School Life #21: School's Out 2021

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(C)Copyright 2021, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

You can't just go breaking a fourth wall like that!

The more things change, the more they'll stay the same. It's a weird cosmic, temporal vortexual balancing act. With generic building blocks.

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.

Come back often for more funny math and geeky comics.

Leftovers: Chat Box

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(C)Copyright 2021, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

I'm too sexy for this school, too sexy for this school, too sexy it's cruel.

Not that this is in any way semi-autobiographical, the number of times I said "in the Chat box" was too darn high. So, yes, this ear worm popped up at some point. But that's all right, said me.

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.

Come back often for more funny math and geeky comics.

Leftovers: Faces

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(C)Copyright 2021, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

The year is over, but I'd like to see my students!

It was a fact of life for many of my colleagues this year that the students just would not (unless required or mandated in some manner) turn on their screens. And many of them wouldn't speak either. While being muted was helpful for parts of the lesson, limiting oneself to just the Chat box was problematic because there are times that the teacher is presenting something, and thus taking over the screen, where the Chat box wouldn't be visible.

As a result, many students only unmuted and spoke up when they were frustrated about something. Unfortunately, that frustrated attitude could be catchy if a teacher couldn't ameloriate the situation. Basically, it was a trying year with remote learning, even when we were in the classroom.

But I'm sure the students understood the little shot Mr. Ibsen threw out there. Even if only ten of them appear to be there.

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.

Come back often for more funny math and geeky comics.

Algebra Problems of the Day (Integrated Algebra Regents, January 2013)

While I'm waiting for new Regents exams to come along (AND THEY ARE COMING SOON!), I'm revisiting some of the older NY Regents exams.

More Regents problems.

Administered January 2013

Part III: Each correct answer will receive 3 credits. Partial credit can be earned.


34. In a game, a player must spin each spinner shown in the diagram below once.

Draw a tree diagram or list a sample space showing all possible outcomes.

Determine the number of outcomes that consist of a prime number and a letter in the word “CAT.”

Answer:

A sample space is a list of all the possible outcomes. There are five outcomes on the first spinner and three on the second, so there are 15 possible outcomes, and you can list them in order very quickly:

1A 1B 1C 3A 3B 3C 5A 5B 5C 7A 7B 7C 9A 9B 9C

You can write those in any order, but I would suggest you put the number first. (It shouldn't matter, but I strongly suggest it.) You could've written 1A 3A 5A ... etc, and then B and C, and you would have been fine.

A tree diagram is more visual and is more helpful where there are a lot of outcomes to list, particularly when you list words (e.g., red hat, blue shirt, and striped shorts).

If you have drawn a tree diagram or listed a sample space, then you only need to supply a numerical answer to the last part of the question. The first question would be considered the "work" for the second. You can look at your first response and count the positive results. If you skipped the first portion of this question, you would have to show some kind of work, how you approached the problem and got your answer.

Obviously, only the letters C and A appear on the second spinner. There are 3 prime numbers, 3, 5, and 7, on the first spinner. According to the Counting Principle 2 * 3 = 6 outcomes.

Instead of the Counting Principle, you could have just circled the six positive outcomes you were looking for, but that wouldn't have been necessary.

35. The cost of three notebooks and four pencils is $8.50. The cost of five notebooks and eight pencils is $14.50. Determine the cost of one notebook and the cost of one pencil.

[Only an algebraic solution can receive full credit.]

Answer:

Let N = the price of one notebook and P = the price of one pencil
You could use x and y, but meaningful variables are better to use.
Then set up a system of equations to solve.

3N + 4P = 8.50
5N + 8P = 14.50
Double the first equation, and then subtract.
6N + 8P = 17.00
5N + 8P = 14.50

N = 2.50
Substitute into the original equation to find the cost of a pencil.
3(2.50) + 4P = 8.50
7.50 + 4P = 8.50
4P = 1.00
P = 0.25

Write your answers in sentences. Don't leave it as N= and P=. We made those up!

Now if you defined your variables at the top of the page, you're all set! You already wrote the sentences you need!

Let N = the price of one notebook = 2.50 and P = the price of one pencil = 0.25

36.Wendy measures the floor in her rectangular bedroom for new carpeting. Her measurements are 24 feet by 14 feet. The actual measurements are 24.2 feet by 14.1 feet.

