Monday, August 31, 2015

Substitute Teacher School Assignment Chart

(Click on the comic if you can't see the full image.)
(C)Copyright 2015, C. Burke.

There are reasons why some schools have vacancies, not all of them maternity or jury duty.

Just to clarify, in New York City, there is something called the Absentee Teacher Reserve Pool, which is comprised of many teachers who were placed "in excess" for a number of reasons. (For example, shrinking enrollment shrinks the school's budget, meaning there is less need and less money for the same number of teachers.) Every month, teachers in the pool get an assignment to report to a certain school as a substitute. That assignment can be any place in the borough (i.e., the county). Not all parts of Brooklyn are easy to travel to as the subway system generally moves people toward Manhattan. Many of the streets in old neighborhoods are very narrow for the amount of traffic they get today. And don't even think about parking.

Moreover, there are many reasons why schools are low achieving, and many of those reasons have little to do with the students themselves, but none of them work well for a substitute in the midst.

Come back often for more funny math and geeky comics.

Friday, August 28, 2015


(Click on the comic if you can't see the full image.)
(C)Copyright 2015, C. Burke.

Shouldn't Volume 6 be three times as meaty as Volume 2? Assuming some books aren't denser than others, of course.

Again, there are times I'm surprised that I haven't done jokes like this one before.

Come back often for more funny math and geeky comics.

Thursday, August 27, 2015

August 2015 Geometry (Common Core) Regents: Parts 3 and 4

Here are the questions, with answers and explanations, for the New York State Geometry (Common Core) Regents exam, Parts 3 and 4. There were 3 questions in Part 3, each worth 4 credits. There were 2 questions in Part 4, each worth 6 credits. Partial credit may be earned. All work must be shown. In general, a correct answer without any work is worth 1 credit, unless that answer is given as a choice and an explanation is required.

Link to Part 1

Link to Part 2

Part 3

32. As shown in the diagram below, a ship is heading directly toward a lighthouse whose beacon is 125 feet above sea level. At the first sighting, point A, the angle of elevation from the ship to the light was 7°. A short time later, at point D, the angle of elevation was 16°.

To the nearest foot, determine and state how far the ship traveled from point A to point D.

Label DC as x and AC as y. To find AD, you need to find (y - x). You have angles and the opposite side. You need to find the adjacent side. Therefore, you need to use tan. Make sure your calculator is in DEGREE mode!

tan(16) = 125/x, x = 125/tan(16) = 435.9268...
tan(7) = 125/y, y = 125/tan(7) = 1018.0433...
Subtract y - x = 1018.0433 - 435.9268 = 582.1165 = 582 feet

33. Triangle ABC has vertices with A(x,3), B(-3,-1), and C(-1,-4). Determine and state a value of x that would make triangle ABC a right triangle. Justify why triangle ABC is a right triangle. [The use of the set of axes below is optional.]

On the graph, plot points B and C. Note that the slope of BC is -3/2. (You can use the formula or just write it out.) Note that the slope of AB must be 2/3 to be perpendicular. Go up 2 over 3, make a point at (0, 1). Repeat up 2 over 3, make a point at (3, 3). Label this A. Draw a line from A to C. You have a right triangle. State that x = 3.

NOTE: you could instead have AC with a slope of 2/3. In that case, point A will not be a whole number. If you use point C and the slope to find the equation of the line, you can find point A, which would be (9.5, 3), making x = 9.5.

34. In the diagram below, AC = DF and points A, C, D, and F are collinear on line ℓ.

Let triangle D'E'F' be the image of triangle DEF after a translation along ℓ, such that point D is mapped onto point A. Determine and state the location of F. Explain your answer.

Let triangle D'E'F' be the image of triangle DEF after a reflection across line ℓ. Suppose that E is located at B. Is triangle DEF congruent to triangle ABC? Explain your answer.

First: Point F' would be at point C after the translation. Because AC = DF and DF = D'F', if D' is at point A then F' must be at point C.

Second: If E'' lays on point B, then by SSS ABC = D''E''F''. Because the translation and the reflection are rigid motions which do not affect the size or shape of the figure, D''E''F'' = DEF. Therefore ABC = DEF. (Note: Every equal sign (=) in this paragraph should be a "congruent" symbol.)

Part 4

35. In the diagram of parallelogram ABCD below, BE ⊥ CED, DF ⊥ BFC, and CE = CF.

Prove ABCD is a rhombus

You can write this as a two-column proof or a paragraph. Look at triangles BEC and DEF. (Highlight them if is makes it easier to see.) Each triangle has a right angle. CF is congruent to CE (given). Angle C is congruent to itself (reflexive). By ASA, triangle BEC = DFC. Therefore, BC is congruent to CD by CPCTC. By definition, a parallelogram with two consecutive congruent sides is a rhombus.

36. Walter wants to make 100 candles in the shape of a cone for his new candle business. The mold shown below will be used to make the candles. Each mold will have a height of 8 inches and a diameter of 3 inches. To the nearest cubic inch, what will be the total volume of 100 candles?

Walter goes to a hobby store to buy the wax for his candles. The wax costs $0.10 per ounce. If the weight of the wax is 0.52 ounce per cubic inch, how much will it cost Walter to buy the wax for 100 candles?

If Walter spent a total of $37.83 for the molds and charges $1.95 for each candle, what is Walter’s profit after selling 100 candles?

Volume for 100 candles: 100 V = (100)(1/3)(pi)(r)2(h) = (100)(1/3)(pi)(1.5)2(8) = 1884.9555... = 1885 in3.
Cost is $0.10/oz and the weight is 0.52/in3. Weight = 1885(0.52) = 980.2 ounces Cost = 980.2(.10) = $98.02.

Profit = revenue - cost. He made 100(1.95) = $195. His costs were 37.83 + 98.02 = $135.85. Profit = 195 - 135.85 = $59.15.

End of exam.

Wednesday, August 26, 2015

August 2015 Geometry (Common Core) Regents: Part 2, Open-Ended

Here are the questions, with answers and explanations, for the New York State Geometry (Common Core) Regents exam, Part 2. There were 7 questions, each worth 2 credits. Partial credit may be earned for correct work on a problem without a solution, or for a problem with a solution that contains one computational or conceptual error. All work must be shown. In general, a correct answer without any work is worth 1 credit, unless that answer is given as a choice and an explanation is required.

Link to Part 1

Part 2

25. A wooden cube has an edge length of 6 centimeters and a mass of 137.8 grams. Determine the density of the cube, to the nearest thousandth.
State which type of wood the cube is made of, using the density table below.

Density = mass / Volume. Volume of a cube = (edge length)3.
Therefore, Density = mass / (edge)3 = 137.8 / (6)3 = 0.63796...

This is the approximate Density of Ash wood, according to the table.

26. Construct an equilateral triangle inscribed in circle T shown below. [Leave all construction marks.]

Using the straightedge, make a diameter in circle T. Label one of the points. (I used S.) Use the compass to measure radius ST. Make a circle around point S. Label the two points of intersection (A and B). Use the compass to measure the distance from A to B, make a small arc. Use the straightedge to draw chord AB. Label the other end of the diameter (C). Without changing the compass, put one end at A and make a small arc that crosses C. Without changing the compass, put one end at B and make a small arc that crosses C. Use the straightedge to draw AC and BC.

(Image to come.)

27. To find the distance across a pond from point B to point C, a surveyor drew the diagram below. The measurements he made are indicated on his diagram.

Use the surveyor’s information to determine and state the distance from point B to point C, to the nearest yard.

Make a proportion: 230/120 = (230 + 85) / x
230 x = (120)(315)
230 x = 37800
x = 164.3478...
x = 164 yards

28. In parallelogram ABCD shown below, diagonals AC and BD intersect at E.

Prove: ∠ACD = ∠CAB

Statement: Reason
ABCD is a parallelogram: given
DC || AB, DA || CB: Opposite sides of a parallelogram are parallel.
Angle ACD = Angle BAC: If two parallel lines are cut by a transversal, the alternate interior angles are congruent.

