Happy Tau Day

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(C)Copyright 2014, C. Burke.

I'm more of a pi guy, but I wanted to use Tau-er of power in a comic.

Comic Sites You'll Love -- Hey, I'm Listed.

I don't remember if I posted a link to this before, but I didn't see one in a quick search.

Guillermo Bautista has an entry in Math and Multimedia, entitled 7 Funny Math Comic Sites You Will Love.

(x, why?) made the list. Well, actually, this blog made the list, but I like to give a plug to the comics-only site every now and then. These days the blog is starting to get more traffic, but without the other site, some of the comics wouldn't be readable.

What else is on the list? Well, there's . . . ah, that would spoil it. Go check it out.

End-of-Year Review

(Click on the cartoon to see the full image.)
(C)Copyright 2014, C. Burke.

Because, let's face it, I'm just a flippin' genius!

June 2014 Geometry Regents, Parts 2

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I really can't do Parts 3 and 4 until I take the time to create the necessary illustrations. Sorry. So here's Part 2. If you missed Part 1, it's here.

Part 2

29. The coordinates of the endpoints of BC are B(5, 1) and C(-3, -2). Under the transformation R₉₀, the image of BC is B'C'. State the coordinates of points B' and C'.

Remember, if they don't state which direction you are rotating, the default is counterclockwise. That might seem counter-intuitive, but you are moving from quadrant I to II, from II to III, etc.

Under R90, a point P(x, y) becomes P'(-y, x). So these points will become B'(-1, 5) and C'(2, -3). Interestingly enough, according to the key, you didn't need to say which was which, but you did need to include the parentheses.

30. As shown in the diagram below [COMING SOON], AS is a diagonal of trapezoid STAR, RA || ST, m<ATS = 48, m<RSA = 47, and m<ARS = 68.
Determine and state the longest side of triangle SAT.

You needed to calculate that m<AST = 65 degrees. You can get this using triangle RAS or knowing that the same-side interior angles are supplementary.
Once you know that, you can use 65 + 48 + x = 180 to find m<SAT, which is 67.

Side ST is the longest side because it is opposite the largest angle. You needed to give the side AND show some kind of work that indicates that you didn't go eeny-meeny... Seriously.
ST without any work -- even just angle sizes after you did the subtraction in your head or on your calculator -- is worth 0 points.

31. In right triangle ABC shown below [I'LL GET TO IT!], altitude BD is drawn to hypotenuse AC. If AD = 8 and DC = 10, determine and state the length of AB.

There's a short way and a long way to do this. Both are okay for full credit.

First, remember that there are three right triangles in this diagram. Both of the smaller ones are similar (and therefore proportional) to the larger one and to each other.

You could write a proportional comparing ( base / hypotenuse) = ( base / hypotenuse) of triangle BAD and triangle ABC.
You would have ( 8 / x ) = ( x / 18 ). (It's 18 because it is 10+8.)
Cross-multiply: x² = 144.
x = 12, which is the length of AB.

The longer method: If you forgot about that proportionality, but remembered the Altitude Rule, you would have found that altitude (BD)² = (AD)(DC), so y² = (8)(10) or 80. So the altitude is the square root of 80. Okay, that's good for partial credit, but we aren't done.

Now you can use Pythagorean Theorem: 8² + 80 = x². Remember: (the square root of 80) squared is just 80.
64 + 80 = x²
144 = x²
x = 12, again.

Either method is valid and will get full credit. I saw quite a few of each type. The only problem with the longer method was that some students started estimating an answer in the middle and that introduced an error into the rest of the calculations.

32. Two prisms with equal altitudes have equal volumes. The base of one prism is a square with a side length of 5 inches. The base of the second prism is a rectangle with a side length of 10 inches. Determine and state, in inches, the measure of the width of the rectangle.

I graded a lot of these. Two things: First, I can't believe how many Geometry students can do so well and still not know the Area of a rectangle or the Volume of a rectangular prism.
Second: This is a good case of Less is More. What do I mean by that? A lot of students talked or worked themselves out of a 1 or 2 points. Seriously.

Despite the fact that the word prism is used repeatedly, students draw pyramids, which by themselves, might not have been work and could've been overlooked. But then they used formulas for Volume that included 1/3 in them. That's a conceptual error. What makes this worse is that the (1/3)s cancel out and don't affect the outcome -- just the final score. I had to call over an arbitrator on this one because I didn't want to make that call.

Another error that was made repeatedly was that students assumed that the first prism was a cube even though there is nothing supporting this. They substituted 5 for the height! There is nothing about 5 being the height. This led to a discussion about whether that could be a case of "guess and check" and not an actual "conceptual error". Again, the mistake factors out.

But the worst part of it all is that because the heights were the same, the formula for Volume wasn't even needed! You only needed to compare the Area of the two bases.

This became confusing as well. People did perimeter of one and Area of the other. Or they assume that both bases were squares or other crazy stuff.

If was a very simple problem. How simple?

5² = 10x.

That simple: x = 2.5.

Funniest answer: Someone wrote 2.5in = w, with a little bit of cursive and I thought it said "2 Sin w". Okay, it was funny to me.

33. As shown in the diagram below, BO and tangents BA and BC are drawn from external point B to circle O. Radii OA and OC are drawn. If OA = 7 and DB = 18, determine and state the length of AB.

A very similar question was asked in August 2011, but that was multiple choice.

Again, there are TWO methods to solve this: Pythagorean Theorem, and Tangent-Secant.

First, realize that OD is also a radius, with length 7. That means that OB is 25. OA is perpendicular to the tangent line BA. Therefore, OAB is a right triangle with OB as its hypotenuse and OA, a radius, as one of its legs. So 7² + (AB)² = 25².
Solving the equation, we find that AB has a length of 24.

The other way: Draw diameter DOE (make point E on the circle opposite from D). BE is now a secant line with a length of 18 + 7 + 7 = 32.
Tangent-secant rule: (BA)² = (BD)(BE) = 18 X 32 = 576. AB = 24.

