(x, why?) Mini: Shoes

(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?).

The fact that I used this joke for my 200th "Mini" just isn't right.

They don't look too comfortable. They must be dress shoes.

On a side note, I used to wonder when I got to comics numbered in the 1900s (the century I was born in, don't you know) if I'd start trying to match up comics to significant events from the corresponding year. Now that I'm here, I realize that there are so many significant things that happened every year, so I'll cop out and say "Hey, this isn't a History comic ... except for those few times when it is!"

So ... basically, I figure I'll plan something for 1999, 2000 and 2001. And possibly 1969, being the sci-fi nut that I am, especially if it occurs around July.

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, August 2022)

This exam was adminstered in August 2022.

More Regents problems.

August 2022 Geometry Regents

Part I

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


16. In right triangles ABC and RST, hypotenuse AB = 4 and hypotenuse RS = 16. If △ABC ∼ △RST, then 1:16 is the ratio of the corresponding

(1) legs
(2) areas
(3) volumes
(4) perimeters

Answer: (2) areas

If the two hypotenuses are 4 and 16, then the ratios of those lengths is 1:4.

The legs and the perimeters also have ratios of 1:4.

The areas have a ratio of 1:4² or 1:16.

The volumes don't make sense because they are triangles and not prisms. However, if they were prisms, then the ratios of the volumes would have been 1:4³ or 1:64.

The correct answer is Choice (2).

17. Parallelogram ABCD with diagonal DB is drawn below. Line segment EF is drawn such that it bisects DB at M.

Which triangle congruence method would prove that △EMB ≅ △FMD?

(1) ASA, only
(2) AAS, only
(3) both ASA and AAS
(4) neither ASA nor AAS

Answer: (3) both ASA and AAS

Lines AB and CD are parallel. These lines are crossed by two transversals, EF and DB, and each of these creates alternate interior angles. The two transversal intersect, creating vertical angles. So anyone could pick any two angles they wanted to to prove congruence.

Since DB is bisected at M, the DM = MB. Depending upon which angles where used for your proof, you may have either ASA, if the side is included, or AAS, if the side is not included between the two pairs of angles.

The correct answer is Choice (3).

18. In the diagram below of circle O, chords AD and BC intersect at E, and chords AB and CD are drawn.

Which statement must always be true?

(1) AB ≅ CD
(2) AD ≅ CD
(3) ∠B ≅ ∠C
(4) ∠A ≅ ∠C

Answer: (4) ∠A ≅ ∠C

Angles A and C are both inscribed angles and they intercept the same arc BD. Therefore both angles, A and C, must have a measure of 1/2 the size of arc BD.

There is no reason for chords AB nd CD to be congruent. They could be if they are equidistant from point O, but there is no information that they are.

Similarly, there is no reason for AD and BC to be congruent. There are rules about there products of their parts, but not about the chords themselves.

For angles B and C, they would be congruent only if AB || CD. There is nothing to suggest that those chords are parallel.

The correct choice is Choice (4).

19.What are the coordinates of the center and length of the radius of the circle whose equation is x² + y² - 12y - 20.25 = 0?

(1) center (0,6) and radius 7.5
(2) center (0,-6) and radius 7.5
(3) center (0,12) and radius 4.5
(4) center (0,-12) and radius 4.5

Answer: v

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

To find h,k, and r, you need to Complete the Square to rewrite the equation in the proper form.

HOWEVER, look at the four choices. There are four different y co-ordinates, meaning that once you have that, you know the correct choice.

To complete the square, you need to halve the coefficent of the y term. Half of -12 is -6. However, the formula have a minus sign in it. So if (y - 6)² is in the equation, then the center is (0,6), which is Choice (1).

To prove it:

x² + y² - 12y - 20.25 = 0
x² + y² - 12y = 20.25
x² + y² - 12y + 36 = 20.25 + 36
x² + (y - 6)² = 56.25
x² + (y - 6)² = 7.5²

The center is (0,-6) and the radius is 7.5.

20. In the diagram below, ABCD is a rectangle, and diagonal BD is drawn. Line ℓ, a vertical line of symmetry, and line m, a horizontal line of symmetry, intersect at point E.
Which sequence of transformations will map △ABD onto △CDB?

