Monday, July 04, 2022

Enjoy the Fourth!

(Click on the comic if you can't see the full image.)

Sorry for the placeholder comic, but family demands this past weekend trump other concerns. This will be updated by tomorrow.

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

I hope everyone had a pleasant and safe Fourth of July/Independence Day! As you can see, I did not have time to complete today's comic, but you can get the gist of it.

This is that odd time of year when summer vacation starts and so many things that had to wait until summer vacation suddenly have to get done. On top of that, things have sort of snowballed for the last couple of weeks with new things popping up. That's life. I'm hoping for a boring July.



I also write Fiction!


You can now preorder 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.
Preorder the softcover or ebook at Amazon.

Also, check out In A Flash 2020, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides.
Available in softcover or ebook at Amazon.

If you enjoy it, please consider leaving a rating or review on Amazon or on Good Reads.





Come back often for more funny math and geeky comics.



Sunday, July 03, 2022

June 2022 Algebra 2 Regents, Part II



This exam was adminstered in June 2022. These answers were not posted until they were unlocked on the NY Regents website or were posted elsewhere on the web.

More Regents problems.

Algebra 2 June 2022

Part II: Each correct answer will receive 2 credits. Partial credit can be earned. One mistake (computational or conceptual) will lose 1 point. A second mistake will lose the other point. It is sometimes possible to get 1 point for a correct answer with no correct work shown.


25. Does the equation x2 - 4x + 13 = 0 have imaginary solutions? Justify your answer.

Answer:


You can determine the number of solutions by using the discriminant, or by graphing and stating the number of x-intercepts. A negative discriminant means imaginary solutions. No x-intercepts means imaginary solutions.

The discriminant is b2 - 4ac = (-4)2 - 4(1)(13) = (-4)2 - 4(1)(13) = 16 - 52, which is < 0. So the discriminant is negative and the solutions are imaginary.

If graphed, the range of the function would be entirely above the x-axis, so there would only be imaginary solutions.





26. The initial push of a child on a swing causes the swing to travel a total of 6 feet. Each successive swing travels 80% of the distance of the previous swing. Determine the total distance, to the nearest hundredth of a foot, a child travels in the first five swings.

Answer:


You can use the formula for the sum of a finite series, or you can manually calculate 80% for the second through the fifth swings. Either is acceptable but the latter leaves more room for errors to creep in. Two errors is 0 credits, regardless of the work done.

Using the formula from the back of the booklet:

Sn = a1(1 - (r)n) / (1 - r)

S5 = (6)(1 - (0.80)5)) / (1 - 0.80) = 20.1696

The total amount is about 20.17 feet.

Doing this in steps:

S1 = 6
S2 = 6(.8) = 4.8
S3 = 4.8(.8) = 3.84
S4 = 3.84(.8) = 3.072
S5 = 3.072(.8) = 2.4576

S1 + S2 + S3 + S4 + S5 = 6 + 4.8 + 3.84 + 3.072 + 2.4576 = 20.1696 = 20.17





27. Solve algebraically for n: 2/n2 + 3/n = 4/n2

Answer:


Excuse me for saying that this is ridiculously easy, but this question could appear on an Algebra 1 exam. You don't even need to worry about n = 0 in the problem unless you somehow introduce 0 as a possible solution.

2/n2 + 3/n = 4/n2

(n2(2/n2 + 3/n) = (4/n2)n2

2 + 3n = 4>

3n = 2

n = 2/3

That's it. Two credits.





28. Factor completely over the set of integers:

-2x4 + x3 + 18x2 - 9x


Answer:


Remove the GCF, which is x (or -x), then factor by grouping. Then check if anything else can be factored.

Personally, I prefer for the leading coefficient to be positive when I'm factoring, so I would factor out (-1) along with the GCF of x.

-2x4 + x3 + 18x2 - 9x

(-x) (2x3 - x2 - 18x + 9)

(-x) ((x2)(2x - 1) - 9(2x - 1))

(-x) (x2 - 9)(2x - 1)

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

Anything equivalent is acceptable. For example, (x)(x + 3)(x - 3)(-2x + 1).





29. The relative frequency table shows the proportion of a population who have a given eye color and the proportion of the same population who wear glasses.

Wear
Glasses
Don't Wear
Glasses
Blue Eyes 0.140.26
Brown Eyes 0.110.24
Green Eyes 0.100.15

Given the data, are the events of having blue eyes and wearing glasses independent? Justify your answer.