29. Triangles RST and XYZ are drawn below. If RS = 6, ST = 14, XY = 9, YZ = 21, and ∠S = ∠Y, is triangle RST similar to triangle XYZ? Justify your answer.

If the triangles are similar than 6/14 = 9/21. Cross-multiply you get 126 = 126. (Alternatively, you could reduce both sides to 2/7.) Because a pair of corresponding sides are proportional and the included angles between them are congruent, then by SAS, RST ~ XYZ.

30. In the diagram below, triangle ABC and triangle XYZ are graphed.

Use the properties of rigid motions to explain why triangle ABC = triangle XYZ.

In a rotation of 180 degrees, the image of point (x, y) is (-y, -x). This transformation is a rotation and when rotating size and shape are preserved. So ABC = XYZ

31. The endpoints of DEF are D(1,4) and F(16,14). Determine and state the coordinates of point E, if DE:EF = 2:3.

16 - 1 = 15. 15 * (2/5) = 6. 1 + 6 = 7
14 - 4 = 10. 10 * (2/5) = 4. 4 + 4 = 8. E(7,8).

End of Part II

Tuesday, August 25, 2015

(x, why?) Mini: CSI

(Click on the comic if you can't see the full image.)
(C)Copyright 2015, C. Burke.

If that were an isosceles triangle, the bisector would definitely be the perp.

Come back often for more funny math and geeky comics.

Saturday, August 22, 2015

August 2015 Algebra 1 (Common Core) Regents: Parts 3 and 4

Here are the questions, with answers and explanations, for the New York State Algebra 1 (Common Core) Regents exam, Parts 3 and 4. There were 4 questions in Part 3, each worth 4 credits. There is one question in Part 4, worth 6 credits. Partial credit may be earned for correct work on a problem without a solution, or for a problem with a solution that contains computational or conceptual errors. All work must be shown. In general, a correct answer without any work is worth 1 credit, unless that answer is given as a choice and an explanation is required.

Link to Part 1
Link to Part 2

Part 3

33. Graph f(x)= x2 and g(x) = 2x for x ≥ 0 on the set of axes below.
State which function, f(x) or g(x), has a greater value when x = 20. Justify your reasoning.

They did not specify a scale, but it you used each box is one, it will be hard to read. Make sure you label it clearly. You might want to include your Table of Values from your calculator. If you use a scale of 1, you will reach the top of the graph by x = 4.

f(20) = (20)2 = 400; g(20) = (2)20 = 1048576. g(20) > f(20).
You could also justify it from the graph if the graph clearly shows that g(x) is increasing more steeply than f(x) for x > 4.
An alternate graph, which is more obvious:

34. Solve for x algebraically: 7x - 3(4x - 8) ≤ 6x + 12 - 9x
If x is a number in the interval [4,8], state all integers that satisfy the given inequality. Explain how you determined these values.

7x - 3(4x - 8) ≤ 6x + 12 - 9x
7x - 12x + 24 ≤ -3x + 12
-5x + 24 ≤ -3x + 12
-2x ≤ - 12
x > 6

The integers 6, 7, 8 satisfy the condition because there are greater than or equal to 6 and in the interval [4, 8].

Seriously, I'm not sure what kind of explanation is needed if you solved the inequality, which explains the answer. They are in the solution set of the equation.

35. The volume of a large can of tuna fish can be calculated using the formula V = πr2h. Write an equation to find the radius, r, in terms of V and h.
Determine the diameter, to the nearest inch, of a large can of tuna fish that has a volume of 66 cubic inches and a height of 3.3 inches.

Look at the work in the image below: r = the square root of (V divided by (i times h))

Plug is V = 66 and h = 3.3 and solve. Volume divided by (pi times height) is approximately 6.366. The square root of that is approximated 2.52, so to the nearest inch, the diameter is 2.5 * 2 = 5 inches.

Don't forget to double the radius. I'm shocked that this was to the nearest inch when the height was given to the nearest tenth of an inch.

36. The table below shows the attendance at a museum in select years from 2007 to 2013.

State the linear regression equation represented by the data table when x = 0 is used to represent the year 2007 and y is used to represent the attendance. Round all values to the nearest hundredth.
State the correlation coefficient to the nearest hundredth and determine whether the data suggest a strong or weak association.

Subtract 2007 from all the years to get 0, 1, 2, 4, and 6. Put those numbers and the second row of data into two lists on the calculator. Perform a linear regression. You should get y = 0.16x + 8.27. The correlation coefficient is 0.97, with is a strong association. (Anything close to 1 or close to -1 is a strong association.)

Part 4

37. 7 A rectangular picture measures 6 inches by 8 inches. Simon wants to build a wooden frame for the picture so that the framed picture takes up a maximum area of 100 square inches on his wall. The pieces of wood that he uses to build the frame all have the same width.

Write an equation or inequality that could be used to determine the maximum width of the pieces of wood for the frame Simon could create.

Explain how your equation or inequality models the situation.

Solve the equation or inequality to determine the maximum width of the pieces of wood used for the frame to the nearest tenth of an inch.

You have to love that they won't tell you whether to use an equation or an inequality. (Okay, you don't really have to love it at all.)

The wood has width x, and is added to both sides of the picture, plus the top and bottom. Area = L X W.
So (2x + 6)(2x + 8) < 100
4x2 + 28x + 48 < 100

Explain: the wood adds 2x to the length of the picture and the width of the picture. The area cannot be more than 100 square inches.

Solve: first, divide by 4: x2 + 7x + 12 < 25
subtract 25 from both sides: x2 + 7x - 13 < 0
From here you need to use the quadratic formula or complete the square. Let's complete the square: x2 + 7x + 49/4 - 13 < 49/4

x2 + 7x + 49/4 < 12.25 + 13
(x + 7/2)2 < 25.25
x + 7/2 < 5.024 or x + 7/2 > -5.024. Discard the negative answer.
x < 1.524
The width cannot be more than 1.5 inches.

Okay, I probably should've gone with the quadratic formula. Probably would've been more straightforward, and you wouldn't have had all those halves and quarters.

End of exam.

Friday, August 21, 2015

August 2015 Algebra 1 (Common Core) Regents: Part 2, Open-Ended

Here are the questions, with answers and explanations, for the New York State Algebra 1 (Common Core) Regents exam, Part 2. There were 8 questions, each worth 2 credits. Partial credit may be earned for correct work on a problem without a solution, or for a problem with a solution that contains one computational or conceptual error. All work must be shown. In general, a correct answer without any work is worth 1 credit, unless that answer is given as a choice and an explanation is required.

Link to Part 1

Part 2

25. Each day Toni records the height of a plant for her science lab. Her data are shown in the table below

The plant continues to grow at a constant daily rate. Write an equation to represent h(n), the height of the plant on the nth day.

The plant is growing at a constant rate, so use any two points to find the slope. Let's use (1, 3.0) and (2, 4.5). m = (4.5 - 3)/(2 - 1) = 1.5/1 = 1.5
Solve for b: 3 = 1.5(1) + b; b = 3 - 1.5 = 1.5
The function is h(n) = 1.5n + 1.5

Even if the slope or the y-intercept were obvious to you, show the work anyway. It helps.

26. On the set of axes below, graph the inequality 2x + y > 1.

Things they will be looking for: the correct slope and y-intercept, a broken line, proper shading.
In slope-intercept form: 2x + y > 1 becomes y > -2x + 1. Shade above the line.

27. Rachel and Marc were given the information shown below about the bacteria growing in a Petri dish in their biology class.

Rachel wants to model this information with a linear function. Marc wants to use an exponential function. Which model is the better choice? Explain why you chose this model.

