(x, why?): March 2023

Tuesday, March 28, 2023

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

This exam was adminstered in January 2023.

January 2023 Geometry Regents

Part I

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

17. In the diagram below of circle O, AC and BC are chords, and m∠C = 70°.

If OA = 9, the area of the shaded sector AOB is

(1) 3.5π
(2) 7π
(3) 15.75π
(4) 31.5π

Answer: (4) 31.5π

The area of the sector is equal to the fraction of the circle that the sector represents times the area of the entire circle. The fraction is determined by the central angle.

The inscribed angle is half the size of the central angle, so the central angle is 70 times 2 = 140 degrees. This means that the fraction of the circle in the sector is 140/360.

Therefore, the area of the sector is

A = (140/360)(9)²π = 31.5π

This is Choice (4).

18. Quadrilateral BEST has diagonals that intersect at point D. Which statement would not be sufficient to prove quadrilateral BEST is a parallelogram?

(1) BD ≅ SD and ED ≅ TD
(2) BE ≅ ST and ES ≅ TB
(3) ES ≅TB and BE || TS
(4) ES || BT and BE || TS

Answer: (3) ES ≅TB and BE || TS

There are several ways to show that that a quadrilateral is a parallelogram.

The opposite sides are parallel. This is Choice (4). Eliminate it.

The opposite sides are congruent. This is Choice (2). Eliminate it.

The diagonals bisect each other, which means that each one is split into two congruent segments. This is Choice (1). Eliminate it.

One pair of sides is BOTH parallel and congruent. This is NOT Choice (3). Choice (3) says that one pair of sides is congruent but a different pair is parallel. This could be true in an isosceles trapezoid. So Choice (3) is the correct answer.

19. The equation of line t is 3x - y = 6. Line m is the image of line t after a dilation with a scale factor of 1/2 centered at the origin. What is an equation of line m?

(1) y = 3/2 x - 3
(2) y = 3/2 x - 6
(3) y = 3x + 3
(4) y = 3x - 3

Answer: (4) y = 3x - 3

Rewrite the equation for line t in slope-intercept form. If m is 1/2 of the original, then the y-intercept of the image is half the distance from the origin. The slope remains unchanged.

3x - y = 6
- y = -3x + 6
y = 3x - 6

Half of -6 is -3, so the equation of m is y = 3x - 3, which is Choice (4).

20. A cylindrical pool has a diameter of 16 feet and height of 4 feet. The pool is filled to 1/2 foot below the top. How much water does the pool contain, to the nearest gallon? [1 ft³ = 7.48 gallons]

(1) 704
(2) 804
(3) 5264
(4) 6016

Answer: (3) 5264

If the pool is 4 feet high and is filled to 1/2 foot below then top then it is the height of the water is 3.5 feet. The number of gallons will be the Volume in cubic feet times 7.48 gallons.

Remember that the radius is HALF of the diameter.

The Volume of a cylinder is V = π r² h

V = π (8)² (3.5) = 703.7167...

Multiply 703.717 * 7.48 = 5263.80316, or 5264, which is Choice (3).

21.The area of △TAP is 36 cm². A second triangle, JOE, is formed by connecting the midpoints of each side of △TAP. What is the area of △JOE, in square centimeters?

(1) 9
(2) 12
(3) 18
(4) 27

Answer: (1) 9

By connecting the midsegments of the triangle, you will divide it into four congruent triangles. So the area will be one quarter of TAP.

One quarter of 36 is 9. This is Choice (1).

22. On the set of axes below, the endpoints of AB have coordinates A(-3,4) and B(5,2).

If AB is dilated by a scale factor of 2 centered at (3,5), what are the coordinates of the endpoints of its image, A'B'?

(1) A'(-7,5) and B'(9,1)
(2) A'(-1,6) and B'(7,4)
(3) A'(-6,8) and B'(10,4)
(4) A'(-9,3) and B'(7,-1)

Answer: (4) A'(-9,3) and B'(7,-1)

Points A and B will be moved twice as far from the point (3,5) then they already are.

B is 2 units to the right and 3 units below (3,5). If you double that distance then B' will be located at (5+2, 2-3), which is (7,-1). This is Choice (4).