(1) a reflection over line ℓ followed by a 180° rotation about point E
(2) a reflection over line ℓ followed by a reflection over line m
(3) a 180° rotation about point B
(4) a reflection over DB

Answer: (2) a reflection over line ℓ followed by a reflection over line m

A reflection over line ℓ followed by a reflection over line m will move A to B then to C, move B to A then to D, and move D to C then to B, so ABD goes to CDB. This is Choice (2).

In Choice (1), a rotation of 180 degrees about point E would by itself map ABD onto CDB. However, the reflection of line l first messes this up. Visualize it like this: the reflection will change the diagonal from a positive slope to a negative slope, and a rotation will leave it with a negative slope (from upper left to lower right).

In Choice (3), rotating about point B will put the rectangle somewhere in the upper right of the graph. It won't map it onto itself.

In Choice (4), a reflection of DB would change the orientation of the rectangle, so it would be impossible for one triangle to map onto the other.

More to come. Comments and questions welcome.

More Regents problems.

Geometry Problems of the Day (Geometry Regents, August 2022)

This exam was adminstered in August 2022.

More Regents problems.

August 2022 Geometry Regents

Part I

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


11. A plane intersects a cylinder perpendicular to its bases.

This cross section can be described as a

(1) rectangle
(2) parabola
(3) triangle
(4) circle

Answer: (1) rectangle

If the plane intersects perpendicular to the bases, then it is slicing through vertically. If you slice it vertically, you will get a rectangle.

You would get a circle if you cut horizontally.

If you cut at an angle you would get either an ellipse or a parabola.

You cannot get a triangle with a single cut.

12. An equation of line p is y = 1/3 x + 4. An equation of line q is y = 2/3 x + 8. Which statement about lines p and q is true?

(1) A dilation of 1/2 centered at the origin will map line q onto line p.
(2) A dilation of 2 centered at the origin will map line p onto line q.
(3) Line q is not the image of line p after a dilation because the lines are not parallel.
(4) Line q is not the image of line p after a dilation because the lines do not pass through the origin.

Answer: (3) Line q is not the image of line p after a dilation because the lines are not parallel.

A dilation of a line doesn't chage its orientation, which means that its slope will be unchanged.

The slope of line p is 1/3 and the slope of line q is 2/3 . These lines are not parallel, so one cannot be the image of the other after a dilation. This is Choice (3).

Neither line passes through the origin, but that has nothing to do with lines being a dilation or not.

13. The coordinates of the endpoints of SC are S(-7,3) and C(2,-6). If point M is on SC, what are the coordinates of M such that SM:MC is 1:2?

(1) (-4,0)
(2) (0,4)
(3) (-1,-3)
(4) (-5/2, -3/2)

Answer: (1) (-4,0)

Add 1 + 2 = 3. Point M is 1/3 of the way from S to C.

The distance from S to C is 2 - -7 = 9 and -6 - 3 = -9. One-third of 9 is 3 and onet-third of -9 is -3.

Point C is (-7 + 3, 3 - 3) = (-4, 0), which is Choice (1).

Choice (3) is at 2:1, or 2/3 of the way.

Choice (4) is the midpoint of SC, which is a 1:1 ratio.

14. On the set of axes below, rectangle WIND has vertices with coordinates W(-4,2), I(4,0), N(3,-4), and D(-5,-2).

What is the area of rectangle WIND?

(1) 17
(2) 31
(3) 32
(4) 34

Answer: (4) 34

Because it's a rectangle, you can find the lengths of DW and WI and multiply them. You can find the lengths using the distance formula or Pythagorean Theorem (which are the same thing!). Leave your answers in radical form.

DW = SQRT(4² + 1² = SQRT(17).

WI = SQRT(2² + 8² = SQRT(68).

SQRT(17) * SQRT(68) = SQRT(1156) = 34

or SQRT(17 * 17 * 2 * 2) = 17 * 2 = 34

Choice (4) is the answer.

15. In parallelogram ABCD shown below, EB bisects ∠ABC.

If m∠A = 40°, then m∠BED is

(1) 40°
(2) 70°
(3) 110°
(4) 140°

Answer: (3) 110°

If m∠A = 40°, then ∠C = 40° and m∠ABC = 140°.

If m∠ABC is bisected, then ∠ABE = 70° and m∠CBE = 70°.