The better choice is exponential because the function isn't growing at a constant rate.
If you find the rate of change for each pair of numbers, you will get the following: 60, 70, 80, 90, 110, 140, 170, 210, 270, 340. Not only is it NOT constant, it's growing with each passing hour.

If you divide B(2)/B(1), you get 1.27... Divide B(3)/B(2), you get 1.25. B(4)/B(3), you get 1.257.
You probably don't need to go all the way to 10 -- it's only a 2-point question -- but you can see that the growth factor is approximately 25%.

28. A driver leaves home for a business trip and drives at a constant speed of 60 miles per hour for 2 hours. Her car gets a flat tire, and she spends 30 minutes changing the tire. She resumes driving and drives at 30 miles per hour for the remaining one hour until she reaches her destination.

On the set of axes below, draw a graph that models the driver’s distance from home.

Graph a line with a slope of 60 from time 0 to time 2. Graph a horizontal line (slope 0) from 2 to 2.5. Graph a line with a slope of 30 from time 2.5 to 3.5.

29. How many real solutions does the equation x2 - 2x + 5 = 0 have? Justify your answer.

None. If you complete the square, you will get (x - 1)2 = -4, which has no real roots.

Or you could find the discriminant (of the quadratic formula):

Alternatively, I don't know if you would get credit for just saying that you graphed y = x2 - 2x + 5 and it didn't have any roots, or it didn't cross the x-axis. On the other hand, if you show that the minimum point for the parabola is above the x-axis and, therefore, has no roots, that might have been acceptable.

30. The number of carbon atoms in a fossil is given by the function y = 5100(0.95)x, where x represents the number of years since being discovered.
What is the percent of change each year? Explain how you arrived at your answer.

There is a 5% decrease each year. The decay factor is .95, and 1.00 - .95 = .05, which is 5%.
If you left it at .05, you probably lost a point because it asked for percent. If you didn't say "decrease", you probably lost a point.

31. A toy rocket is launched from the ground straight upward. The height of the rocket above the ground, in feet, is given by the equation h(t)= 16t2 + 64t, where t is the time in seconds.
Determine the domain for this function in the given context. Explain your reasoning.

The domain of the function is 0 < t < 4.
Time cannot be negative. At t=0, h(0) = -16(0)2 + 64(0) = 0 + 0 = 0
At t=1, h(1) = -16(1)2 + 64(1) = -16 + 64 = 48
At t=1, h(2) = -16(2)2 + 64(2) = -64 + 128 = 64
At t=1, h(3) = -16(3)2 + 64(3) = -144 + 192 = 48
At t=1, h(4) = -16(4)2 + 64(4) = -256 + 256 = 0. The rocket hits the ground.
If t > 4, the rocket would have negative height, which is impossible.

32. Jackson is starting an exercise program. The first day he will spend 30 minutes on a treadmill. He will increase his time on the treadmill by 2 minutes each day. Write an equation for T(d), the time, in minutes, on the treadmill on day d.

Find T(6), the minutes he will spend on the treadmill on day 6.

First part: T(d) = 30 + 2(d - 1), 30 minutes for the first day, plus 2 more each additional day.
Second part: T(6) = 30 + 2(6 - 1) = 30 + 2(5) = 30 + 10 = 40 minutes,

Matrix Problems

(Click on the comic if you can't see the full image.)
(C)Copyright 2015, C. Burke.

Afterthought: I'm thinking about the poor guy who searched Matrix Problems and found this.

It'd probably be as amusing as that guy who complained about my Definition: Polygon comic years ago.

Or his anonymous friend who backed him up a month later.

Yeah, like I wouldn't see through that. Hell-ooo! Teacher.

Come back often for more funny math and geeky comics.

Thursday, August 20, 2015

August 2015 Geometry Regents -- NOT Common Core: Part 1, Multiple-Choice

Here are the questions, with answers and explanations, for the New York State Geometry Regents exam. This is the old exam, which is being retired not the Common Core exam, which can be found here.

There were 28 questions, each worth 2 credits. No partial credit. No work needed to be shown (but it would still be a good idea to work out the answers, even if no one will see it). Twenty correct answers on this part -- which is roughly two-thirds of the questions -- results in 40 credits, which curves to a grade of 65 (which is roughly two-thirds of the points). This is slightly lower than previous exams, but is a higher threshold than the Common Core exam.

Images will be added when I have the resources available. Sorry, but there are a lot of images in this exam!

Part 1

1. In triangle ABC shown below with ADC, AEB, CFE, and BFD, triangle ACE = triangle ABD. (image omitted)
Which statement must be true?

(4) <AEF = <ADF. Corresponding parts of congruent triangles are congruent.

2. In a circle whose equation is (x - 1)2 + (y + 3)2 = 9, the coordinates of the center and length of its radius are

(3) (1, -3) and r = 3. (x - h)2 + (y - k)2 = r2. h = 1, k = -3, r = 3.

3. Parallel secants FH and GJ intersect circle O, as shown in the diagram below. (image omitted)
If m(arc)FH = 106 and m(arc)GJ = 24, then m(arc)FG equals

(2) 115. Arcs FG and HJ are congruent, and FG + HJ + 106 + 24 = 360. FG + HJ = 230, FG = 115.

4. What are the coordinates of P', the image of point P(x, y) after translation T4,4?

(2) (x + 4, y + 4). Definition of a translation.

5. The statement "x > 5 or x < 3" is false when x is equal to

(4) 4. For an "OR" statement (disjunction) to be false, both parts have to be false. Four is neither greater than 5 nor less than 3.

6. Triangle JTM is shown on the graph below. (Image omitted)
Which transformation would result in an image that is not congruent to triangle JTM?

(4) D2. Dilations preserve the shape and orientation but not the size of the original shape. The image is similar but not congruent.

7. In the diagram below of triangle ABC, with CDEA and BGFA, EF || DG || CB. (image omitted)
Which statement is false?

(3) AE/AD = EC/AC. The three triangles are similar, so their corresponding sides are proportional. In Choice (3) the proportion is not set up correctly. The sides do not correspond to each other.

8. Which pair of edges is not coplanar in the cube shown below? (image omitted)

(1) EH and CD. EH sort of goes front to back while CD sort of goes left to right. They aren't in the same plane. They couldn't be lines, for example, on a single sheet of paper.

9. What is an equation of the line that passes through the point (-2,1) and is parallel to the line whose equation is 4x - 2y = 8?

(3) y = 2x + 5. Parallel means the same slope. Converting the original equation

4x - 2y = 8
- 2y = -4x + 8
y = 2x - 4
The slope of the line is 2, so eliminate choices (1) and (2). Plug in (-2, 1) into the other choices, and choice (3) is correct.

10. In triangle JKL, JL = KL. If m<J = 58, then m<L is

(2) 64. Triangle JKL is an isosceles triangle. Angle L is the vertex angle, angles J and K are the base angles, which are congruent to each other. So 58 + 58 + L = 180. L = 64.

11. The corresponding medians of two similar triangles are 8 and 20. If the perimeter of the larger triangle is 45, what is the perimeter of the smaller triangle?

(2) 18. The scale factor is 8/20, or 2/5. Multiply (45)(2/5) = 18. Note: Perimeter is one-dimensional so the change is proportional. Had it been Area, which is two dimensions, you would have had to multiply by the square of the scale factor.

12. Which construction of parallel lines is justified by the theorem "If two lines are cut by a transversal to form congruent alternate interior angles, then the lines are parallel"?

(3). Choice (3) is the only one where the alternate interior angles are marked off with a construction.

13. Given: "If a polygon is a triangle, then the sum of its interior angles is 180°."
What is the contrapositive of this statement?

