Thursday, February 11, 2016

(x, why?) Mini: Flowers

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(C)Copyright 2016, C. Burke.

Thinking of Spring and flowers . . .

And, oddly, I thought I could reuse old artwork for a quick comic. And I hated it, so I replaced the old artwork with new artwork. These "minis" are sometimes as time-consuming as the regular ones.

Come back often for more funny math and geeky comics.

Wednesday, February 10, 2016

Music Venn Diagram

(Click on the comic if you can't see the full image.)
(C)Copyright 2016, C. Burke.

The views expressed in the diagram reflect the opinions of the comic makers in so far as there are four possibilities expressed in the comic.

The fourth possibility is left as an exercise for the reader (but should be really, really obvious).

Come back often for more funny math and geeky comics.

Tuesday, February 09, 2016

January 2016 New York Geometry (Common Core) Parts 3 and 4

Below are the questions with answers and explanations for Parts 3 and 4 of the Geometry (Common Core) Regents exam for January 2016. Part I questions appeared in a here. Part II questions appeared in a Here.

Part III

Question 32 was answered in the post for Part II. It is reprinted here because it was actually a Part III question.

32. The aspect ratio (the ratio of screen width to height) of a rectangular flat-screen television I s16:9. The length of the diagonal of the screen is the television’s screen size. Determine and state, to the nearest inch, the screen size (diagonal) of this flat-screen television with a screen height of 20.6 inches.

This can be solved using ratios and Pythagorean Theorem, or by using Trigonometric Ratios.

First, set up a proportion 16/9 = x/20.6 and cross-multiply.
9x = 329.6, x = 36.6
20.62 + 36.62 = c2
424.36 + 1339.56 = c2
1763.02 = c2
C = 41.999 = 42 inches

Second, the ratio 16:9 represents the opposite over the adjacent, which is the tangent of the top angle of the set. So tan(y) = 16/9 and y = tan-1(16/9) = 60.64 degrees.

Using the adjacent and the hypotenuse and cosine, we get the following:
cos(60.64) = 20.6/x, so x = 9/cos(60.64) = 42.0155… = 42 inches.

33. Given the theorem, "The sum of the the measures of the interior angles of a triangle is 180o," complete the proof for this theorem.

Fill in the missing reasons below.

This problem caused students a bit of trouble. First, many students had no idea what to write as a reason for Statement (2). What they wrote might have been a correct statement, but not a correct reason for the statement that was made. There is a difference between the two.

Another problem is that many students gave the reason that "The sum of the angles of a triangle is 180 degrees." While this is a true statement, it cannot be used as a Reason when that is exactly what you are trying to prove!

Most of the papers I graded score only 2 of the 4 points. (One point for each correct reason. "Given" was preprinted.)

(1) Triangle ABC. Given
(2) Through point C, draw DCE parallel to AB. Through any line and a point not on the line, there is exactly one line passing through that point parallel to the line.
(3)m<l = m<ACD, m<3 = m<BCE When two parallel lines are cut by a transversal, alternate interior angles are congruent.
(4) m<ACD + m<2 + m<BCE = 180° The sum of angles creating a straight line is 180 degrees.
(5) mLl + mL2 + mL3 = 180° Substitution Property

I saw many students who didn't label angles as "alternate interior", leaving out one word or the other. And a number of them mentioned supplementary angles in statement (4), but "supplementary" applies to two angles, not three.

34. Triangle XYZ is shown below. Using a compass and straightedge, on the line below, construct and label triangle ABC, such that triangle ABC = triangle XYZ. [Leave all construction marks.]
Based on your construction, state the theorem that justifies why triangle ABC is congruent to triangle XYZ.

Sorry, but I can't illustrate constructions very well. I promise to get back to it.

Simple steps: From point X, measure the distance to Y. On the line, make point A and mark the distance to with an arc, and label where the arc intersects the line as B. Go back to X and measure the distance to Z. Go to X and make another arc. Go back to Y and measure the length to Z. Put the compass on B and make an arc. Where the two arcs intercept, label that point C. Using the straightedge, make lines AC and BC. The theorem used was SSS.

Note: It is possible that you could have used SAS to create the triangles. Whichever theorem you stated, the construction marks had to match it to get full credit.

You also could have gotten 1 point for saying SSS, SAS, or ASA without any construction at all.

Part IV

35. Given: Parallelogram ANDR with AW and DE bisecting NWD and REA at points W and E, respectively

Prove that triangle ANW = triangle DRE.
Prove that quadrilateral AWDE is a parallelogram.

There are many variations of this proof which will be acceptable. When using SAS to prove two triangles congruent, be sure that you have included statements regarding the two pairs of sides and the included pairs of angles.

(1) Parallelogram ANDR with AW and DE bisecting NWD and REA. Given
(2) AE = RE and DW = NW Definition of segment bisector.
(3) RA = DN and RD = NA Opposite sides of a parallelogram are congruent.
(4) RE = NW Halves of congruent segments are congruent.
(5) Angle R = Angle N Opposite angles of a parallelogram are congruent.
(6) Triangle DRE = Triangle ANW SAS
(8)AE = DW Halves of congruent segments are congruent.
(9) AWDE is a parallelogram If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram

Parallelograms have many properties that apply only to them. Proving any of those would be sufficient proof that AWDE was a parallelogram.

36. Cathy wants to determine the height of the flagpole shown in the diagram below. She uses a survey instrument to measure the angle of elevation to the top of the flagpole, and determines it to be 34.9°. She walks 8 meters closer and determines the new measure of the angle of elevation to be 52.8°. At each measurement, the survey instrument is 1.7 meters above the ground.