Since ∠C = 40° and m∠CBE = 140° then angle BED, which is exterior to triangle BEC, must be 70 + 40 = 110°. This is Choice (3).

More to come. Comments and questions welcome.

More Regents problems.

Algebra Problems of the Day (Algebra 1 Regents, August 2022)

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.


21. A swimmer set a world record in the women’s 1500-meter freestyle, finishing the race in 15.42 minutes. If 1 meter is approximately 3.281 feet, which set of calculations could be used to convert her speed to miles per hour?

1) 1500 meters / 15.42 min * 60 min / 1 hour * 1 meter / 3.281 feet * 1 mile / 5280 feet
2) 1500 meters / 15.42 min * 60 min / 1 hour * 3.281 feet / 1 meter * 1 mile / 5280 feet
3) 1500 meters / 15.42 min * 3.281 feet / 1 meter * 1 mile / 5280 feet
4) 1500 meters / 15.42 min * 60 min / 1 hour * 1 mile / 5280 feet

Answer: 2) 1500 meters / 15.42 min * 60 min / 1 hour * 3.281 feet / 1 meter * 1 mile / 5280 feet

You are starting with meters on top and minutes on the bottom. You want miles on top and hours on the bottom.

That means that you need to divide by meters so that the units will cancel (reduce to 1) and multiply the top by min so that the units will cancel (reduce to 1).

Choice (1) gives a result of meters * miles / (feet * feet). Eliminate Choice (1).

Choice (2) gives a result of miles) / hour. Choice (2) is the correct answer.

Choice (3) gives a result of miles / min, which isn't quite right. Eliminate Choice (3).

Choice (4) gives a result of (Meters * miles) / (hour * feet). Eliminate CHoice (4).

22. The diagram below shows the graph of h(t), which models the height, in feet, of a rocket t seconds after it was shot into the air.

The domain of h(t) is

1) (0,4)
2) [0,4]
3) (0,64)
4) [0,64]

Answer: 2) [0,4]

The domain is the set of all the x values, and the range is the set of all the y values. We care about x, not y. Or in this case, we want t, not h(t). Eliminate Choices (3) and (4).

THis would be a good time to mention that the choices are INTERVLS. They are NOT points on a coordintae plane.

The interval (0,4), with round brackets, or parentheses, means that the number is NOT included in the interval. It is only a boundary, like an open circle at the end of an inequality graph. The interval [0,4] with square brackets means that the values shown are included in the interval.

In the graph, t=0 and t=4 are valid values, and are included in the set. So Choice (2) is the correct answer.

10(x - 5) = -15

x - 5 = -15/10

x = 5 - 15/10 = 5 - 3/2 = 7/2 or 3 1/2 or 3.5

4 + 2(x - 2) = 9

2(x - 2) = 5

x - 2 = 5/2

x = 2 + 5/2 = 9/2 or 4 1/2 or 4.5

1/3 x = 3/2

x = 3(3/2) = 9/2 or 4 1/2 or 4.5

23. The table below shows the time, in hours, spent by students on electronic devices and their math test scores. The data collected model a linear regression.

What is the correlation coefficient, to the nearest hundredth, for these data?

1) -0.98
2) -0.95
3) 0.98
4) 0.95

Answer: 1) -0.98

You need a calculator that can handle lists and can do linear regression. However, you can tell from looking at the data that the correlation is negative -- as the input gets higher, the output gets lower. So you can eliminate Choices (3) and (4).

When you put it in the calculator and run the linear regression, you will see that the linear regression is -0.98.

24. The volume of a trapezoidal prism can be found using the formula V = 1/2 a(b + c)h. Which equation is correctly solved for b?

1) b = V / (2ah) + c
2) b = V / (2ah) - c
3) b = (2V) / (ah) + c
4) b = (2V) / (ah) - c

Answer: 4) D and B

Use inverse operations to isolate b.

V = 1/2 a(b + c)h.

2V = a(b + c)h

(2v) / ah = b + c

(2V) / ah - c = b

This is Choice (4).

End of Part I.

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.

Geometry Problems of the Day (Geometry Regents, August 2022)

This exam was adminstered in August 2022.

More Regents problems.