(4). "If the sum of the interior angles of a polygon is not 180°, then it is not a triangle." Definition of contrapositive: IF A THEN B. ==> IF NOT B THEN NOT A.
Choice (2) is a biconditional (if and only if). Choice (3) is the inverse. Choice (4) is the converse. Note that all four statements have the same truth value.

14. In the diagram below, point P is not on line L. (image omitted)
How many distinct planes that contain point P are also perpendicular to line L?

(1) 1. Given a line and point not on that line, there is only one line parallel to the first one that goes through the point.

15. The image of triangle ABC after the transformation ry-axis is triangle A'B'C'. Which property is not preserved?

(2) orientation. A reflection will change the direction of an object. The object and its image will be congruent to each other.

16. The equations y = 2x + 3 and y = -x2 - x + 1 are graphed on the Use this space for same set of axes. The coordinates of a point in the solution of this system of equations are

(4) (-2, -1). You can plug the points into both equations and see which one is a solution to both equations. Choice (1) is right out: 2(0) + 3 =/= 1. Or you can solve the quadratic equation:

-x2 - x + 1 = 2x + 3
-x2 - 3x - 2 = 0
x2 + 3x + 2 = 0
(x + 2)(x + 1) = 0
x = - 2 or x = -1

Substitute y = 2(-1) + 3 = -2 + 3 = 1, (-1, 1)
y = 2(-2) + 3 = -4 + 3 = -1, (-2, -1)
NOTE: The fact that one x value was -1 and one y value was -1 IS A CO-INCIDENCE.

17. Which quadrilateral has diagonals that are always perpendicular bisectors of each other?

(1) square. That is a property of rhombuses, and a square is a rhombus. If it were true for (2) or (4), it would also be true for (1), so those should be eliminated immediately.

18. As shown in the diagram below, AB is a diameter of circle 0, and chord AC is drawn. (image omitted)
If m<BAC = 70, then m(arc)AC is

(1) 40. Arc AB is a semicircle, which is 180 degrees. Arc BC is twice as big as the inscribed angle, which is 2 * 70 = 140 degrees. Subtract 180 - 140 = 40 degrees.

19. In parallelogram JKLM, m<L exceeds m<M by 30 degrees. What is the measure of <J?

(2) 105o. Angle J is congruent to angle L, which it is across from. Angles L and M are supplementary because they are consecutive angles (and same-side interior angles of a transversal across two parallel lines). L + M = 180 and L = M + 30, which means L - 30 = M. So L + L - 30 = 180. 2L = 210. L = 105.

20. Which equation represents the circle shown in the graph below?

(2) (x + 5)2 + (y - 3)2 = 1. The equation of a circle, again. It has its center at (-5, 3) and a radius of 1. Flip the signs on x and y, and square r.

21. What is the measure of each interior angle in a regular octagon?

(2) 135o. First of all, you should have seen problems like this so often that you should just know the answers for equilateral triangle, square, pentagon, hexagon, octagon and decagon. (Yes, I skipped a couple, because you don't see them as much.)
(n-2) * 180 gives you the total number of degrees: 6 * 180 = 1080 degrees. Divide that by the number of angles, which is 8: 1080 / 8 = 135 degrees.

22. Points A and B are on line L, and line L is parallel to line m, as shown in the diagram below. (image omitted)
How many points are in the same plane as L and m and equidistant from L and m, and also equidistant from A and B?

(1) 1. Equidistant from L and m is a parallel line between them, running left to right. Equidistant from A and B is a line perpendicular to line L that intersects it at the midpoint of AB. The vertical line and the horizontal line intersect at exactly one point. (Two lines with different slopes in the same plane will always intersect at exactly one point.)

23. A carpenter made a storage container in the shape of a rectangular prism. It is 5 feet high and has a volume of 720 cubic feet. He wants to make a second container with the same height and volume as the first one, but in the shape of a triangular prism. What will be the number of square feet in the area of the base of the new container?

(3) 144. I read this one twice, looking for the trick. There is none. It is total misdirection. The shape of the Base does not matter. If the height is the same and the Volume is the same then the Area of the Base has to remain the same. V = Bh, so B = V/h = 720/5 = 144.

24. In triangle ABC, m<B < mltA < m<C. Which statement is false?

(1) AC > BC. The smallest angle is opposite the smallest side. The largest side is opposite the largest angle. AC < BC < AB.

25. In the diagram below of circle O with radius OA, tangent CA and secant COB are drawn. (image omitted)
If AC = 20 cm and OA = 7 cm, what is the length of OC, to the nearest centimeter?

(3) 21. Very straightforward Pythagorean Theorem problem. No tricks, no hunting for information. The tangent forms a right angle with the radius. That makes OC the hypotenuse of a right triangle. 72 + 202 = OC2. You should know that 7-20-21 is a Pythagorean Triple.
The answer is "obviously" 21 because OC is the hypotenuse, so 19 and 20 are out, and 20-7-27 is not a valid triangle, let alone a right triangle.

26. In the diagram below of triangle ABC, point H is the intersection of the three medians. (image omitted)
If DH measures 2.4 centimeters, what is the length, in centimeters, of AD?

(3) 7.2. Three medians meet at a centroid. The centroid divides median DA into the following relationships: DH is half the size of HA. DH is one-third the size of DA. And HA is two-thirds the size of DA. Multiply 2.4 * 3 = 7.2.

27. Which set of numbers could be the lengths of the sides of an isosceles triangle?

(2) {3, 3, 5}. First, eliminate (3) because it isn't even isosceles. Choices (1) and (4) do not form triangles because the sum of the smaller sides is not greater than the longest side.

28. In the diagram below of right triangle ABC, CD is the altitude to hypotenuse AB, AD = 3, and DB = 4. (image omitted)
What is the length of CB?

(3) 2*Sqrt(7). There are three right triangles in the picture. The large one, which is divided into two smaller ones. They are all similar, so their corresponding sides are proportional. Therefore, you can write 4/x = x/(3+4). So x2 = 4*7 = 28, and x = sqrt(28) = sqrt(4)*sqrt(7) = 2*sqrt(7).

That's the end of Part 1. Part 2 will hopefully be uploaded soon, and there will be a link here.

Wednesday, August 19, 2015

August 2015 Geometry (Common Core) Regents: Part 1, Multiple-Choice

Here are the questions, with answers and explanations, for the New York State Geometry (Common Core) Regents exam. There were 24 questions, each worth 2 credits. No partial credit. No work needed to be shown (but it would still be a good idea to work out the answers, even if no one will see it). According to the conversion chart for this exam, 33 credits are needed to achieve a scaled score of 65. While that number is a little higher than the Common Core Algebra 1 exam, it is significantly lower than the raw score required on recent non-Common Core Geometry Regents. So seventeen correct out of the twenty-four questions asked would be enough to pass the exam.

Part 1

1. A parallelogram must be a rectangle when its

(2) diagonals are congruent. Choices (1) and (4) is not true for all rectangles. Choice (3) applies to all parallelograms.

2. If triangle A'B'C' is the image of triangle ABC, under which transformation will the triangles not be congruent?

(3) dilation centered at the origin with scale factor 2. The center doesn't matter. When you dilate, the size changes. Therefore, the image cannot be congruent to the original.

3. If the rectangle below is continuously rotated about side w, which solid figure is formed? (image omitted)

(4) cylinder. Although the rectangle is flat on the paper, it is being rotated into 3-D space, and the figure formed will be a three-dimensional solid. Imagine the rectangle is a little flag and side w is attached to the flag pole. If you spin the pole, the flag goes around in a circle. The circle on top will be the same as the circle on the bottom, directly below it. The space between them is a cylinder.

4. Which expression is always equivalent to sin x when 90 < x < 90o?

(1) cos(90o - x). Sine and cosine are complementary. In a right triangle the sine of one acute angle is the same as cosine of the other acute angle. Consider the definitions opposite/hypotenuse and adjacent/hypotenuse. What is opposite on angle is adjacent to the other.

