Fair is Foul

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

Mike has brushed up his Shakespeare!

But I think Michele is one woman he will not wow.

This is a leftover from summer time. It should've been created at least a month ago, but as the Bard might've said, yada yada real life yada. That and the fact that I had to look up real quotes and not just what I thought might've been quotes. I would've like to have cursed the umpire's eyes or something, but nothing appropriate was jumping out at me (aside from all the websites wanted to let me know about love being blind, for example).

I'm curious what team Michele might root for, and if it's even in the major leagues. It's not like I've stated what state this takes place in. On the other hand, not a lot of minor league games get televised. Thankfully, I don't need to make that call. If I did, some of you might hate Michele for liking the wrong baseball team.

... instead of hating her for being an English teacher!

FYI: the filename on my hard drive for this image is fairisfoulball.png.

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.

(x, why?) Mini: Halfway

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

I would guess that he's a teacher.

I briefly flirted with the idea of using C, J, and B for the points, but that might've been confusing.

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.

School Life #36: Lost ID

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

Sven will make a great Dad some day.

And we're back. "By for how long, Mr. Spock? For ... how ... long?

Well, that I can't answer. Life has been crazy for me, and honestly, when I've had time to work on comics, I haven't had the drive. When I posted the last one, about three weeks ago, I had jotted down notes for two more. I haven't started either one yet. This "dad joke" wasn't one of those.

But I don't minder having a couple of summer leftovers. I would have labeled this one, except that it already has the School Life tag on it.

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.

(x, why?) Mini: Box Plot

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

That'd be a gallon of data.

This could be a valid chart, if someone had a use for it. You'd need a Six-Number Summary based on quintiles, instead of quartiles. If someone wanted just the 10% above and below the median, that would be the middle box. If for some reason, you wanted the central 60% instead of 50%, that would be the entire box containing the three smaller boxes.

And if anyone is looking for outliers, have a dram at the bar.

As for as the whiskey bottles go, the labels apper to be three flowers or butterflies or possibly an ancient family crest. That's my story and I'm sticking to it.

Cheers!

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, June 2023)

This exam was adminstered in June 2023.

More Regents problems.

June 2023 Geometry Regents

Part I

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


17. The measure of one of the base angles of an isosceles triangle is 42°. The measure of an exterior angle at the vertex of the triangle is

(1) 42°
(2) 84°
(3) 96°
(4) 138°

Answer: (2) 84°

If one base angle is 42° then the second base angle is also 42°. According to the Remote Angle Theorem, the measure of the exterior angle is equal to the sum of the two remote angles, which in the case of the vertex angle, the remote angles would be the two base angles. That sum is 84°, so Choice (2) is the correct answer.

If you forgot about the Remote Angle Teheorem, you could have used the sum of 84° to find the size of the vertex angle, which is 180 - 84 = 96°. Then the exterior angle would be supplementary to the vertex angle, so 180 - 96 = 84°. You not only got the same answer, but you just demonstrated the Remote Angle Theorem.

Two supplementary angles, A and B, add up to 180 degrees. The three angles of a triangle. B, C, and D, add up to 180 degrees. This means that A + B = B + C + D, so if you remove B (which in this problem is the vertex angle), you are left with A = C + D. This is Remote Angle Theorem.

18. In the diagram below, AFKB || CHLM, FH ≅ LH, FL ≅ KL, and LF bisects ∠HFK.

Which statement is always true?

(1) 2(m∠HLF) = m∠CHE
(2) 2(m∠FLK) = m∠LKB
(3) m∠AFD = m∠BKL
(4) m∠DFK = m∠KLF

Answer: (4) m∠DFK = m∠KLF

Look at the choices one at a time.

Angle HLF is one of two base angles and is congruent to angle HFL. Angle FHL (and therefore angle CHE) will equal 180 - 2(m∠HLF), not 2(m∠HLF). Eliminate Choice (1).

Angle FLK is the vertex angle of isosceles triangle FLK. LKB is an exterior angle. That means that, according to the Remote Angle Theorem that the sum of KFL and FLK equals the measure of LKB. For Choice (2) to be true, angle KFL must be congruent to angle FLK and there is no reason to suggest that they always will be. Eliminate Choice (2).

Angles AFD and BKL have no relationship between them DE and KL are not given to be parallel. Eliminate Choice (3).

Angles DFK and KLF look like they have no relationship, but they do. You may want to mark the illustration with this information. Angles HFL and LFK are congruent becuase LF bisects HFK. Angles LFP and LKF are congruent because triangle FKL is isosceles (with FL ≅ LK). This means that 2(LFK) + DFK = 180 degrees because that's the measure of a striaght line. And 2(LFK) + KLF = 180 because that's the measure of a traingle. Therefore, DFK must be congurnet to KLF. A little convoluted for a multiple-choice queston, but the correct answer is Choice (4).