Determine and state, to the nearest tenth of a meter, the height of the flagpole.

Personally, I hate problems that force you to add 1.7 meters at the end for no good reason. It's just an added step for people to forget about and lose a point.

One also wonder why Cathy didn't walk to the pole and measure that distance rather than walking an extra 8 meters away from it. But this is the problem we are given.

We are looking for the height of the pole. That is the opposite side to both angles. We have some information about the distance along the ground. That is the adjacent side to the angles. We have no information about the hypotenuse, nor are we looking for it.

Opposite and adjacent means that we are using tangent.

Let h be the height and x be the distance along the ground to the first measurement.
tan 52.8 = h / x
so h = x (tan 52.8)
And tan 34.9 = h / (x + 8)
so h = (x + 8)(tan 34.9)
This means that x (tan 52.8) = (x + 8)(tan 34.9). We need to isolate x.
Distribute: x (tan 52.8) = x (tan 34.9) + 8(tan 34.9)
Subtract: x (tan 52.8) - x (tan 34.9) = 8(tan 34.9)
Factor: x (tan 52.8 - tan 34.9) = 8(tan 34.9)
Divide: x = 8(tan 34.9) / (tan 52.8 - tan 34.9)
Calculate: x = 9.00371 = 9 meters
Use x to get h:
h = 9.00371(tan 52.8) = 11.8619
And the height of the survey instrument: 11.86 + 1.7 = 13.56 = 13.6 meters.

An amazing amount of work, but it was worth 6 points.
You lost credit if you forgot the 1.7 meters, used the wrong functions, or rounded in the middle of the problem so that your answer didn't round to 13.6

Oddly, if you multiplied 1.7 * 8 you get EXACTLY 13.6. If you wrote 13.6 with NO WORK WHATSOEVER, you got one point. If you wrote 1.7 * 8 = 13.6, you got ZERO POINTS for a totally incorrect response or a correct response found through a totally incorrect method. Those are the breaks.


How did you do?

Monday, February 08, 2016

January 2016 New York Geometry (Common Core) Part 1

Sorry for the delays. These things happen.

Below are the questions with answers and explanations for Part 1 the Geometry (Common Core) Regents exam for January 2016. Part II questions appeared in a another post.

Part I

1. William is drawing pictures of cross sections of the right circular cone below. (image omitted)
Which drawing can not be a cross section of a cone?

(1) the square. You can make a slice through that cone and get an oval (or circle), a semicircle with a diameter on the bottom or even triangle if you split it vertically. You can't get a square.

2. An equation of a line perpendicular to the line represented by the equation y = -(1/2)x - 5 and passing through (6, -4) is

(4) y = 2x - 16. A line perpendicular to a line with a slope of -1/2 would have a slope of 2, so (1) and (2) are out. If you plug in 6 for x, -4 = 2(6) + b, b = -16.

3. In parallelogram QRST shown below, diagonal TR is drawn, U and V are points on TS and QR, respectively, and UV intersects TR at W.

If m<S = 60°, m<SRT = 83°, and m<TWU = 35°, what is m< WVQ?

(3) 72°. Look at quadrilateral QTWV, which has 360°. Angle Q = S = 60. Angle QTV = SRT = 83. Angle TWV = 180 - TWU = 180 - 35 = 145. 60 + 83 + 145 = 288. 360 = 288 = 72°.

4. A fish tank in the shape of a rectangular prism has dimensions of 14 inches, 16 inches, and 10 inches. The tank contains 1680 cubic inches of water. What percent of the fish tank is empty?

(2) 25. The volume of the tank is 14 * 16 * 10 = 2240. 2240 - 1680 = 560 gallons empty. 560 / 2240 = .25 = 25%

5. Which transformation would result in the perimeter of a triangle being different from the perimeter of its image?

(3) (x,y)--> (4x,4y). A dilation would change the distance between the vertices, and therefore the perimeter. Reflections and translations do not affect the size, so the image would have the same perimeter.

6. In the diagram below, FE bisects AC at B, and GE bisects BD at C.

Which statement is always true?

(1) AB = DC. Because of bisecting AB = BC and BC = CD. Therefore, AB = CD. We have no information about where points F or G are, so you cannot make any assumptions about those lines being bisected or the segments being equal.

7. As shown in the diagram below, a regular pyramid has a square base whose side measures 6 inches. (image omitted)
If the altitude of the pyramid measures 12 inches, its volume, in cubic inches, is

(2) 144. The Volume is (1/3)(Area of the Base)(height) = (1/3)(6 * 6)(12) = (12)(12) = 144.

8. Triangle ABC and triangle DEF are graphed on the set of axes below.

Which sequence of transformations maps triangle ABC onto triangle DEF?

(1) a reflection over the x-axis followed by a reflection over the y-axis. It is also a 180 degree rotation about the origin, but it would NOT be followed by a reflection in y = x.

9. In triangle ABC, the complement of <B is <A. Which statement is always true?

(4) sin <A =cos <B. The sine of one complementary angle is the cosine of the other in a right triangle. The side opposite angle A will be adjacent to angle B.

10. A line that passes through the points whose coordinates are (1,1) and (5,7) is dilated by a scale factor of 3 and centered at the origin. The image of the line

(2) is parallel to the original line. As long as the line does not go through the origin, its dilation will be a parallel line (having the same slope). If the line went through the origin, it would be the same line. The slope of the line is 6/4 = 3/2. The line y = 3/2x does not go through (1, 1), so the line being dilated does not go through the origin.

