Friday, March 17, 2023

Happy St. Paddy's Day!

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(C)Copyright 2023, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

Watching those Eyes in March!

Or something like that.

Today we celebrate having someone to watch your back. It's all about the eyes, right?

I'm guessing the History class mentioned that movie Mark was asking about.



I also write Fiction!


You can now order 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.
Order 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, March 15, 2023

School Life #34: Beware

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(C)Copyright 2023, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

Watching those Eyes in March!

Or something like that.

Today we celebrate having someone to watch your back. It's all about the eyes, right?

I'm guessing the History class mentioned that movie Mark was asking about.



I also write Fiction!


You can now order 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.
Order 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.



Geometry Problems of the Day (Geometry Regents, January 2023)



This exam was adminstered in January 2023.

More Regents problems.

January 2023 Geometry Regents

Part I

Each correct answer will receive 2 credits. No partial credit.


9. Which polygon does not always have congruent diagonals?

(1) square
(2) rectangle
(3) rhombus
(4) isosceles trapezoid

Answer: (3) rhombus


Rectangles have congruent diagonals, so squares do as well. Isosceles trapezoids can be proven to have congruent diagonals using a similar proof to the one for rectangles.

Unless a rhombus is a square, one diagonal is longer than the other, which becomes even more obvious as the rhombus is "squished" with two small angles and two large ones.

The corrent answer is Choice (3).





10. If the circumference of a standard lacrosse ball is 19.9 cm, what is the volume of this ball, to the nearest cubic centimeter?

(1) 42
(2) 133
(3) 415
(4) 1065

Answer: (2) 133


Find the radius from the circumference and then use this to find the volume. Note that since this is a multiple-choice test, the most "obvious" mistakes will use diameter instead of radius. If I were to guess from looking at the choices, I would think that at least one of them will be surface area if you picked the wrong formula.

C = 2πr = 19.9, so r = 19.9/(2π) = 3.167... Leave a few decimal places to avoid rounding errors.

V = 4/3 π r3 = 4/3 π (3.167)3 = 133.055... or 133, which is Choice (2).





11. Which polygon always has a minimum rotation of 180° about its center to carry it onto itself?


Answer: (1) rectangle


The key is that it says "minimum" rotation. A rectangle rquires 180 degrees of rotation to carry onto itself, but a square only requires 90 degrees, even though 180 degrees always works.

An isosceles trapezoid requires a full 360 degrees to carry onto itself.

A regular pentagong carries onto itself with a 72-degree rotation. And 180 degrees would NOT carry it onto itself.

The correct answer is Choice (1).





12. Circle O is drawn below with secant BCD. The length of tangent AD is 24.


If the ratio of DC:CB is 4:5, what is the length of CB?

(1) 36
(2) 20
(3) 16
(4) 4

Answer: (2) 20


Use the formula: AD2 = (CD)(BD). Note that it is (part)(whole), not (part)(part). Since the ratio is 4:5, we have to use 4x and (4+5)x.

(4x)(9x) = 242
36x2 = 576
x2 = 16
x = 4

So CD = 4(4) = 16 and CB = 5(4) = 20, which is Choice (2).

Note the incorrect choices: x = 4, CD = 16, and BD = 36.





13.The equation of a line is 3x - 5y = 8. All lines perpendicular to this line must have a slope of

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

Answer: (4) -5/3


Find the slope of the given line. A perpendicular line will be the inverse reciprocal.

3x - 5y = 8
-5y = -3x + 8
y = 3/5x - 8/5

The slope of the given line is 3/5, so the perpendicular slope is -5/3, which is Choice (4).





14. What are the coordinates of the center and length of the radius of the circle whose equation is x2 + y2 + 2x - 16y + 49 = 0?

(1) center (1,-8) and radius 4
(2) center (-1,8) and radius 4
(3) center (1,-8) and radius 16
(4) center (-1,8) and radius 16

Answer: (2) center (-1,8) and radius 4


You have to rearrange the terms and then complete the squares to find the center and the radius.

x2 + y2 + 2x - 16y + 49 = 0

x2 + 2x + y2 - 16y + 49 = 0

x2 + 2x + y2 - 16y = -49

x2 + 2x + 1 + y2 - 16y + 64 = -49 + 1 + 64

x2 + 2x + 1 + y2 - 16y + 64 = 16

(x + 1)2 + (y - 8)2 = 42

So the center of the circle is (-1, 8) because the signs are flipped, and the radius is 4, not 16.