August 2022 Geometry Regents

Part I

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


6. On the set of axes below, △RST is the image of △ABC after a dilation centered at point P.

The scale factor of the dilation that maps △ABC onto △RST is

(1) 1/3
(2) 2
(3) 3
(4) 2/3

Answer: (1) 1/3

AB has a length of 3 and RS has a length of 1. Going from ABC to RST, the triangle is reduced by a factor of 3, so the dilation is 1/3, which is Choice (1).

Mapping RST onto ABC would be a dilation of 3.

7. In the diagram of △ABC below, m∠C = 90°, CB = 13, and AB = 16.

What is the measure of ∠A, to the nearest degree?

(1) 36°
(2) 39°
(3) 51°
(4) 54°

Answer: (4) 54°

We know the opposite side from angle A and we know the hypotenuse. So we need to use the inverse of the sine ratio to find the size of the angle.

sin^-1(13/16) = 54.3, or about 54 degrees.

If you did cosine instead, you would have gotten 36 degrees.

If you did tangent instead, you would have gotten 39 degrees.

If you did tangent and swapped the two side lengths, you would have gotten 51 degrees.

8. The Pyramid of Memphis, in Tennessee, stands 107 yards tall and has a square base whose side is 197 yards long.

What is the volume of the Pyramid of Memphis, to the nearest cubic yard?

(1) 751,818
(2) 1,384,188
(3) 2,076,212
(4) 4,152,563

Answer: (2) 1,384,188

The Volume of a square pyramid is given by the formula V = (1/3)s²h.

V = (1/3)s²h
= (1/3)(197)²(107)
= 1384187.66...

This is Choice (2).

If you forgot the (1/3), you would've gotten Choice (4), which is incorrect.

If you used (1/2), you would've gotten Choice (3), which is also incorrect.

9. A square is graphed on the set of axes below, with vertices at (1,2), (1,2), (3,2), and (3,2).

Which transformation would not carry the square onto itself?

(1) reflection over the y-axis
(2) reflection over the x-axis
(3) rotation of 180 degrees around point (1,0)
(4) reflection over the line y = x - 1

Answer: (1) reflection over the y-axis

Reflecting (3,2) over the y-axis maps it to (-3,2), which is not part of the square. So Choice (1) is correct.

In Choice (2), reflecting over the x-axis will switch the signs of the y-coordinate. -2 will map to 2 and 2 will mapt to -2. This parts are still on the square.

In Choice (3), the key is that the center is (1,0), which is the center of the square. Rotating a square 180 degrees around its center will map it onto itself.

In Choice (4), the line y = x - 1 is the diagonal of the square. (You can graph it if you don't believe me.) Reflecting a square across its diagonal will make it onto itself.

Squares have diagonals that are perpendicular and congruent, so this is incorrect.

Rhombuses (rhombi) have diagonals that are perpendicular. If they are also congruent, then the rhombus is also a square, but they do NOT have to be congruent.

A rectangle has diagonals that are congruent. If they are also perpendicular, then the rectangle will be a square.

An isosceles trapezoid has congruent diagonals.

10. If scalene triangle XYZ is similar to triangle QRS and m∠X = 90°, which equation is always true?

(1) sin Y = sin S
(2) cos R = cos Z
(3) cos Y = sin Q
(4) sin R = cos Z

Answer: (4) sin R = cos Z

Since the triangle is scalene, the acute angles are not equal to each other. Therefore the sines of those angles cannot equal each other and the cosines of those angles cannot equal each other.

Angles Q and X are right angles.

The sine of R = the sine of Y, and the cosine of R = the cosine of Y.

The sine of S = the sine of Z, and the cosine of S = the cosine of Z.

The sine of R = the cosine of Z, and the cosine of R = the sine of Z.

The sine of S = the cosine of Y, and the cosine of S = the sine of Y.

The correct answer is Choice (4).

More to come. Comments and questions welcome.

More Regents problems.

Getting Me Down

(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?).

Then rainy Mondays must be the worst! Especially if you argue with someone who's playing you like a song.

This comic was inspired by a rainy Monday, and the fact that I think I hear snippets of this song on the radio more on rainy Mondays than any other time. And that's silly.