Answer:


Two events are independent if P(A and B) = P(A)P(B).

Let A be the probability of Wearing Glasses. Let B be the probability of Blue Eyes.

P(A) = 0.14 + 0.11 + 0.10 = 0.35.

P(B) = 0.14 + 0.26 = 0.40

P(A)P(B) = (0.35)(0.40) = 0.14

P(A and B) = 0.14 from the table.

Since the amounts are the same, the events are independent.





30. For x ≠ 0 and y ≠ 0, ∛(8lx15y9) = 3ax5y33. Determine the value of a.

Answer:


You can simplify the cube root or you can focus on the 81 portion of it.

∛(8l) = ∛(3 * 3 * 3 * 3) = 3 4/3 = 3 a

Therefore, a = 4/3.





31. Graph y = 2cos(1/2 x) + 5 on the interval [0,2π], using the axes below.


Answer:


The midline is 5 and the amplitude is 2, so the range is from 3 to 7. The period of cos(nx) is 2π/n, which is 2π/(1/2) = 4π. So over the range of [0,2π], only half of the wave will be seen.

Your graph should look something like this:

Do NOT continue past 2π outside of the interval you were given.

Do NOT put Arrows on either end of the line because that indicates going beyond the interval.

DO number the axes, especially the X-axis.

DO make it look like a curve and NOT a linear function.

Remember that this is only a TWO credit problem, and you lose one point for each graphing error. That means that 2 (or more) "little" mistakes means you will get zero credit, regardless of the amount of work you point into it.





32. A cup of coffee is left out on a countertop to cool. The table below represents the temperature, F(t), in degrees Fahrenheit, of the coffee after it is left out fort minutes.

Based on these data, write an exponential regression equation, F(t), to model the temperature of the coffee. Round all values to the nearest thousandth.

Answer:


Put the information into L1 and L2 in your calculator. Run an Exponential Regression. Write the answer using F(t) and t. Do NOT use y or x in your final answer.

If you entered the data correctly, you should have gotten a = 169.136 and b = 0.971. This means that your equation should have been:

F(t) = 169.136(0.971)t

I scored this problem on literally hundreds of exams. The biggest mistake among those who did the work correctly was to use y and x, or F(t) and x, either of which cost a point. This mistake was far more common than rounding to the wrong number of decimals, or even misentering the data and getting a and b values which were slightly off.




End of Part III

How did you do?








More to come. Comments and questions welcome.

More Regents problems.

I also write Fiction!


You can now preorder 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.
Preorder 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.



Saturday, July 02, 2022

June 2022 Geometry Regents, Part 4

The following are some of the multiple questions from the recent January 2020 New York State Common Core Geometry Regents exam.

June 2022 Geometry, Part IV

A correct answer is worth up to 6 credits. Partial credit can be earned. Work must be shown or explained.


32. The coordinates of the vertices of D.ABC are A(-2,4), B(-7, -1), and C(-3, -3). Prove that D.ABC is isosceles.
[The use of the set of axes on the next page is optional.]

State the coordinates of D.A 'B 'C', the image of △ABC, after a translation 5 units to the right and 5 units down.

Prove that quadrilateral AA'C'C is a rhombus.
[The use of the set of axes below is optional.]

Answer:
The breakdown was 2 points from proving the triangle is isosceles, with one point for finding the lengths and one point for a statement saying it was isosceles and why. One point was given for finding all three set of coordinates of the image. The final three points were for proving that the object was a rhombus, including the work and a statement. If the student only showed that it was a parallelogram but not enough for a rhombus, that was one credit.

Points could be earned on the last portion proving that it was NOT a rhombus if incorrect coordinates for the image were found. In this case, supporting work still had to have been shown and a proper concluding statement needed to be given.

In the first part, you could use the distance formula, or you could use the graph and show it through the Pythagorean Theorem.

AB = √((5)2 + (5)2) = √(50)
AC = √((1)2 + (7)2) = √(50)

Triangle ABC is isosceles because AB ≅ AC.

To translate the pre-image to the image, add 5 to each x coordinate and subtract 5 from each y coordinate. You didn't have to show this, just the response.

A(-2,4) --> A'(3,-1)
B(-7,-1) --> B'(-2,-6)
C(-3,-3) --> C'(2, -8)

You didn't need to show your work on the graph. However, the visual is a great aid for solving the problem.