19. The line whose equation is 6x + 3y = 3 is dilated by a scale factor of 2 centered at the point (0,0). An equation of its image is

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

Answer: (2) y = -2x + 2

When you dilate a line, the image will either be the same line or a line parallel to the original line. The slope will not change. The y-intercept will only change if the center of dilation is a point that is not on the pre-image.

First, find the slope of the original line. Since it is in Standard Form, Ax + By = C, the slope is given by the formula m = -A/B = -6/3 = -2.

If you didn't remember how to find the slope that way, then you have to rewrite it in slope-intercept form.

6x + 3y = 3
3y = -6x + 3
y = -2x + 1

Elimnate Choice (3) and (4).

So (0,1) is a point on the pre-image. If that point is dilated by a factor of 2, centered on the origin, the image will be at (0,2), which will be the y-interecept of the image. Then the correct answer is Choice (2).

20. Which figure will not carry onto itself after a 120-degree rotation about its center?

(1) equilateral triangle
(2) regular hexagon
(3) regular octagon
(4) regular nonagon

Answer: (3) regular octagon

Consider that 120 degrees is 1/3 of a full rotation.

If an equilateral triangle rotates 1/3 of the way, it will carry onto itself because it has 3 sides.

If a regular hexagon rotates 1/3 of the way, two sides will have rotated around and it will still look the same.

If a regular nonagon (which has nine sides) rotates 1/3 of the way, three sides will have rotated and it will still look the same.

An octagon, however, has eight sides, so if it rotates 1/3 of the way, it will not look the same because 1/3 * 8 is not a whole number.

The correct choice is Choice (3).

21.Triangle ADF is drawn and BC || DF.

Which statement must be true?

(1) AB/BC = BD/DF
(2) BC = 1/2 DF
(3) AB:AD = AC:CF
(4) ∠ACB ≅ ∠AFD

Answer: (4) ∠ACB ≅ ∠AFD

Because BC || DF, the small triangle is similar to the larger triangle and the corresponding sides are proportional.

Choice (1) and (3) show incorrect proportions because in each case there is are non-corresponding sides. In Choice (1), if BD said AD, or in Choice (3), if CF said AF, the proportions would have been correct. Eliminate Choice (1) and (3).

Choice (2) looks like a good answer EXCEPT it is only true if B and C are the midpoints of AD and AF, respectively, which would make BC a midsegment. We are not given this piece of information and we cannot assume it. Eliminate Choice (2).

Since the two triangles have the same shape, their corresponding angles are congruent. The correct answer is Choice (4).

22. In △ABC, M is the midpoint of AB and N is the midpoint of AC. If MN = x + 13 and BC = 5x - 1, what is the length of MN?
Which statement is always true?

(1) 3.5
(2) 9
(3) 16.5
(4) 22

Answer: (4) 22

The thing to remember here is that BC is NOT congruent to MN. MN is a midsegment, so it is HALF the length of BC. That is, BC is TWICE the length of MN.

Write that as an equation. Then substitute and solve.

BC = 2(MN)
5x - 1 = 2(x + 13)
5x - 1 = 2x + 26
3x = 27
x = 9

If x = 9, which is NOT the final answer, then MN = x + 13 = 9 + 13 = 22. That is Choice (4), which is the correct answer.

If you forgot to multiply by 2, then you would have gotten 3.5 for x and 16.5 for MN. Incorrect.

23. In the diagram below of isosceles trapezoid STAR, diagonals AS and RT intersect at O and ST || RA, with nonparallel sides SR and TA.
Which pair of triangles are not always similar?

(1) △STO and △ARO
(2) △SOR and △TOA
(3) △SRA and △ATS
(4) △SRT and △TAS

Answer: (3) △SRA and △ATS

The legs of the isosceles trapezoid are congruent and the diagonals are congruent. The bases are not congruent. The bases are parallel, which creates alternate interior angles that are congruent. And the diagonals form vertical angles that are congruent.

In Choice (1), the top and bottom triangles are always similar because of the alternate interior angles and the vertical angles.

In Choice (2), the left and right can be show to be not only similar but congruent using AAS.

In Choice (3), the left and the bottom together is not similar to the top and right together. They cannot be. Because SR ≅ TA, then ST must be congruent to AR for the corresponding sides to be proportional. However, we know that ST is NOT congruent to AR, because then we would have a parallelogram instead of a trapezoid. Choice (3) is the correct answer.

In Choice (4), these two triangles can be shown to be congruent (and therefore similar) through SAS.

24. The endpoints of AB are A(0,4) and B(–4,6). Which equation of a line represents the perpendicular bisector of AB?

(1) y = -1/2 x + 4
(2) y = -2x + 1
(3) y = 2x + 8
(4) y = 2x + 9

Answer: (4) y = 2x + 9

The perpendicular bisector has a slope that is the inverse reciprocal to the line connecting (0,4) and (-4,6). It also must contain the midpoint of AB, which is (-2,5), which is half the distance from A to B.

(6 - 4) / (-4 - 0) = 2/-4 = -1/2
Slope of AB = -1/2
Perpendicular slope = 2

The slope of the bisector is 2, so y = mx + b, and 5 = (2)(-2) + b.