To explain what's going on in the Venn diagram, the intersection of Rainy Days and Mondays are Rainy Mondays. However, Karen Carpenter quite clearly sings that "rainy days and Mondays always get me down" not that "rainy Mondays get me down". That means if it's a "rainy day", it gets her down, and if it's a "Monday", it gets her down. It could be one OR the other, OR it could be both. (Indeed, once you have one, you don't need to check the other because the condition has been fulfilled.)

So was Ms. Carpenter NOT down on Not Rainy days that were other days of the week?

We actually don't know that. We only know that either of these two conditions got her down. That's not to say that there aren't other conditions that would've gotten her down which did not fit into the lyrics of the song. However, it is safe to say that she might have had a bummer of sunny Thursday.

What gets me down is that I didn't get this comic up on Monday. And the amount of time between comics, of course.

Final question: Did you think from the title that this was going to be a gravity joke? Just wondering.

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.

Algebra Problems of the Day (Algebra 1 Regents, August 2022)

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.


16. When 3x² + 7x - 6 + 2x³ is written in standard form, the leading coefficient is

1) 7
2) 2
3) 3
4) -6

Answer: 2) 2

The leading term in standard form is 2x³, so the leading coefficient is 2, which is Choice (2).

17. Which of the equations below have the same solution?

I. 10(x - 5) = -15
II. 4 + 2(x - 2) = 9
III. 1/3 x = 3/2

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

Answer: 3) II and III, only

Since the want to know the percent of all students, you need to find how many (males and females) are in the survey and the total that prefer Pizza.

10(x - 5) = -15

x - 5 = -15/10

x = 5 - 15/10 = 5 - 3/2 = 7/2 or 3 1/2 or 3.5

4 + 2(x - 2) = 9

2(x - 2) = 5

x - 2 = 5/2

x = 2 + 5/2 = 9/2 or 4 1/2 or 4.5

1/3 x = 3/2

x = 3(3/2) = 9/2 or 4 1/2 or 4.5

18. In an organism, the number of cells, C(d), after d days can be represented by the function C(d) = 120 • 2^3d. This function can also be expressed as

1) C(d) = 240^3d
2) C(d) = 9600 • 2^d
3) C(d) = 120 • 6^d
4) C(d) = 120 • 8^d

Answer: 4) C(d) = 120 • 8^d

The laws of exponents tell us that 2^3d = (2³)^d.

And 2³ evaluates to 8.

So C(d) = 120 • 2^3d = C(d) = 120 • 8^d, which is Choice (4).

CHoice (3) shows s³ = 6, which is incorrect.

Choices (1) and (2) break the laws of exponents and the rules for the Order of Operations.

19. In the process of solving the equation 10x² - 12x - 16x = 6, George wrote 2(5x² - 14x) = 2(3), followed by 5x² - 14x = 3. Which properties justify George’s process?

A. addition property of equality
B. division property of equality
C. commutative property of addition
D. distributive property

1) A and C
2) A and B
3) D and C
4) D and B

Answer: 4) D and B

First, George used the distributive property, in reverse, factoring out 2 from each expression. Then he used the division property, and divided both sides of the equation by 2.

Comibining -12x and -16x into -28x, which then became 2(-14x) is NOT the addition property of equality. That was just combining like terms.

20. a sequence is defined recursively by

a₁ = -2
a_n = 3a_{n - 1} + 1

What is the value of a₄?

1) -41
2) -14
3) 22
4) 67

Answer: 1) -41

Use the formula, Luke!

a₂ = 3a₁ + 1 = 3(-2) + 1 = -5

a₃ = 3a₂ + 1 = 3(-5) + 1 = -14

a₄ = 3a₃ + 1 = 3(-14) + 1 = -41

The correct 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.

Geometry Problems of the Day (Geometry Regents, August 2022)

This exam was adminstered in August 2022.

More Regents problems.

August 2022 Geometry Regents

Part I

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


1. In the diagram below, △ABC is reflected over line ℓ to create △DEF.

If m∠A = 40° and m∠B = 95°, what is m∠F?

(1) 40°
(2) 45°
(3) 85°
(4) 95°

Answer: (2) 45°

In this reflection A maps to D, B maps to E, and C maps to F. So the measure of angle F is the same as the measure of angle C.

If A measures 40 and B measure 95, and 40 + 95 = 135, then the measure of angle C is 180 - 135 = 45 degrees, which is Choice (2).

