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.

How Many Lunches?

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

What? No appetizers or antipasto?

For those unable to read the fine print on the menu (sorry, but it had to shrink to fit), the questions asks, "How many combinations of a main course, a vegetable, a side and a dessert can be made from this menu?"

The Counting Principle tells us that the number of possible combinations of one item from each list is equal to the product of the number of items in each list. I did a video on this for a grad school class. When I showed it a several years later, my AP remarked that I looked the same. When I showed it in a different school three years later, my coteacher said, "Mr. Burke, you look so young!" I thnk that was meant as a compliment.

In this particular instance, however, there are 3 choices for a main course, despite what you think of the pizza or the tuna, 2 choices for a vegetable, which no one said had to taste good to be good for you, 4 desserts, and 3 beverages. According to the Counting Principle, that means that there are 3 * 2 * 4 * 3 = 72 choices.

If you found this page by googling a question on your state exam, please not that I altered the question to suit this comic. Your teacher may be expected a different answer than I have provided.

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.


11. Which verbal expression is represented by 2(x + 4)?

(1) twice the sum of a number and four
(2) the sum of two times a number and four
(3) two times the difference of a number and four
(4) twice the product of a number and four

Answer: (1) twice the sum of a number and four

Twice means you are multiplying something by two. Sum means that you are adding two things. Twice the sum means that you have to add something first before multiplying it.

Choice (2) would be 2x + 4.

Choice (3) would be 2(x - 4).

Choice (4) would be 2(4x).

12. How many cubes with 5-inch sides will completely fill a cube that is 10 inches on a side?

(1) 50
(2) 25
(3) 8
(4) 4

Answer: (3) 8

You can fit two smaller cubes side by side and front to back. That means you can have a layer of four cubes. You also have a second layer of four cubes on top of that for a total of 8.

Mathematically, (2)³ = 8.



13. A school newspaper will survey students about the quality of the school’s lunch program. Which method will create the least biased results?

(1) Twenty-five vegetarians are randomly surveyed.
(2) Twenty-five students are randomly chosen from each grade level.
(3) Students who dislike the school’s lunch program are chosen to complete the survey.
(4) A booth is set up in the cafeteria for the students to voluntarily complete the survey.

Answer: (2) Twenty-five students are randomly chosen from each grade level.

You want to choose students at random from across the entire student population.

Choice (1) specifically targets vegetarians, who would likely have a bias toward the school lunch program.

Choice (3) is selecting people who dislike the menu in a survey about the quality of the menu, and that is a bias.

Choice (4) is self-selecting, which is biased because people with stronger feelings are more likely to take part in it.

14. The vertex of the parabola y = x² + 8x + 10 lies in Quadrant

(1) I
(2) II
(3) III
(4) IV

Answer: (3) III

You can quickly graph it and see that it is in Quadrant III.

Or you could find that the Axis of Symmetry is x = -b/(2a) = -8/2 = -4. And when x = -4, y = (-4)² + 8(-4) + 10 = -6. The vertex is (-4, -6), which is in Quadrant III.



15.In the figure below, ABCD is a square and semicircle O has a radius of 6.

What is the area of the figure?

(1) 36 + 6π
(2) 36 + 18π
(3) 144 + 18π
(4) 144 + 36π

Answer: (3) 144 + 18π

The area of the semicircle is 1/2 π r², which is 1/2 π (6)² = 1/2 π (36) = 18π. At this point, you can eliminate (1) and (4).

The area of the square is s² and the side is equal to the diamater of the semicircle, which is twice the radius. So A = (12)² = 144.






More to come. Comments and questions welcome.

More Regents problems.

Doing Some Writing: Writing Prompts

Occasionally, I troll reddit's subgroup for "Writing Prompts", which I've mentioned in the past. I've even blogged about it before. I'm writing more of them now, but the same issue remains. By the time I respond, the prompts have slipped from view, so my efforts don't (to my knowledge) get read a lot, which means that they don't get a lot of votes.