Determine the relative error in calculating the area of her bedroom. Express your answer as a decimal to the nearest thousandth.

Answer:

Relative error and percent of error used to be questions that the Regents asked about a lot. The biggest different between the two of them is that relative error is a decimal and percent of error is a percentage, and if you gave a percentage for relative error, you would lose a point. Annoying, but it wasn't the answer that they asked for.

Relative error is the different between the actual and the estimate divided by the actual amount.

The estimated area is 24 * 14 = 336.

The acutal area is 24.2 * 14.1 = 341.22

The relative error is (341.22 - 336) / 341.22 = 0.0152..., which is 0.015 to the nearest thousandth. (It rounds down because 0 - 4 rounds down.)

Be careful that you enter it into the calculator correctly. If you got 340.235 as an answer, it was because you forgot the parentheses and the Order of Operations did you in.

End of Part III






More to come. Comments and questions welcome.

More Regents problems.

Leftovers: Shoe

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(C)Copyright 2021, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

The year is over, but I'll shoehorn this one in!

I have a few leftovers, mostly from downshifting the past couple of weeks. I'll use them this week rather than hold them for fall.

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.

Come back often for more funny math and geeky comics.

Hot Tub Teacher Machine

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(C)Copyright 2021, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

This is what school is, after hours, in the minds of some of my students.

Obviously, this is the reason students are kept out of the teacher room.

I took most of this last week of school off from creating comics just because I was a little tired. The comics I was going to create will be presented a little out of sequence in the coming days.

To all the teachers out there, Enjoy your summer!

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.

Come back often for more funny math and geeky comics.

Algebra Problems of the Day (Integrated Algebra Regents, January 2013)

While I'm waiting for new Regents exams to come along (AND THEY ARE COMING SOON!), I'm revisiting some of the older NY Regents exams.

More Regents problems.

Administered January 2013

Part II: Each correct answer will receive 2 credits. Partial credit can be earned.


31. Express in 4 SQRT(75) simplest radical form.

Answer:

Factor the 75 into 3 * 5 * 5. The square root of (5 * 5) is simply 5. So we can remove the two factors of 5 from beneath the radical and put one factor of 5 in front of it.

So 4 * SQRT(75) = 4 * SQRT(3 * 5 * 5) = 4 * 5 * SQRT(3) = 20 SQRT(3)

Note 1: Write this with a radical sign. Don't use the letters "SQRT". I'm a little limited on this blog.

Note 2: If you wrote a decimal for an answer, you scored 0 points, even if you're answer was accurate to a dozen places.

32. Factor completely: 5x³ - 20x² - 60x

Answer:

A general rule of thumb is that when the question says "Factor completely" then there will be more than one step involved. In this case, each term has a factor of x that can be removed, and each coefficient is a multiple of 5. So start by factoring 5x from each term.

5x³ - 20x² - 60x
= 5x(x² - 4x - 12)
= 5x(x - 6)(x + 2)

You can check your answer by multiplying. You should get the original expression back.



33. On the set of axes below, graph y = 2|x + 3|. Include the interval 7 < x < 1.

Answer:

First, an absolute value graph is V shaped, assuming that you graph its vertex. Second, if it says to show the interval then you must only show that interval, closed at the ends. No arrows. Arrows would be a graphing error.

You can use the graphing calculator to see what it looks like, but you might realize from the equation that the vertex will be at (-3, 0) and the slopes of the two sides will be -2 and 2.

More to come. Comments and questions welcome.

More Regents problems.

Algebra Problems of the Day (Integrated Algebra Regents, January 2013)

While I'm waiting for new Regents exams to come along, I'm revisiting some of the older NY Regents exams.

More Regents problems.

Administered January 2013

Part I: Each correct answer will receive 2 credits.


26. Given:

A = {perfect square integers from 4 to 100, inclusive}
B = {16, 36, 49, 64}
The complement of set B in the universal set A is

(1) {9, 25, 81}
(2) {4, 9, 25, 81, 100}
(3) {1, 4, 9, 25, 81, 100}
(4) {4, 16, 36, 49, 64, 100}

Answer: (2) {4, 9, 25, 81, 100}

If A is the set of perfect square integers between 4 and 100, inclusive (meaning 4 and 100 are part of the set), then the complement of B in set A would be all the perfect squares between 4 and 100 that are NOT in set B.