To prove that the shape was a rhombus, you had to show that four sides were congruent, OR you could show that the diagonals bisected each other. If you showed that the diagonals bisected each other, that only showed that it was a parallelogram. In that case, you still needed to show two consecutive sides were congruent.

Conversely, if you only showed two consecutive sides, then you still needed to show that it was a parallelogram.

The four sides of the rhomus are AA', A'C', C'C, and CA.

Length of AC = √(50). Shown above, you don't need to do it again..

Length of A'C' = √(50) because a translation is a rigid motion that preserves distance. This must be stated, or you could use the distance formula again.

Length of AA' = √( (5)2 + (5)2 ) = √(50)

Length of C'C = √( (5)2 + (5)2 ) = √(50)

All four sides are congruent, therefore AA'C'C is a rhombus.

Alternatively,

The slope of AC' is (-8 - 4)/(2 - -2) = -12/4 = -3

The slope of A'C is (-1 - -3)/(3 - -3) = 2/6 = 1/3

The slopes of the two diagonals are inverse reciprocals, so the diagonals are perpendicular. Therefore, the quadrilateral is a rhombus.

Other methods were possible.

End of Part Exam

How did you do?

Questions, comments and corrections welcome.

I also write Fiction!


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.

Thank you.



Friday, July 01, 2022

June 2022 Geometry Regents, Part 3

The following are some of the multiple questions from the recent January 2020 New York State Common Core Geometry Regents exam.

June 2022 Geometry, Part III

Each correct answer is worth up to 4 credits. Partial credit can be given. Work must be shown or explained.


32. As modeled below, a projector mounted on a ceiling is 3.74 m from a wall, where a whiteboard is displayed. The vertical distance from the ceiling to the top of the whiteboard is 0.41 m, and the height of the whiteboard is 1.17 m.

Determine and state the projection angle, θ, to the nearest tenth of a degree.

Answer:
Find the angle in the large right triangle, and the angle in the skinny right triangle. Subtract those and you have θ. In both cases, you have the opposite and adjacent sides, so you need to use tangent.

tan x = 0.41/3.74
x = tan-1(0.41/3.74) = 6.2561...

tan x = (0.41+1.17)/3.74
x = tan-1 ((0.41+1.17)/3.74) = 22.9021...

θ = 22.9021 - 6. 2561 = 16.646 = 16.6 to the nearest tenth of a degree.




33. Given: Parallelogram PQRS, QT ⟂ PS, SU ⟂ QR

Prove: PT ≅ RU

Answer:

Answer:
You need to make a two-column proof. You can also write a paragraph proof, if you want, but you still need to have all the correct statements and reasons/justifications. The scorer will look to see that all the points are there.

You can prove this by showing that the two triangles are congruent and using CPCTC (corresponding parts of congruent triagnles are congruent). Looking at those triangles, you know that they are right triangles because the lines are perpendicular. And you know that the hypotenuses are congruent because they are the opposite sides of a parallelogram. And you know that angle P is congruent to angle R because its a parallelogram. This is what you need to show.

Note that the above paragraph isn't a paragraph proof because I didn't fully state the reasons and properties. It was meant as a guideline for a two-column proof.

Update: Proof added. Formatting takes time.

Statement Reason
1. Parallelogram PQRS, QT ⟂ PS, SU ⟂ QR Given
2. RS ≅ QP Opposite sides of a parallelogram are congruent.
3. ∠P ≅ ∠R Opposite angles of a parallelogram are congruent.
4. ∠PTQ and ∠RUS are right angles Definition of perpendicular lines
5. ∠PTQ ≅ ∠RUS All right angles are congruent.
6. △PTQ ≅ △RUS AAS Theorem
7. PT ≅ RU CPCTC (Corresponding Parts of Congruent Triangles are Congruent)

The first is required but not worth any points. Lines 2 and 3 are each worth 1 point. Lines 4-6 together are worth 1 point. Line 7 is worth the final point. Basically, proving AAS is three points, but you'll lose one point for each pair of "A" or "S" that you didn't show to be congruent, or for not stating AAS.

You can also prove PT = RU by showing that QUST is a parallelgragm (a rectangle even), with opposite sides congruent. Since QR = PS because it's a paralellogram, you can use the subtraction property to show PT = RU.