If you're interested in some of my creative writing, here are some of my more recent attempts. Feel free to comment here or there. If you're not a "redditor", you'll need to join before voting or commenting. (It's free.) Comments on my blog are always welcome -- unless they're spam, of course.

Recent Writing Prompts on reddit

The Boss of Me
[WP] You find yourself in a room of "you"s at every age from 1 to now. On the wall it says "Who's in Charge? To end the game, all must agree, by either force or diplomacy." You all turn to face each other. 5 year old you pulls at your shirt, "So what happens now?" https://www.reddit.com/r/WritingPrompts/comments/nyv6ua/wp_you_find_yourself_in_a_room_of_yous_at_every/h1nh04l/?utm_source=reddit&utm_medium=web2x&context=3

Cartoon Physics ... featuring a very generic mighty nimrod of a hunter
[SP] "That isn't how physics work" https://www.reddit.com/r/WritingPrompts/comments/nwi880/sp_that_isnt_how_physics_work/h19uyue/?utm_source=reddit&utm_medium=web2x&context=3

What the Key Unlocks -- I extended it because I was asked to, but going beyond that would require a complete overhaul
[WP] There's a door with a single key hole - it will open regardless of what key is used. All keys open this door, but what's on the other side, however, entirely depends on the key. https://www.reddit.com/r/WritingPrompts/comments/nwk8t4/wp_theres_a_door_with_a_single_key_hole_it_will/h1a66oh/?utm_source=reddit&utm_medium=web2x&context=3

Unable to Find Myself. Oddly enough, I've had ideas about finding parallel worlds which were alike, but all the others had this one different that ours doesn't have.
[WP] Scientist have created a machine that allow people a window into alternate realities. It becomes mainstream and people talk about alternate versions of themselves. Finally you decide to take a look only to discover that there are no alternate versions of you. You're the only you in existence. https://www.reddit.com/r/WritingPrompts/comments/nqhyz7/wp_scientist_have_created_a_machine_that_allow/h0c2p0a/?utm_source=reddit&utm_medium=web2x&context=3

Making the Most of a Small Sacrifice. This one got some legs because I deliberately included a phrase that people could latch onto or mock. It got their attention.
[WP] The ritual calls for 100 sacrifices, but reading carefully you realize it never specified they had to be human. Deciding to be a smartass, you got a petri dish full of bacteria and sacrificed that instead. https://www.reddit.com/r/WritingPrompts/comments/nv3y8v/wp_the_ritual_calls_for_100_sacrifices_but/h11xdfa/?utm_source=reddit&utm_medium=web2x&context=3

The Four-Color Peacock. I like my superhero stories
[WP] In a world of super villians who never monologue about their plans and heroes that never wait until the last moment to save the day, you are the only one to have any sense of theatrics. https://www.reddit.com/r/WritingPrompts/comments/nld7sw/wp_in_a_world_of_super_villians_who_never/gzixoyg/?utm_source=reddit&utm_medium=web2x&context=3

If you want to see more, LET ME KNOW! My tagline for this blog years ago was "I THRIVE ON FEEDBACK". Maybe I should resurrect that.

(x, why?) Mini: Vertical-Line Test

(Click on the comic if you can't see the full image.)

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

Yes, a vertical line will automatically fail the test named for it. And a North wind blows South, so there you go.

Happy Name Day to all the Anthonys and Antoinettes and other variations on this fine Feast Day of St. Anthony. If you ever lost something, you say a prayer to St. Anthony to help you find it. Except for a cause -- the patron saint of lost causes is St. Jude.

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.


6. The expression 100n² - 1 is equivalent to

(1) (10n + 1)(10n - 1)
(2) (10n - 1)(10n - 1)
(3) (50n + 1)(50n - 1)
(4) (50n - 1)(50n - 1)

Answer: (1) (10n + 1)(10n - 1)

The Difference of Squares Rules: when you have two terms which are perfect squares with a subtraction operation, then the expression factors into two conjugates where each term is a square root of one of the terms in the original expression. "Conjugates" means that the two binomials will be the same, except that one will have addition and the other will have subtraction. The result of this is when you multiply two conjugates, the "inner" and "outer" terms will create a zero pair and go away, leaving only two terms in the quadratic expression.