Choice (1) does not include 4 and 100, so it is eliminated.

Choice (3) includes 1, but 1 is not an element of set A, so it is eliminated.

Choice (4) includes all the members of set B, along with 4 and 100. Eliminate it.

27. The expression (2x² + 10x - 28) / (4x + 28) is equivalent to

(1) (x - 2) / 2
(2) x - 1
(3) (x + 2) / 2
(4) (x + 5) / 2

Answer: ((1) (x - 2) / 2

Factor the numerator and denominator and cross out common factors. (2x² + 10x - 28) / (4x + 28)
= ( (2)(x² + 5x - 14) ) / ( (4)(x + 7) )
= ( (2)(x + 7)(x - 2) ) / ( (4)(x + 7) )
= ( (x - 2) ) / ( (2) )



28. Which value of x is the solution of the equation 1/7 + 2x / 3 = (15x - 3) / 21

(1) 6
(2) 0
(3) 4 / 13
(4) 6 / 29

Answer: (1) 6

You can combine the fractions and cross-multiply, or you can multiply the entire equation by 21 to get rid of all the denominators: [ 1/7 + 2x / 3 = (15x - 3) / 21 ] (21)
(21)(1/7) + (21)(2x/3) = ((15x - 3) / 21 ) (21)
3 + 14x = 15x - 3
6 = x

You could all try the choices to see which one works. It's fairly obvious that 0 incorrect ( 1/7 =/= -1/7), and 6 is the correct choice. The only problem is if the answer had been one of the fractions -- then plugging in values might've been a problem.

29. Which statement is true about the data set 4, 5, 6, 6, 7, 9, 12?

(1) mean = mode
(2) mode = median
(3) mean < median
(4) mode > mean

Answer: (2) mode = median

The median and the mode are both 6, because 6 is the middle number of the 7 listed, and there are two of them.

The mean is the Sum / N, and N = 7. The Sum is 49, and 49 / 7 = 7, so the mean = 7.

The mean =/= mode. Eliminate (1)

The mode = median. This is the answer.

The mean is greater than the median, not less. Elinimate (3).

The mode is less than the mean, not greater. Eliminate (4).

Vertical lines, which are parallel to the y-axis, will have equations in the form x = a, like choices (1) and (2).

30. How is the graph of y = x² + 4x + 3 affected when the coefficient of x² is changed to a smaller positive number?

(1) The graph becomes wider, and the y-intercept changes.
(2) The graph becomes wider, and the y-intercept stays the same.
(3) The graph becomes narrower, and the y-intercept changes
(4) The graph becomes narrower, and the y-intercept stays the same.

Answer: (2) The graph becomes wider, and the y-intercept stays the same.

The coefficient of x² determines whether the parabola opens up or down and how narrow or wide the parabola will be. It has nothing to do with the y-intercept, which is determined by the constant (in this case, + 3).

Making the coefficient smaller, but still positive, means that the parabola will become wider as the y values will not increase as quickly.


End of Part I.



More to come. Comments and questions welcome.

More Regents problems.

Writing: What Was the Inspiration for "Familiar Feeling"?

The following appeared on my blog on my Good Reads Author page.

A common question from readers and writing articles is: where do you get your ideas, and what inspired you to write that?

I've been thinking about those questions as I try to approach my next Big Idea. When you're writing for yourself, you can scribble a few paragraphs whenever an idea hits you and revisit it whenever you want. But if I want to sell something that I write, I have to be a little more focused with the ideas I develop.

And so I thought I could revisit some of my prior stories. Where did they come from?

The lead-off story in my book In A Flash 2020 is "Familiar Feeling", which takes place at the Carrowmore School for Magic and Wizardry. Honestly, I wasn't trying to recreate Harry Potter or Hogwarts. On the other hand, I did want enough details that you could imagine walking up to the front steps or strolling down to the lake.

The story came from the title as much as the other way around. Many years ago, playing Dungeons & Dragons, a group of us started a new campaign. Just about everyone was a multi-class character, so there were several magic-users. Probably in the first session, one character said he was summoning a familiar, and before you knew it, every mage had a bird or brownie or something tagging along.

And then there wasn't much more to it, and they didn't get mentioned a lot after.