Given that PQRS is a parallelogram, then RQ ≅ PS because the opposite sides of a parallelogram are congruent. QU || ST because the opposite sides of a parallelogram are parallel. Angle QUS is a right angle because of the definition of perpendicular lines. Angle UST is a right angle because the same-side interior angles of a transversal between parallel lines are supplementary. Angles UST and STQ are supplementary because they add up to 180 degrees. So US || QT because the same-side angles are supplementary. QUST is a parallelogram because its oppisite sides are parallel. QU ≅ ST because the opposite sides of a parallelogram are congruent. Therefore, PT &cong RU by the Subtraction Property.

Okay, I'll admit that this took longer than I thought it would because I misread the given and didn't realize that I still had a couple extra things to prove to get where I wanted to go. HOWEVER, I knew I could get there, and I couldn't get the idea out of my head, so I went with it.




34. A concrete footing is a cylinder that is placed in the ground to support a building structure. The cylinder is 4 feet tall and 12 inches in diameter. A contractor is installing 10 footings.

If a bag of concrete mix makes ; of a cubic foot of concrete, determine and state the minimum number of bags of concrete mix needed to make all 10 footings.

Answer:
First, convert the inches to feet. Then find the radius from the diameter. Then find the volume of ONE footing. Multiply it by 10 to get the volume of TEN footings. Then DIVIDE by 2/3 (which means multiply by 3/2) to find the number of bags of concrete. Then, and this is Important, ROUND UP to the next full bag. If you round down, you won't have enough concrete mix to finish the tenth footing.

d = 12 inches = 1 foot, so r = 1/2 foot

V = π r2h = π (.5)2(4) = π = 3.14159....

10V = 10(3.14159...)

Divide 10(3.14159)/(2/3) = 47.12385

48 bags of concrete mix are needed.

Basically, you got a point for the Volume of one footing, a point for the Volume of 10 footings, a points for the amount of concrete mix needed and a point for the final number of bags. You didn't have to show those numbers to get full credit (the test didn't ask for any in-between answers), but they could give you partial credit if your final answer was incorrect. For example, you could have written one big equation to find the answer in one step. As long as the work was there, you got full credit.

End of Part III

How did you do?

Questions, comments and corrections welcome.

I also write Fiction!


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.

Thank you.



Thursday, June 30, 2022

June 2022 Geometry Regents, Part 2

The following are some of the multiple questions from the recent January 2020 New York State Common Core Geometry Regents exam.

June 2022 Geometry, Part II

Each correct answer is worth up to 2 credits. Partial credit can be given. Work must be shown or explained.


25. The Leaning Tower of Pisa in Italy is known for its slant, which occurred after its construction began. The angle of the slant is 86.03° from the ground. The low side of the tower reaches a height of 183.27 feet from the ground.

Determine and state the slant height, x, of the low side of the tower, to the nearest hundredth of afoot.

Answer:
First off, this isn't your typical Regents trigonometry ratio question, but it is still just a trigonometry question.

You have a right triangle, a leg (the height), a base angle that is opposite the leg, and you're looking for the hypotenuse. That means that you need to use sine.

sin 86.03 = 183.27 / x
x = 183.27 / sin 86.03
x = 183.7108...

To the nearest hundredth of a foot, the slant height is 183.71.




26. In the diagram below, quadrilateral ABCD is inscribed in circle 0, and mCD : mDA : mAB : mBC = 2:3:5:5. (refering to the arcs)

Determine and state m∠B.

Answer:
The sum of the four arcs is 360 degrees. Write an equation from the proportion:

2x + 3x + 5x + 5x = 360
15x = 360
x = 24

Since ∠B is an inscribed angle, it is half the size of the arc that it intercepts. It intercepts arc CDA. So ∠B = 1/2( 2(24) + 3(24)) = 60.




27. In the diagram below, a right circular cone has a diameter of 10 and a slant height of 13. Determine and state the volume of the cone, in terms of ℼ.

Answer:
The volume of a cone is 1/3 of the volume of a cylinder with the same base and height.

The radius is half of 10, which is 5. The height can be found using Pythagorean Theorem, with the radius being the other leg and the slant height being the hypotenuse of the right traingle.

I constantly remind students that you should know and recognize 3-4-5 an 5-12-13 triangles when you see them.

If you didn't, then 132 - 52 = h2, so 169 - 25 = 144 = h2, and h = 12.

V = 1/3 ℼ r2 h = 1/3 ℼ (5)2(12) = 1/3 ℼ (25)(12) = 100ℼ




28. In the diagram below, parallelogram EFGH is mapped onto parallelogram IJKH after a reflection over line l.


Use the properties of rigid motions to explain why parallelogram EFGH is congruent to parallelogram IJKH.

