - 6. 2561 = 16.646 = 16.6 to the nearest tenth of a degree.




33. Given: Parallelogram PQRS, QT ⟂ PS, SU ⟂ QR

Prove: PT ≅ RU

Answer:

Answer:
You need to make a two-column proof. You can also write a paragraph proof, if you want, but you still need to have all the correct statements and reasons/justifications. The scorer will look to see that all the points are there.

You can prove this by showing that the two triangles are congruent and using CPCTC (corresponding parts of congruent triagnles are congruent). Looking at those triangles, you know that they are right triangles because the lines are perpendicular. And you know that the hypotenuses are congruent because they are the opposite sides of a parallelogram. And you know that angle P is congruent to angle R because its a parallelogram. This is what you need to show.

Note that the above paragraph isn't a paragraph proof because I didn't fully state the reasons and properties. It was meant as a guideline for a two-column proof.

Update: Proof added. Formatting takes time.

Statement Reason
1. Parallelogram PQRS, QT ⟂ PS, SU ⟂ QR Given
2. RS ≅ QP Opposite sides of a parallelogram are congruent.
3. ∠P ≅ ∠R Opposite angles of a parallelogram are congruent.
4. ∠PTQ and ∠RUS are right angles Definition of perpendicular lines
5. ∠PTQ ≅ ∠RUS All right angles are congruent.
6. △PTQ ≅ △RUS AAS Theorem
7. PT ≅ RU CPCTC (Corresponding Parts of Congruent Triangles are Congruent)

The first is required but not worth any points. Lines 2 and 3 are each worth 1 point. Lines 4-6 together are worth 1 point. Line 7 is worth the final point. Basically, proving AAS is three points, but you'll lose one point for each pair of "A" or "S" that you didn't show to be congruent, or for not stating AAS.

You can also prove PT = RU by showing that QUST is a parallelgragm (a rectangle even), with opposite sides congruent. Since QR = PS because it's a paralellogram, you can use the subtraction property to show PT = RU.

Given that PQRS is a parallelogram, then RQ ≅ PS because the opposite sides of a parallelogram are congruent. QU || ST because the opposite sides of a parallelogram are parallel. Angle QUS is a right angle because of the definition of perpendicular lines. Angle UST is a right angle because the same-side interior angles of a transversal between parallel lines are supplementary. Angles UST and STQ are supplementary because they add up to 180 degrees. So US || QT because the same-side angles are supplementary. QUST is a parallelogram because its oppisite sides are parallel. QU ≅ ST because the opposite sides of a parallelogram are congruent. Therefore, PT &cong RU by the Subtraction Property.

Okay, I'll admit that this took longer than I thought it would because I misread the given and didn't realize that I still had a couple extra things to prove to get where I wanted to go. HOWEVER, I knew I could get there, and I couldn't get the idea out of my head, so I went with it.




34. A concrete footing is a cylinder that is placed in the ground to support a building structure. The cylinder is 4 feet tall and 12 inches in diameter. A contractor is installing 10 footings.

If a bag of concrete mix makes ; of a cubic foot of concrete, determine and state the minimum number of bags of concrete mix needed to make all 10 footings.

Answer:
First, convert the inches to feet. Then find the radius from the diameter. Then find the volume of ONE footing. Multiply it by 10 to get the volume of TEN footings. Then DIVIDE by 2/3 (which means multiply by 3/2) to find the number of bags of concrete. Then, and this is Important, ROUND UP to the next full bag. If you round down, you won't have enough concrete mix to finish the tenth footing.

d = 12 inches = 1 foot, so r = 1/2 foot

V = π r2h = π (.5)2(4) = π = 3.14159....

10V = 10(3.14159...)

Divide 10(3.14159)/(2/3) = 47.12385

48 bags of concrete mix are needed.

Basically, you got a point for the Volume of one footing, a point for the Volume of 10 footings, a points for the amount of concrete mix needed and a point for the final number of bags. You didn't have to show those numbers to get full credit (the test didn't ask for any in-between answers), but they could give you partial credit if your final answer was incorrect. For example, you could have written one big equation to find the answer in one step. As long as the work was there, you got full credit.

End of Part III

How did you do?

Questions, comments and corrections welcome.

I also write Fiction!


Check out In A Flash 2020, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides.
Available in softcover or ebook at Amazon.

If you enjoy it, please consider leaving a rating or review on Amazon or on Good Reads.

Thank you.