11. Quadrilateral ABCD is graphed on the set of axes below.

When ABCD is rotated 90° in a counterclockwise direction about the origin, its image is quadrilateral A' B 'C 'D'. Is distance preserved under this rotation, and which coordinates are correct for the given vertex?

(4) yes and B'(-3,4) . Yes, distance is preserved, (1) and (2) are eliminated. A(-2, 6) will move to A'(-6, -2). Choice (3) would have been correct for a clockwise rotation.

12. In the diagram below of circle 0, the area of the shaded sector LOM is 2(pi) cm2.

If the length of NL is 6 cm, what is m<N?

(3) 40°. The diameter is 6 cm, so the radius is 3 cm. That makes the Area of the entire circle = (pi)(3)2 = 9pi.
if the shaded sector is 2pi, then that sector and that arc is 2/9 of the circle, which is 2/9(360) = 80 degrees. Angle N is an inscribed angle that intercepts arc LM, so it is half of 80 degrees, or 40 degrees.

13. In the diagram below, triangle ABC ~ triangle DEF.

If AB = 6 and AC = 8, which statement will justify similarity by SAS?

(1) DE = 9, DF = 12, and <A = <D. Angles A and D are the included angles, so (3) and (4) are eliminated. 6/8 = .75 and 9/12 = .75, so they are proportional.

14. The diameter of a basketball is approximately 9.5 inches and the diameter of a tennis ball is approximately 2.5 inches. The volume of the basketball is about how many times greater than the volume of the tennis ball?

(3) 55. The radius of the basketball is 4.75. The radius of the tennis ball is 1.25. Volume requires cubing the radii. Divide (4.753 / 1.253) = 54.872, which is approximately 55.

15. The endpoints of one side of a regular pentagon are (-1,4) and (2,3). What is the perimeter of the pentagon?

(2) 5*SQRT(10). Using the Distance formula (or the Pythagorean Theorem), we can find the length of the segment joining those two endpoints to be SQRT(32 + 12), which is SQRT(10). Since there are five sides to a pentagon, the perimeter is 5*Sqrt(10). [5 radical 10]

16. In the diagram of right triangle ABC shown below, AB = 14 and AC= 9. (image omitted)
What is the measure of <A, to the nearest degree?

(3) 50. AC is the adjacent side and AB is the hypotenuse, so cos A = 9/14. That means <A = cos-1(9/14) = 49.99... degrees.

17. What are the coordinates of the center and length of the radius of the circle whose equation is x2 + 6x + y2 - 4y = 23?

(4) (-3,2) and 6 You need to complete the squares to find the equation for the circle. (I could make a comic out of that sentence.)
Half of +6 is +3, and half of -4 is -2, but you have to flip the signs to find (h, k). Remember: (x - h)2 + (y - k)2.
So the center is at (-3, 2) and you eliminate (1) and (2).
To complete the square you need to add (3)2 add (-2)2 to both sides of the equations. That adds 9 + 4 to 23, giving 36, which is r2. So the radius is 6 (not 36).

18. The coordinates of the vertices of triangle RST are R(-2, -3), S(8,2), and T(4,5). Which type of triangle is triangle RST?

(1) right. Eliminate (4) equiangular, because equiangular/equilateral triangles are always acute, and you can't have two correct answers.
If the triangle is right, then two of the sides will have perpendicular slopes -- that is, they will be negative reciprocals. If that is not true, then you have to find the lengths of the three sides to determine if the triangle is acute or obtuse.

Slope of RS = (2 - -3)/(8 - -2) = 5/10 = 1/2. Slope of ST = (5 - 2)/(4 - 8) = 3/(-4) = -(3/4). Slope of TR = (-3 - 5)/(-2 - 4) = -8/-6 = 4/3. ST is perpendicular to TR because (-3/4)(4/3) = -1. It is a right triangle.

19. Molly wishes to make a lawn ornament in the form of a solid sphere. The clay being used to make the sphere weighs .075 pound per cubic inch. If the sphere's radius is 4 inches, what is the weight of the sphere, to the nearest pound?

(2) 20. Multiply the Volume of the sphere by .075, so .075(4/3)(pi)(4)3 = 20.1..., which is about 20.

20. The ratio of similarity of triangle BOY to triangle GRL is 1:2. If BO = x + 3 and GR = 3x - 1, then the length of GR is

(4) 20. Because GR is twice as big as BO, start with 2(x + 3) = 3x - 1,
So 2x + 6 = 3x - 1
and 7 = x. (note that this is choice (2).)
The length of GR is 3(7) - 1 = 20.

21. In the diagram below, DC, AC, DOB, CB, and AB are chords of circle O, FDE is tangent at point D, and radius AO is drawn. Sam decides to apply this theorem to the diagram: "An angle inscribed in a semi-circle is a right angle."

Which angle is Sam referring to?

(3) <DCB. DOB is a diameter of the triangle and arc DAB is a semi-circle. Angle DCB is an inscribed angle that intercepts the semi-circle. Since it is half the measure of the 180 degrees, it must be 90 degrees, making it a right angle.

22. In the diagram below, CD is the altitude drawn to the hypotenuse AB of right triangle ABC. (image omitted)
Which lengths would not produce an altitude that measures 6*SQRT(2)?