7. In right triangle ABC shown below, what is the value of cos A?

(1) 12/20
(2) 16/20
(3) 20/12
(4) 20/16

Answer: (2) 16/20

Cosine can never be greater than 1, so Choices (3) and (4) are eliminated immediately.

Since the Cosine is the side adjacent to the angle divided by the hypotenuse (the longest side, which makes the fraction less than 1), the ratio is 16/20.



8. A bag contains five green gumdrops and six red gumdrops. If Kim pulls a green gumdrop out of the bag and eats it, what is the probability that the next gumdrop she pulls out will be red?

(1) 5/11
(2) 5/10
(3) 6/11
(4) 6/10

Answer: (4) 6/10

There had been 11 gumdrops in the bag. One green was removed and not replaced, (at least we hope it wasn't replaced since it was eaten). That leaves 10 gumdrops in the bag, 6 of which are red.

The probability of taking out a red one next is therefore 6/10.

Choice (1) is the probability of green if the first green had been replaced.

Choice (2) is the probability of red if a red gumdrop had been removed and not replaced first.

Choice (3) is the probability of red if the first gumdrop had been replaced.

9. Which graph represents a function?

(see image)

Answer: (3)

A function can only have one y value for any given x value. That means that a function must past the Vertical Line Test -- if a Vertical Line can be drawn so that it passes through two points on the line, then the line cannot be a function.

Choice (1) is a circle which fails the Vertical Line Test.

Choice (2) is an actual Vertical Line, so it obviously fails the test.

Choice (4) shows every point from Quadrant II reflected in Quadrant III, so it fails the test.



10. The current population of a town is 10,000. If the population, P, increases by 20% each year, which equation could be used to find the population after t years?

(1) P =10,000(0.2)^t
(2) P =10,000(0.8)^t
(3) P =10,000(1.2)^t
(4) P =10,000(1.8)^t

Answer: (3) P =10,000(1.2)^t

If the population is increasing, that is growth, which requires a rate of greater than 1, so Choices (1) and (2) are eliminated because they show decay.

If it is growing an additional 20%, then next year there will be 120% of the current population, and 120% is 1.2 as a decimal.

Choice (4) is just there to match up with Choice (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.


1. The number of hours spent on math homework during one week and the math exam grades for eleven students in Ms. Smith’s algebra class are plotted below.

Based on the plotted data, what is the correlation between the time spent on homework and the exam grade?

(1) positive
(2) negative
(3) no correlation
(4) cannot be determind

Answer: (1) positive

The data is going up and to the right. If you draw a trend line through the data, it would have a positive slope.

2. A car uses one gallon of gasoline for every 20 miles it travels. If a gallon of gasoline costs $3.98, how much will the gas cost, to the nearest dollar, to travel 180 miles?

(1) 9
(2) 36
(3) 45
(4) 80

Answer: (2) 36

One gallon of gas for every 20 miles, then 180 miles would require 180 / 20 = 9 gallons of gas.

Each gallon is $3.98, which is approximately $4, and 4 * 9 = 36, which is Choice 2.



3. If Angelina’s weekly allowance is d dollars, which expression represents her allowance, in dollars, for x weeks?

(1) dx
(2) 7dx
(3) x + 7d
(4) d / x

Answer: (1) dx

Her allowance is per week and the number 7 (days in a week) plays not part in this problem. She earns d dollars in 1 week, 2d dollars in 2 weeks, and so on until she has xd, or dx, dollars after x weeks.

4. What is the solution of the system of equations shown in the graph below?

(1) (1, 0) and (-3, 0)
(2) (0, -3) and (0, -1)
(3) (-1, -2)
(4) (-2, -1)

Answer: (3) (-1, -2)

Choice (1) are the two x-intercepts. Choice (2) are the two y-intercepts. Choice (3) is the point where the two lines intersect, which is the solution to the system of equations. Choice (4) is the point with the reverse co-ordinates of the solution if you got the x and y order incorrect.