2. The diagram below shows triangle ABC with point X on side AB and point Y on side CB.

Which information is sufficient to prove that △BXY ∼ △BAC?

(1) ∠B is a right angle.
(2) XY is parallel to AC
(3) triangle ABC is isosceles.
(4) AX ≅ CY

Answer: (2) XY is parallel to AC

Similarity is proven is two pairs of corresponding angles are congruent. This makes the third pair congruent as well. Having the same size angles means the triangle has the same shape. Congruent sides are not neceassry for similarity. In fact, sides most likely will NOT be congruent because they it will not longer be a similarity problem but a congruency problem.

Angle B is congruent to itself by the Reflexive Property. It doesn't matter what the size of Angle B is. Eliminate Choice (1).

If the lines are parallel, then the corresponding angles are congruent. So if angle ACB is congruent angle XYB, that, along with angle B being congruent to itself, is enough to prove AA Similarity. CHoice (2) is the correct answer.

Choice (3) would be good IF we also knew that triangle BXY was also isosceles, because then we could write equal expressions for the sizes of all the angles, even if we didn't know the sizes of the angles themselves. However, only knowing that one triangle is isosceles is not sufficient.

In Choice (4), knowing that AX = CY doesn't help because we don't know the relationship between BY and BX to set up a proportion, and use SAS similarity instead of AA. Eliminate Choice (4).

3. Quadrilateral MATH is congruent to quadrilateral WXYZ. Which statement is always true?

(1) MA = XY
(2) m∠H = m∠W
(3) Quadrilateral WXYZ can be mapped onto quadrilateral MATH using a sequence of rigid motions.
(4) Quadrilateral MATH and quadrilater WXYZ are the same shape, but not necessarily the same size.

Answer: (3) Quadrilateral WXYZ can be mapped onto quadrilateral MATH using a sequence of rigid motions.

If they are congruent, then they are the came size, so Eliminate Choice (4).

Congruent polygons can be mapped onto each other using a series of rigid motions, such as translations, reflections or rotations. But no dilations. Choice (3) is the correct answer.

Choice (1) is not true. By defualt, MA = WX. I say by default because I have seen math teachers who are a little sloppy with this when naming their polygons (particularly with triangles). But the first letter maps to the first letter, the second to the second, etc., unless otherwise specified.

Choice (2) is incorrect as well for the same reasons as Choice (1).

4. A quadrilateral has diagonals that are perpendicular but not congruent. This quadrilateral could be

(1) a square
(2) a rhombus
(3) a rectangle
(4) an isocelese trapezoid

Answer: (2) a rhombus

The shape is a rhombus.

Squares have diagonals that are perpendicular and congruent, so this is incorrect.

Rhombuses (rhombi) have diagonals that are perpendicular. If they are also congruent, then the rhombus is also a square, but they do NOT have to be congruent.

A rectangle has diagonals that are congruent. If they are also perpendicular, then the rectangle will be a square.

An isosceles trapezoid has congruent diagonals.

5. Which regular polygon has a minimum rotation of 36° about it center that carries the polygon onto itself.

(1) pentagon
(2) octagon
(3) nonagon
(4) decagon

Answer: (4) decagon

For a regular polygon to carry onto itself, it must rotate a multiple of 360 degrees divided by the number of sides that the polygon has.

A pentagon rotates 360 / 5 = 72 degrees.

An octogon rotates 360 / 8 = 45 degrees.

A nonagon rotates 360 / 4 = 40 degrees.

A decagon rotates 360 / 10 = 36 degrees. CHoice (4) is correct.

More to come. Comments and questions welcome.

More Regents problems.

Algebra Problems of the Day (Algebra 1 Regents, August 2022)

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.


11. What would be the order of these quadratic functions when they are arranged from the narrowest graph to the widest graph?

f(x) = -5x², g(x) = 0.5x², h(x) = 3x²

1) f(x), g(x), h(x)
2) g(x), h(x), f(x)
3) h(x), f(x), g(x)
4) f(x), h(x), g(x)

Answer: 4) f(x), h(x), g(x)

Ignore the sign of the coefficient. We only care about how big that value is absolutely. The narrowest graph with have the largest coefficient (absolutely) making it jump higher faster. The closer the coefficient is to 0, the wider it will be.

So |-5| > |3| > |0.5|, makined f(x) the narrowest and g(x) the widest.