5. In the diagram below, a square is graphed in the coordinate plane. (image omitted).
A reflection over which line does not carry the square onto itself?

(1) x = 5. In choice (1), the side of the square on the vertical line x = -1 would end up at x = 11. Choices (3) and (4) are the diagonals of the square; flipping over these will cause the square to remain in place. Choice (2) is the horizontal line that divides the square in half; reflecting over this line will put the square back into its original place.

6. The image of triangle ABC after a dilation of scale factor k centered at point A is triangle ADE, as shown in the diagram below. (image omitted)
Which statement is always true?

(4) BC || DE. In a dilation, size changes, but orientation does not. Therefore, the corresponding parts of the two objects will have the same slopes, making them either parallel or part of the same line. Choices (1) and (3) are true only when the scale factor is 2. Choice (2) could only be true if we knew that AB is perpendicular to BC, but that is not stated.

7. A sequence of transformations maps rectangle ABCD onto rectangle A"B"C"D", as shown in the diagram below. (image omitted)
Which sequence of transformations maps ABCD onto A'B'C'D' and then maps A'B'C'D' onto A"B"C"D"?

(1) a reflection followed by a rotation. First step, reflect over the x-axis. Second step, rotate 90 degrees (counterclockwise) about the origin. If it had bee a translation, A' would have been above D' (for example).

8. In the diagram of parallelogram FRED shown below, ED is extended to A, and AF is drawn such that AF = DF. (image omitted)
If m<R = 124°, what is m<AFD?

(3) 68°. AF = DF means that it is an isosceles triangle and the base angles are the same. Angle FDE is congruent to angle R, so it is also 124°. Subtract 180 - 124 = 56. Solve x + 56 + 56 = 180, x = 68.

9. If x2 + 4x + y2 - 6y - 12 = 0 is the equation of a circle, the length of the radius is

Complete the squares:

x2 + 4x + y2 - 6y - 12 = 0
x2 + 4x + 4 + y2 - 6y + 9 - 12 = 0 + 4 + 9
(x + 2)2 + (y - 3)2 - 12 + 12 = 13 + 12
(x + 2)2 + (y - 3)2 = 25
Thus r2 = 25, therefore r = 5.

10. Given MN shown below, with M(-6,1) and N(3,-5), what is an equation of the line that passes through point P(6,1) and is parallel to MN? (image omitted)

(1) y = (-2/3)x + 5. Parallel means that it has the same slope, which in this case is negative. Only two choices are negative and they have the same slope, so that is it. You can substitute (6, 1) in both equations and you will see that choice (1) is a solution and choice (2) is not.

OR Calculate the slope of the line, either with the formula or counting the boxes, (-5 - 1)/(3 - -6) = -6/9 = -2/3. Eliminate choices (3) and (4). Use (6, 1) to solve this equation

(1) = (-2/3)(6) + b
1 = -4+ b
b = 5

11. Linda is designing a circular piece of stained glass with a diameter of 7 inches. She is going to sketch a square inside the circular region. To the nearest tenth of an inch, the largest possible length of a side of the square is

(2) 4.9. The largest square inside a circle will be one that is inscribed in the circle (vertices on the circle), which will have a diameter for its diagonal. That diagonal forms two right triangles with legs that are equal in length and the hypotenuse equal to the diameter, which is 7.

So s2 + s2 = 72
2s2 = 49
s2 = 24.5
s = 4.9497...

12. In the diagram shown below, AC is tangent to circle 0 at A and to circle P at C, OP intersects AC at B, OA = 4, AB= 5, and PC= 10. (image omitted)
What is the length of BC?

(3) 12.5. Triangles OAB and PCB are similar by AA because they have two pairs of congruent angles: the right angles and the vertical angles. Therefore, the corresponding sides are proportional: 4/10 = 5/x. x = 50/4 = 12.5.

13. In the diagram below, which single transformation was used to map triangle A onto triangle B? (image omitted)

(2) rotation. B is the image of A after a 90o counterclockwise rotation.

14. In the diagram below, triangle DEF is the image of triangle ABC after a clockwise rotation of 180° and a dilation where AB = 3, BC = 5.5, AC= 4.5, DE= 6, FD= 9, and EF = 11. (image omitted)
Which relationship must always be true?

(4) m<B/m<E = m<C/m<F. The scale factor is 2, but that has to do with the sides of the triangle, not the angles. The angles are congruent. Choice (3) does not relate corresponding angles in the proportion.

15. In the diagram below, quadrilateral ABCD is inscribed in circle P. (image omitted)
What is m<ADC?

(3) 108o. The opposite angles of an inscribed quadrilateral are supplementary because they will intercept two arcs (one major, one minor; or two semicircles) that form a complete circle of 360 degrees. Since angle ABC is 72 degrees, then ADC is 108. Likewise, angle BCD is 70.

16. A hemispherical tank is filled with water and has a diameter of 10 feet. If water weighs 62.4 pounds per cubic foot, what is the total weight of the water in a full tank, to the nearest pound?

(1) 16,336. The radius is half the diameter; half of 10 is 5. Use the formula for Volume of a sphere and halve it to find a hemisphere: V = (1/2)(4/3)(pi)(r)3 = 261.799...
Multiply 261.799 * 62.4 = 16336.28.., which rounds down. If you're off by a little bit (but didn't get one of the other answers, it's because you didn't use enough decimal places.
Note: Choice (2) is if you forget the 1/2 for the hemisphere. Choice (3) is if you used 10, not 5. Choice (4) is if you made both mistakes. Given this, you could have reasoned out the correct answer through approximation.

17. In the diagram below, triangle ABC ~ triangle ADE. (image omitted)
Which measurements are justified by this similarity?

(4) AD = 2, AB = 6, AE = 5, and AC =15. AD/AB = AE/AC. The corresponding sides are proportional. The scale factor is 3.

18. Triangle FGH is inscribed in circle O, the length of radius OH is 6, and FH = OG. (image omitted)
What is the area of the sector formed by angle FOH?

(3) 6*pi. FH = OG. OG = OH. OH = 6. Therefore, FH = 6. Also, OF = 6, because it is another radius. Triangle FOH is equilateral, which means that all of its angles are 60 degrees. The central angle of 60 degrees marks off 60/360 = 1/6 of the circle. The area of 1/6 of the circle is (1/6)pi*r2 = (1/6)(6)2*pi = 6 pi.

19. As shown in the diagram below, AB and CD intersect at E, and AC || BD. (image omitted)
Given triangle AEC ~ triangle BED, which equation is true?

(2) (AE/BE) = (AC/BD). Given that the triangles are similar, their corresponding sides are proportional. Choice (2) has the corresponding sides in the correct order.

20. A triangle is dilated by a scale factor of 3 with the center of dilation at the origin. Which statement is true?

(1) The area of the image is nine times the area of the original triangle. The area of a triangle is one half times base times height: (1/2) b h. The area of the image is (1/2) (3b) (3h) = (9)(1/2) b h.

21. The Great Pyramid of Giza was constructed as a regular pyramid with a square base. It was built with an approximate volume of 2,592,276 cubic meters and a height of 146.5 meters. What was the length of one side of its base, to the nearest meter?

(4) 230. Volume is (1/3)(Area of the base)(height).

2,592,276 = (1/3)(Area)(146.5)
Area = 53084.150...
Side = square root (53084.150)
Side = 230.400...

22. A quadrilateral has vertices with coordinates (-3,1), (0,3), (5,2), and (-1,-2). Which type of quadrilateral is this?

(4) trapezoid. Find the slopes of the lines. One pair of equal slopes means trapezoid. Two pairs means some type of parallelogram (the other three choices). First line is (3-1)/(0-(-3)) = 2/3. Second line is (2-3)/(5-0) = -1/5. Third line is (-2-2)/(-1-5)= -4/-6 = 2/3. Fourth line is (1-(-2))/(-3-(-1)) = 3/-2 = -3/2. One pair of parallel sides is a trapezoid.