The correct choice is (2).





15. In the diagram below of right triangle MDL, altitude DG is drawn to hypotenuse ML.


If MG = 3 and GL = 24, what is the length of DG?

(1) 8
(2) 9
(3) &sqrt;(63)
(4) &sqrt;(72)

Answer: (4) &sqrt;(72)


The Right Triangle Altitude Theorem states that (MG)(GL) = (DG)2. Substitute and solve.

(3)(24) = x2
x = &sqrt;(72)

This is Choice (4).





16. Segment AB is the perpendicular bisector of CD at point M. Which statement is always true?

(1) CB ≅ DB
(2) CD ≅ AB
(3) &9651;ACD ~ &9651;BCD
(4) &9651;ACM ~ &9651;BCM

Answer: (1) CB ≅ DB


If AB is the perpendicular bisector of CD then AC = AD and BC = BD, but AC =/= BC and AD =/= BD.

So Choice (1) is the correct answer.

In Choice (2), just because AB bisects CD, it doesn't mean that the two segments must be congruent.

In Choices (3) and (4), the triangles aren't similar because the AC =/= BC but CD = CD and CM = CM, so the sides of the triangles cannot be corresponding.

So AD/AB = AE/AC = DE/BC

Choice (1): If AD/AB = DE/BC then AD/DE = AB/BC, not DB/BC. Eliminate Choice (1).

Choice (2): If AD/AB = DE/BC then AD/DE = AB/BC. THis is the correct answer.

In Choices (3) and (4): AD/BC is not a proper ratio. It compares one side of the small triangle with a different leg of the larger triangle. These are not corresponding sides. Eliminate these two choices.





More to come. Comments and questions welcome.

More Regents problems.

Tuesday, March 14, 2023

Pi Piper

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(C)Copyright 2023, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

Happy Pi Day!

Update: Why, yes, I did post a nearly but not quite finished comic earlier this afternoon. I was afraid I wouldn't get back to it if something else came up as I left work or any time after (and things have been coming up). And this is a timely comic that can't wait for tomorrow. Thankfully, I got to put the final tweeks in place.

For all the Pi jokes I've used, most seem to fall in the pie or pirate category, along with "pi rates", not to mention Sherlock Pi (P.I.). So I guess it's understandable how I haven't touch on this one in the past 15 years.



I also write Fiction!


You can now order 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.
Order 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, March 13, 2023

Geometry Problems of the Day (Geometry Regents, January 2023)



This exam was adminstered in January 2023.

More Regents problems.

January 2023 Geometry Regents

Part I

Each correct answer will receive 2 credits. No partial credit.


1. In the diagram below, a line reflection followed by a rotation maps △ABC onto △DEF.


Which statement is always true?

(1) BC ≅ EF
(2) AC ≅ DE
(3) ∠A ≅ ∠F
(4) ∠B ≅ ∠D

Answer: (1) BC ≅ EF


A rotation preserves size and shape, so the two triangles are congruent and their corresponding parts are congruent. So the answer is the choice that lists two corresponding parts.

In Choice (1), BC correspondes to EF, so this is the correct answer.

In Choice (2), AC corresponds to DF, no DE. Eliminate (2).

In Choice (3), ∠A corresponds to ∠D, not F. Eliminate (3).

In Choice (4), ∠B corresponds to ∠E, not D. Eliminate (4).





2. A circle is continuously rotated about its diameter. Which threedimensional object will be formed?

(1) cone
(2) prism
(3) sphere
(4) cylinder

Answer:(3) sphere


If spun around its diameter, a sphere would be formed in three dimensions.

The other choices are not possible, regardless of the axis of rotation.





3. In the diagram below of △CER, LA || CR.


If CL = 3.5, LE = 7.5, and EA = 9.5, what is the length of AR, to the nearest tenth?

(1) 5.5
(2) 4.4
(3) 3.0
(4) 2.8

Answer: (2) 4.4


The corresponding sides are proportional: CL/LE = AR/EA

3.5/7.5 = AR/9.5

AR = (9.5)(3.5)/(7.5) = 4.433..., or 4.4

Since EA is bigger than LE, AR must be bigger than CL, so Choices (3) and (4) could be eliminated.