5. The solution of the equation 5 - 2x = -4x - 7 is

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

Answer: (4) -6

5 - 2x = -4x - 7
-2x = -4x - 12
2x = -12
x = -6

You also could have plug in the values to find the correct one, but in this case, you would have had to have tried all four numbers to get the correct one.

5 - 2(1) = 3, -4(1) - 7 = -11

5 - 2(2) = 1, -4(2) - 7 = -15

5 - 2(-2) = 9, -4(-2) - 7 = 1

5 - 2(-6) = 17, -4(-6) - 7 = 17.






More to come. Comments and questions welcome.

More Regents problems.

Logged In

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

LoggedIn would be a great numerical social network. Ln for short.

Did you ever notice that whenever no one can log in, there's always someone who knows something and doesn't have a problem.

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, June 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 June 2013

Part IV: Each correct answer will receive 4 credits. Partial credit is possible.


37. Solve algebraically:

2 / 3x + 4 / x = 7 / (x + 1)

[Only an algebraic solution can receive full credit.]

Answer:

You need to eliminate the denominators however you can. You could multiply each side by (3x) and (x + 1), for instance. After that, you will have a simpler equation to solve.

You can also combine the two fractions on the left, leaving you with a proportion, and then you can cross-multiply, which amounts to the same thing as above.

2 / 3x + 4 / x = 7 / (x + 1)
2 / 3x + 12 / 3x = 7 / (x + 1)
14 / 3x = 7 / (x + 1)
(cross-multiply)
21x = 14 (x + 1)
21x = 14x + 14
7x = 14
x = 2

Checking your answer: (not required, but a good idea)
2 / (3(2)) + 4 / 2 ?= 7 / (2 + 1)
2/6 + 2 = 7/3
1/3 + 2 = 7/3
7/3 = 7/3 (check)

35.A jar contains five red marbles and three green marbles. A marble is drawn at random and not replaced. A second marble is then drawn from the jar.

Find the probability that the first marble is red and the second marble is green.

Find the probability that both marbles are red.

Find the probability that both marbles are the same color.

Answer:

The total number of marbles for the first draw is 8 and for the second draw is 7. These are dependent events.

P(R, G) = 5 / 8 * 3 / 7 = 15/56

P(R, R) = 5/ 8 * 4 / 7 = 20/56

P(both the same) = P(R, R) + P(G, G) = 20/56 + 3/8 * 2/7 = 20/56 + 6/56 = 26/56.

The second and third answers could be simplified, but didn't need to be. Also, tree diagrams would have been acceptable for showing work.

39.In the diagram below of rectangle AFEB and a semicircle with diameter CD, AB = 5 inches, AB = BC = DE = FE, and CD = 6 inches. Find the area of the shaded region, to the nearest hundredth of a square inch.

Answer:

First, since it is a rectangle, then side AF = BC + CD + DE = 5 + 6 + 5 = 16. Second, the radius of the semicircle is half of CD, which would be 3.

The area of the semicircle is 1/2 π r² = 1/2 (3.141592)(3)² = 14.137164.

The area of the rectangle is (5)(16) = 80

The shaded region is the area of the rectangle minus the area of the semicircle: 80 - 14.137164 = 65.862836, or 65.86 to the nearest hundredth.

Note that if you used 3.14 for pi, you would not have had enough accuracy. Use the pi key on your calculator.

End of Part IV



More to come. Comments and questions welcome.

More Regents problems.

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

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

More Regents problems.

Administered June 2013

Part III: Each correct answer will receive 3 credits. Partial credit is possible.


34. The menu for the high school cafeteria is shown below.

Determine the number of possible meals consisting of a main course, a vegetable, a dessert, and a beverage that can be selected from the menu.

Determine how many of these meals will include chicken tenders.