Both methods are valid.

34. Triangle RST is similar to triangle XYZ with RS = 3 inches and XY = 2 inches. If the area of triangle RST is 27 inches, determine and state the area of triangle XYZ, in square inches.

Short way: The ratio of the areas of two similar triangles is the square of the ratio of their sides. The scale factor of the sides is 3:2, so the scale factor for the areas is 9:4. If the bigger one has area 27, then ( 9 / 4 ) = ( 27 / x ).
Cross-multiply and 9x = 108, and x = 12.

LONG WAY: If you forgot about that, you could have done the following: if the large triangle has Area 27 and base 3, and A = 1/2 b h, then 27 = (1/2)(3)(h), and h = 18.
Then to find the height of the smaller triangle, ( 3 / 2 ) = ( 18 / h ). Then 3h = 36 and h = 12 inches. But that isn't the answer -- that's just the height.
A = (1/2) b h = (1/2) (2) (12) = 12 sq inches. That's the answer. Even though both numbers are 12, if you didn't complete it, you lost a point.

35.

June 2014 Geometry Regents, Part 1

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I just got home from grading the Geometry Regents exams in Staten Island, so I finally had a good look at the exam. Students and teachers alike had both described it as a "fair" exam (sort of like the Algebra test), and I'd have to agree.

Here are the multiple-choice questions from Part I. Keep in mind, that I have to type all of these, so the rest of the test may not show up on my blog as quickly as you may like. Questions are always welcome. Likewise, because I've been asked to hurry with this, there are no diagrams included. They may get added at a later time.

1. Plane P is parallel to plane Q. If plane P is perpendicular to line l, then plane Q...

(3) is perpendicular to line l. Think of a fire pole at a fire station going through the second floor to the first floor.

2. In the diagram below [Diagram Omitted], quadrilateral ABCD has vertices A(-5,1), B(6, -1), C(3, 5) and D(-2, 7). What are the coordinates of the midpoint of diagonal AC?

The midpoint is the average of the two x-values and the average of the two y-values. ( (-5+3)/2, (1+5)/2 ), which is (-1, 3).

3. In the diagram below, transversal TU intersects PQ and RS at V and W, respectively. If m<TVQ = 5x - 22, and m<VWS = 3x + 10, for which value of x is PQ || RS?

You want the value of x that makes 5x - 22 = 3x + 10, because corresponding angles are congruent when parallel lines are cut by a transversal. Solve for x and you get 16.

4. The measures of the angles of a triangle are in the ratio 2:3:4. In degrees, the measure of the largest angle of the triangle is:

Take the ratio and write the following equation: 2x + 3x + 4x = 180. So 9x = 180, and x = 20. The largest angle is 4x, which is 4(20) = 80 degrees.

5. The diamter of the base of a right circular cylinder is 6 cm and the height is 15 cm. In square centimeters, the lateral area of the cylinder is

The lateral area is the circumference of the base times the height, or (pi)(d)(h), which is (pi)(6)(15), which is 90*pi.

6. When the system of equations y + 2x = x² and y = x is graphed on a set of axes, what is the total number of points of intersection?

Substitute y = x into the other equation and you get x + 2x = x²
This can be rewritten as x² - 3x = 0.
You don't need to solve it to know that there are two distinct solutions. You could have graphed the equations as well. Note: A parabola and a straight line can have 0, 1 or 2 intersection points. They cannot have three.

7. The vertex angle of an isosceles triangle measures 15 degrees more than one of its base angles. How many degrees are there in a base angle of the triangle?

The sum of the angles is x + x + x + 15 = 180. So 3x + 15 = 180, 3x = 165 and x = 55, which is the measure of one base angle.

8. Circle O is graphed on the set of axes below. [Diagram omitted] Which equation represents circle O?

The correct form is (x - h)² + (y - k)² = r², where (h, k) is the center of the circle and r is the radius.
In the graph, the center is (-1, 3) and the radius is 3.
So the equation is (x + 1)² + (y - 3)² = 9.

9. In the diagram of the circle shown below [Diagram omitted], chords AC and BD intersect at Q and chords AE and BD are parallel. Which statement must always be true?

Arcs AB and DE are congruent because the arcs formed by two parallel chords are always congruent.

10. In the diagram below [Diagram omitted], triangle AEC is congruent to triangle BED. Which statement is not always true?

Angle ACE does not correspond to angle DBE, so they will not always be congruent although they could be in an isosceles or equilateral triangle.

11. What is the length of RS with R(-2, 3) and S(4, 5)?

Distance formula, a.k.a. Pythagorean Theorem. The square root of (6² + 2² is the square root of 40, which simplifies to 2 times square root of 10.

12. What are the truth values of the statement "Two is prime" and its negation?

Two is a prime number so the statement is True, which means the negation is False. Choices 2 and 3 are just silly and should have been eliminated immediately.

13. A regular polygon has an exterior angle that measures 45 degrees. How many sides does the polygon have?

If you didn't know just from constant repetition of the problem in class, remember that 360/n = 45, so n = 360/45 = 8 sides.

14. In rhombus ABCD with diagonals AC and DB, AD = 10. if the length of diagonal AC is 12, what is the length of DB?

If one side is 10, all sides are 10. The diagonals are perpendicular bisectors. This means that there are four little right triangles with hypotenuse 10. One leg of each triangle is 6 (12/2). Using Pythagorean Theorem, or just knowing the triple, the other leg of each triangle is 8. That makes the length of the entire diagonal DB 16.

15. If the surface area of a sphere is 144*pi square centimeters, what is the length of the diameter of the sphere, in centimeters?

The surface area of a sphere (given in the back of the booklet) is 4*pi*r² = 144 * pi. Divide both sides by 4*pi, and r² = 36, so r = 6.
The radius is half the diameter, which is 12.