(2) AD = 3 and AB = 24 Square (6*SQRT(2)) and you get 36 * 2 = 72 as the altitude. The product of AD and DB must be 72 for the altitude to be 6*SQRT(2). Read the choices carefully. Two of the choices give you AB, not DB. You need to subtract AD from AB to get DB. While 3 * 24 equals 72, you are supposed to be multiplying 3 * 21, which is only 63.

23. A designer needs to create perfectly circular necklaces. The necklaces each need to have a radius of 10 cm. What is the largest number of necklaces that can be made from 1000 cm of wire?

(1) 15. If the radius is 10 cm, then the length of the wire to produce one necklace, (2)(10)(pi), is approximately 62.83 cm. Divide 1000/62.83 and you get 15.9. However, you cannot round up because you don't have enough wire to finish the 16th necklace.

24. In triangle SCU shown below, points T and O are on SU and CU, respectively. Segment OT is drawn so that <C = <OTU.

If TU = 4, OU = 5, and OC = 7, what is the length of ST?

(3) 11. Triangles OUT and CUS are similar, and the sides are proportional, but make sure you set up the proportion correctly. OT is NOT parallel to CS. The correct proportion is 4/(7 + 5) = 5/(4 + x), so 16 + 4x = 60.
4x = 44, x = 11.


Year of the Monkey

(Click on the comic if you can't see the full image.)
(C)Copyright 2016, C. Burke.

For the record, Xiu is *not* an immortal. That I know of.

He still gets around very well, and he might outlast me. And ... hmmmm....

Happy New Year 4713!

Come back often for more funny math and geeky comics.

Sunday, February 07, 2016

Super Bowl 50

(Click on the comic if you can't see the full image.)
(C)Copyright 2016, C. Burke.

Let the discussion of Ordinal v. Cardinal begin now.

Seriously, I get comments from people who get it backward and/or think I got it backward.

As for our buddy, L in the comic, he is refering to last year's comic.

Come back often for more funny math and geeky comics.

Sunday, January 31, 2016

January 2016 New York Algebra I (Common Core) Parts 3 and 4

Below are the questions with answers and explanations for Part 3 and 4 of the Algebra I (Common Core) Regents exam for January 2016. The multiple-choice questions appeared in a previous post. Part II questions appeared in a another post.

As always, in order to get this thread up quickly, the images have been omitted. They will be added at a later date.

Each question in Part 3 is worth 4 credits, for a total of 16 credits. Partial credit will be given. The one question in Part 4 is worth 6 credits.

January 2016 Algebra 1 (Common Core) Regents, Part 3

33. Let h(t) = -16t2 + 64t + 80 represent the height of an object above the ground after t seconds. Determine the number of seconds it takes to achieve its maximum height. Justify your answer.

State the time interval, in seconds, during which the height of the object decreases. Explain your reasoning.

The maximum height is at the vertex, and the vertex is on the axis of symmetry.
The axis of symmetry is x = -b/(2a) = -64/(2(-16)) = -64/-32 = 2 seconds.
h(2) = -16(2)2 + 64(2) + 80 = -16(4) + 128 + 80 = -64 + 128 + 80 = 144. It will be 144 feet off the ground.

It will decrease in height from 2 seconds until it hits the ground, which is when h(t) = 0.
Solve -16t2 + 64t + 80 = 0
Divide by -16: t2 - 4t - 5 = 0
Factor: (t - 5)(t + 1) = 0
Solve t = 5 or t = -1.
Discard the negative time value. The object hits the ground at 5 seconds.

The object is descending from 2 < t < 5.
At 5 seconds, it is no longer descending. Before 2, it was still rising.

34. Fred's teacher gave the class the quadratic function f(x) = 4x2 + 16x + 9.
a) State two different methods Fred could use to solve the equation f(x) = 0.
b) Using one of the methods stated in part a, solve f(x) = 0 for x, to the nearest tenth.

a) Pick any two methods for solving are factoring, completing the square and the quadratic formula. There is also graphing, but that won't be helpful if the answer is not an integer.

Note that factoring may not be possible. If it turns out that it isn't, cross that one out and use the other two.
4x2 + 16x + 9 = 0 -- 4 and 9 are the squares of 2 and 3 and (2)(2)(3) = 12, not 16, so it is (2x + 3)2. (One could hope it would be easy.)

Personally, I prefer the Quadratic formula is Completing the Square is going to involve fractions anyway, so look at the illustration:

So the solutions are {-3.3, -.7}

35. Erica, the manager at Stellarbeans, collected data on the daily high temperature and revenue from coffee sales. Data from nine days this past fall are shown in the table below. (image omitted)

State the linear regression function, f(t) that estimates the day's coffee sales with a high temperature of t. Round all values to the nearest integer.

State the correlation coefficient, r, of the data to the nearest hundredth. Does r indicate a strong linear relationship between the variables? Explain your reasoning.

You need to put the data into LISTs in the graphing calculator and then do a Linear Regression. Before the exam, your calculator should have had its memory reset. After that, the two things that should have happened were that it was put back into Degree Mode, and DiagnosticsON should have been executed. It's a function in the CATALOG. With that on, the correlation coefficient will appear on the screen when you run a Linear Regression.

Put the temperatures in L1, and the sales in L2. Double check your work. (I had one typo, and that would have skewed my answer!)
Hit Stat, go to the Calc menu, and press 4: Linear Regression. Press ENTER.

f(t) = -58t + 6182 -- use f(t) and t. Don't use y and x.