4. Right triangle ABC is shown below.


Which trigonometric equation is always true for triangle ABC?

(1) sin A = cos C
(2) cos A = sin A
(3) cos A = cos C
(4) tan A = tan C

Answer: (1) sin A = cos C


The sine of one angle is equal to the cosine of the complementary angle, which is Choice (1).

In Choice (2), cos A will only equal sin A if A was 45 degrees.

In Choice (3), cos A does not equal cos C unless both angles are 45 degrees.

In Choice (4), tan A does not equal tan C. Tan A = 1/(tan C). They are reciprocals.





5. In the diagram of △ABC below, AE bisects angle BAC, and altitude BD is drawn.


If m∠C = 50° and m∠ABC = 60°, m∠FEB is

(1) 35°
(2) 40°
(3) 55°
(4) 85°

Answer: (4) 85°


You have to work your way around the triangle. If m∠C = 50° and m∠ABC = 60°, then m∠BAC = 180 - 50 - 60 = 70 degrees.

AE bisects ∠BAC, so m∠BAE = 35 degrees.

This means m∠AEB, which is also ∠FEB, is 180 - 60 - 35 = 85 degrees. This is Choice (4).

Alternatively, m∠C = 50, and m∠DAF = 35. According to the Remote Angle Theorem, m∠AEB = 85, which is the same as m∠FEB.





6. A jewelry company makes copper heart pendants. Each heart uses 0.75 in3 of copper and there is 0.323 pound of copper per cubic inch. If copper costs $3.68 per pound, what is the total cost for 24 copper hearts?

(1) $5.81
(2) $21.40
(3) $66.24
(4) $205.08

Answer: (2) $21.40


Multiply the pounds/in3 by the number of in3 times the $/pound. Note the units. If written correctly, the units will cancel out, leaving only $ in the numerator.

0.323 * 0.75 * 3.68 = 0.89148 per heart

0.89148 * 24 = 21.39552 = 21.40, which is Choice (2).





7. In right triangle LMN shown below, m∠M = 90°, MN = 12, and LM = 16.


The ratio of cos N is

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

Answer: (1) 12/20


The cosine ratio is the adjacent angle over the hypotenuse. Since the hypotenuse is the longest side, the cosine must ALWAYS be less than 1. Eliminate Choice (4).

You aren't given the cosine but it is even to figure out. First, the only options in the choices are 12, 16 and 20. So the hypotenuse must be 20.

Second, 3-4-5 triangles are scalable. Multiply 3-4-5 by 4 and you get 12-16-20.

Third, you could do Pythagorean Theorem and find out that LN = 20.

The leg adjacent to N has a length of 12. So cos N = 12/20, which is Choice (1).





8. In △ABC below, DE is drawn such that D and E are on AB and AC, respectively.


If DE || BC, which equation will always be true?

(1) AD/DE = DB/BC
(2) AD/DE = AB/BC
(3) AD/BC = DE/DB
(4) AD/BC = DE/AB

Answer: (2) AD/DE = AB/BC


If DE is parallel to the third side of a triangle, then the smaller triangle will be similar to the larger triangle and the corresponding sides will be proportional.

So AD/AB = AE/AC = DE/BC

Choice (1): If AD/AB = DE/BC then AD/DE = AB/BC, not DB/BC. Eliminate Choice (1).

Choice (2): If AD/AB = DE/BC then AD/DE = AB/BC. THis is the correct answer.

In Choices (3) and (4): AD/BC is not a proper ratio. It compares one side of the small triangle with a different leg of the larger triangle. These are not corresponding sides. Eliminate these two choices.





More to come. Comments and questions welcome.

More Regents problems.

Sunday, March 12, 2023

Algebra Problems of the Day (Algebra 1 Regents, January 2023)



The following questions appeared on the August 2022 Algebra 1 Regents Exam

More Regents problems.

Algebra 1 Regents, August 2022

Part I: Each correct answer will receive 2 credits.


17. In a geometric sequence, the first term is 4 and the common ratio is -3. The fifth term of this sequence is

1) 324
2) 108
3) -108
4) -324

Answer: 1) 324


Each term is (-3) times the term before it.

The first five terms are 4, -12, 36, -108, 324.