Answer:
A reflection is a rigid motion where size (distance) and shape (angle measure) are preserved, so the image will be congruent to the original.


29. Izzy is making homemade clay pendants in the shape of a solid hemisphere, as modeled below. Each pendant has a radius of 2.8 cm.


How much clay, to the nearest cubic centimeter, does Izzy need to make 100 pendants?

Answer:
The volume of a hemisphere is 1/2 the volume of a sphere. So the volume is V = 1/2 (4/3 ℼ r3). You are given the radius, so there's no reason to divide it. Use the ℼ key on the calculator to get enough decimals to avoid a rounding error. DO NOT USE 3.14. (And forget about 22/7!)

V = 1/2 (4/3 ℼ r3)
V = 1/2 (4/3) (3.141592) (2.8)3
V = 45.976

The Volume for 100 pendants is 4598.




30. Determine and state the coordinates of the center and the length of the radius of the circle whose equation is x2 + y2 + 6x = 6y + 63.

Answer:
The equation must be put into the standard form for a circle. That means getting all the x and y terms onto the left side and then Completing the Squares.

x2 + y2 + 6x = 6y + 63

x2 + y2 + 6x - 6y = 63

x2 + 6x + y2 - 6y = 63

x2 + 6x + 9 + y2 - 6y + 9 = 63 + 9 + 9

(x + 3)2 + (y - 3)2 = 81

The center of the circle is (-3, 3) and the radius is √(81) = 9.

Remember to flip the signs and to take the square root of 81.




31. 1 Use a compass and straightedge to construct a line parallel to AB through point C, shown below. [Leave all construction marks.]

Answer:
To make a parallel line, draw line CA. Construct an angle congruent to CAB with C as its vertex.

End of Part II

How did you do?

Questions, comments and corrections welcome.

I also write Fiction!


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.

Thank you.



Tuesday, June 28, 2022

Happy Tau Day 2022

(Click on the comic if you can't see the full image.)
(C)Copyright 2022, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

Its proponents will always swear that its better but the general public just isn't interested.

This was something that I typed on Facebook (and possibly Twitter) six years ago. It came up today as a memory. So I shared it again. It got a couple of comments, so I made it this year's comic. (It make be evening, but it's still Tau Day!)

I've had a few different Tau characters. Most likely they'll never appear during the rest of the year. I'm not one of the proponents of tau. I teach Algebra to high school students. Let the Ultra Math Nerds use tau. (I don't count myself among their numbers even if my family and friends might.)



I also write Fiction!


You can now preorder 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.
Preorder 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.



June 2022 Algebra 1 Regents, Part IV



This exam was adminstered in June 2022. These answers were not posted until they were unlocked on the NY Regents website or were posted elsewhere on the web.

More Regents problems.

June 2022

Part IV: A correct answer will receive 6 credits. Partial credit can be earned.


37. At an amusement park, the cost for an adult admission is a, and for a child the cost is c. For a group of six that included two children, the cost was $325.94. For a group of five that included three children, the cost was $256.95. All ticket prices include tax.

Write a system of equations, in terms of a and c, that models this situation.

Use your system of equations to determine the exact cost of each type of ticket algebraically.

Determine the cost for a group of four that includes three children.


Answer:


If you don't get the two equations correct, the rest will be difficult. Important: Keep going. You can get partial credit for consistent work but only if you finish the problem and get an answer. Yes, the person scoring the exam will go through your work.

This is the first time that I remember a question saying "a group of six including two children" meaning that there are only 4 adults.

The equation for the first group is 4a + 2c = 325.94

The equation for the second group is 2a + 3c = 256.95

You can solve this system using elimination because 2a is a factor of 4a.

4a + 2c = 325.94
2a + 3c = 256.95

4a + 2c = 325.94
-4a - 6c = -513.90
-4c = -187.96

c = 46.99

4a + 2(46.99) = 325.94

4a + 93.98 = 325.94

4a = 231.96

a = 57.99

An adult ticket costs $57.99 and a child's ticket costs $46.99.

You can answer the last question through substitution into the expression a + 3c, or you use the amount provided in the quation for 2a + 3c and subtract c.

Method 1: 57.99 + 3(46.99) = 198.96

Method 2: 256.95 - 57.99 = 198.96




End of Exam

How did you do?








More to come. Comments and questions welcome.

More Regents problems.