The correlation coefficient is -.94 (to the nearest hundredth). This indicates a strong negative relationship because the number is close to -1, meaning that it is almost a straight line.

36. A contractor has 48 meters of fencing that he is going to use as the perimeter of a rectangular garden. The length of one side of the garden is represented by x, and the area of the garden is 108 square meters.

Determine, algebraically, the dimensions of the garden in meters.

P = 2x + 2w = 48; 2w = 48 - 2x; w = 24 - x
A = L * W = x (24 - x) = 108
24x - x2 = 108
0 = x2 - 24x + 108
0 = (x - 6)(x - 18)
x = 6 or x = 18

The dimensions of the garden are 6 m X 18 m.
Check: 6 * 18 = 108. 2(6) + 2(18) = 12 + 36 = 48. Check.

January 2016 Algebra 1 (Common Core) Regents, Part 4

37. The Reel Good Cinema is conducting a mathematical study. In its theater, there are 200 seats. Adult tickets cost $12.50 and child tickets cost $6.25. The cinema's goak is to sell at least $1500 worth of tickets for the theater.

Write a system of linear inequalities that can be used to find the possible combinations of adult tickets, x, and child tickets, y, that would satisfy the cinema's goals.

Graph the solution to this system of inequalities on the set of axes on the next page. Label the solution S.

Marta claims that selling 30 adult tickets and 80 child tickets will result in meeting the cinema's goal. Explain whether she is correct or incorrect, based on the graph drawn.

The graph will be coming shortly. Please be patient.

The system of inequalities is:

x + y < 200
12.50x + 6.25y >1500

When you graph them, both will have solid lines. The first inequality (# of tickets) will be shaded below. The second inequality (money from the sales) will be shaded above the line. The area shaded twice will be get the S.

Marta is incorrect. (If you graphed correctly) If you look at the graph, (30, 80) is not in the section with the S, so it is not a solution to the system of inequalities. (You needed to refer to the graph, so just plugging the numbers into both inequalities is not sufficient.)

End of Test

How'd you do?

Saturday, January 30, 2016

January 2016 New York Algebra I (Common Core) Part 2

Below are the questions with answers and explanations for Part 2 of the Algebra I (Common Core) Regents exam for January 2016. The multiple-choice questions appeared in a previous post. The rest of the questions will appear in a later post.

As always, in order to get this thread up quickly, the images have been omitted. They will be added at a later date.

Each question in Part 2 is worth 2 credits, for a total of 16 credits. Partial credit will be given. Basically, you can have one computational, conceptual, graphing or rounding error, but as long as you have a consistent answer, you can still get a point. Two different mistakes, and there is no credit for the answer.

January 2016 Algebra 1 (Common Core) Regents, Part 2

25. The function, t(x) is shown in the table below. (image omitted)
Determine whether t(x) is linear or exponential. Explain your answer.

The function t(x) is linear because the slope is consistent. Take any pair of points and you will find the slope is -2.5/2 or -1.25. Use a couple of pairs of points as examples to prove your point.

If you're curious, the function is t(x) = -1.25x + 6.25, but that isn't necessary for the question. In fact, that answer isn't any good unless you had the work to back it up.

26. Marcel claims that the graph below represents a function. (image omitted)
State whether Marcel is correct. Justify your answer.

Marcel is incorrect. It is not a function because when the graph does not pass the vertical line test. The line x = 2 goes through two points on the graph. Both are closed circles.

27. Solve the equation for y.

(y - 3)2 = 4y - 12

Square the binomial: y2 - 6y + 9 = 4y - 12
Move everything to the left: y2 - 10y + 21 = 0
Factor: (y - 7)(y - 3) = 0
solve y = 7 or y = 3.

You could have also completed the square or used the quadratic formula after putting it in standard form. If you made one computational error, but continued until the end and gave an answer, you would've gotten one point.

28. The graph below shows the variation in the average temperature of Earth's surface from 1950-2000, according to one source. (image omitted)
During which years did the temperature variation change the most per until time? Explain how you determined your answer.

The largest change was between 1960 and 1965 when the slope of the graph was -.15/5. It is the steepest part of the graph. The increase from 1975 to 2000 is a constant .1/5.

Be careful with the fractions and decimals. I just typed them incorrectly, but I caught the mistake before I posted them. (Had the decimal point in the wrong position.)

29. The cost of belonging to a gym can be modeled by C(m) = 50m + 79.50, where C(m) is the total cost for m months of membership.

State the meaning of the slope and y-intercept of this function with respect to the costs associated with the gym membership.

It costs $79.50 to joint the gym. That is a one-time fee that you pay even if you go for zero months. $50 is the monthly fee, which is paid for the number of months, m.

Note that this was just a definition question with nothing to work out. Common Core is doing a lot of that.

30. A statistics class surveryed some students during one lunch period to obtain opinions about television programming preferences. The results of the survey are summarized in the table below. (image omitted)
Based on the sample, predict how many of the schools 351 males would prefer comedy. Justify your answer.

70 out of (70 + 35) males preferred comedy. That is 70/105 or 2/3.

Multiply (2/3)(351) = 234 males.

31. Given that a > b, solve for x in terms of a and b.

b(x - 3) > ax + 7b

Follow the steps:

bx - 3b > ax + 7b
bx > ax + 10b
bx - ax > 10b
x(b - a) > 10b
x < 10b/(b - a)

Because a > b, that makes (b - a) a negative number, and when you divide by a negative number, the inequality symbol has to flip around.
(Also, since a > b, (b - a) cannot equal zero, so it is okay to divide by it.)