Choice (4) is the correct answer.

12. At Berkeley Central High School, a survey was conducted to see if students preferred cheeseburgers, pizza, or hot dogs for lunch. The results of this survey are shown in the table below.

Based on this survey, what percent of the students preferred pizza?

1) 30
2) 37
3) 44
4) 74

Answer: 2) 37

Since the want to know the percent of all students, you need to find how many (males and females) are in the survey and the total that prefer Pizza.

There are 44 + 30 = 74 who prefer pizza.

There are 32 + 36 + 44 + 30 + 24 + 34 = 200 students.

74/200 = 37%, which is Choice (2).

Choice (1) is the number of Males, and not a percent.

Choice (3) is the number of Females, and not a percent.

Choice (4) is the total number, but not a percent.

13. Which situation could be modeled by a linear function?

1) The value of a car depreciates by 7% annually
2) A gym charges a $50 initial fee and then $30 monthly
3) The number of bacteria in a lab doubles weekly.
4) The amount of money in a bank account increases by 0.1% monthly.

Answer: 2) A gym charges a $50 initial fee and then $30 monthly

In Choice (2), there is an initial value and a constant monthly rate. This is the same as a y-intercept and a constant slope. Choice (2) is the correct answer.

Choice (1) shows exponential decay because 7% of one year will be smaller than 7% of the year before. It is not a constant change.

Choice (2) shows exponential growth because double last week is not the same increase as doubling the week before that.

Choice (4) shows exponential growth even if it is very slow growth.

14. Which function has the smallest y-intercept value?

Answer: 2) [See Image]

The y-intercept value occurs when x = 0 (at the y-axis).

In Choice (1), the y-intercept is 1.

In Choice (2), the y-intercept is -3, so eliminate Choice (1).

In Choice (3), the y-intercept is -2. Eliminate Choice (3).

In Choice (4), the y-intercept is -1. Eliminate Choice (4).

Choice (2) is the correct answer because h(0) = -3.

15.When solving x² - 10x - 13 = 0 by completing the square, which equation is a step in the process?

1) (x - 5)² = 38
2) (x - 5)² = 12
3) (x - 10)² = 38
4) (x - 10)² = 12

Answer: 1) (x - 5)² = 38

When you complete the square, the value of b (the coefficient in front of the x term) is halved. So Eliminate Choices (3) and (4).

Half of 10 is 5, and 5 sqaured is 25.

x² - 10x + 25 - 13 = 25

x² - 10x + 25 = 38

(x - 5)² = 38

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, August 2022)

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.


6. The expression 1/3 x(6x² − 3x + 9) is equivalent to

1) 2x² − x + 3
2) 2x² + 3x + 3
3) 2x³ − x² + 3x
4) 2x³ + 3x² + 3x

Answer: 3) 14

Multiplying the polynomial by 1/3 decreases all the coefficients by a factor of 3. Multiplying the polynomial by x increases the exponent on each term (with the constant term getting x¹, which is simply x.)

1/3 x(6x² − 3x + 9) = 1/3 (6x³ − 3x² + 9x) = 2x³ − x² + 3x.

This is Choice (3).

7. The graphs below represent four polynomial functions. Which of these functions has zeros of 2 and -3?

Answer: 3) [See Image]

If the function has zeroes of 2 and -3, then the function crosses the x-axis at x = 2 and x = -3.

Only Choice (3) shows this.

Choice (1) has zeroes of -2 and 3. Choice (2) has zeroes of -2 and -3. Choice (4) has zeroes of 2 and 3.

8. What is the constant term of the polynomial 4d + 6 + 3d²?
1) 6
2) 2
3) 3
4) 4

Answer: 1) 6

The constant term has no variable. It's just a number. In this case that number is 6.

2 is the exponent of the variable in the leading term, and the highest power in the polynomial.

3 is the leading coefficient.

4 is a coefficient of x, but not the leading coefficient. The polynomial is not written in standard form, which would have the highest power first.

9. Emily was given $600 for her high school graduation. She invested it in an account that earns 2.4% interest per year. If she does not make any deposits or withdrawals, which expression can be used to determine the amount of money that will be in the account after 4 years?