16. Which numbers could represent the lengths of a the sides of a triangle?

Not a right triangle, just a triangle. The rule for the sides of a triangle is that the sum of the two smaller sides must be more than the length of the largest side. 5 + 9 = 14, 7 + 7 < 15, 1 + 2 < 4, 3 + 6 > 8.

17. The equation of a line is 3y + 2x = 12. What is the slope of the line perpendicular to the given line?

The slopes of two perpendicular lines are inverse reciprocals. (That is, change the sign, and flip the fraction over.)
First you have to find the slope of the given line. Subtract 2x from both sides and divide everything by 3 (the coefficient of y). The slope of the given line is -2/3.
That makes the perpendicular slope 3 / 2. CORRECTION.

18. In the diagram below, point K is in plane P. How many lines can be drawn through K, perpendicular to plane P?

Similar to the other question, only one line perpendicular to a plane can be drawn at any one point. Think of a pencil through a hole in your loose leaf.

19. In the diagram below, AB and CD are bases of trapezoid ABCD. (Not drawn to scale) If m<B = 123 and m<D = 73, what is the m<C?

Because it isn't an isosceles trapezoid, <C and <D can not be assumed to be congruent. However, <B and <C are supplementary. So the answer is 57.

20. What is the equation of a line passing through the point (4, -1) and parallel to the line whose equation is 2y - x = 8?

Rewrite 2y - x = 8 as y = (1/2) x + 4, slope is 1/2. So the parallel line must have the equation y = (1/2) x + b.
Plug is 4 for x and -1 for y and solve for b. You will find that b = -3, so the answer is y = (1/2) x - 3.

21. The image of rhombus VWXY preserves which properties under the transformation T_{2, -3}?

Translations do not change shape nor orientation, so the correct answer is both parallelism and orientation.

22. The equation of a circle is (x - 3)² + y² = 8. The coordinates of its center and the length of its radius are

Using the form given above, the center is (3, 0) and the radius is the square root of 8, which is 2 times the square root of 2.

23. Which statement has the same truth value as the statement, "If a quadrilateral is a square, then it is a rectangle"?

Note that it didn't ask what the truth value of the statement was. That isn't important. What is important is that the statement has the same truth value as its contrapositive, which is "If a quadrilateral is not a rectangle, then it is not a square."

24. The three medians of a triangle intersect at a point. Which measurements could represent the segments of one of the medians?

The centroid occurs 2/3rds the way down the length of the median. In other words, the two segments will have a ratio of 1:2, with each segment being 1/3 or 2/3 the length of the median itself. The correct choice is 3 and 6.

25. In the diagram of triangle PQR shown below {NO, IT ISN'T], PR is extended to S, m<P = 110, m<Q = 4x and m<QRS = x² + 5x. What is m<Q?

The sum of the measures of angles P and Q is equal to the exterior angle R. The equation to write is x² + 5x = 4x + 110.
This becomes a quadratic equation, x^{2 + x - 110 = 0.
This factors in (x + 11)(x - 10) = 0, so x = -11 (discard the negative) or x = 10.
HOWEVER, they want the size of the angle, not of x. The size of the angle is 4x = 4(10) = 40.}

26. Triangle PQT with RS || QT is shown below [DONT HOLD YOUR BREATH]. If PR = 12, RQ = 8 and PS = 21, what is the length of PT?

You can set up a proportion (21 / x) = (12 / 8), which will find the length of ST, which is 14. PT = 21 + 14, which is 35.

27. In the diagram of line segment WXYZ below, WY is congruent to XZ. Which reasons can be used to prove VW is congruent to YZ?

The reflexive property (XY is congruent to XY) and subtraction postulate (WY - XY is congruent to XZ - XY).

28. The coordinates of the endpoints of the diamter of a circle are (2, 0) and (2, -8). What is the equation of the circle?

This is the third question about the equation of a circle. The center of the circle is (2, -4). The radius is 4.
The equation of the circle is (x - 2)² + (y + 4)² = 16.

* * *

So how did everyone do? More importantly, how did I do? I mean, I did rush this. Mistakes happen.

June 2014 NY Integrated Algebra Regents, Thread 2

Parts 2, 3 and 4 of the Algebra test were open-ended and you had to show your work to get full credit. I don't know the raw score required for passing, but every point here, even partial credit helps. In general, over the last few years, a score of 30-32 is a passing grade. That means you got at least half of the multiple-choice questions correct. However, if you were able to get that many of those questions correct, you have no excuse for not showing some work on these problems.

To answer those who asked on yesterday's Part 1 thread and on Twitter, for the past year I worked at Tottenville High School in Staten Island, and before that I was at William E. Grady High School in Coney Island, Brooklyn, with some time at Boys High School in Bed-Stuy.

Okay, so here we go. Illustrations may get added later, but students seem to be in a hurry for answers:

June 2014 Integrated Algebra Regents, Parts 2, 3 and 4

31. A patio consisting of two semicircles and a square is shown in the diagram below. [ A SQUARE with a semicircle on top and right sides.] The lenght of each side of the square region is represented by 2x. Write an expression for the area of the entire patio, in terms of x and pi.

In terms of x and pi means that you aren't solving for x and you aren't replacing pi with 3.14 or anything else.

Two semicircles make one whole circle. The area of a circle is pi*r². The radius is half the diameter. The diameter is the side of a square, which is 2x. So the radius is 1x. The area of a square is s².
The area of the entire patio is (2x)² + (pi)(x)², which is 4x² + (pi)(x)². That should be enough for full credit.
You could also represent it as (4 + pi)(x)², but that shouldn't be necessary.

32. Clayton is performing some probability experiments consisting of flipping three coins. What is the probability that when Clayton flips the three coins, he gets two tails and one heads?

I did a very similar problem in class recently. Hopefully, you made a tree diagram or a sample space because there aren't that many possible outcomes: 2 X 2 X 2 = 8 outcomes.

On your diagram you'll see the only possibilities are that the first is H and the others tails, the second is H and the others tails or the third is H and the others tails. Therefore there are 3 positive outcomes, so the probability is 3/8.