A is 6 units to the left and 1 unit below (3,5). If you double that distance then A' will be located at (-3-6, 4-1), which is (-9,3). This is still Choice (4), and now we've double-checked it.

23. In the circle below, AD, AC, BC, and DC are chords, EDF is tangent at point D, and AD || BC.

Which statement is always true?

(1) ∠ADE ≅ ∠CAD
(2) ∠CDF ≅ ∠ACB
(3) ∠BCA ≅ ∠DCA
(4) ∠ADC ≅ ∠ADE

Answer: (2) ∠CDF ≅ ∠ACB

Since AD || BC, then arcs AB and CD are congruent. Angles ACB is half the size of arc AB, and CDF is half the size of arc CD.

Since they are both half the size of congruent arcs then the angles are congruent.

Choice (2) is the correct answer.

Choice (1) ∠ADE ≅ ∠CAD would only work if AC || EF, but would not always be true.

Choices (3) and (4) cannot always be true. All you need to do is slide point A along the circle and you can see how the relative sizes of the listed angles will change.

24. In the diagram below of △ABC, D and E are the midpoints of AB and AC, respectively, and DE is drawn.

I. AA similarity
II. SSS similarity
III. SAS similarity

Which methods could be used to prove △ABC ~ △ADE?

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

Answer: (4) I, II, and III

DE is a midsegment of triangle ABC. It is parallel to BC and 1/2 the size of BC.

Angle A is congruent to itself through the Reflexive Property.

Angle B is congruent to angle ADE, and angle C is congruent to angle AED because of corresponding angles when transversals cross parallel lines.

Therefore AA similarlity can be used to prove the triangles are similar.

Because D and E are midpoints and angle A is congruent to itself, SAS can be used to prove that they are similar.

And because the midsegment is half the size of BC, SSS similarity can be used to prove that they are similar.

So the answer is I, II, and III, which is Choice (4).

More to come. Comments and questions welcome.

More Regents problems.

Friday, March 24, 2023

(x, why?) Mini: Polykite

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Read all about it in the aperiodical!

I admit that I don't know much about aperiodical tiling. Unlike tessellations, aperiodical tiling doesn't repeat. If you take kites and darts and make Penrose tiling, you can start with a nice flower (call it a rose or a penrose, if you will) and expand outward, but the patterns will never repeat.

The tridecagon shown in the comic is a "new" shape, recently discovered, that doesn't tessellate. It creates tiles that do not repeat. This is of interest because unlike kites and darts, this is a single polygon. That's something new.

It's called a polykite and is referred to as a "hat". I used both references, although I didn't manage a "pass the hat" quip. And, yes, like Polywhirl and Polycarp, I think that Polykite sounds like it sound evolve into another shape.

Or, since it's Geometry, maybe it should "transform".

Such a transformation likely wouldn't be a rigid motion. But the tiling definitely 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 22, 2023

Conway Circle

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Hello, Darlin'.

This was a math calendar inspired comic, with the answer, 22, being the date.

There isn't much more to say about Conway circles that isn't already stated in the comic. The incenter of the triangle will be the center of the outer circle.

I thought about putting the circle around my triangle, but it would be too big to use. I also tried to pixelate a picture of Conway Twitty to use in creating a face for this triangle. It was too freaky to use, so I went with something neutral.

I also write Fiction!

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.

Friday, March 17, 2023

Happy St. Paddy's Day!

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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!

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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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!

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) √(63)
(4) √(72)

Answer: (4) √(72)

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

(3)(24) = x²
x = √(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) △ACD ~ △BCD
(4) △ACM ~ △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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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!

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.

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(T_f - T_i), where C is the specific heat capacity of the substance. If its initial temperature is T_i, an equation to find its final temperature, T_f , 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(T_f - T_i)
Q / (mC) = (T_f - T_i)
Q / (mC) + T_i = T_f

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
a_n = -4 + a_{n - 1}

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

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Friday, March 10, 2023

Vocabulary Initiative

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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.

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Thursday, March 09, 2023

Coffee Tips

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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.

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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.

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