5 = -4 + b, so b = 9.

End of Part I. How did you do?


Comments and questions welcome.

More Regents problems.

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

This exam was adminstered in June 2023.

More Regents problems.

June 2023 Geometry Regents

Part I

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


9. An equation of circle M is x² + y² + 6x - 2y + 1 = 0. What are the coordinates of the center and the length of the radius of circle M?

(1) center (3, -1) and radius 9
(2) center (3, -1) and radius 3
(3) center (-3, 1) and radius 9
(4) center (-3, 1) and radius 3

Answer: (4) center (-3, 1) and radius 3

You have to complete the squares for the x² term and the y² term. Thankfully, the choices give you hints.

Rewrite the equation grouping the x terms and the y terms and then move the constant to the right side of the equation.

x² + 6x + y² - 2y + 1 - 1 = 0 - 1

x² + 6x + y² - 2y = -1

To complete the squares, take have of 6, which is 3, and then square it, which is 9. Take half of -2, which is -1, and square it, which is 1. Add 9 + 1 to both sides of the equation.

x² + 6x + 9 + y² - 2y + 1 = -1 + 10

(x + 3)² + (y - 1)² = 9

The equation of a circle is given by the formula (x - h)² + (y - k)² = r², where (h,k) is the center of the circle and r is the radius.

So the center of the circle is (-3,1), because the signs flip to make 0, and the radius is 3, because r² is 9. Choice (4) is the correct answer.

10. Parallelogram BETH, with diagonals BT and HE, is drawn below.

Which additional statement is sufficient to prove that BETH is a rectangle?

(1) BT ⊥ HE
(2) BE || HT
(3) BT ≅ HE
(4) BE ≅ ET

Answer:(3) BT ≅ HE

If a parallelogram has diagonals that are congruent then it is a rectangle. Choice (3) is the correct answer.

Perpendicular diagonals make a parallelogram a rhombus, not a rectangle. Eliminate Choice (1).

Parallel sides are part of the definition of a parallelogram. Eliminate Choice (2).

Consecutive sides being congruent makes a parallelogram into a rhombus, not a rectangle. Eliminate Choice (4).

11. A gardener wants to buy enough mulch to cover a rectangular garden that is 3 feet by 10 feet. One bag contains 2 cubic feet of mulch and costs $3.66. How much will the minimum number of bags cost to cover the garden with mulch 3 inches deep?

(1) $3.66
(2) $10.98
(3) $14.64
(4) $29.28

Answer: (3) $14.64

Volume of a cone is V = L * W * H, but make sure that you are using the correct units, which must be the same.

V = 3 * 10 * .25 = 7.5. Since one bag covers 2 cubic feet, he needs 4 bags (because three isn't enough and they won't sell half a bag.)

The cost is then $3.66 * 4 = $14.64, which is Choice (3).

12. In the diagram below, &9651;DOG ~ &9651;CAT, where ∠G and ∠T are right angles.

Which expression is always equivalent to sin D?

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

Answer: (1) cos A

Sin D = sin C, but that isn't one of the choices.

However, the sine of one complementary angle is equal to the cosine of the other complementary angle in a right triangle.

That means that sin D = cos O = cos A. Cos A is Choice (1), which is the correct answer.

13. On the set of axes below, nDEF is the image of nABC after a dilation of scale factor 1/3.

The center of dilation is at

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

Answer: (2) (2,-3)

Take a straightedge and draw line AD, extending it past D. Do the same for BE and CF.

The point of concurrence is (2,-3), which is the center of dilation. Choice (2) is the correct answer.

You can see that from (2,-3) to point F is a translation of -2,+1, and from (2,-3) to C is -6,+3, which is 3 times farther. Or F is only 1/3 of the distance to C.

You can check the distance to D and A and you will find that D is 1/3 of the way to A. Likewise, E is 1/3 of the way to B.

14. In the diagram below of isosceles triangle AHE with the vertex angle at H, CB ⊥ AE and FD ⊥ AE.
Which statement is always true?

(1) AH/AC = EH/EF
(2) AC/EF = AB/ED
(3) AB/ED = CB/FE
(4) AD/AB = BE/DE

Answer: (2) AC/EF = AB/ED

Because AHE is isosceles, angles A is congruent to angle E. Angle ABC and EDF are right angles, so they are congruent to each other. Therefore, by AA, triangles ABC and EDF are similar (not congruent), and that makes the corresponding sides proportional.

Choice (1) uses point H, comparing part of the side to the length of the entire side of triangle AHE. This would imply that AC ≅ EF, which isn't necessarily true. (It doesn't look it.) Eliminate Choice (1).

Choice (2) compares the ratio of the two hypotenuses to the ratio of the two bases. Choice (2) is the correct answer.

Choice (3) is incorrect because the first ratio compares the bases of the two triangles but the second ratio compares the vertical leg of one triangle to the hypotenuse of the other.