This was much easier than a Common Core two-point question!

33. Ross is installing edging around his pool, which consists of a rectangle and a semicircle, as shown in the diagram below. [RECTANGLE WITH A SEMICIRCLE ON TOP, WITH A DIAMETER THE WIDTH OF THE SMALLER END OF THE RECTANGLE.] Determine the length of edging, to the nearest tenth of a foot, that Ross will need to go completely around the pool.

Two irregular shapes in one test? That's new. And both contain semicircles. However, this is a perimeter/circumference problem, not an area problem.

The three sides of the rectangle that need edging are 30 + 15 + 30, which is 75 feet. The semicircle is (1/2)(pi)(d), which is (1/2)(3.141592...)(15), which is 23.56..., which rounds to 23.6.

The total is 75 + 23.6 = 98.6 feet.

Part 3

34. Solve the following system of equations algebraically for all values of x and y. y = x² + 2x - 8; y = 2x + 1

Algebraically means that you can't use your graphing calculator to find the answer and then use "guess and check" on the paper.

Using substitution, you get x² + 2x - 8 = 2x + 1
Subtract 2x and subtract 1 from both sides and you have x² - 9 = 0
This is a Difference of Squares and can be factored as: (x + 3)(x - 3) = 0
So either x + 3 = 0 or x - 3 = 0
Your solution set for x is {-3, 3}

DO NOT STOP HERE. You need the values of y as well.

If x = -3, y = 2(-3) + 1 = -6 + 1 = -5. (-3, -5)
If x = 3, y = 2(3) + 1 = 6 + 1 = 7, (3, 7)

If you plug these into the other equations, they will check.

35. A storage container in the form of a rectangular prism is measured to be 12 inches by 8 inches by 4 inches. Its actual measurements are 11.75 inches by 7.75 inches by 4 inches. Find the relative error in calculating the volume of the container, to the nearest thousandth.

Volume = length X width X height.
Relative error = (difference in the values) / (actual value)

So this problem is ( (12)(8)(4) - (11.75)(7.75)(4) ) / ( (11.75)(7.75)(4) ). If you are using a graphing or scientific calculator, and doing this in one step, those extra parentheses are needed.

This becomes ( 384 - 364.25 ) / 364.25 = 19.75 / 364.25 = 0.054221..., which is 0.054 to the nearest thousandth.
DO NOT make it into a percentage.

36. Perform the indicated operations and express the answer in simplest radical form.

This is new, using the Distributive Property with radicals. Good luck.
Keep in mind, this is how I would do it. As long as you did similar steps, possibly in the same order, and achieved the same solution, you're fine. This would've taken longer if I hadn't seen a number pattern and multiplied the sevens before finding the largest perfect squares.

Again, you didn't have to do it my way, as long as you multiplied, factored, added and came up with 189 times the square root of 2.

If I made a mistake here, feel free to point it out and I'll correct it.

Part 4

37. During its first week of business, a market sold a total of 108 apples and oranges. the second week, five times the number of apples and three times the number of oranges were sold. A total of 452 apples and oranges were sold during the second week. Determine how many apples and how many oranges were sold the first week. [Only an algebraic solution can receive full credit.]

We'll let x = the number of apples sold the first week and y = the number of oranges sold the first week. I won't use A and O, because O looks like 0 and that can be confusing.

We know that x + y = 108 and 5x + 3y = 452. Multiply (x + y = 108) by (-3) and you get -3x + -3y = -324.
Combine 5x + 3y = 452 and -3x + -3y = -324 and you get 2x = 128. The y (number of oranges) is eliminated.
If 2x = 128, then x = 64.

64 + y = 108, so y = 44.

Check: 5(64) + 3(44) = 452?
320 + 132 = 452?
452 = 452 (check!)

38. On the set of axes below, solve the following systems of inequalities graphically. Label the solution set S.

2x + 3y < -3; y - 4x > 2

If you want to use your calculator as a guide, you need to rewrite these equations in slope-intercept form.

y < (-2/3)x - 1; y > 4x + 2

Note that the first line will be a broken or dashed line. The second inequality has a solid line.

A picture of the graph will have to wait until later but, you will shade below the first line and above the second line. The solution set will be the section on the left side of the graph between the two lines. That's where the S goes.

39. During the last 15 years of his baseball career, Andrew hit the following number of home runs each season: 35, 24, 32, 36, 40, 32, 40, 38, 36, 33, 11, 20, 19, 22, 8. State and label the values of the minimum, 1st quartile, median, 3rd quartile and maximum. Using the line below, construct a box-and-whisker plot for this set of data.

First, put the numbers in order to make life easier. Your graphing calculator can do this. (It can also make the box-and-whisker plot, too!)

The numbers, in order, are: 8, 11, 19, 20, 22, 24, 32, 32, 33, 35, 36, 36, 38, 40, 40.

There are 15 numbers, so the 8th number is the median (32). The middle of the first 7 numbers, the 4th number, is Q1 (20). The middle of the back 7 numbers, the 12th number, is Q3 (36). The minimum is 8 and the maximum is 40.

I would use a scale of 2 on the number line, which should allow the numbers from 0 to 50 if you wanted. Plot the five numbers we found above. Draw the box around the middle three and the extend the whiskers to the first and last.

New Rubric

(Click on the cartoon to see the full image.)
(C)Copyright 2014, C. Burke.

What part of ''Show Your Work'' are you still not getting? And ''well, duh!'' is NOT a valid explanation!

June 2014 NY Integrated Algebra Regents, Thread 1

Today, Friday, was both the Integrated Algebra and the Geometry Regents exams. Integrated Algebra is over as I type this (although it won't be posted until later, after all extended time and conflict rooms have completed). I taught more Algebra students than Geometry students this year and the Algebra examis the one required for graduation, so I'll start my review with this test.

