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

This exam was adminstered in August 2022.

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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

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(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.

Happy New Years Eve!

Happy New Years Eve!
Prepartations were being made today, and I was watching my nephew, so I didn't get to the comic that I wanted to. Watch this space. It'll be here. Hopefully, with the New Year.

Merry Christmas 2022

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(C)Copyright 2022, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

Merry Christmas!

Fire D Cannon on D orders of D Major Johann Pachelbel.

Credit to Trans-Siberian Orchestra for the lyrics.

I'm sure I've heard this played in recent years, but this year it seemed to be in heavy rotation on the Internet station I was listening to at work. The funny thing about it is that when I hear it, it reminds me of something from a long time before TSO put their version on any album. I think this was used for TV commercials, specifically, diamonds and diamond rings, but I couldn't tell you what store. (They're probably not around any more.)

I only put the refrain in the comic because that's all I know and remember of the song. I looked up the rest, but I figured, this is what others might recognize, even if they didn't know the name of the tune was "Christmas Canon", based on Johann Pachelbel's Canon in D Major.

Again, have a Merry Christmas.

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.

Happy Christmas Eve

i've been out all day (stii am -- at the in-laws) so the comic I wanted to do didn't get done. watch this space, maybe they'll be two tomorrow

If not, I hope you have a happy holiday, with family or friends or pets or memories.

God Bless and Merry Christmas

---

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(C)Copyright 2022, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

Merry Christmas

At the in-laws, Christmas Eve has always been a fish dinner. It's called 7 fishes, but I've never counted them to know if it referred to the number of fishes or the number of fishes dishes. I've heard references to the 7 Sacraments and to the 7 Hills of Rome.

For the record, I eat some of them, but some members of the family have plain pasta with any clam sauce.

The family tried to recreate my FIL's tradition as best as they could. New traditions will be in the offing in the years ahead.

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.

Long Lay The World...

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(C)Copyright 2022, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

Pining for the fjords. Or for presents. Or something.

My apologies if I've done this one before. I know I've thought it before.

It's a crazy time. I've come up with a couple of good ones, but the fact that I don't remember them a day later tells me that they weren't that memorable. I've been listening to Christmas music radio, so anything inspired by a song would reoccur to me should I hear it again. Much like the above comic.

I do have one more comic that I might start for next year because it'll take time that I don't have. Plus I have a couple other thoughts that I need to use this year.

Merry Christmas and a Happy Holiday Season and a Happy New Year to everyone out there who has kept me going. I plan on updating again before Sunday, but if I don't make it, I want it printed now.

Thank you everyone. I appreciate you whether you post or not. Especially those who post when I don't make math errors, but when I do, too.

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.

O Christmas Tree

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(C)Copyright 2022, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

I said shapes because my hand-drawn triangles didn't look like triangles. But they are similar!

I'm learning to code Scratch, even as I facilitate a class where the students are learning it as well. I doodled this Christmas tree so I could make an animated of a tree dancing around.

This is a learner's version of it, not a professional version. That is to say, the animation and music synch up sometimes, but it could synch up a lot more.

Dancing Xmas Tree.

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.

It's Beginning ...

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(C)Copyright 2022, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

Beginning of a new season. Hopefully, there will be more updates in the next 9 days!

The thought of those updates obviously horrifies Michele.

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.

Problems

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(C)Copyright 2022, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

It's good when you can pinpoint what your problem is. Or "problems are".

My problem is limited amounts of time. More to do at work and longer tirips home eat away at the time I have to do other things. And one problem I have is that I'm accumulating "other things".

I didn't realize that two weeks had gotten away from me since the last update.

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.