18. The amount of energy, Q, in joules, needed to raise the temperature of m grams of a substance is given by the formula Q = mC(Tf - Ti), where C is the specific heat capacity of the substance. If its initial temperature is Ti, an equation to find its final temperature, Tf , is


Answer: 2) [See image]


Inverse operations must be used to isolate Tf. Divide both sides by mC, and then add Ti to both sides.

That gives you Choice (2).

Q = mC(Tf - Ti)
Q / (mC) = (Tf - Ti)
Q / (mC) + Ti = Tf





19. When using the method of completing the square, which equation is equivalent to x2 - 12x - 10 = 0?

1) (x + 6)2 = -26
2) (x + 6)2 = 46
3) (x - 6)2 = -26
4) (x - 6)2 = 46

Answer: 4) (x - 6)2 = 46


When completing the square, the number inside the binomial will be HALF of the "b" term in the original polynomial. Since b = -12, the binomial must be (x - 6), not (x + 6). Eliminate Choices (1) and (2).

62 = 36, so add 36 to both sides to complete the square. Then add 10 to each side to remove the -10 from the left side.

x2 - 12x - 10 = 0
x2 - 12x + 36 - 10 = 0 + 36
x2 - 12x + 36 + 10 - 10 = 36 + 10
x2 - 12x + 36 = 46
(x - 6)2 = 46





20. Which quadratic function has the smallest minimum value?

Answer: 1) [SEE IMAGE]


The minimum value of a quadratic function occurs at the vertex, which is on the line of symmetry.

The vertex in j(x), Choice (4), is (0,0). The vertex of h(x) is not shown in the table but it is slightly less than zero. You could work out the equation, but it isn't necessary.

Choice (3), g(x), is written in Vertex form, and it's vertex is (2,-2).

Choice (1), f(x), has its y-intercept at -2, but that is NOT the vertex, which would be even lower on the graph. You have enough information to know that Choice (1) is correct.

The vertex of f(x) is at x = -(5)/(2*6) = -5/12.
f(-5/12) = -3.041666...





21. Which representation yields the same outcome as the sequence defined recursively below?

a1 = 3
an = -4 + an - 1


1) 3, 7, 11, 15, 19,…
2) 3, 1, 5, 9, 13,…
3) an = -4n - 1
4) an = 4 - n

Answer: 2) 3, 1, 5, 9, 13,…


Each number in the sequence, after the first, is 4 less than the one before it because of the -4 being added to the sequence. So Choice (1) is eliminated because it gets bigger, not smaller.

Choice (2) is a sequence of integers decreasing by 4, so this is the correct choice.

In Choice (3), a(1) = 4(1) - 1 = 3, but a(2) = 4(2) - 1 = 7. This is the same as Choice (1), which was eliminated. If it had -4 instead of 4, it would have been correct.

Choice (4) goes 3, 2, 1, 0, ... which is decreased by 1, not 4. Eliminate it.





22. If the zeros of the function g(x) are {-3,0,4}, which function could represent g(x)?

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

Answer: 3) g(x) = x(x + 3)(x - 4)


The zeroes of the function means that g(-3) = 0, g(0) = 0 and g(4) = 0. There will be three terms in the function.

In Choices (1) and (2), g(0) = -12. Eliminate (1) and (2).

In Choice (3), g(-3) = (-3)(0)(-7) = 0, g(0) = 0(3)(-4) = 0, and g(4) = (4)(7)(0) = 0. This is the correct answer.

In Choice (4), g(-3) = (-3)(-6)(-7) = -126. This is incorrect. There's no need to check the other two.

In Factored Form, the zeroes follow "(x - " in each binomial. This example should have been:

(x - -3)(x - 0)(x - 4)

But (x - 0) is just x, and (x - -3) is (x + 3).





23. Morgan read that a snail moves about 72 feet per day. He performs the calculation

72 feet / 1 day * 1 day / 24 hours * 1 hour / 60 minutes * 12 inches / 1 foot
to convert this rate to different units. What are the units for the converted rate?

1) hours / inch
2) minutes / inch
3) inches / hour
4) inches / minute

Answer: 4) inches / minute


Units can be multipled and canceled like any other factors in math (and science).

The days cancel. The hours cancel. Feet cancels foot.

You are left with inches on top and minutes on the bottom.

This is Choice (4).