If a student chooses pizza, corn or carrots, a dessert, and a beverage from the menu, determine the number of possible meals that can be selected.

Answer:

There are three parts to this question and each is worth a point. Some work needs to be shown. If you just write the three numbers, you will only get 1 point total.

Part 1: Use the Counting Principle to find the number of possible meals, selecting one item from each category. That is, multiply the number of items in each column. So the total is 5 * 3 * 5 * 3 = 225.

Part 2: If we only include chicken tenders from the firt column, then we would multiply 1 * 3 * 5 * 3 = 45.

Part 3: There is one main course, two side dishes, five desserts and three beverages, so 1 * 2 * 5 * 3 = 30.

Note that if you wrote the answer as a decimal instead of a radical, you would have scored 0 points.

35.A man standing on level ground is 1000 feet away from the base of a 350-foot-tall building. Find, to the nearest degree, the measure of the angle of elevation to the top of the building from the point on the ground where the man is standing

Answer:

The distance along the ground is adjacent to the angle of elevation. The height of the building is opposite to the angle of elevation. Opposite and adjacent means that we need to use the Tangent ratio.

Tan x = opp / adj
Tan x = 350 / 1000
x = tan^-1 (350 / 1000)
x = 19.29...

The angle of elevation is 19 degrees.

Make sure the calculator is measuring degrees and not radians (which is the default).

36. Express √25 - 2√3 + √27 + 2√9 in simplest radical form.

Answer:

Take the square root of the terms which are perfect squares. Find the largest perfect square which is a factor of the other terms so you can simplify. √25 - 2√3 + √27 + 2√9
= 5 - 2√3 + √27 + 2(3)
= 5 - 2√3 + √9√3 + 6
= 5 - 2√3 + 3√3 + 6
= 11 + √3





End of Part III



More to come. Comments and questions welcome.

More Regents problems.

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

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

More Regents problems.

Administered June 2013

Part II: Each correct answer will receive 2 credits. Partial credit is possible.


31 Solve the inequality -5(x - 7) < 15 algebraically for x.

Answer:

-5(x - 7) < 15
(x - 7) > -3
x > 4

Note that you could have used the Distributive Property in the first line, but that would have added an extra step. Also when you divide by a negative number, you flip the direction of the inequality symbol.

One mistake would cost 1 of the 2 credits. Two mistakes score 0. A solution without work is only 1 credit.

32. Oatmeal is packaged in a cylindrical container, as shown in the diagram below.

The diameter of the container is 13 centimeters and its height is 24 centimeters. Determine, in terms of π, the volume of the cylinder, in cubic centimeters.

Answer:

The Volume of a cylinder is V = πr²h
V = π (6.5)²(24) = 1014π

The diameter is 13, so the radius is half that, which is 6.5.

One mistake, such as forgetting to halve the denominator, would cost 1 of the 2 credits. Two mistakes score 0. A solution without work is only 1 credit.

33. The distance from Earth to Mars is 136,000,000 miles. A spaceship travels at 31,000 miles per hour. Determine, to the nearest day, how long it will take the spaceship to reach Mars.

Answer:

Divide the distance (miles) by the speed (miles per hour) to get the number of hours. Divide that by 24 hours/day to get the number of days.

136000000 / 31000 / 24 = 182.79...

The trip would take 183 days.

One mistake, such as forgetting to divide by 24, would cost 1 of the 2 credits. Two mistakes score 0. A solution without any work shown is only 1 credit.

End of Part II



More to come. Comments and questions welcome.

More Regents problems.

Bonus Comic: Calendar Math

Welcome to everyone who found my blog through the daily math calendar. My answer for the June 6, 2021 puzzle appeared on Twitter today. (Link is coming -- I can't do Twitter at work, except on my phone. And I can't update the blog on my phone.)

For my blog readers, this is the comic that appeared today. The question being asked in the puzzle is in the first panel. The answer is the date (or, I should say, yesterday's date).