1) 600(1 + 0.24)⁴
2) 600(1 - 0.24)⁴
3) 600(1 + 0.024)⁴
4) 600(1 - 0.024)⁴

Answer: 3) 600(1 + 0.024)⁴

Interest adds to your balance, so subtraction is silly. Eliminate Choices (2) and (4).

The interest rate of 2.4% is written as 0.024, in decimal form. This is shown in Choice (3), which is the Correct answer.

Choice (1) shows 24% interest, which would be great if you can get it!

The only possibility here is Choice (4), 0.

10. Different ways to represent data are shown below.

Which data representations have a median of 2?

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

Answer: 1) I and II, only

The median in the box and whisper plot is the line over the number 2, so II is correct. That eliminates Choice (2).

There are 15 dots in graph I, so the median is the 8th dot, which is over the number 2, so I is correct. This eliminates choice (3).

In the bar chart, there are 1 + 3 + 5 + 3 + 4 + 5 + 2 = 23 pieces of data. The medain is the 12th piece of data. 1 + 3 + 5 = 9. The 12th piece of data is NOT over the number 2. Eliminate Choice (4).

Choice (1) is the correct answer.

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, August 2022)

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. If f(x) = (3x + 4)/2, then f(8) is

1) 21
2) 16
3) 14
4) 4

Answer: 3) 14

This is a simple substitution and evaluation question. However, on the test it was written as a fraction without parentheses. Be careful entering it into your calculator.

(3(8) + 4)/2 = 28/2 = 14, which is Choice (3).

If you didn't use the parentheses, they you got 24 + 4/2 = 26. Luckily, that isn't one of the choices.

Offhand, I couldn't tell you where they got the incorrect choices from.

2. If x ≠ 0, then the common ratio of the sequence x, 2x², 4x³, 8x⁴, 16x⁵, ... is

1) 2x
2) 2
3) x
4) 1/2 x

Answer: 1) 2x

Divide 2x² by x. The quotient is 2x. Repeat for (4x³)/(2x². You will get the same result: 2x.

3. The expression 36x² - 9 is equivalent to
1) (6x - 3)²
2) (18x - 4.5)²
3) (6x + 3)(6x - 3)
4) (18x + 4.5)(18x - 4.5)

Answer: 3) (6x + 3)(6x - 3)

The Difference of Two Perfect Squares Rule states that the two factors will be conjugates. That is, the binomials will be the same except that one has a plus sign and one has a minus sign. Also the coefficient will be the square roots of the original, not the numbers divided by 2.

They like to use "divided by 2" in thiese kinds of problems. However, as long as my students make this mistake (along with other students across the state), they'll continue to use this. It makes the question easy to answer for everyone else!

The correct answer is Choice (3), (6x + 3)(6x - 3).

Expended, it becomes 36x² - 18x + 18x - 36. The two middle terms are a zero pair.

4. Given the relation R = {(-4,2), (3,6), (x,8), (-1,4)}
Which value of x would make this relation a function?

1) -4
2) -1
3) 3
4) 0

Answer: 4) 0

For the relation to be a function, no x value can be repeated. This eliminates -4, 3, and -1. Any other number could stand in for x except those three.

The only possibility here is Choice (4), 0.

5. If the point (K,-5) lies on the line whose equation is 3x + y = 7, then the value of K is

1) -8
2) -4
3) 22
4) 4

Answer: 4) 25

Substitute y = -5 and solve for x.

3x + y = 7
3x + -5 = 7
3x = 12
x = 4

Choice (4) is 4, which is the correct answer.

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.

Quality Scale

(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?).

You can't rush greatness, except with lots of money. You can't rush whatever this is, either, because there is no money.

I was going to change it to Time vs Effort, and leave money out of it, but effort takes time.

And for the past couple of weeks, when I've had the time, I didn't have the energy (okay, maybe that and time would've worked). I did have ideas for 12/31/22 and 1/1/23 comics but by the time I got to do them, I didn't want to back-date them, even with 2023 copyright dates on them.

Hopefully more this week. I'd like to see this strip make it to 2000 before I stopped. No, 2001 would be more appropriate.

Thanks for being here. You are here, right? Feel free to comment if you're here. Unless you're selling carpets or test prep classes or something like that.

Also, at some point, I need to upload my last for Math Calendar solutions for the people who aren't on the blog. I might adapt one or two of them into comics (which I've done before).

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.