I wanted to write a story about a wizard and his or her familiar, but then I thought about the experience of summoning one, and how there could be more to it. It should be a big deal. And it would be a bigger deal if a group of mages were to have a special summoning ceremony.

From this came the idea of having it of having it at a school, the details of which were still to come. I thought about a shy boy named Thomas who summons something really special and shades of Rudoph when all of the others like him now (maybe nothing that extreme). And in creating this, I mentioned some of the other animals, including one girl's fawn (with a "w", baby dear).

Suddenly, I was more interested in her, and rewrote the story of Diana, and her uncle who had been a forest ranger with a bear cub familiar, and the pieces were falling into place.

If you haven't read the story, don't worry. The above isn't a spoiler. In the end, after all the rewriting, the fawn was changed in favor of something a little more, let's say, "magical".

But now I have this school named after a group of monoliths in County Sligo, Ireland, along with a bunch of professors who are more than just pieces of cardboard with names. This is definitely a place I want to revisit, telling more stories about the students who pass through its doors, or one great novel about some major event that happens there one year.

And whenever that happens, you'll know what inspired that!

Algebra Problems of the Day (Integrated Algebra Regents, January 2013

While I'm waiting for new Regents exams to come along, I'm revisiting some of the older NY Regents exams.

More Regents problems.

Administered January 2013

Part I: Each correct answer will receive 2 credits.


21. If x = -3, what is the value of |x - 4| - x²?

(1) -8
(2) -2
(3) 7
(4) 16

Answer: (2) -2

|-3 - 4| - (-3)² = | -7 | - 9 = 7 - 9 = -2

Remember your Order of Operations. Absolute Value is a grouping symbol, like parentheses.

If you got 16 as your answer, you forgot to put parentheses around the -3 before you squared it.

22. Which equation represents a line parallel to the line whose equation is 2x - 3y = 9?

(1) y = 2/3 x - 4
(2) y = -2/3 x + 4
(3) y = 3/2 x - 4
(4) y = -3/2 x + 4

Answer: (1) y = 2/3 x - 4

Parallel lines have the same slope, and the choices are all written in slope-intercept form. So find the slope of the given line.

The given line is in Standard Form Ax + By = C, and the slope of a line written this way is -A/B.

If you weren't aware of that, or didn't remember, you could just rewrite the equation, solving for y in terms of x.

2x - 3y = 9
-3y = -2x + 9
y = 2/3 x - 3

The slope is 2/3.

Note that Choice (4) would be a perpendicular line. The other two would intersect the given line, but are not perpendicular to it.

23. Which ordered pair is in the solution set of the system of inequalities y < 3x + 1 and x - y > 1?

(1) (-1, -2)
(2) (2, -1)
(3) (1, 2)
(4) (-1, 2)

Answer: (2) (2, -1)

You could either plug these in, or put both inequalities into the graphing calculator. (You have to rewrite the second inequality.)

-2 ?< 3(-1) + 1 = -2, true, but -1 - (-2) = 1 ?> 1 is false.

-1 ?< 3(2) + 1 = 7, true, and 2 - (-1) = 3 ?> 1 is true. This is the solution.

2 ?< 3(1) + 1 = 4, true, but 1 - (2) = -1 ?> 1 is false.

2 ?< 3(-1) + 1 =-2, is false, and -1 - (2) = -3 ?> 1 is false.

24. Which equation represents the line that passes through the point (-3,4) and is parallel to the x-axis?

(1) x = 4
(2) x = -3
(3) y = 4
(4) y = -3

Answer: (3) y = 4

Lines parallel to the x-axis are horizontal and have a slope of 0, which means that they have an equation of the form y = b. Since it goes through (-3, 4), it must be y = 4.

Vertical lines, which are parallel to the y-axis, will have equations in the form x = a, like choices (1) and (2).

25. A cube with faces numbered 1 through 6 is rolled 75 times, and the results are given in the table below.

(1) P(odd) <P(even)
(2) P(3 or less) < P(odd)
(3) P(even) < P(2 or 4)
(4) P(2 or 4) < P(3 or less)

Answer: (4) P(2 or 4) < P(3 or less)

Calculate the experimental probabilities and compare the results.

P(odd) = (7 + 14 + 20) / 75 = 41 / 75, which is more than half, so there is no reason to calculate P(even) which must be 1 - 41/75. Eliminate Choice (1).