32. Jacob and Jessica are studying the spread of dandelions. Jacob discovers that the growth over t weeks can be defined by the funtion f(t) = (8)*2t. Jessica finds that the growth function over t weeks is g(t) = 2t+3.

Calculate the number of dandelions that Jacob and Jessica will each have after 5 weeks.

Based on the growth from both function, explain the relationship between f(t) and g(t).

f(t) = 8(2)5 = 8(32) = 256.
g(t) = 25+3 = 28 = 256.

Based on the growth, the two functions are the same.
This is because g(t) = 2t+3 = 2t * 23 = 2t * 8 = f(t).

I'll be honest here. I have no clue what they are going for in this last question. Based on only one data point, you can only conclude that the are the same function for that one input. It isn't enough to say the functions are the same. It is easy to prove that they are the same (as I showed above) but that isn't what they asked.

End of Part 2
How did you do?

Friday, January 29, 2016

January 2016 New York Algebra I (Common Core) Part 1: Multiple-Choice

Below are the questions with answers and explanations for Part 1 of the Algebra I (Common Core) Regents exam for January 2016, the multiple-choice questions. The open-ended questions will be posted separately.

As always, in order to get this thread up quickly, the images have been omitted. They will be added at a later date.

Each question in Part 1 is worth 2 credits, for a total of 48 credits. Usually, a score of 30 credits on the entire test is worth a score of 65. The curve is steep after that. To achieve a final score of, say, 75, you will need roughly 55 credits. (The exact curve will not be revealed for a few days.)

January 2016 Algebra 1 (Common Core) Regents, Part 1

1. In the function f(x) = (x - 2)2 + 4, the minimum value occurs when x is

(2) 2. The vertex is at the point (h, k) taken from the general form, f(x) = (x - h)2 + k.

2. The graph below was created by an employee at a gas station. (image omitted)
Which statement can be justified by using the graph?

(2) For every gallon of gas purchased, $3.75 was paid. (1) and (3) can be eliminated by checking the graph. Neither (10, 35) nor ((2, 5) are points on the line. You can't see (1, 3.75) but if you multiply by 4, you will see (4, 15) on the graph. Choice (4) is just ridiculous -- the graph has nothing to do with miles driven.

3. For a recently released movie, the function y = 119.67(0.61)x models the revenue earned, y, in millions of dollars each week, x, for several weeks after its release.

Based on the equation, how much more money, in millions of dollars, was earened in revenue for week 3 than for week 5?

(3) 17.06. Substituting 3 for x in the function gives us about 27.16 million, and substitution 5 gives us 10.11, with a difference of 17.05. The difference with the answer is a minor rounding error.

4. Given the following expressions:
I. -(5/8) + (3/5); II. (1/2) + sqrt(2); III. (sqrt(5))*(sqrt(5)); IV. 3*(sqrt(49))
Which expression(s) result in an irrational number?

(1) II only. The first is the sum of two rationals, which is rational. The second is the sum of a rational and an irrational, which is irrational. The third squares the square root of 5, which is 5, a rational number. The last is three times the square root of 49, which equals 3 * 7 which is 21, a rational number.

5. Which inequality is represented by the graph below?

(2) y > 2x - 3. The y-intercept is -3. The slope is 2. The graph is shaded above the line. (The line is also solid, but that doesn't matter for the choices given.)

6. Michael borrows money from his uncle, who is charging him simple interest using the formula I = Prt. To figure out what the interest rate, r, is, Michael rearranges the formula to find r. His new formula is r equals

(3) I/Pt. Divide both sides of the equation by P and t to isolate r.

7. Which equation is equivalent to y - 34 = x(x - 12)?

(4) y = (x - 6)2 + 2.
Distribute the x and you get: y - 34 = x2 - 12x.
Add 34 to both sides: y = x2 - 12x + 34.
There are no integer factors of 34 that have a sum of -12. (Eliminate choices 1 and 2.) This means completing the square, OR working backward from the other choices.
Squaring (-6) gives us +36. To make 36 into 34, we need to subtract 2. Choice (4).

8. The equation A = 1300(1.02)7 is being used to calculate the amount of money in a savings account. What does 1.02 represent in this equation?

(4) 2% growth. It's greater than 1, so it is growth. (And if it is a savings account, it better be growth, or why have the account?) The decimal .02 is 2% as a percentage.

9. The zeros of the function f(x) = 2x2 - 4x - 6 are

(1) 3 and -1. Plug in the numbers and see what gives you 0. This is probably quicker than factoring.
2x2 - 4x - 6 = 0
x2 - 2x - 3 = 0
(x - 3)(x + 1) = 9
x = 3 or x = -1

10. When (2x - 3)2 is subtracted from 5x2, the result is

(3) x2 + 12x - 9.
5x2 - (4x2 - 12x + 9) = 5x2 - 4x2 + 12x - 9 = x2 + 12x - 9.

11. Joe has a rectangular patio that measures 10 feet by 12 feet. He wants to increase the area by 50% and plans to increase each dimension by equal lengths, x. Which equation could be used to determine x?

(2) (10 + x)(12 + x) = 180. Area = Length * Width = (10)(12) = 120. Increasing 120 by 50% means 120 + .5(120) = 180.

12. When factored completely, x3 - 13x2 - 30x is

(3) x(x + 2)(x - 15)
x(x2 - 13x - 30) = x(x - 15)(x + 2)

13. The table below (image omitted) shows the cost of mailing a postcard in different years. During which time interval did the cost increase at the greatest average rate?