Choice (4) uses line segment BD, which is not part of either of the two similar triangles. Eliminate Choice (4).

15. Rectangle ABCD has two vertices at coordinates A(-1,-3) and B(6,5). The slope of BC is

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

Answer: (1) -7/8

Sides AB and BC must be perpendicular to each other, so the slopes of the two sides must be inverse reciprocals. That is, flip the fraction over, and switch the sign.

The slope of AB is (-3 - 5) / (-1 - 6) = -8/-7 = 8/7. That means that the negative reciprocal is -7/8, which is Choice (1).

16. In right triangle ABC, m∠A = 90^o, m∠B = 18^o, and AC = 8. To the nearest tenth, the length of BC is

(1) 2.5
(2) 8.4
(3) 24.6
(4) 25.9

Answer: (4) 25.9

AC is Opposite of angle B, and BC is the hypotenuse of the triangle. Opposite and hypotenuse means that we need to use the sine ratio.

sin 18 = 8 / x
so x = 8 / (sin 18) = 25.88...

This rounds to 25.9, so Choice (4) is the correct answer.

More to come. Comments and questions welcome.

More Regents problems.

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

This exam was adminstered in June 2023.

More Regents problems.

June 2023 Geometry Regents

Part I

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


1. A square pyramid is intersected by a plane passing through the vertex and perpendicular to the base.
Which two-dimensional shape describes this cross section?

(1) square
(2) triangle
(3) pentagon
(4) rectangle

Answer: (2) triangle

If this were a cake and you were to slice it straight down the middle so that you cut through the highest point (the vertex) and sliced straight down, when you separated the two pieces, you would have a triangle slice, which is Choice (2).

If you missed the vertex, any other vertical cut would result in a trapezoid (which is not one of the choices).

If the plane was parallel to the base, the cross section would have been a square. If the was not parallel but passed through all four sides, it would result is a rectangle. Note that since a square is a rectangle, you know that square MUST BE incorrect because rectangle would also be correct.

There is no way for a place to intersect the pyramid and result in a pentagon.

2. Trapezoid ABCD is drawn such that AB||DC. Trapezoid A'B'C'D' is the image of trapezoid ABCD after a rotation of 110^o counterclockwise about point P.

Which statement is always true?

(1) ∠A ≅ ∠D'
(2) AC ≅ B'D'
(3) A'B' || D'C'
(4) B'A' ≅ C'D'

Answer:(3) A'B' || D'C'

The iamge after the rotation will have the same shape. That means that parallel lines are still parallel. The image is still a trapezoid. Therefore, A'B' || D'C'. Choice (3) is the correct choice.

∠A doesn't correspond to ∠D', so there is no reason for the two angles to be congruent. Eliminate Choice (1).

Since it is not stated that this is an isosceles trapezoid, the diagonals (not drawn) are not necessarily congruent. Eliminate Choice (2).

In a trapezoid, the bases do not have to be congruent. In fact, in standard HS Geometry, the bases cannot be congruent, because that would make it a parallelogram instead of a trapezoid. (When you get to advanced mathematics, parallelograms will become special cases of trapezoids the way that equilateral triangles are also isosceles.)

3. What is the volume of a right circular cone that has a height of 7.2 centimeters and a radius of 2.5 centimeters, to the nearest tenth of a cubic centimeter?

(1) 37.7
(2) 47.1
(3) 113.1
(4) 141.4

Answer: (2) 47.1

The Volume of a cone is V = 1/3 π r² h.

V = 1/3 π (2.5)² (7.2) = 47.1239..., which is approximately 47.1. Choice (2) is the correct answer.

I think the other two choices come from using (2.5)(2) instead of (2.5)², with and without the 1/3 in front.

4. In the diagram below of right triangle SUN, where ∠N is a right angle, SU = 13.6 and SN = 12.3.
What is m∠S, to the nearest degree?

(1) 25^o
(2) 42^o
(3) 48^o
(4) 65^o

Answer: (1) 25^o

You are given the hypotenuse and the side adjacent to the angle, so you need to use the cosine ratio.

Cos x = 12.3/13.6, so x = cos^-1 (12.3/13.6) = 25.26 degrees, so Choice (1) is the correct answer.

The other choices are for sine and tangent (using the hypotenuse instead of the opposite side) and tangent flipping the order of the numbers.

5. In the diagram below of circle O, diameter AOB and chord CB are drawn, and m/B = 28°

What is mBC (arc BC)?

(1) 56°
(2) 124°
(3) 152°
(4) 166°

Answer: (2) 124°

Angle B is an inscribed angle, so it intercepts an arc that is equal to twice the size of its angle. AOB is a diameter which marks off two semicircles, each having arcs of 180 degrees. Arc BC is therefore 180 - 2(28), which is 124 degrees, which is Choice (2).

Choice (1) is the size of arc AC, not BC. Choice (3) is 180 - 28, instead of 2(28). Choice (4) is 180 - 1/2(28).