I also write Fiction!


You can now preorder 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.
Preorder 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.



Monday, June 27, 2022

Integrity of the Interval

(Click on the comic if you can't see the full image.)
(C)Copyright 2022, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

I have no idea where that phrase came from. I just popped out of my mouth.

Is it something that Bro. Steve might've said? Maybe. But I'm fairly certain that he didn't. Not that I remember.

A question did come up regarding a rubric about graphing errors. If a specific interval is given, then going beyond that interval is a graphing error because it didn't follow directions. If the graphs is within the interval but has arrows at the endpoints, that implies that it continues beyond the interval. So it is an error as well.

Why are the arrows such a big deal? I stated the above without even thinking about it. I was just giving a rationale, by the way, not necessarily endorsing it. I hate when questions are set up in ways that make losing points easy. I don't mind thoughtful problems, but I don't like "gotcha" problems.

Are arrows on the endpoints a "gotcha"? The prrof is left as an exercise to the reader.

One other thing: for my newer readers, Bro. Steve appeared once before back at a time when most of the male characters were tall and thin with big feet. I found it amusing back then, but started altering the characters after a number of years.



I also write Fiction!


You can now preorder 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.
Preorder 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.



June 2022 Algebra 1 Regents, Part III



This exam was adminstered in June 2022. These answers were not posted until they were unlocked on the NY Regents website or were posted elsewhere on the web.

More Regents problems.

June 2022

Part III: Each correct answer will receive 4 credits. Partial credit can be earned. One mistake computational will lose 1 point. A conceptual error will lost half credit, or 2 points. It is sometimes possible to get 1 point for a correct answer with no correct work shown.


33. The graph below models the height of Sma's kite over a period of time.

Explain what the zeroes of the graph represent in the context of the situation.

State the time interval over which the height of the kite is increasing.

State the maximum height, in feet, that the kite reaches.


Answer:


The zeroes of the graph represent when the kite is on the ground. Its height is zero.

There are two intervals where the kite's height is only increasing and doesn't decrease.. They are 0 < t < 1/2 and 1 < t < 2.

The kites's maximum height is 60 feet, which happens at t = 2 minutes.

This problem doesn't require any work to be shown because it assumes that you are gathering information from the graph provided.





34. On the set of axes below, graph f(x) = x2 - 1 and g(x) = 3x.

Based on your graph, for how many values of x does f(x) = g(x)? Explain your reasoning.

Answer:


Graph and label both equations. Label the points of intersection.

According to the graph, there is only on value of x where f(x) = g(x), which is the point where the graphs intersect.

Note: if you make a graphing error, you should answer this question based on your graph. If your graph shows, e.g., 0 or 2 points of intersection, then you should answer that number and explain that your graph says so. You will lose credit for the graphing error but you will NOT lose credit for the final question if it is consistent with your graph.





35. An insurance agent is looking at records to be determine if there is a relationshop between a driver's age and percentage of accidents cause by speeding. The table below shows his data.

State the linear regression that models the relationshop between the driver's age, x, and the percentage of accidents caused by speeding, y. Round all values to the nearest hundredth.

State the value of the correlation coefficient to the nearest hundredth. Explain what this means in the context of the problem.



Answer:


Put the information into lists L1 and L2. Use the Linear regression fuction to find y = ax + b and the value of r.

To the nearest hundredth, a = -0.96, b = 64.74, and r = -0.98. (Ignore r2. We don't need that.)

So y = -0.96x + 64.74 and r = -0.98.

In the context of this problem, there is a strong correlation between the age of the driver and the number of accidents caused by speeding.





36. Solve the system of inequalities graphically on the set of axes below. Label the solution set S.
2x + 3y < 9
2y > 4x + 6

Determine if the point (0,3) is a solution to this system of inequalities. Justify your answer.


Answer:


To use your graphing calculator, you need to rewrite the inequalities in slope-intercept form. Note that you must use the Original equations when labeling the graph.

2x + 3y < 9

3y < -2x + 9

y < -2/3 x + 3



2y > 4x + 6

y > 2x + 3



Remember that "<" means a dotted, dashed or broken line and you must shaded under the line. The line itself is NOT part of the solution.

The > means use a solid line and shade above the line. A solid line IS part of the solution set.

Make a large "S" in the section of the graph that is shaded twice.

The point (0,3) is not a solution because it is on the broken line and the broken line is not a part of the solution.