(4) 2006-2012. Find the average rate for each interval. From 1898-1971, the change was 5/73. From 1971-1985, it was 8/14. From 1985-2006, it was 10/21. From 2006-2012, it was 11/6. Choice (4) is the only one greater than 1, so it is obviously the greatest.

If you eliminate choice 1 as obviously very flat, you can sketch out the other points and see that 2006-2012 is the steepest line.

14. When solving the equation x2 - 8x - 7 = 0 by completing the sqaure, which equation is a step in the process?

(2) (x - 4)2 = 23. Half of 8 is 4, so eliminate choices (3) and (4). (-4)2 = +16. You have to add 16 to both sides AND add 7 to both sides to get the 7 to the other side of the equation. 16 + 7 = 23.

15. A construction company uses the function f(p), where p is the number of of people working on a project, to model the amount of money it spends to complete a project. A reasonable domain for this function would be

(1) positive integers. The number of people has to be a counting number. It cannot be negative nor a fraction. People can also be zero, but in the context of the problem, if zero people are working on a project, the company won't make any money. (None of the options include zero.)

16. Which function is shown in the table below? (image omitted)

(4) f(x) = 3x. Substitute 0 into the functions and only choice (4) works. For that matter, choice (4) is the only one that can produce fractions as output when the input is integers.

17. Given the functions h(x) = 1/2x + 3 and f(x) = |x|, which value of x makes h(x) = f(x)?

(1) -2. It is quicker to work backward from the answers given. 1/2(-2) + 3 = -1 + 3 = 2 = |-2|

18. Which recursively defined function represents the sequence 3, 7, 15, 31 ... ?

(3) f(1) = 3, f(n + 1) = 2f(n) + 1. Each term after the first is one more than twice the previous term.

Note: I don't know if there are typos in choices (1) and (2) or if it was intended to write f(n) as an exponent. It's just odd-looking. As an exponent or not, the answers are incorrect, so it doesn't matter.

19. The range of the function defined as y = 5x is

(2) y > 0. A positive number to any exponent will be positive, never zero or negative.

20. The graph of y = f(x) is shown below. (image omitted)
What is the graph of y = f(x + 1) - 2?

(1). The graph will move 1 space to the left and 2 down because h = -1 and k = -2.

21. Which pair of equations could not be used to solve the following equations for x and y?

4x + 2y = 22
-2x + 2y = -8

(4) 8x + 4y = 44; -8x + 8y = -8. In choices (1) - (3), one or both of the equations is multiplied by some constant. In (4), the -8 was not multiplied by 4 but the left side of the equation was.

22. The graph representing a function is shown below. (image omitted)
Which function has a minimum that is less than the one shown in the graph?

The graph has a minimum at (3, -7). This eliminates choices (2) (-3, -6) and (4) (8, 2).
In choice (1), the axis of symmetry is 3, and the y-coordinate of the vertex is y = (3)2 - 6(3) + 7 = 9 - 18 + 7 = -2
In choice (3), the axis of symmetry is 1, and the y-coordinate of the vertex is y = (1)2 - 2(1) - 10 = 1 - 2 - 10 = -11.
Alternatively, you could have put these into your graphing calculator and just observed the correct answer.

23. Grisham is considering the three situations below.
I. For the first 28 days, a sunflower grows at a rate of 3.5 cm per day.
II. The value of a car depreciates at a rate of 15% per year after it is purchased.
III. The amount of bacteria in a culture triples every two days during an experiment.
Which of the statements describe a situation with an equal difference over an equal interval?

(1) I, only. The first situation is a linear function, growing the same amount every day. (Constant slope.) The other two are exponential functions with the amount changing from interval to interval, even if the percentage remains the same.

24. After performing analyses on a set of data, Jackie examined the scatter plot of the residual values for each analysis. Which scatter plot indicates the best linear fit for the data? (images omitted)

(3). The residual plot should contain randomly-scattered points above and below the x-axis. It should not have a pattern to it. Choices (1), (2) and (4) show curve-like patterns to the plotting of their residuals. These indicate a poor fit for the data.

End of Part I

Movie Functions Quiz

(Click on the comic if you can't see the full image.)
(C)Copyright 2016, C. Burke.

I wanted to use this as a stumper "__(M + M) = 1939"

I easily had over 50 of these, but they were a bit repetitive after a while, which is why I added the years to the equations. If this is popular, then, as with all good movies and many bad ones, there will be a sequel.

The Answer Key will be up soon -- as soon as I find it. Yes, a couple of them are stumping me now and I wrote the thing!

Come back often for more funny math and geeky comics.

Thursday, January 28, 2016

January 2016 New York Geometry (Common Core) Part 2

It's that time of year again: Regents time. I will be uploading parts of the Regents exams for Geometry and Algebra over the next few days. Please, be patient -- I have to type all this in by hand. And that doesn't count the amount of stuff that I need to scan in.

As always, the images will be scanned in when available, and the entries will be updated.

New York Geometry (Common Core) Part 2

25. Triangle ABC is graphed on the set of axes below. (image omitted) Graph and label triangle A’B’C’, the image of triangle ABC after a reflection over the line x = 1.

There are two parts to this: you need to both graph it and label it to get both of them.
The line x = 1 is the vertical line one unit to the right of the y-axis. So your answer is a triangle with vertices A’ (5, 0), B’(2, 4) and C’(2,0).
The vertices need to be labeled correctly, or you could probably get away with writing the labels and coordinates below the grid.
If you forgot something, you lose a point. If you do some other transformation, including a reflection over a different line, you will lose half-credit, which is one point.