6. In the diagram below of parallelogram ABCD, diagonal BED and EF are drawn, EF ⊥ DFC, m∠DAB = 111°, and m∠DBC = 39°.

What is m∠DEF?

(1) 11°
(2) 51°
(3) 60°
(4) 120°

Answer: (3) 60°

Work your way around the parallelogram. Angle C = 111°. Angle EFC = 90°. So m∠BEF = 360 - (39 + 111 + 90) = 120°.

This makes m∠DEF = 180 - 120 = 60°.

7. In the diagram below of nACT, ES is drawn parallel to AT such that E is on CA and S is on CT.

Which statement is always true?

(1) CE/CA = CS/ST
(2) CE/ED = EA/AT
(3) CE/EA = CS/ST
(4) CE/ST = EA/CS

Answer: (3) CE/EA = CS/ST

Triangle CES is similar to triangle CAT, so the corresponding sides are proportional. Look for the correct proportion from the choices.

Choice (1) is incorrect because it uses ST instead of CT.

Choice (2) is incorrect because it uses EA instead of CA.

Choice (3) compares segments split by parallel lines. Choice (3) is the correct answer.

Choice (4) is inconsisent with the triangles, comparing a part of one triangle with a part of the a different triangle.

8. On the set of axes below, congruent triangles ABC and DEF are drawn.

Which sequence of transformations maps △ABC onto △DEF?

(1) A counterclockwise rotation of 90 degrees about the origin, followed by a translation 8 units to the right.
(2)A counterclockwise rotation of 90 degrees about the origin, followed by a reflection over the y-axis.
(3) A counterclockwise rotation of 90 degrees about the origin, followed by a translation 4 units down.
(4) A clockwise rotation of 90 degrees about the origin, followed by a reflection over the x-axis.

Answer: (1) A counterclockwise rotation of 90 degrees about the origin, followed by a translation 8 units to the right.

Look at each of the choices, one at a time.

Choice (1) moves point A to (-6,-2) and B to (-2,-2). If you translate the triangle 8 units to the right, it will line up with DE. Likewise, you can follow point C and it will move onto point F. Choice (1) is the correct answer.

Choice (2) is incorrect because a reflection over the y-axis would reverse the positions of D and E. Furthermore, the image of point C would be must closer to the y-axis (at (2,-5)).

Choice (3) has a translation down, instead of to the right.

Choice (4) has a clockwise rotation, which will not line up with DEF after a reflection over the x-axis.

More to come. Comments and questions welcome.

More Regents problems.

Algebra Problems of the Day (Algebra 1 Regents, June 2023 Part I)

The following questions appeared on the une 2023 Algebra 1 Regents Exam

More Regents problems.

Algebra 1 Regents, June 2023

Part I: Each correct answer will receive 2 credits.


17. Some adults were surveyed to find out if they would prefer to buy a sports utility vehicle (SUV) or a sports car. The results of the survey are summarized in the table below.

Of the number of adults that preferred sports cars, approximately what percent were males?

1) 15.8
2) 45.2
3) 64.4
4) 82.6

Answer: 2) 45.2

84 adults preferred sports cars, and 38 of them were male, which is less than half, so eliminate Choices (3) and (4).

Divide 38/84 = 0.452..., which is about 45.2%, which is Choice (2).

18. The solution to 2x² = 72 is

1) {9,4}
2) {-4. 9}
3) {6}
4) {+6}

Answer: 4) {+6}

You could test 4, 6 and 9 and see which numbers work.

Or you can solve it as follows:

2x² = 72
x² = 36
x = +6.

The correct answer is Choice (4).

While (4)(9) = 36, neither 4² nor 9² equal 36.

It's important to remember the negative solution. Positive 6 isn't the ONLY solution.

19. Three quadratic functions are given below.

Which of these functions have the same vertex?

1) I and II, only
2) II and III, only
3) I and III, only
4) I, II, and III

Answer: 3) I and III, only

The vertex for function III can be seen to be (-2,5). Function I is written in vertex form, and the vertex for I is (-2,5). So I and III are the same. Eliminate Choices (1) and (2).

Next, determine if the vertex for function II is also (-2,5). You can see in the table that (-2,5) is a point in the function. HOWEVER, it is NOT the vertex. There is another point at (-1,5). That means that the Axis of Symmetry for function II occurs at x = -1.5. And the vertex of the function is on the Axis of Symmetry. Eliminate Choice (4).

The correct answer is Choice (3).

20. The domain of the function f(x) = x² + x - 12 is

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

Answer: 2) (-∞,∞)

The domain is the set of all x values that are valid for the function. In this example, every value of x is valid, from negative infinity to positive infinity, which is Choice (2).

If the funtion contained, for example, a variable in the denominator of a fraction, or a variable under a square root symbol, then there would be values of x for which the function would not make sense and they would be outside of the function's domain.

The function has ZEROES at x = -4 and x = 3, but that is not the domain.

Choices (1) and (4) are portions of the function where the range (the y value) is non-negative. But these have nothing to do with the domain.