Choice (2) is upside down. Choices (1) and (3) are incorrect because hour(s) are canceled because they form a multiplicative unit.





24. During summer vacation, Ben decides to sell hot dogs and pretzels on a food cart in Manhattan. It costs Ben $0.50 for each hot dog and $0.40 for each pretzel. He has only $100 to spend each day on hot dogs and pretzels. He wants to sell at least 200 items each day. If h is the number of hot dogs and p is the number of pretzels, which inequality would be part of a system of inequalities used to determine the total number of hot dogs and pretzels Ben can sell?
1) h + p ≤ 200
2) h + p ≥ 200
3) 0.50h + 0.40p ≥ 200
4) 0.50h + 0.40p ≤ 200

Answer: 2) h + p ≥ 200


This was is so simple that I read it twice to make sure I hadn't misread it the first time.

The variables h and p stand for the number of items that he's going to sell. He wants to sell at least 200 of these total. So the sum of h and p must be greater than or equal to 200, but not less than 200.

This is Choice (2).

Choices (3) and (4) tell us how much money Ben must spend to buy the product that he sells, and this amount must be less than or equal to $100, not 200. These choices can be eliminated.




More to come. Comments and questions welcome.

More Regents problems.

I also write Fiction!


You can now order 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.
Order 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.



Friday, March 10, 2023

Vocabulary Initiative

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(C)Copyright 2023, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

We all hope for dice, but don't let them kid you -- it's really a cult.

Mark was going to say that the other students thought it might be some sort of cult or secret society. (Not that I had room to squeeze in other students.) But it needed a little more setup and Mike's reaction wouldn't ahve fit in a single panel.

And let's face it, the real language word here is "initiative", at least in the sense that it is being used.

This is going on at my school, and they even gave me four sheets of paper to hang in the room. I found out yesterday that the words have been chaning weekly, but they haven't distributed oany more flyers for the past couple weeks.



I also write Fiction!


You can now order 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.
Order 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, March 09, 2023

Coffee Tips

(Click on the comic if you can't see the full image.)
(C)Copyright 2023, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

Field tested! This works every time!

I almost didn't do this because I knew I'd agonize over the details of the coffee pot but I finished it. Then I decided to link it to the teachers room. Which meant that my coffee pot was now the wrong color and I had to change it.

I nearly posted a picture of the coffee machine in my classroom, but I wanted to avoid advertising. And then I tried doodling over a photo in the background, and that was terrible even as a doodle. But here it is.



I also write Fiction!


You can now order 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.
Order 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, March 01, 2023

Algebra Problems of the Day (Algebra 1 Regents, January 2023)



The following questions appeared on the August 2022 Algebra 1 Regents Exam

More Regents problems.

Algebra 1 Regents, August 2022

Part I: Each correct answer will receive 2 credits.


9. Which expression is equivalent to (x + 4)2(x + 4)3?

1) (x + 4)6
2) (x + 4)5
3) (x2 + 16)6
4) (x2 + 16)5

Answer: 2) (x + 4)5


The bases are the same, so keep the base and add the exponents.

(x + 4)2(x + 4)3 = (x + 4)5.





10. Caitlin graphs the function f(x) = ax2, where a is a positive integer. If Caitlin multiplies a by -2, when compared to f(x), the new graph will become

1) narrower and open downward
2) narrower and open upward
3) wider and open downward
4) wider and open upward

Answer: 1) narrower and open downward


If a is positive, then -2a is negative. If the coefficient is negative, then the parabola opens downward. Eliminate Choices (2) and (4).

As the coefficient gets larger, the value of f(x) increases faster, so the parabola will by narrower.

The correct choice is Choice (1).





11. Sunny purchases a new car for $29,873. The car depreciates 20% annually.
Which expression can be used to determine the value of the car after t years?

1) 29,873(.20)t
2) 29,873(20)tJessica, only
3) 29,873(1 - .20)t
4) 29,873(1 + .20)t

Answer: 3) 29,873(1 - .20)t


The worth of a depreciated object after one year is 100% of the value minus the rate of depreciaton. This is shown in Choice (3).

Choice (1) actually shows an 80% depreciation because 1.00 - .80 = .20. Eliminate Choice (1).

Choice (2) shows exponential growth of 2000% per year. (Thats 20 * 100%.) Eliminate Choice (2).