Note 1: If your graph is incorrect, answer according to what is on your graph if, for example, (0,3) is not on the dotted line.

Note 2: If you didn't draw the graph, you could still answer this last part algebraically for a point. Solving counts as justification. You must have justification -- "Yes" or "No" by itself will NOT get any credit.




End of Part III

How did you do?








More to come. Comments and questions welcome.

More Regents problems.

I also write Fiction!


You can now preorder 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.
Preorder 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.



Sunday, June 26, 2022

June 2022 Algebra 1 Regents, Part II



This exam was adminstered in June 2022. These answers were not posted until they were unlocked on the NY Regents website or were posted elsewhere on the web.

More Regents problems.

June 2022

Part II: Each correct answer will receive 2 credits. Partial credit can be earned. One mistake (computational or conceptual) will lose 1 point. A second mistake will lose the other point. It is sometimes possible to get 1 point for a correct answer with no correct work shown.


25. Is the product of √(1024) and -3.4 rational or irrational? Explain your answer

Answer:


The product of √(1024) and -3.4 is -108.8, which is a rational number. All numbers with terminating decimals are rational.

If you know binary, you might have recognized that 1024 is 210. The square root of 210 is 25, which is a rational number. The product of two rational numbers is always rational.

Note that you didn't actually have to have the product written to get full credit, so long as you stated that the radical was a rational number and the product was rational.





26. Describe the tranformations performed on the graph of f(x) = x2 to obtain the graph of g(x) when g(x) = (x - 3)2 - 4.

Answer:


The transformation is 3 units to the right and 4 units down.

If you weren't sure, you could have put both equations in your graphing calculator and looked at the graphs and the tables of values.

The vertex of f(x) is (0,0), and the vertex of g(x) is (3,-4).





27. The total profit earned at a garage sale during the first five hours is nodeled by the graph shown below.

Determine the average rate of change, in dollars per hour, over the interval 1 < x < 4.

Answer:


Look at the dollar values for x = 1 and x = 4. The two points are (1, 40) and (4, 100).

The average rate of change is (100 - 40) / (4 - 1) = 60 / 3 = 20

$20 per hour.





28. Subtract 3x(x - 2y) from 6(x2 - xy) and express your answer as a monomial.

Answer:


A monomial is one term. This means that something in the problem is going to cancel because the terms are not alike.

When subtracting, remember that the "from" value goes FIRST.

6(x2 - xy) - 3x(x - 2y)

6x2 - 6xy - 3x2 + 6xy

3x2

If you made one algebraic or arithmetic mistake, you could still get one credit. The word "monomial" was a big hint though.





29. A function is graphed on the set of axes below.

State the domain of the funtion.

State the range of the function.


Answer:


The domain of the function refers to all the possible x values. It is all real numbers, from negative infinity to positive infinity.

The range of the function refers to all the possible y values. It is y > 3. The range cannot be any value less than 3, which is the minimum of the function.





30. Solve 6x2 + 5x - 6 = 0 algebraically for the exact values of x.

Answer:


Usually when they do to the trouble to state "exact values", it generally means that they don't want you to round any decimal solutions. This might mean the answer is irrational or that there is a repeating decimal.

If you aren't sure how to factor, you can use the Quadratic Formula.

I like using the Reverse Area model, and then Factor by Grouping. Full disclosure: I would not have said that 10 years ago. We live and learn and grown.

Using the standard form, ax2 + bx + c = 0, multiply (a)(c) = (6)(-6) = -36.

What two factors of -36 have a sum of +5? A quick check of the factors of -36 will give you -9 and +4.

So 6x2 + 5x - 6 = 6x2 + 9x - 4x - 6

6x2 + 9x - 4x - 6 = 0

3x(2x + 3x) - 2(2x + 3) = 0

(3x - 2)(2x + 3) = 0

(3x - 2) = 0 or (2x + 3) = 0

3x = 2 or 2x = -3

x = 2/3 or x = -3/2

You must write 2/3 as a fraction or use notation to indicate that it is a repeating decimal. If you write x = .7 or x = .67, etc., you would lose a point.

Quadratic Formula solution will be added soon.





31. Factor the expression x4 - 36x2 completely.

Answer:


When you see "completely", that's a hint that there's going to be more than one step.