21. A father makes a deal with his son regarding his weekly allowance. The first year, he agrees to pay his son a weekly allowance of $10. Every subsequent year, the allowance is recalculated by doubling the previous year’s weekly allowance and then subtracting 8. Which recursive formula could be used to calculate the son’s weekly allowance in future years?

1) a_n = 2n - 8
2) a_n = 2(n + 1) - 8
3) a₁ = 10, a_n+1 = 2a_n - 8
4) a₁ = 10, a_n+1 = 2(a_n - 8)

Answer: 3) a₁ = 10, a_n+1 = 2a_n - 8

A recursive formula needs a beginning, which is a₁. Each value after that is calculated from the one before it. Eliminate Choice (1) and (2).

The difference between Choices (3) and (4) is that in (3) the value is doubled and then 8 is subtracted. In (4), 8 is subtracted first and then the amount is doubled.

The definition of the formula agrees with Choice (3), which is the correct answer.

22. AWhat is the solution to the inequality below?

4 - 2/5 x > 1/3 x + 15

1) x < 11
2) x > 11
3) x < -15
4) x > -15

Answer: 3) x < -15

You could put each half of the equation into your graphing calculator to find where they are equal and see where the left side is greater than the right side.

4 - 2/5 x > 1/3 x + 15
-2/5 x > 1/3 x + 11
-2/5 x - 1/3 x > 11
-6/15 x - 5/15 x > 11
-11/15 x > 11
1x < (-15/11)11
x < -15

Choice (3) is the correct answer.

Remember to multiply by the reciprocal of the fraction. Remember that when you multiply an inequality by a fraction, you have to flip the inequality symbol the other direction.

23. Which statement is correct about the polynomial 3x² + 5x -2?


1) It is a third-degree polynomial with a constant term of 22.
2) It is a third-degree polynomial with a leading coefficient of 3.
3) It is a second-degree polynomial with a constant term of 2.
4) It is a second-degree polynomial with a leading coefficient of 3.

Answer: 4) It is a second-degree polynomial with a leading coefficient of 3.

The degree of the polynomial has to do with the highest exponent in the expression, not the number of terms it contains. The leading coefficient is the coefficient of the term with the highest exponent. The constant is the term without a variable and the sign in front of it is part of the constant.

The polynomial 3x² + 5x -2 is second degree with a leading coefficient of 3 and a constant of -2. The only choice that does not contradict any of this is Choice (4).

24. A store manager is trying to determine if they should continue to sell a particular brand of nails. To model their profit, they use the function p(n), where n is the number of boxes of these nails sold in a day. A reasonable domain for this function would be

1) nonnegative integers
2) rational numbers
3) real numbers
4) integers

Answer: 1) nonnegative integers

The store manager cannot sell fractions of boxes of nails and he can't sell negative boxes of nails. Irrational boxes is right out.

He can sell zero boxes.

Nonnegative integers is the reasonable domain for this function. The correct answer is choice (1).

End of Part I. How did you do?

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.

Algebra Problems of the Day (Algebra 1 Regents, June 2023 Part I)

The following questions appeared on the une 2023 Algebra 1 Regents Exam

More Regents problems.

Algebra 1 Regents, June 2023

Part I: Each correct answer will receive 2 credits.


9. A company ships an average of 30,000 items each week. The approximate number of items shipped each minute is calculated using the conversion

Answer: 2) [See Image]

To get items per minute, you need to have items in the numerator and minutes in the denominator. Every other unit needs to cancel out (reduce to 1) by appearing once in each of the numerator and the denominator. You can cancel units like factors.

Choice (1) has minutes in the numerator, and two weeks in the denominator, which would give you "weeks²", which is meaningless. Eliminate Choice (1).

Choice (2) has items in the numerator and minutes in the denominator. It also has weeks, days, and hours in both the numerator and the denominator, so they they cancel out. Choice (2) is the correct answer.

Choices (3) and (4) have items in the denominator. Eliminate Choices (3) and (4).

10. A function is graphed below.

A possible equation for this function is

1) f(x) = (x + 2)(x - 3)
2) f(x) = (x - 2)(x + 3)
3) f(x) = (x - 2)²(x + 3)
4) f(x) = (x - 2)(x + 3)(x - 12)

Answer:
3) f(x) = (x - 2)²(x + 3)

The zeroes of the function are at x = -3 and x = 2, which makes (x + 3) and (x - 2) factors. This is a cubic function, so you no that there is a third factor. (In reality, those fact checking me will tell you that it's obviously a function with an odd number exponent greater than 1, not an even number exponent. We aren't dealing with those graphs in Algebra 1, so don't worry about that right now.)

Basically, it's not a parabola, so it's not (1) or (2).

Choice (4) uses the y-intercept for no particular reason except to confuse you. There is no zero at x = 12. The graph indicates that it continues upward.

The fact that there is a minimum point is (2,0) is the reason that there are two factors of (x - 2), which is (x - 2)². If you multiply (-2)(-2)(3), you will get +12, which is the expected y-intercept on the graph.