Part 1 was "fair" to use a term that a student or two used. Most of the ones I saw in the hallway after the exam, returning their calculators as I was resetting a box of them for the next exam, said that they think they did well. Only one said she'd done poorly, and I hope (and I believe) that she's wrong because I know how intelligent she really is. She's underestimating herself. I hope.

Keep in mind that I had to type all of this. There was no cutting and pasting involved, so I left out most of the "choices", and illustrations will have to wait. Given all that then, without further ado....

NOTE: Thread 2, with the rest of the questions, is here.

June 2014 Integrated Algebra Regents

1. The product of 6x³y³ and 2x²y is

Multiply the coefficients, add the exponents, and don't forget that the exponent on the last y is 1. 12x⁵y⁴

2. Which set of data is qualitative?

My class (as I'll speak of them often) just went over this on an old exam. Qualitative is descriptive, like the swimmer's favorite swimsuit colors.

3. It takes a snail 500 hours to travel 15 miles. At this rate, how many hours will it take the snail to travel 6 miles.

Simple proportion: 500/15 = x/6 (or similar). Cross-multiply and get 200 hours, which is 2/5ths of 500.

4. The equation y = ax² + bx + c is graphed on the set of axes below. [GRAPH OMITTED] Based on the graph, what are the roots of the equation ax² + bx + c?

The roots of the equation are where the parabola crosses the x-axis. As you can see plainly from the graph (if you happen to have the test in front of you) is that it crosses at 1 and 5.

5. When solving for the value of x in the equation 4(x - 1) + 3 = 18, Aaron wrote the following lines on the board [ILLUSTRATION TO BE ADDED LATER]
Which property was used incorrectly when going from line 2 to line 3?

They made it easier by telling you where the mistake would be. Aaron didn't distribute the 4, writing - 1, instead of - 4.

6. What is the solution of 4x - 30 > -3x + 12?

Add 3x to both sides. Add 30 to both sides. You get 7x > 42. That becomes x > 6.

7. A local government is planning to increase the fee for use of a campsite. If a survey were taken, which group woulb be most biased in their opposition to the increase?

Even if you didn't cover "bias", it should be obvious that of the groups listed, the ones who would be against the tax are the campers.

8. An example of an algebraic equation is

They ask one of thee every test. An equation has an equal sign, an expression does not. 5x = 7 is an equation, even if its solution is a fraction. (That doesn't matter, but I'm sure it bothered someone.)

9. What is the value of x in the solution of the system of equations 3x + 2y = 12 and 5x - 2y = 4?

Add the two equations together and y is eliminated. 8x = 16, so x = 2.

10. What is the slope of a line that passes through the points (-2. -7) and (-6, -2)?

Change in y divided by change in x. (-2 - -7) / (-6 - -2), which is - (5 / 4).

11. Which notation is equivalent to the inequality -3 < x < 7?

As I've told my students, if you don't remember which is ( and which is [, the first thing you need to realize is that the two symbols have to be different. So you've eliminated two choices. The second thing to ask yourself is this: which is more likely to be the square bracket -- less than or less than or equal to?

The correct answer is (-3, 7].

12. What is the value of the expression 3a² - 4|a| + 6 when a = -3?

Simple substitution. Don't forget parentheses around the (-3) when squaring it on your calculator! 3(9) - 4(3) + 6 = 27 - 12 + 6 = 21

13. Which relation is a function?

In a function, the x values can not be repeated. The y values can be repeated. Option 1 is correct. {(2, 1), (3, 1), (4, 1), (5, 1)}

14. When 6x² - 4x + 3 is subtracted from 3x² - 2x + 3, the result is

"Subtracted from". You need to do 3x² - 6x², which gives -3x². The answer is -3x² 2x . The +3 and +3 cancel out.

15. The lengths of the sides of a right triangle can be

Use Pythagorean Theorem on the choices, or just recognize 9-12-15 because it gets used so often. You should recognize some of the basic Pythagorean Triples.

16. Which equation represents a line that is parallel to the y-axis?

The y-axis is x = 0. A line parallel to that would be in the form x = a. The correct answer is (1) x = 5. There is no y in the equation.

17. In right triangle ABC, shown below, AC = 12, BC = 16 and AB = 20. Which equation is not correct?

Remember SOHCAHTOA to remember which sides go with which ratios. You'll notice that tangent doesn't use the hypotenuse. The correct choice is tan B = 16/20, which is an incorrect statement.

18. Three times the sum of a number and four is equal to five times the number, decreased by two. If x represents the number, which equation is a correct translation of the statement?

Taking it one piece of the statement at a time, we get 3(x + 4) = 5x - 2. Note that the comma before "decreased by two" reinforces the fact that the previous multiplication only applies to the variable and not to the difference of the variable and 2.

19. What is the equation of the line that passes through the point (3, -7) and has a slope of -4/3?

Obviously, it has to start with y = -4/3 x. Is the y-intercept +3 or -3? Plug is 3 for x, and see which y-intercept will give you 7 for y. The answer is (2) y = -4/3 x - 3.

20. Which parabola has an axis of symmetry of x = 1?

Look at the graph. Which has a vertex where x = 1? Choice (1) does.

21. When factored completely, the expression 3x² - 9x + 6 is equivalent to

Factor out the 3 and you get x² - 3x + 2, which factors into (x - 1)(x - 2). So the answer is 3(x - 1)(x - 2).

22. The equation P = 0.0089t² + 1.1149t + 78.4491 models the United States population, P, in millions since 1900. If t represents the number of years after 1900, then what is the estimated population in 2025 to the nearest tenth of a million?

t = 2025 - 1900 = 125. Plug the equation into your calculator, and you get 356.9.

23. Which graph represents an absolute value equation?

Absolute value graphs are shaped like the letter "V" or an upside-down "V". Choice (2).

24. The expression (a / b) - (1 / 3) is equivalent to

The least common denominator between b and 3 is 3b. You have to multiply the first fraction by (3/3) and the second fraction by (b/b). This gives you (3a / 3b) - (1b / 3b), which is (3a - b) / 3b.