23. In the diagram below, triangle ABE is the image of triangle ACD after a dilation centered at the origin. The coordinates of the vertices are A(O,O), B(3,0), C(4.5,0), D(0,6), and E(0,4). (image omitted)
The ratio of the lengths of BE to CD is

(1) 2/3. You don't need to find the lengths of BE or CD to solve this. You only need the scale factor. AE/AD = 4/6 = 2/3. That's it.

24. Line y = 3x - 1 is transformed by a dilation with a scale factor of 2 and centered at (3,8). The line's image is

(4) y = 3x - 1. Because (3, 8) is a solution to the equation, it is a point on the line. If a line is dilated but goes through the center of the dilation, the line is unchanged. A line segment would have its length changed, but a line is infinite in length, so it is unaffected.

That's the end of Part 1. Part 2 will hopefully be uploaded soon, and a link will be added here.

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.


(Click on the comic if you can't see the full image.)
(C)Copyright 2015, C. Burke.

I'm a firm believer that children should have household chores. My kids aren't believers though.

Come back often for more funny math and geeky comics.

Tuesday, August 18, 2015

August 2015 Algebra 1 (Common Core) Regents: Part 1, Multiple-Choice

Here are the questions, with answers and explanations, for the New York State Algebra 1 (Common Core) Regents exam. There were 24 questions, each worth 2 credits. No partial credit. No work needed to be shown (but it would still be a good idea to work out the answers, even if no one will see it). Fifteen correct answers on this part -- which is roughly two-thirds of the questions -- results in 30 credits, which curves to a grade of 65 (which is roughly two-thirds of the points).

Part 1

1. Given the graph of the line represented by the equation f(x) = -2x + b, if b is increased by 4 units, the graph of the new line would be shifted 4 units

(2) up. Adding 4 increasing the value of f(x), which is graphed as y. So the graph is higher.

2. Rowan has $50 in a savings jar and is putting in $5 every week. Jonah has $10 in his own jar and is putting in $15 every week. Each of them plots his progress on a graph with time on the horizontal axis and amount in the jar on the vertical axis. Which statement about their graphs is true?

(3) Jonah's graph has a steeper slope than Rowan's. Rowan's has a slope of 5 while Jonah has a slope of 15, which is greater. The two graphs will intersect at some point, so neither graph will always be above the other.

3. To watch a varsity basketball game, spectators must buy a ticket at the door. The cost of an adult ticket is $3.00 and the cost of a student ticket is $1.50. If the number of adult tickets sold is represented by a and student tickets sold by s, which expression represents the amount of money collected at the door from the ticket sales?

(4) 3.00a + 1.50s. The number of adult tickets X the price of the adult tickets plus the number of the student tickets X the prices of the student tickets.

4. The graph of f(x) is shown below. (diagram omitted)
Which function could represent the graph of f(x)?

(1) f(x) = (x + 2)(x2 + 3x - 4). The graph has zeros at -4, -2, and 1, which gives us the factors (x + 4)(x + 2)(x - 1). Remember: you flip the sign because (-4) + 4 = 0. You can eliminate choices (2) and (4) immediately. Multiplying the other two factors, you will get a constant of -4, not +4, so the answer is 1. Note: You could also have solved this by factoring the choices and finding the correct zeroes.

5. The cost of a pack of chewing gum in a vending machine is $0.75. The cost of a bottle of juice in the same machine is $1.25. Julia has $22.00 to spend on chewing gum and bottles of juice for her team and she must buy seven packs of chewing gum. If b represents the number of bottles of juice, which inequality represents the maximum number of bottles she can buy?

(4) 0.75(7) + 1.25b < 22. This is similar to question 3. The price of the gum times the number of packs (7, in this case) plus the price of the juice times the number of bottles must be less than or equal to $22 because that's all that Julia has.

6. Which graph represents the solution of y < x + 3 and y > -2x - 2? (image omitted)

(3). The lines are all graphed correctly, so you don't need to check slopes or y-intercepts. The only difference is the shading. The first inequality (less than) is shaded below the line. The second (greater than) is shaded above the line. That is choice (3).

7. The country of Benin in West Africa has a population of 9.05 million people. The population is growing at a rate of 3.1% each year. Which function can be used to find the population 7 years from now?

(3) f(t) = (9.05 x 106)(1 + 0.031)7. All four operations show the correct formula for exponential growth and decay. The question is about growth, so eliminate the choices with minus signs (decay). You need to turn 3.1% into a decimal, and that is 0.031, not 0.31, which is 31%.
Fun fact: there actually is a country in West Africa named Benin, and it's current population is about 10.3 million, so I guess the question was written a couple years ago.

8. A typical cell phone plan has a fixed base fee that includes a certain amount of data and an overage charge for data use beyond the plan. A cell phone plan charges a base fee of $62 and an overage charge of $30 per gigabyte of data that exceed 2 gigabytes. If C represents the cost of g represents the total number of gigabytes of data, which equation could represent this plan when more than 2 gigabytes are used?

(4) C = 62 + 30(g - 2). The fixed rate is 62, which doesn't change. The overage charge, 30, gets the variable. However, since the first 2 gigs are free, you have to subtract 2 from the total. Note: if you subtract g from 2, (2 - g), you will get a negative number. Also note that the question stated that this equation only makes sense if you used more than 2 gigabytes.

9. Four expressions are shown below:

I. 2(2x2 - 2x - 60L
II. 4(x2 - x - 30>
III. 4(x + 6)(x - 5)
IV. 4x(x - 1) - 120

The expression 4x2 - 4x - 120 is equivalent to

(3) I, II, and IV. Multiply the factors and three of the expressions are correct. Expression III has +4x as its middle term.

10. Last week, a candle store received $355.60 for selling 20 candles. Small candles sell for $10.98 and large candles sell for $27.98. How many large candles did the store sell?

(2) 8. The set of equations to solve is:

S + L = 20
10.98S + 27.98L = 355.60
Note: because each term has two decimal places, you can remove the decimal points if you want to.
Because this a multiple-choice problem, you have the option to work backward from the answer. That is, you can plug is (14, 6), (8, 12), (10, 10), and (8, 12) for (S, L) and see which works. If you do this, my advice is to start with 10 and 10 -- it's quickest and you can see if you are over and need fewer Large candles or if it's under and you need more. (Or if it's correct, which in this case, it isn't.)
10.98(20 - L) + 27.98L = 355.60
219.60 - 10.98L + 27.98L = 355.60
219.60 + 17L = 355.60
17L = 136
L = 8

11. Which representations are functions? (image omitted)

(2) II and IV. Option I is not a function because 2 appears twice in the x column. Option III is not a function because it fails the vertical-line test.

12. (Image omitted) If f(x) = (the square root of (2x + 3)) over (6x - 5), then f(1/2) =

(3) -1. This looks overly complicated but it's a straight substitute and evaluate problem. It's also one that's difficult to put in most calculators is you aren't careful, so work some of it out before going to a calculator. Once you start, you might not even need it.
Two times (1/2) + 3 = 1 + 3 = 4. The square root of 4 is 2. Six times (1/2) - 5 is 3 - 5 = -2. 2/-2 = -1.

13. The zeros of the function f(x) = 3x2 - 3x = 6 are

(4) -1 and 2. Once again, you can work backward from the choices, substituting and evaluating. Or you can factor:

3x2 - 3x - 6 = 0
3(x2 - x - 2) = 0
3(x - 2)(x + 1) = 0
x = 2 or x = -1

14. Which recursively defined function has a first term equal to 10 and a common difference of 4?

(1) f(1) = 10, f(x) = f(x - 1) + 4. Choices (3) and (4) are silly as they give the first term as 4, instead of 10. The common difference means that each term is 4 more than the previous one, not four times more, so choice (3) is out.