Choice (3) is the correct answer.

As with similar problems, you could put all four choices into your graphing calculator and see which one looks like the given graph.

11. If g(x) = -x² - x + 5, then g(-4) is equal to?

1) -15
2) -7
3) 17
4) 25

Answer: 4) 25

Put the function in your calculator and look at the table of values for x = -4. Or just enter (-4)² - (-4) + 5. Make sure that you use parentheses.

(-4)² - (-4) + 5 = 16 + 4 + 5 = 25, which is Choice (4).

12. A movie theater’s popcorn box is a rectangular prism with a base that measures 6 inches by 4 inches and has a height of 8 inches. To create a larger box, both the length and the width will be increased by x inches. The height will remain the same. Which function represents the volume, V(x), of the larger box?

1) V(x) = (6 + x)(4 + x)(8 + x)
2) V(x) = (6 + x)(4 + x)(8)
3) V(x) = (6 + x) + (4 + x) + (8 + x)
4) V(x) = (6 + x) + (4 + x) + (8)

Answer: 2) V(x) = (6 + x)(4 + x)(8)

Volume equals length times width times height, so eliminate Choice (3) and (4).

The height of the box doesn't change, so there is no reason to add x to the 8 inches of the original box. Eliminate Choice (1).

Choice (2) show the length and the width increased by the same x inches while the height remains its orignal 8 inches. Choice (2) is the correct answer.

13. The expression 300(4)^{x + 3} is equivalent to

1) 300(4)^x(4)³
2) 300(4^x)³
3) 300(4^x) + 300(4³)
4) 300^x(4³)

Answer: 1) 300(4)^x(4)³

If you mutliply two terms with the same base, you add the exponents. So multiplying (4)^x(4)³ will give you (4)^{x + 3}. Choice (1) is the correct answer.

If you don't believe that, you can try it on your calculator using any number in place of x, or enter both expressions as graphs and look at the matching tables of values.

Choice (2) can be rewritten as 300(4)^3x, which isn't what we are looking for.

Choice (3) can be rewritten as 300(4^x + 4³), which isn't what we want.

Choice (4) is just nutty. It's there for people who don't evenread the questions before they choose an answer. Why would 300 suddenly gain an exponent of x?

14. Ashley only has 7 quarters and some dimes in her purse. She needs at least $3.00 to pay for lunch. Which inequality could be used to determine the number of dimes, d, she needs in her purse to be able to pay for lunch?

1) 1.75 + d > 3.00
2) 1.75 + 0.10d > 3.00
3) 1.75 + d < 3.00
4) 1.75 + 0.10d < 3.00

Answer: 2) 1.75 + 0.10d > 3.00

"At least" means that you want to have an amount that is greater than or equal to $3.00. Eliminate Choice (3) and (4).

Ashley doesn't want to add the number of dimes to the $1.75. She needs to add the value of those dimes, which is $0.10 times the number of dimes, d, or 0.10d.

Choice (2) is the correct answer.

15. The formula for the area of a trapezoid is A = 1/2(b₁ + b₂)h.
The height, h, of the trapezoid may be expressed as

Answer: 4) [See Image]

You can solve a literal equation in the same manner that you solve an equation with one variable. The only difference is that you most likely won't combine any "like terms".

To solve for h, isolate the variable by using inverse operations.

A = 1/2(b₁ + b₂)h

2A = (b₁ + b₂)h

2A / (b₁ + b₂) = h

This is Choice (4).

16. The function f(x) = |x| is multiplied by k to create the new function g(x) = k|x|. Which statement is true about the graphs of f(x) and g(x) if k = 1/2?

1) g(x) is a reflection of f(x) over the y-axis
2) g(x) is a reflection of f(x) over the x-axis.
3) g(x) is wider than f(x).
4) g(x) is narrower than f(x).

Answer: 3) g(x) is wider than f(x).

As with a parabola, if the coefficient is between 0 and 1, the function will be wider than f(x) = |x|. This is Choice (3).

If k = -1, then g(x) would be a reflection of f(x) over the x-axis.

A reflection of f(x) = |x| over the y-axis would be equal to f(x) itself.

If k > 1 (or k < -1) then g(x) would be narrower than f(x).

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.

Algebra Problems of the Day (Algebra 1 Regents, June 2023 Part I)

The following questions appeared on the une 2023 Algebra 1 Regents Exam

More Regents problems.

Algebra 1 Regents, June 2023

Part I: Each correct answer will receive 2 credits.


1. The expression 9m² - 100 is equivalent to

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

Answer: 1) (3m - 10)(3m + 10)

The given expression is a difference of two perfect squares, which factors into two conjugates, two expressions with the same terms, but one has a plus and the other has a minus.

If you look at the choices, only one has a pair of constants that will have a product of -100 when multiplied, and that is Choice (1).

Choice (2) has a product of 9m² - 60m + 100.
Choice (3) has a product of 9m² - 2500.
Choice (4) has a product of 9m² - 300m + 2500.