Choice (4) shows 20% appreciation, not depreciation. It is gaining 20% of value every year.





12. If f(x) = x2 + 2x + 1 and g(x) = 7x - 5, for which values of x is f(x) = g(x)?

1) -1 and 6
2) -6 amd -1
3) -3 and -2
4) 2 and 3

Answer: 4) 2 and 3


Set the two expressions equal to each other, and solve for x by factoring.

x2 + 2x + 1 = 7x - 5
x2 - 5x + 6 = 0
(x - 3)(x - 2) = 0
(x - 3) = 0 or (x - 2) = 0
x = 3 or x = 2

This is Choice (4).





13. Skyler mows lawns in the summer. The function f(x) is used to model the amount of money earned, where x is the number of lawns completely mowed. A reasonable domain for this function would be

1) real numbers
2) rational numbers
3) irrational numbers
4) natural numbers

Answer: 4) natural numbers


The domain is the input and the range is the output.

The input to f(x) is the number of lawns. It is reasonable to believe that the number of lawns should be a natural or whole number. It is reasonable to assume that if he mows half a lawn, he will not get paid for the job (not even half of the job). So the correct answer is Choice (4).

The range for f(x) could be rational numbers because it may deal in dollars and cents.

Irrational or real numbers (which include irrational numbers) aren't reasonable for either.





14. Which expression is equivalent to 2x2 + 8x - 10?

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

Answer: 1) 2(x - 1)(x + 5)


Factor out the 2 and then factor the polynomial into two binomials.

2x2 + 8x - 10

2(x2 + 4x - 5)

2(x + 5)(x - 1)

This is Choice (1).





15. Ian throws a ball up in the air and lets it fall to the ground. The height of the ball, h(t), is modeled by the equation h(t) = -16t2 + 6t + 3, with h(t) measured in feet, and time, t, measured in seconds. The number 3 in h(t) represents

1) the maximum height of the ball
2) the height from which the ball is thrown
3) the number of seconds it takes for the ball to reach the ground
4) the number of seconds it takes for the ball to reach its maximum height

Answer: 2) the height from which the ball is thrown


The 3 is the value of f(0), the starting value. This is Choice (2).

The maximum height of the ball if f(x) when x = -(6)/(2(-16)). Eliminate Choice (1).

The number of seconds it takes for for the ball to reach the ground is the value of x when f(x) = 0. Eliminate Choice (3).

The number of seconds it takes for the ball to reach its maximum height is -6/-32 = 3/16 of a second. Eliminate Choice (4).





16. IThirty-two teams are participating in a basketball tournament. Only the winning teams in each round advance to the next round, as shown in the table below.


Which function type best models the relationship between the number of rounds completed and the number of teams remaining?

1) absolute value
2) exponential
3) linear
4) quadratic

Answer: 2) exponential


If you look at the f(x) values, each is half of the number before it. This is an exponential function, which is Choice (2).

Specifically, it is f(x) = 32(1/2)x.

It is not absolute value because the rate of change would have to be equal, and then change and become linear with the rate of change being additive inverse of the previous slope.

It is not linear becaue the rate of change would have to be equal, but it is decreasing.

It is not a quadratic pattern because then the second differences would be equal. However, the second differences are also decreasing.




More to come. Comments and questions welcome.

More Regents problems.

I also write Fiction!


You can now order 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.
Order 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, February 28, 2023

Algebra Problems of the Day (Algebra 1 Regents, January 2023)



The following questions appeared on the August 2022 Algebra 1 Regents Exam

More Regents problems.

Algebra 1 Regents, August 2022

Part I: Each correct answer will receive 2 credits.


1. When the expression 2x(x − 4) − 3(x + 5) is written in simplest form, the result is

1) 2x2 − 11x − 15
2) 2x2 − 11x + 5
3) 2x2 − 3x − 19
4) 2x2 − 3x + 1

Answer: 1) 2x2 − 11x − 15


Multiply and then combine like terms.

2x(x − 4) − 3(x + 5)
2x2 − 8x − 3x - 15
2x2 − 11x - 15

Choice (1) is the correct answer.





2. The point (3,w) is on the graph of y = 2x + 7. What is the value of w?

1) -2
2) -4
3) 10
4) 13

Answer: 4) 13


What is the value of y when x = 3?