25. Which value of x is the solution of the equation 2(x - 4) + 7 = 3?

2x - 8 + 7 = 3
2x - 1 = 3
2x = 4
x = 2.

26. Given: M = {green, red, yellow, black} and N = {blue, green, yellow}
Which set represents M U N? (union)

The Union of two sets is every element contained in either or both sets. Don't repeat them in the final set.

{green, red, yellow, blue, black}

27. Which situation describes a correlation that is not a causal relationship?

The more miles you walk, the more calories you burn. The more hours a TV is on, the more electricity is uses. The higher the speed of a car, the fewer hours is takes to travel a given distance. The census happens every ten years and it measures the population of the country. A rise in the population does not speed up when the census takes place. When I first read the question, my first response was "What about the census taken every ten years?"

28. A school offers three classes of math and two classes of science, all of which meet at different times. What is the total number of ways a student can take a math class and a science class?

The Counting Principle! My students should've gotten this. Again, it wasn't that long ago.

The number of items in the first list times the number of items in the second list: 3 X 2 = 6.

29. The expression (x - 7) / (9 - x²) is undefined when x is

A fraction is undefined when the denominator is equal to 0 and at no other time. (The numerator doesn't matter for this.) When is 9 - x² = 0? When x = 3 or -3.

30. What is the product of (1.5 X 10²) and (8.4 X 10³) expressed in scientific notation?

Multiplying with give you 12.6 X 10⁵, but that's not scientific notation. You have to move the decimal one place to the left and raise the exponent by 1. The answer is 1.26 X 10⁶.

Algebra 2/Trigonometry Regents for the Algebra 1 Student & Teacher

Today was the New York State Algebra 2/Trigonometry Regents exam. I don't teach this course, so I won't comment personally on how good a test it was for Trig students, other than to say that a couple of colleagues called it a "fair exam". What I can say about this exam is this: Algebra 1 teachers can use many of these multiple choice questions in their own classes with little to no adjustments. If I might so boldly and "arrogantly" claim, the top students in my Algebra 1 class could have solved 8 of the first 9 problems. An above average student would've gotten at least five of those correct.

With this in mind, I'd like to once again go over the Algebra 2 problems which I believe Algebra 1 students could handle, even if only as challenge problems.

Algebra 2/Trigonometry

1. Which survey is least likely to contain bias?
1. surveying a sample of people leaving a movie theater to determine which flavor of ice cream is the most popular
2. surveying the members of a football team to determine the most-watched TV sport
3. surveying a sample of people leaving a library to determine the average number of books a person reads in a year
4. surveying a sample of people leaving a gym to determine the average number of hours a person exercises per week

Not having my students the entire year, I didn't get to "bias" in Common Core Algebra (I believe the previous teacher should have touched on it). I know it was covered in the Integrated Algebra course. The second, third and fourth choices are going to places to ask a question pertaining to the place where the questions are asked; e.g., readers at a library. Only the first one goes to a place where you will find different types of people, not just ice cream lovers. Could there be bias in Choice 1? Of course, it could. Not all people go to movies. But it is still less biased than the other three.

2. The expression (2a)^-4 is equivalent to ...?

If you know that a negative exponent means to (basically) take the reciprocal, then you'll get 1/(16a⁴ as your answer.

Question 3 is a trigonometry question. We'll skip that.

4. Expressed in its simplest form,

is

This could easily be used in Algebra 1 without the negatives under the radicals. It could be used as an extension if there's time. Some of my students knew about imaginary numbers, even if they weren't sure exactly what they were. And they knew they had something to do with square roots.

It's also easy to reason out the answer from the choices. Once you realize that i is involved in both radicals and can be factored out, you've eliminated choices (1) and (2). Realizing that you're subtracting a bigger number from a smaller number indicates that the answer will be negative, eliminating choice (4). (3) is the answer.

5. Theresa is oomparing the graphs of y = 2^x and y = 5^x. Which statement is true?

First of all, both graphs have a y-intercept of (0, 1). Choices (1) and (4) are silly. (Really, "neither graph has a y-intercept"?) Of the two, y = 5^x is steeper. You can check this in your graphing calculator if you weren't sure.

6. The solution set of the equation

is

For Algebra 1 students (and some Trig students), the fastest method is to plug in the choices. Trying -2 doesn't work. Trying 2 does work. Only one solution set contains 2. It also contains 4, which also works.

How are you supposed to solve this? Square both sides and solve the resulting quadratic equation. For multiple choice, plugging in is much faster.

7. The expression is equivalent to

(2)(2) = 4; (-3)(x)^.5 X (-3)(x)^.5 = 9x; (2)(2)(-3)(x)^.5 = -12(x)^.5
The correct choice is (3).

8. Which step can be used when solving x² - 6x - 25 = 0 by completing the square.

Okay, I never did completing the square in Integrated Algebra. It might've been there in the textbook, but it wasn't covered in the curriculum, and it wasn't on the Algebra Regents. That said, it was in the Common Core Algebra this year, and my students picked it up pretty easily. (Well, most of them did.)

To complete the square, you need to halve the -6, getting -3, and then squaring that, getting 9. So +9 is added to each side of the equation and +25 is also added to each side of the equation to get rid of the -25 on the left. The correct choice is (1).

9. Which graph represents a function?

Seriously? This is an Algebra 1 question. If there aren't two y values for the same x-value, then it is a function. Choice (1).

Question 10 is a trigonometry question. We'll skip that.

11. What is the common difference of the arithmetic sequence below?
-7x, -4x, -x, 2x, 5x, . . .

Algebra students should recognize the pattern and deduce that the "common difference" is 3x.

Jumping ahead...

14. What is the product of the roots of the quadratic equation 2x² - 7x = 5?

I should include questions like this. There's no reason not to, and it will get an extra step of them. First solve the quadratic equation, and then multiply the roots. The only problem I have with this -- and maybe it isn't a problem at all -- is that the most common mistake my students make in solving quadratics in flipping the sign. If they flipped both signs and then multiply the answer, then the mistakes will cancel out.