15. Firing a piece of pottery in a kiln takes place at different temperatures for different amounts of time. The graph below shows the temperatures in a kiln while firing a piece of pottery after the kiln is preheated to 200oF. (image omitted)
During which time interval did the temperature in the kiln show the greatest average rate of change?

(1) 0 to 1 hour. The greatest change will have the steepest slope. The slope is decreasing from point to point. You can check that using the Slope formula.
(700 - 200)/(1 - 0) = 500
(900 - 700)/(1.5 - 1) = 400
(1640 - 1300)/(5 - 2.5) = 136
(1800 - 1640)/8 - 5) = 53.3

16. Which graph represents f(x) = |x|, x < 1 and f(x) = Sqrt(x), x > 1? (image omitted)

(3). This could be put into a graphing calculator and you would see the result.
The left side is the absolute value graph, which is a straight line, decreasing to zero. The right side looks like half of a sideways parabola.

17. If f(x) = x2 - 2x - 8 and g(x) = (1/4)x - 1, for which values of x is f(x) = g(x)?

(2) -1.75 and 4. Again, you have a calculator and you have TWO options: graph them or substitute. You can tell from the constants that they will NOT be the same at 0, so you can eliminate (3) and (4) immediately.
If you want to solve the equations, set them equal to each other and make a quadratic equation:

x2 - 2x - 8 = (1/4)x - 1
x2 - 2x - 8 - (1/4)x + 1 = 0
x2 - 2.25x - 7 = 0
Because completing the square would be a bit messy at this point (and if you use the calculator to complete the square, you might as well just graph it!), I'm going to multiply by 4 to get rid of the decimal.
4x2 - 9x - 28 = 0
(4x + 7)(x - 4) = 0
x = -7/4 or x = 4

18. Alicia has invented a new app for smart phones that two companies are interested in purchasing for a 2-year contract.
Company A is offering her $10,000 for the first month and will increase the amount each month by $5000.
Company B is offering $500 for the first month and will double their payment each month from the previous month.
Monthly payment are made at the end of the each month. For which monthly payment will company B's payment first exceed company A's payment?

(8) The first equation is 5000x + 10000, and the payments are 10000, 15000, 20000, 25000, 30000, 35000, 40000, 45000, 50000, 55000, 60000, 65000.
The second equation is 500 * 2(n-1), and the payments are 500, 1000, 2000, 4000, 8000, 16000, 32000, 64000, 128000, 256000, 512000.
At the 8th month, $128,000 is greater than $55,000. Note that Alicia hasn't earned more money total at this point, but which the second plan, she'll earn over a million dollars in a very short time!

19. The two sets of data below represent the number of runs scored by two different youth baseball teams over the course of a season.

Team A: 4, 8, 5, 12, 3, 9, 5, 2
Team B: 5, 9, 11, 4, 6, 11, 2, 7

Which set of statement about the mean and standard deviation is true?

(1) mean A < mean B; standard deviation A > standard deviation B.
TIP: Learn how to use LISTs and 1-Var Stats on your calculator. Calculating standard deviation by hand is a pain in the butt.
First off, the mean of A is 6 and the mean of B is 6.875, so (2) and (4) are out. Using the calculator, the std dev of A is about 3.16 and B is about 3.06. That's (1).

Look at the differences from the mean. In A, it's 2, 2, 1, 6, 3, 3, 1, 4. In B, if we use 7, we get 2, 2, 4, 3, 1, 4, 5, 0. Eliminating the numbers that are the same, A has 1, 6, 3 and B has 4, 5, 0. Because we need to deal with the squares of these differences, we can see that the sum of the squares in set A will be higher than the sum of the squares in set B.

20. If Lylah completes the square for f(x) = x2 - 12x + 7 in order to find the minimum, she must write f(x) in the general form f(x) = (x - a)2 + b. What is the value of a for f(x)?

(1) 6. To complete the square, the first step is to take half of the x term. Half of -12 is -6. However, the formula already has the minus sign in it, so it's just 6. You don't need to work out the rest of the equation nor find b.

21. Given the following quadratic functions:

g(x) = -x2 - x + 6
x, -3, -3, -1, 0, 1, 2, 3, 4, 5
n(x), -7, 0, 5, 8, 9, 8, 5, 0, -7
Which statement about these functions is true?

(4) The sum of the roots of n(x) = 0 is greater than the sum of the roots of g(x) = 0.
In choice (1), the rate of change for g(x) is 2/2 = 1; the rate of change for n(x) is 4/2 = 2, which is greater. In choice (2), g(0) = 6, n(0) = 8, so g(0) is not greater. In choice (3), g(x) has a maximum value when x = 1/2 and g(x) is a little more than 6, but n(1) = 9, so this statement is not true, either.

22. For which value of P and W is P + W a rational number?

(2) (image omitted) P = 1/SQRT(4) and W = 1/SQRT(9). The only way that the sum of two positive numbers will be a rational number is if both of the numbers are rational. (If two irrational numbers which are inverses will add up to zero.) Choice (2) is the only choice with two rational numbers in the denominators of the fractions. The sum of 1/2 and 1/3 is 5/6, which is rational.

23. The solution of (x + 3)2 = 7 is

(3) -3 + Sqrt(7). Inverse operations: take the square root of both sides, then subtract 3.

24. Which trinomial is equivalent to 3(x - 2)2 - 2(x - 1)?

(4) 3x2 - 14x + 14.

3(x - 2)2 - 2(x - 1)
3(x2 - 4x + 4) - 2x + 2
3x2 - 12x + 12 - 2x + 2
3x2 - 14x + 14

That's the end of Part 1. Part 2 will hopefully be uploaded soon, and there will be a link here.

Link to Part 2

(x, why?) Mini: Nachos

(Click on the comic if you can't see the full image.)
(C)Copyright 2015, C. Burke.

Bring the next bowl! I heartily 8!

I almost made the last bowl blue salsa because I ran out of colors! (Bean dip, brown or black, wouldn't have shown up well.) Then I went for a light cheddar than in the second bowl.

Come back often for more funny math and geeky comics.

Friday, August 14, 2015

Shadow Height

(Click on the comic if you can't see the full image.)
(C)Copyright 2015, C. Burke.

Come on. Hit me with your best shot!

This was supposed to go up in May when I did The Distance Around and Three and Four.

I guess it was put off until after Memorial Day and then I just didn't get back to it. I'm glad I saw it in my notes! (I'm glad I keep notes!)

Come back often for more funny math and geeky comics.

Thursday, August 13, 2015

Code a Movie Quote #1 Answers

First of all, thank you for the tremendous response to my first Code a Movie Quote comic. I never expected it because this has become, far and away, my most popular strip, beating out Coffee Logic and almost topping my Irregular Webcomic tie-in strip (but that one was a bit of a flash in a pan from IWC readers).

This started as an exercise in logic, but I morphed it when a couple of coding ideas came to me. You can see that in the second Casablanca quote, which is basically meta-code of some type, not any real language.

As for the actual languages, I mixed it up a bit. I haven't experimented a lot with them since a few college classes on different languages. Most are basically the same, and I varied them because I didn't want anyone to get hung up on syntax. (Thus, the semicolons and the periods doing the same thing.)

Answer Key to #1

As noted twice now, I'm referring to this as #1 because I have 20,000 reasons to do it again. Maybe I'll wait a month or so, and post another ten. (What do you think?)

1. Field of Dreams. "If you build it, he will come."

2. Casablanca. "You played for her, you can play it for me!" (Bogey never actually said, "Play it again, Sam.", except maybe in an old Bugs Bunny cartoon.)

3. The Wizard of Oz. "There's no place like home."

4. Batman. "I'm Batman." Duh. (Seriously, I included one obvious thing for the humor value. I don't need 100 similar submissions!)