2. Which expression represents an irrational number?

1) √(16) + √(1)
2) √(25) + √(4)
3) √(36) + √(7)
4) √(49) + √(9)

Answer: 3) √(36) + √(7)

The sum of two rational numbers is always rational. The sum of a rational and in irrational number is always irrational.

If you take the square root of a perfect square, you will get a rational number. Out of all the numbers which appear in the choices above, only √(7) is irrational. So the correct answer is Choice (3).

3. Which linear equation represents a line that passes through the point (-3,-8)?

1) y = 2x - 2
2) y = 2x - 8
3) y = 2x + 13
4) y = 2x - 14

Answer: 1) y = 2x - 2

You could put all four equations into your graphing calculator and check the Table of Values for the correct equation for x = -3. Or you can just substitute x = -3 into each of this equations. Since they all have the same slope, at most one of the equations will work.

y = 2(-3) - 2 = -8. This is correct, so the answer is Choice (1).
y = 2(-3) - 8 = -14, so (-3,-8) is not on this line.
y = 2(-3) + 13 = 7, so (-3,-8) is not on this line.
y = 2(-3) - 14 = -20, so (-3,-8) is not on this line.

4. The expression (5x² - x + 4) - 3(x² - x - 2) is equivalent to

1) 2x² - 2x + 2
2) 2x² + 2x + 10
3) 2x⁴ - 2x² + 2
4) 2x⁴ - 2x² + 10

Answer: 2) 2x² + 2x + 10

When subtracting polynomials, the exponents DO NOT change. Eliminate Choices (3) and (4) immediately.

When you use the Distributive Property on the second expression, make sure you carry the minus sign. You are multiplying by NEGATIVE three.

(5x² - x + 4) - 3(x² - x - 2)
5x² - x + 4 - 3x² + 3x + 6
2x² + 2x + 10

Choice (2) is the correct answer.

5. 5 The 24th term of the sequence 25, 211, 217, 223, … is

1) -149
2) -143
3) 133
4) 139

Answer: 2) -143

Each given term is negative and the value is decreasing. So the 24th term will be a negative number. Eliminate Choices (3) and (4).

Each term in the sequence is 6 less than the one before it. The common difference is -6.

If the first term is -5, and the common difference is -6, then the 24th term will be -5 - 6(23), because you have to subtract -6 twenty-three times to get to the 24th term.

-5 - 6(23) = -143, which is Choice (2).

If you made a mistake and multiplied by 24, you would've gotten -149, which is Choice (1), which would be an incorrect response.

6. When completing the square for x² - 18x + 77 = 0, which equation is a correct step in this process?

1) (x - 9)² = 4
2) (x - 3)² = 2
3) x = +13
4) (x - 9) = +9

Answer: 1) (x - 9)² = 4

When you Complete the Square, you need half of the coefficient of the middle term (when the x² term has a coefficient of 1). So (x - 3)² is incorrect, and can be eliminated.

To Complete the Square, do the following:

x² - 18x + 77 = 0
x² - 18x = -77
(half of -18 is -9, and (-9)² = +81)

x² - 18x + 81 = -77 + 81
x² - 18x + 81 = 4
(x - 9)² = 4

This is Choice (1).

If you expand (x - 9)² = 4, you get
x² - 18x + 81 = 4, which when you subtract 4 from both sides becomes,
x² - 18x + 77 = 0.

7. Which function will have the greatest value when x > 1?

1) g(x) = 2(5)^x
2) f(x) = 2x + 5
3) h(x) = 2x² + 5
4) k(x) = 2x³ + 5

Answer: 1) g(x) = 2(5)^x

You could put all four equations into your graphing calculator and look at the results on the screen and in the table of values.

It should be obvious that quadratic functions will rise faster than linear functions when x > 1. Also, cubic functions will rise faster than quadratic functions when x > 1. Finally, exponential functions will rise faster than cubic functions when x > 1.

The correct answer is Choice (1).

When x = 1, f(1) = h(1) = k(1) = 7, and g(1) = 10.

8. Mike uses the equation b = 1300(2.65)^x to determine the growth of bacteria in a laboratory setting. The exponent represents

1) the total number of bacteria currently present
2) the percent at which the bacteria are growing
3) the initial amount of bacteria
4) the number of time periods

Answer: 4) the number of time periods

The exponent is how many times you are going to multiply by 2.65.

The total number of bacteria is b.

The percent at which the bacteria are growing is 165%. (That is, 1 + 1.65 = 2.65.)

The initial amount of bacteria is 1300.

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.

Know Your Axis

(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?).

They go in any which direction they wish.

Except when it's raining.

This is one of those jokes that when it occurs to me, I find myself wondering how I didn't think of it 15 years ago. Better late than never. And the later you get there, the more the meter has run up, which is only fare.

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.

Tall-People Problems

(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?).

My pants get wet, and my shoes get squishy.

When your umbrella is higher than everyone else's, your runoff falls onto their umbrellas. And their runoff falls onto you.

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.