P(3 or less) = (7 + 22 + 14) / 75 = 43 / 75, which is more than P(odd), or 41/75. Eliminate (2).

P(even) = 1 - 41/75 = 34/75. P(2 or 4) = (22 + 6) / 75 = 28 / 75. Eliminate (3).

P(2 or 4) = 28/75. P(3 or less) = 43/75. This is the correct choice.






More to come. Comments and questions welcome.

More Regents problems.

How Many Launches?

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(C)Copyright 2021, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

Assume a giant space amoeba doesn't swallow up the entire system first.

Solutions are left as an exercise for the reader.

Nah, that's not a copout. I'm just trying to get more comments.

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.

Come back often for more funny math and geeky comics.

Algebra Problems of the Day (Integrated Algebra Regents, January 2013)

While I'm waiting for new Regents exams to come along, I'm revisiting some of the older NY Regents exams.

More Regents problems.

Administered January 2013

Part I: Each correct answer will receive 2 credits.


16. What is 24x²y⁶ - 16x⁶y² + 4xy² divided by 4xy²?

(1) 6xy⁴ + 4x⁵
(2) 6xy⁴ + 4x⁵ + 1
(3) 6x²y³ + 4x⁶y
(4) 6x²y³ + 4x⁶y + 1

Answer: (2) 6xy⁴ + 4x⁵ + 1

When you divide a trimonimal by a monomial, the result will still be a trinomial. You will not reduce the number of terms. So Choices (1) and (3) are eliminated.

When you divide 4xy² by 4xy², the result is 1, not 0, so the answer must end with + 1.

The rule for division of variables with exponents is that you subtract the exponent values: if you have six factors of y and you divided by two factors of y, then four factors of y would remain, which would be y⁴.

You do NOT divide the exponents as is the case in Choices (3) and (4).

17. Which expression can be used to change 75 kilometers per hour to meters per minute?

Answer: (3) (75 km) / (1 hr) X (1,000m) / (1 km) X (1 hr) / (60 min)

Units can be canceled in fractions the same way that factors can. If the same unit appears in both the numerator and the denominator, it is divided by itself with a result of 1 (that is, it goes away). On the other hand, if the same unit appear twice on the top or the bottom of a fraction, it gets multiplied and becomes squared (that is, it doesn't go away).

You want kilometers and hours to go away, leaving you with just meters on top and minutes on the bottom.

Choices (1) and (2) have km on top twice and meters on the bottom. Eliminate them.

Choice (4) has hr on the bottom twice and minutes on the top. Eliminate it.

18. The inequality -2 < x < 3 can be written as

(1) (-2, 3)
(2) [-2, 3)
(3) (-2, 3]
(4) [-2, 3]

Answer: (4) [-2, 3]

If should be obvious that since the inequality symbol < was used twice that the two brackets should be the same.

Since they both say less than or equal to, the square brackets are used to indicate that the numbers shown are part of the solution.

19. The expression (6 X 10^-7) / (3 X 10^-3) is equivalent to

(1) 2 X 10⁴
(2) 2 X 10¹⁰
(3) 2 X 10^-4
(4) 2 X 10^-10

Answer: (3) 2 X 10^-4

Did you notice the parentheses I added to the problem so I could type it out without any ambiguity to the problem? You need those parentheses if you are using a calculator to solve this but its Operating System does not have actual fractions in the display window.

If you didn't include the parentheses in the denominator, you would have gotten an incorrect solution becuase the Order of Operations would take over, and you might never realize the mistake you made.

That said, there is no reason to use a scientific or graphing calculator for this problem!

The factors are lined up nicely for you: 6 / 3 = 2 and 10^-7 / 10^-3 = 10^{(-7 - -3)}. The rule is to subtract the exponents, and -7 - -3 = -4, so it will be 10^-4.

20.The roots of the equation x² - 14x + 48 = 0 are

(1) -6 and -8
(2) -6 and 8
(3) 6 and -8
(4) 6 and 8

Answer: (4) 6 and 8

The factors of x² - 14x + 48 are (x - 6) and (x - 8).

If x - 6 = 0 then x = 6.

If x - 8 = 0 then x = 8.

Note that the final term is + 48, so you should have known immediately that the two roots were both positive or both negative and eliminated Choices (2) and (3).






More to come. Comments and questions welcome.

More Regents problems.