5. Apocalypse Now. "I love the smell of napalm in the morning."

6. M*A*S*H. "Suicide is painless. It brings on many changes. And I can take or leave them if I please." Lyric.
Note: Not having Google 40 years ago, I always thought (until just about the time I typed this) that the line was "I could take all evening if I please." Thank you to everyone for NOT pointing that out. No, seriously, I'm not alone in that misconception, but then those were other kids who misheard the same thing. Or just believed what someone else said were the words.
What kills me is that now that I do have search engines, I usually check that sort of thing!

7. Planet of the Apes (1968) "Damn you all to Hell". (Yes, the line is actually "God damn", but it works without taking my Lord's name in vain.)

8. Casablanca. "Maybe not today. Maybe not tomorrow, but soon" (and for the rest of your life). The entire quote was on Twitter, but I snipped it for the comic.
This was one of the earlier "meta" code examples, but I didn't want to lead off with it.

9. Forrest Gump. "Life was like a box of chocolates. You never know what you're gonna get." So it's random.

10. A Few Good Men. "You can't handle the truth!" Okay, I thought that was a clever way of coding it.

11. EDIT: If It's Tuesday, This Must Be Belgium. The quote is the title. I haven't seen this film, so I don't know if that's actually a line of dialogue from the film or just the movie's title. Thanks to William Ricker for posting it and allowing me to use it.

12. Annie. “The Sun will come out tomorrow." Lyric. Thanks to Evan Weinberg for posting it, and allowing me to use it.

13. Braveheart. “They may take our lives, but they'll never take our freedom!” (Originally, on Twitter, I didn't include the "may" because that seemed to be another "If" condition which wouldn't code in a way that made sense.)

14. Citizen Kane. "Rosebud." If you don't get this, I can't explain it without spoiling the movie. Well, that' not really true. It's the last word that Kane utters before dying. It's his end, so I thought that would be a good end to these quotes.

Wednesday, August 12, 2015


(Click on the comic if you can't see the full image.)
(C)Copyright 2015, C. Burke.

And half of a sixteenth is a "dirty-tooth"!

Some numbers transcend normal arithmetic problems.

Come back often for more funny math and geeky comics.

Tuesday, August 11, 2015

Code a Movie Quote

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

UPDATE: The solutions (and a major correction) are posted here.

(C)Copyright 2015, C. Burke.

Every now and then I like to hash things out.

This started online from a couple suggestions which were more "logic" oriented. I can't find them now because it was a lot of tweets ago and they didn't have a hashtag. I suggested a few ... and then started coding. It didn't really catch on. There were only two entries that weren't mine, and when I stopped for the evening, everyone was thankfu-- I mean, when I stopped, the hashtag died.

I didn't start it the following morning, but I figured that they shouldn't go to waste.

Special thanks to William Ricker (‏@n1vux) and Evan Weinberg ‏(@emwdx) for participating and for allowing me to use their entries.

You can add yours in the Comments section, or on Twitter with #CodeAMovieQuote.

Come back often for more funny math and geeky comics.

Monday, August 10, 2015


(Click on the comic if you can't see the full image.)
(C)Copyright 2015, C. Burke.

Maybe I was pickpocketed several times for twenty bucks apiece.

Little expenses -- the kind that might think too small to charge -- add up after a while. Especially after two weeks.

Adding up all those tiny amounts ... a Summation problem, I guess.

Come back often for more funny math and geeky comics.

Saturday, August 08, 2015

How I Would Have Handled the First Debate: A Mathy Solution

Disclosure: I was either in an airport or in the air when the Debate and the "Also-Ran" Debate were held earlier this week. I have seen neither of them, and from the quips and comments, I didn't miss much that wasn't rehashed later on.

Disclosure: I didn't write Full Disclosure because then I'd actually have to say something about my politics, and I'm leaving that out of the equation. But I will talk about equations.

My main objection to this debate, and any debate of this format, is that there are so many people that they don't really have time for one important thing: An Actual Debate.

There will be the grilling of the candidates (which can be done on any interview show) with softball or "gotcha" questions by moderators who generally have their own axe to grind, or at the very least, their own favorite candidate. And, sometimes, that candidate isn't even on the stage or in that party. And it shows. (Again, this goes back through years of watching these things, not specifically this past week. Which I didn't see.)

And then there are the group questions which don't merit more than a few second answer, a sound bite, an applause line, or a witty retort. One debate four years asked ridiculous questions and then, on the issues asked for a show of hands. I kid you not. Now, I hate the term "nuance" and I don't like my messages "massaged", either, but there's more to complicated issues than a "Yea" or "Nay". And, frankly, how well-defined are the terms of that five-second inquiry?

After this debate, I saw complaints about issues not talked about -- and for that, you need to fault the moderators, not the candidates. Now, candidates will speak on whatever they want to, and can ignore the questions, but too much grandstanding works against them. So they have to keep to the line of questioning for the most part. After all, it'll be over in a couple of minutes due to time constraints and forty-seven bazillion other people on the stage!


On the other hand, there are problems with asking ten candidates about important issues. At the top of the list: it's boring. Any important issue will not have ten positions to stake out. After the third person speaks, there probably won't be much more to say without having to agree with, or take a shot at, one of the prior speakers. (And do we really need a second round to respond?)

It almost reminds me of The Dating Game (Yes, I'm going there, and I'm just as surprised as you are.) where the Bachelorette would ask Bachelor #1 a question, Bachelor #2 the same question, but move on to a new question for Bachelor #3. Otherwise, he'd have the most time to think of a witty answer, but it'd be boring because we already heard two "opening lines" on the same topic.

Bringing this to the debate, you could switch the question after every three candidates, but that's still problematic. We want to know where they stand on all the issues, not just the handful they were asked about.

The candidates know where they stand and where their opponents stand. So why not use that.

The Math Solution

Here is how I would structure a debate with 10 speakers:

Each candidates gets to ask each of the other candidates exactly one question with the proviso that they can't ask someone who already asked them a question (Boring!). The questionee then becomes the questioner, which is somewhat interesting and prevents the other candidates from ganging up on any one person. In case of a dead end, the moderator moves it along to someone else. As a bonus, anyone who can't figure out how many questions total the debate will have is automatically disqualified.

I am actually torn on the topic of follow-up questions because I might to hear the candidate expand upon his thoughts on serious issues affecting the country (and I know what I think are the serious issues affecting our country), but I don't want to hear seven questions about last week's tabloid headlines or anything about what some intern tweeted "by accident". I'm inclined to think that the tabloids always win, so no one person should be asked more than once about any particular issue. But then there's the problem of overlap... sigh.

Each candidate is then limited to 15 seconds to ask one opponent a question to limit the build-up/pontificating/damned-if-you-do-or-dont agenda. The other speaker is entitled to one minute and fifteen seconds to respond. Again, anyone who can't figure out the minimum amount of time for everyone to ask all the questions is automatically eliminated -- call up the JV Squad.

I am actually torn on the topic of follow-up questions because I might to hear the candidate expand upon his thoughts on serious issues affecting the country (and I know what I think are the serious issues affecting our country), but I don't want to hear seven questions about last week's tabloid headlines or anything about what some intern tweeted "by accident". I'm inclined to think that the tabloids always win, so no one person should be asked more than once about any particular issue. But then there's the problem of overlap... sigh.

In this scenario, the moderator is restricted to the task of, you know, actually moderating something. They say, "Good evening", "Next question", "Fifteen seconds left", "Time" and "Thank you, Good Night." This has the advantage of encapsulating all the most interesting things any moderator ever says anyway. As noted above, in the case of a logjam, the moderator will pick who asks the next question.


Would it be more interesting, informative or entertaining. Doubt it. But I'd get a kick out of it with some popcorn. Maybe.