You can find more Comics by scrolling down, or clicking on the "Comic" tag.

You can also find them at Comic Genesis but that site is a couple months out of date.

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

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

More Regents problems.

Administered June 2013

Part I: Each correct answer will receive 2 credits.


26. If the roots of a quadratic equation are -2 and 3, the equation can be written as

(1) (x - 2)(x + 3) = 0
(2) (x + 2)(x - 3) = 0
(3) (x + 2)(x + 3) = 0
(4) (x - 2)(x - 3) = 0

Answer: (2) (x + 2)(x - 3) = 0

Since the signs of the two solutions are different, you can eliminate the two choices where the signs are the same. That is, Choices (3) and (4) are gone.

If 3 is a root, then it is a "zero" of the equation and (x - 3) must equal 0. Likewise, -2 implies that (x - (-2)) or (x + 2) = 0

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

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

Answer: (2) x = 4

Parallel to the y-axis means that it is a vertical line. Vertical lines have equations such as x = a. The x-coordinate is 4, so x = 4 is the equation of the line.



28. There are 18 students in a class. Each day, the teacher randomly selects three students to assist in a game: a leader, a recorder, and a timekeeper. In how many possible ways can the jobs be assigned?
(1) 306
(2) 816
(3) 4896
(4) 5832

Answer: (3) 4896

There are 18 * 17 * 16 ways to select the three positions. Order matters, so this is a permutation problem, not a combination problem.

THere are 18 ways to pick the first assistant, 17 choices for the second, and 16 for the third. The Counting Principle says to multiple all three of them.

29. In triangle RST, angle R is a right angle. If TR = 6 and TS = 8, what is the length of RS?

(1) 10
(2) 2
(3) 2 SQRT(7)
(4) 7 SQRT(2)

Answer: (3) 2 SQRT(7)

If R is the right angle, the line TS, which is 8, is the hypoetenuse, and RS, which must be shorter than 8, is another leg.

This is NOT a 6-8-10 right triangle.

6² + RS² = 8²
36 + RS² = 64
RS² = 28
RS = SQRT(28) = SQRT(4)*SQRT(7)
RS = 2 SQRT(7)

If you didn't know how to simplify SQRT(28), you could have gotten a decimal from your calculator, and then checked it against Choices (3) and (4) in your calculator.

Note that 10 is longer than the hypotenuse, but a likely mistake to make. Choice (2) can be eliminated because 2, 6, 8 does NOT make a triangle. It makes a striaght line.

30. How many solutions are there for the following system of equations? y = x² - 5x + 3
y = x - 6

(1) 1
(2) 2
(3) 3
(4) 0

Answer: (1) 1

First of all, you cannot have 3 solutions to a quadratic-linear system of equations. The line can intersect the curve at two points, at one point, or not intersect at all.

You can put the two equations into you graphing calculator and look to see the answer.

Or you can solve the equation. Again, you don't need to know the solutions, just how many there are.

x² - 5x + 3 = x - 6
x² - 6x + 9 = 0
(x - 3)(x - 3) = 0
x - 3 = 0
x = 3
This is one solution.

This was a simple quadratic to solve. However, had it been more complicated, you might have chosen to check the discriminant, b² - 4ac, to see if it was positive, zero, or negative.

In this case, (-6)² - 4(1)(9) = 36 - 36 = 0, so there is one solution.

End of Part I



More to come. Comments and questions welcome.

More Regents problems.

Just Kant

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

Idealistically, maybe you could narrow it down to a few greats. Some just Kant.

One could be fairly certain that Ken had some idea in advance what Mr. Ibsen, the History teacher, would reply before the setup. Mr. Ibsen has appeared a few times before reminds me of a Philosphy professor from college, not that he was designed that way. And if I found a picture of him (from that time period) and compared it, they'd probably look nothing alike, but somehow they're linked in my head.

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.

BOGO

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

I mean, shouldn't it be B1G1?

That would also make it easier to play around with the pricing, according to what the market brings: B2G1, B3G3, etc.