Factor the GCF, which is x2.

x4 - 36x2 = x2(x2 - 36)

Now there is a difference of perfect squares to deal with. (You could have done this first if you wanted to.)

x2(x2 - 36) = x2(x + 6)(x - 6)





32. Determine the exact values of x2 - 8x - 5 = 0 by completing the square.

Answer:


If you use a different method and get the correct answer, you will only receive 1 credit. Again, "exact values" means that you shouldn't round any decimals, and should leave irrational numbers in radical form.

x2 - 8x - 5 = 0

Half of -8 is -4, and (-4)2 is 16, so add a zero pair of (16 - 16) to the left side of the equation.

x2 - 8x + 16 - 16 - 5 = 0

Reduce the perfect square.

(x - 4)2 - 21 = 0

(x - 4)2 = 21

x - 4 = + √(21)

x = 4 + √(21)




End of Part II

How did you do?








More to come. Comments and questions welcome.

More Regents problems.

I also write Fiction!


You can now preorder 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.
Preorder 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.



Thursday, June 23, 2022

Slipped Up

(Click on the comic if you can't see the full image.)
(C)Copyright 2022, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

Slip ups happen all the time. Papers just move.

I wonder if this means that we'll see the same students in the same classes next year. (Ya think?)

Filling in the wrong bubbles on an exam happens, particularly when a student skips a question and forgets to leave that line blank. Sometimes the catch their error, sometimes they do not.

Personally, I hate using bubble sheets for tests if I can avoid them. I've had APs and colleagues encourage their use, and I've had large amounts of students that needed tests graded. On the other hand, in general, I hate multiple-choice questions in math class. They have some uses, but I'd rather see my students work instead of a bunch of circles.



I also write Fiction!


You can now preorder 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.
Preorder 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, June 22, 2022

Geometry Problems of the Day (Geometry Regents, August 2010)



Now that I'm caught up with the current New York State Regents exams, I'm revisiting some older ones.

More Regents problems.

Geometry Regents, August 2010

Part I: Each correct answer will receive 2 credits.


16. In which polygon does the sum of the measures of the interior angles equal the sum of the measures of the exterior angles?

1) triangle
2) hexagon
3) octagon
4) quadrilateral

Answer: 4) quadrilateral


The sum of the external angles of ANY polygon is 360 degrees. The sum of the interior angles of a polygon is the number of sides minus 2, times 180.

A triangle has 180 degrees. A quadrilateral has 360 degrees. A pentagon has 540 degrees. A hexagon has 720 degrees. A heptagon has 900 degrees. An octagon has 1080 degrees. Etc.

The correct answer is Choice (4).





17. In the diagram below of circle O, chords AB and CD intersect at E.


If CE = 10, ED = 6, and AE = 4, what is the length of EB ?

1) 15
2) 12
3) 6.7
4) 2.4

Answer: 1) 15


If two chords intersect, the products of their segments will be equal.

So (CD)(ED) = (AE)(EB)

Which means (10)(6) = (4)(EB)

EB = 60 / 4 = 15, which is Choice (1).





18. In the diagram below of △ABC, medians AD, BE, and CF intersect at G.


If CF = 24, what is the length of FG?

1) 8
2) 10
3) 12
4) 16

Answer: 1) 8


The ratio of CF:FG is 3:1. The centroid splits to the median into two segments at a ratio of 2:1.

One third of 24 is 8, which is Choice (1).





19. If a line segment has endpoints A(3x + 5, 3y) and B(x − 1, −y), what are the coordinates of the midpoint of AB?

1) (x + 3, 2y)
2) (2x + 2, y)
3) (2x + 3, y)
4) (4x + 4, 2y)

Answer: 2) (2x + 2, y)


To find the midpoint, you have to find the middle value of the x-coordinates and of the y-coordinates. To do that, add the expressions together and divide by two.

3x + 5 + x - 1 = 4x + 4. Dividing that by two gives a quotient of 2x + 2, which is Choice (2).

3y - y = 2y. Divide that by 2, and you get just y. That is also Choice (2). (And also Choice (3), which we eliminated already.)





20. If the surface area of a sphere is represented by 144π, what is the volume in terms of π?

1) 36π
2) 48π
3) 216π
4) 288π

Answer: 4) 288π


The surface area of a sphere can be found using the formula SA = 4πr2.

If 4πr2 = 144π then r2 = 36, and r = 6.

The Volume of a sphere can be found using the formula V = (4/3)πr3.

V = (4/3)π(6)3 = V = 288π, which is Choice (4).




More to come. Comments and questions welcome.

More Regents problems.

I also write Fiction!


You can now preorder 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.
Preorder 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.