Quick use of the quadratic formula will get you ... two radical conjugates. Okay, so this goes beyond the scope of Integrated Algebra, but a teacher could modify this one a little. But anyway, the product is one-sixteenth of (49 - 89), which is -5/2.

It's actually simpler than this: the rule for product of roots is c/a, which is -5/2. Introducing this right after doing a long problem might be a good way to make them remember the shortcut. It also reinforces the fact that if you can't remember the formulas and shortcuts, it helps to know where they come from, so you can derive them if you have to.

* * *

Continuing the thread...

15. What is the equation of the circle passing through the point (6, 5) and centered at (3, -4)?

This question gets asked on the Geometry Regents at least 3 or 4 times on every test. The only difference here is that the radius is an irrational number, but big deal. Geometry students need to deal with irrational numbers, and the square of the number is needed anyway. (6 - 3)² + (5 - -4)² = 90. So the equation is
(x - 3)² + (y + 4)² = 90.

16. The formula to determine continuously compounded interest is A = Pe^rt, where A is the amount of money in the account, P is the initial investment, r is the interest rate and t is the time, in years. Which equation could be used to determine the value of an account with an $18,000 initial investment, at an interest rate of 1.25% for 24 months?

As complicated as this looks, this is a simple substitution question. It could be given to my freshmen as an extension, just to see if they really can parse a question. You don't have to explain e yet, if you don't want to be, because it could be considered just any variable for the moment. (I realize that it's a constant, but let's not confuse matters at the moment.) The only "trick" to the problem is to remember that 24 months is 2 years. This trips up some students with I=PRT, too.

Question 17 is interesting. Without the "+ 1", it's a simple proportion that leads to a quadratic equation if you don't factor the difference of squares and multiply the fraction on the right by (x + 3)/(x + 3). The "+ 1" makes the addition a little more interesting. Lots of possibilities with this equation for Algebra students.

18. The graph below shows the average price of gasoline, in dollars, for the years 1997 to 2007. [GRAPH NOT SHOWN] What is the approximate range of this graph?

Seriously? Range measures the y values on the graph. The lowest point appears to be about 1.00 or lower, and the highest point is between 2.00 and 2.50. The correct choice would be 0.97 < y < 2.38. Choices 1 and 2 relate to the domain of the graph.

What are your opinions of all this?

Modeling

(Click on the cartoon to see the full image.)
(C)Copyright 2014, C. Burke.

Because either though I don't know where I'll be or what I'll be doing three days from now, I need that fifth-gen Doppler tech to tell me the weather through the end of next week! Plans need to be made!

On the bright side, the bells and whistles curve is exponential.

Cleanliness v. Bugginess

(Click on the cartoon to see the full image.)
(C)Copyright 2014, C. Burke.

Cleaning the pool always makes me a little buggy ... but, seriously ...

The clearer the pool is, the more dead bugs I have to skim off the surface every day.

Either they see their reflection, or it actually isn't clean and there's something bad they're attracted to.
I'm hoping for the former.

Runs In, Waves "Hi!"

Just back from a weekend family wedding in Denver and am now in the middle of finals, which actually started without me. I'm catching up. Be back soon.

Recursive Function Brewing

(Click on the cartoon to see the full image.)
(C)Copyright 2014, C. Burke.

f(0) is 'No more bottles of beer on the wall', but that's just sad.

You could call this a follow-up to T-test discussion about Guinness or a follow-up to the article about recursive functions. Either way.

And I just want to take a line to remember the 70th anniversary of D-Day. I didn't have anything special planned. It's not that I forgot when D-Day was -- I knew it was coming. I just forgot that today was the sixth of June until I was typing the date on today's comic. And considering that this needed to be short because I haven't finished packing, there wouldn't have been time to work on something worthy of the occasion. (In fact, some of you may be reading this while I'm on a plane to Denver for niece's wedding.)

Enjoy your weekend.

June 2014 Common Core Algebra 1 Regents Exam, Part 3

Update: I now have a Common Core Regents Review books available on Amazon.

This is the second thread dealing with the Common Core Algebra test, and probably the last for a while, as I'll be away from the weekend. The first thread was about Part 2 of the test. This post deals with part 3 and 4.

Algebra 1 (Common Core), Part 3

Questions is Part 3 are worth four points. Again, I won't assume the point distribution for partial credit.

33. Write an equation that defines m(x) as a trinomial where m(x) = (3x - 1)(3 - x) + 4x² + 19. Solve for x when m(x) = 0.

You need to multiply the binonials and then combine all the like terms.
(3x - 1)(3 - x) = 9x - 3x² - 3 + x = -3x² + 10x - 3
To that, add 4x² + 19.
You get m(x) = x² + 10x + 16. Don't forget to write it as an equation!

For the second part of the question, take the equation you found in the first part (they allow for consistent errors) and substitute 0 for x. The answer should be obvious, but I would still show the work -- especially because it says solve

m(0) = (0)² + 10(0) + 16 = 0 + 0 + 16 = 16.

Frankly, I thought this problem was easier (or at least less complicated) than some of the Part 2 problems.

EDIT: As you may have seen in the comments below, in my rush to complete this section before leaving for the weekend, I misread my notes in solving this problem, despite having typed it correctly above. My original instinct was that this problem was too easy, and now I see why. Generally, when I get the feeling that something is too easy, there's usually something I missed, and I go look for it.

You needed to find when m(x) = 0, so x² + 10x + 16 = 0. That's a simple quadratic equation to solve. What two factors of 16 add up to 10? That's 8 and 2.
So (x + 8)(x + 2) = 0
x + 8 = 0 or x + 2 = 0
Then x = -8 or x = -2.