26. In the diagram below (image omitted) of circle O with diameter BC and radius OA, chord DC is parallel to chord BA.

If m <BCD = 30o, determine and state m <AOB.

Because DC is parallel to BA with BC as a transversal, then the alternate interior angles are congruent. So <DCB = <ABC = 30 degrees.
Because OB and OA are both radii, OB = OA, so triangle OAB is an isosceles triangle, with base angles equal to 30 degrees. 30 + 30 = 60. 180 – 60 = 120. m <AOB = 120o

If you solved this using inscribed angles and arcs, more power to you. If you got the correct answer, you will get full credit.

27. Directed line segment PT has endpoints whose coordinates are P(-2, 1) and T(4, 7). Determine the coordinates of point J that divides the segment in the ratio 2 to 1.
The use of the set of axes below is optional.

The difference of the x values is 6. The difference in the y values is also 6. Two thirds of 6 is 4.
Add 4 to the x and y value of P. J(-2 + 4, 1 + 4) = J(2, 5)

28. As graphed on the set of axes below (image omitted), triangle A’B’C’ is the image of triangle ABC after a sequence of transformations.

Is triangle A’B’C’ congruent to triangle ABC? Use the properties of rigid motions to explain your answer.

Triangle A’B’C’ is the image of ABC reflected over the y-axis and also translated down 3 units. Reflections and translations are rigid motions that do not affect the size or shape, so the image is congruent.

29. A carpenter leans an extension ladder against a house to reach the bottom of a window 30 feet above the ground. As shown in the diagram below, the ladder makes a 70o angle with the ground. To the nearest foot, determine and state the length of the ladder.

This used to be an Algebra topic, but now is in Geometry, involving Trigonometry ratios.

The ladder and the wall form a right triangle. You know the bottom angle. You have the length of the opposite side. You need the length of the hypotenuse. Opposite and Hypotenuse means using Sine.

Your equation is sin(70) = 30/x
Multiply by x: x sin(70)= 30
Divide by sin(70): x = 30/sin(70)
Put that in your calculator and get x = 31.925 = 32 feet.

You will lose a point if you use the incorrect ratio, for making an incorrect calculation, or not rounding correctly. Zero if you make more than one mistake.

You will not be penalized twice for a consistent error.

30. During an experiment, the same type of bacteria is grown in two petri dishes. Petri dish A has a diameter of 51 mm and has approximately 40,000 bacteria after 1 hour. Petri dish B has a diameter of 75 mm and has approximately 72,000 bacteria after 1 hour.

(image omitted)

Determine and state which petri dish has the greater population density of bacteria at the end of the first hour.

The population density is the population divided by the Area. You need to find the area of each circle and then divide the population by that area.

Don’t forget to divide the diameters by 2 to get the radii.

AA = (pi)(25.5)2 = 2042.82…
Divide 40000 by 2042.82 = 19.58

AB = (pi)(37.5)2 = 4417.86…
Divide 72000 by 4417.86 = 16.297

Petri dish A has the greater population density.

Important: Normally, you shouldn’t round in the middle of a problem. I only did it here because a) I didn’t need an exact answer, and b) I have to divide by the number that I rounded. I could have written the equations to avoid that, but then I wouldn’t have found the individual areas. Not that they were needed in this problem.

31. Line L is mapped onto line M by a dilation centered at the origin with a scale factor of 2. The equation of line L is 3x – y = 4. Determine and state an equation for line M.

This is a surprisingly simple question to answer, but a little more complicated to explain. The slope of the line will not change. The x-intercept and y-intercepts will double (be twice as far away from the origin).

So the answer is obviously 3x – y = 8 or some variation.

But how do you show it?

You could rewrite the equations in slope intercept form, and then double the y-intercept: So 3x – y = 4 becomes 3x – 4 = y, with the answer of y = 3x – 8.

Or you can state that the slope doesn’t change, so the equation stays 3x – y = C
When x = 0, 3(0) – y = 4 when y = -4. Double -4 to -8
Evaluate 3(0) – (-8) = C, C = 8. So 3x – y = 8.

32. The aspect ratio (the ratio of screen width to height) of a rectangular flat-screen television I s16:9. The length of the diagonal of the screen is the television’s screen size. Determine and state, to the nearest inch, the screen size (diagonal) of this flat-screen television with a screen height of 20.6 inches.

This can be solved using ratios and Pythagorean Theorem, or by using Trigonometric Ratios.

First, set up a proportion 16/9 = x/20.6 and cross-multiply.
9x = 329.6, x = 36.6
20.62 + 36.62 = c2
424.36 + 1339.56 = c2
1763.02 = c2
C = 41.999 = 42 inches

Second, the ratio 16:9 represents the opposite over the adjacent, which is the tangent of the top angle of the set. So tan(y) = 16/9 and y = tan-1(16/9) = 60.64 degrees.

Using the adjacent and the hypotenuse and cosine, we get the following:
cos(60.64) = 20.6/x, so x = 9/cos(60.64) = 42.0155… = 42 inches.

Wednesday, January 27, 2016


(Click on the comic if you can't see the full image.)
(C)Copyright 2016, C. Burke.

The question you have to ask here is how do you have an "age of -3"? You would think that there was something wrong with the domain.

Also, when will we get Ultron squared, if ever?

Come back often for more funny math and geeky comics.