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.

Manifold

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

Many men can be more successful if they learn to stop talking.

I was going to say witchcraft ... but it's not polite to imply that one's spouse a witch unless she (or he) identifies as such. Plus, it didn't fit.

But folding fitted sheets usually requires a wand or a wiggling of one's nose.

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, June 2013)

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

More Regents problems.

Administered June 2013

Part I: Each correct answer will receive 2 credits.


21. Carol plans to sell twice as many magazine subscriptions as Jennifer. If Carol and Jennifer need to sell at least 90 subscriptions in all, which inequality could be used to determine how many subscriptions, x, Jennifer needs to sell?

(1) x > 45
(2) 2x > 90
(3) 2x - x > 90
(4) 2x + x > 90

Answer: (4) 2x + x > 90
If Jennifer needs to sell at least x, and Carol plans to sell twice as many as Jennifer, which would be 2x, and together they needed to sell more than 90, then the total could be written as 2x + x > 90.

Hopefully you noticed that Choices (1) and (2) are equivalent so neither could be the answer. Jennifer isn't selling half of the 90 subscriptions if Carol is selling twice as many. She's only selling one third.

22. When 2x² - 3x + 2 is subtracted from 4x² - 5x + 2, the result is

(1) 2x² - 2x
(2) -2x² + 2x
(3) -2x² - 8x + 4
(4) 2x² - 8x + 4

Answer: (1) 2x² - 2x

If 2x² - 3x + 2 is subtracted from 4x² - 5x + 2, then you need to calculate
(4x² - 5x + 2) - (2x² - 3x + 2) which is
(4x² - 2x²) + (-5x - (-3x)) + (2 - 2) which is
2x² + (-2x)) + 0

If you didn't put the "from" expression first (I took three dollars from the fifteen in my pocket), you would have gotten the addition inverse, Choice (2), which is incorrect. Choices (3) and (4) happen when you add the 2nd and 3rd terms instead of subtract.

23. Which expression represents the number of hours in w weeks and d days?
(1) 7w + 12d
(2) 84w + 24d
(3) 168w + 24d
(4) 168w + 60d

Answer: (3) 168w + 24d

The number of hours in d days is 24d. The number of hours in w weeks is the same as in 7w days. Since there are 24 hours in one day, that becomes 7(24)w hours, which is 168w.

24. Given:

R = {1, 2, 3, 4}
A = {0, 2, 4, 6}
P = {1, 3, 5, 7} What is R ∩ P?

(1) {0, 1, 2, 3, 4, 5, 6, 7}
(2) {1, 2, 3, 4, 5, 7}
(3) {1, 3}
(4) {2, 4}

Answer: (3) {1, 3}

The Intersection consists of all the elements of R that are also elements of P. Only 1 and 3 are present in both sets.

Thankfully, this is no longer on the Algebra 1 regents exam, not until they change the curriculum again.

Choice (1) is the Union of all three sets R ∪ A ∪ P.

Choice (2) is the Union of R ∪ P.

Choice (4) is the Intersection of R ∩ A.

25. Which equation could be used to find the measure of angle D in the right triangle shown in the diagram below?

(1) cos D = 12 / 13
(2) cos D = 13 / 12
(3) sin D = 5 / 13
(4) sin D = 12 / 13

Answer: (4) y < x - 1
This isn't in Algebra 1 any more, but it is in Geometry.

The sine (sin) ratio is the length of the side opposite the angle divided by the length of the hypotenuse. The cosine (cos) ratio is the length of the side adjacent to the angle divided by the length of the hypotenuse.

The side adjacent to angle D has a length of 5, so cos D = 5 / 13, but this is not one of the choices. The side opposite of angle D has a length of 12, so sin D = 12 / 13, which is Choice (4).

Note that the sine and cosine ratios can never be greater than 1 (or equal to 1 for that matter). Therefore, Choice (2) can be quickly eliminated because 13/12 is greater than 1.

More to come. Comments and questions welcome.

More STAAR problems.