2(3) + 7 = 6 + 7 = 13, which is Choice (4).





3. Students were asked to write 2x3 + 3x + 4x2 + 1 in standard form. Four student responses are shown below.

Alexa: 4x2 + 3x + 2x3 + 1
Carol: 2x3 + 3x + 4x2 + 1
Ryan: 2x3 + 4x2 + 3x + 1
Eric: 1 + 2x3 + 3x + 4x2

Which student’s response is correct?

1) Alexa
2) Carol
3) Ryan
4) Eric

Answer: 3) Ryan


Standard for of a polynomial is that the leading term has the highest exponent, the second term has the next highest exponent, and so forth. The coefficients are not relevant in finding the leading term.

Ryan has the exponents in descending order. This is Choice (3).





4. Given f(x) = −3x2 + 10, what is the value of f(−2)?

1) -26
2) -2
3) 22
4) 46

Answer: 3) 22


Substitute x = -2 and evaluate. Remember Order of Operations.

f(-2) = 3(-2)2 + 10 = 3(4) + 10 = 22.

This is Choice (3).





5. Which relation is a function?


Answer: 1) {(1,3,(2,1),(3,1)(4,7)}


In a function, there can only be one y-value for every x-value. There can only be one output for any input. On a graph, this gives us the Vertical Line Test.

In Choice (1), none of the x-values repeat. This is the correct choice.

In Choice (2), the input 7 has two different outputs, so it is not a function.

In Choice (3), the graph fails the vertical line test. A vertical line, such as the y-axis, can be drawn through the circle and touch it in more than one place.

In Choice (4), the relation map shows that 6 in the domain maps to both 5 and 7 in the range. This means that it is not a function.





6. What is the value of the third quartile in the box plot shown below?



1) 18
2) 22
3) 36
4) 46

Answer: 3) 36


The Five Number Summary that creates the Box and Whisker Plot is made up of (in order): the minimum, the first quartile, the median, the third quartile, and the maximum.

The right edge of the box is the third quartile, and it has a value of 36, which is Choice (3).





7. What is the solution to 2 + 3(2a + 1) = 3(a + 2)?

1) 1/7
2) 1/3
3) -3/7
4) -1/3

Answer: 2) 1/3


Isolate the variable. Get all the a terms on the left and all the numbers on the right.

2 + 3(2a + 1) = 3(a + 2)
2 + 6a + 3 = 3a + 6
3a = 1
a = 1/3

This is Choice (2).





8. One Saturday afternoon, three friends decided to keep track of the number of text messages they received each hour from 8 a.m. to noon.
The results are shown below.


Emily said that the number of messages she received increased by 8 each hour.
Jessica said that the number of messages she received doubled every hour.
Chris said that he received 3 messages the first hour, 10 the second hour, none the third hour, and 15 the last hour.

Which of the friends’ responses best classifies the number of messages they received each hour as a linear function?

1) Emily, only
2) Jessica, only
3) Emily and Chris
4) Jessica and Chris

Answer: 1) Emily, only


A linear function has a constant rate of change.

- Emily's messages increased by the same about each hour.

- Jessica's messages doubled, which is exponential, not linear.

- Chris's messages do not change at a constant rate, so it is not linear.

So the answer is Choice (1).




More to come. Comments and questions welcome.

More Regents problems.

I also write Fiction!


You can now order 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.
Order 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.



Crop Circle

(Click on the comic if you can't see the full image.)
(C)Copyright 2023, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

No cardboard or string.

The truth is out there.



I also write Fiction!


You can now order 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.
Order 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, February 26, 2023

Math Bait

(Click on the comic if you can't see the full image.)
(C)Copyright 2023, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

Everyone else has already left MathBook for the next Math big thing!

These fake math "click bait" articles were things that I came up with on Twitter in response to a challenge to create one. I created two, and would've done more daily but I got distracted by the something else on my To Do list.

I tried taking an informal poll of Twitter between Mather, MathSpace and MathBook, but the more I thought of it, I thought that MathSpace should be the dying one and MathBook should be what everyone's parents use. And both Mike and Ken are parents, so there's that.

Those platforms will probably get mentioned again. I can't ignore social media. Even if it mostly ignore me. (sniff).



I also write Fiction!


You can now order 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.
Order 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.