So that would be a Conceptual error for this part of the question, which is more points off than a simple Computational error. It could have been worse. I might have been incredibly wrong and arrogantly displayed a lack of understanding of how functions work, but that would be entirely overstating the mistake made. In any event, I am told that some people like to make damning accusations under the guise of anonymity rather than step out into the light of day. But we don't talk about such things in polite company.

Now to continue.... End of EDITED SECTION

34. A rectangular garden measuring 12 meters by 16 meters is to have a walkway installed around it with a width of x meters, as shown in the diagram below (Note: This will be added later) Together the walkway and the garden have an area of 396 meters.

Write an equation that can be used to find x, the width of the walkway.

Describe how your equation models the situation.

Determine and state the width of the walkway, in meters.

Unlike the previous question, this was a majorly complicated problem, with more binomial multiplication.

The length of the rectangle is (2x + 16). The width of the rectangle is (2x + 12). The area of the rectangle is 396.

So (2x + 16)(2x + 12) = 396
4x² + 24x + 32x + 192 = 396
4x² + 56x - 204 = 0
x² + 14x - 51 = 0
(x + 17)(x - 3) = 0
x + 17 = 0 or x - 3 = 0
x = -17 or x = 3
Discard the negative value and you are left with the width of the walkway being 3 meters wide.

Hopefully, that's enough.

35. Caitlin has a movie rental card worth $175. After she rents the first movie, the card's value is$172.25. After she rents the second movie, its value is $169.50. After she rents the third movie, the car is worth $166.75. Assuming the pattern continues, write an equation to define A(n), the amount of money on the rental card after n rentals.

Caitlin rents a movie every Friday night. How many weeks in a row can she afford to rent a movie using the rental card only? Explain how you arrived at your answer.

First, "Explain" means in words; equations will NOT be enough information. I am not kidding.

Subtract 175-172.25 and you get $2.75. Check the next and the next to be sure, but you see that each rental is $2.75
That makes the equation A(n) = -2.75n + 175.

To answer the second part, make an inequality greater than 0, and find the largest whole number which makes it true. (Or make an equation equal to 0 and drop everything after the decimal.)
-2.75n + 175 > 0
-2.75n > -175
n < 63.63
Each movie costs $2.75 to rent. If you divide $175 by $2.75, you can get 63 rentals. There is money left over, but not enough for another rental using the card only.

36. An animal shelter spends $2.35 per day to care for each cat and $5.50 per day to care for each dog. Pat noticed that the shelter spent $89.50 caring for cats and dogs on Wednesday.

Write an equation to represent to represent the possible numbers of cats and dogs that could have been at the shelter on Wednesday. Pat said that there might have been 8 cats and 14 dogs at the shelter on Wednesday. Are Pat's numbers possible? Use your equation to justify your answer. Later, Pat found a record sowing that there were a total of 22 cats and dogs at the shelter on Wednesday. How many cats were at the shelter on Wednesday?

This is a lot of work for a system of equations. I heard some teachers complaining about it (one of whom has seen the scoring rubric), and I heard students tell me that this was so easy. The students explained to me how they did it, so they might be correct in how "easy" they found it.

We'll use C for the number of cats and D for the number of dogs. That makes 2.35C the amount spent on cats and 5.50D is the amount spent on dogs.

The equation for Wednesday is 2.35C + 5.50D = 89.50.

For the second part, substitute 8 for C and 14 for D, and see if it's a true statement.

2.35(8) + 5.50(14) = 89.50 ?
18.80 + 77.00 = 89.50 ?
95.80 =/= 89.50 X
This is NOT a possible combination.

The final part states that C + D = 22. We now have a system of equations. Multiply this equation by -2.35

-2.35C + -2.35D = -51.70
2.35C + 5.50D = 89.50
combine: 3.15D = 37.80
divide: D = 12
There are 12 Dogs, so there are 22 - 12 = 10 = C, 10 Cats.

* * * * * * * * * * *

Okay, that's enough for me. This is taking a lot out of me.

What are your opinions of all this?

June 2014 Common Core Algebra 1 Regents Exam, Part 2

Update: I now have a Common Core Regents Review books available on Amazon.

Today was the first ever New York State Common Core Algebra 1 Regents. No one knew what to expect. Sure, math is math, and Algebra is Algebra. What questions could they ask, right? Well, it’s not just a matter of knowing the material. Some of this was covered nearly a year ago and not revisited. Not everything in the course scaffolds into new topics; not every new topic spirals back into the old.

And then there’s the question of presentation. You can do practice problems until the kids’ pencils are worn to nubs, but if the test problems are suddenly presented in a different -- particularly in an odd – way, a young teen might freeze up and yield the opportunity to work it out.

A lot of the test came down to vocabulary, and not necessarily math vocabulary, and reading comprehension. If you could figure out what they were asking, you could figure out what the answer might be. Or should I say “is”. It should be “is”, but who can be sure?

Once again, I’ll be reviewing the test. I’m starting with the open-ended. We’ll spiral back to the multiple-choice in the coming days. Part 1 is shorter than the older test and Part 2 makes up for it. Big Time.

Note: I won’t even pretend to guess at how many points you’ll get for writing what, other than to say if it’s perfect, you’ll get full credit. But who can be sure what “perfect” means?

Algebra 1 (Common Core), Part 2

25. Draw the graph of y = SQRT(x) – 1.

If you put this in your calculator, you had to be sure to close the parentheses after the x. Otherwise, the “- 1” would be part of the expression beneath the radical.

The trick to remember here is that the domain is x > 0. You can’t use negative numbers. The y-intercept is (0, -1). You should have, at the least, plotted the points (0, -1), (1, 0), (4, 1) and (9,2) before drawing a curve through them. There should be an arrow on the right side of the curve because it continues to the right. There is no arrow on the left because the line starts with (0, -1).

26. The breakdown of a sample of a chemical compound is represented by the function p(t) = 300(0.5)^t, where p(t) represents the number of milligrams of the substance and t represents the time, in years. In the function p(t), explain what 0.5 and 300 represent.

I don’t know just how specific an answer they are looking for here.

• 0.5 is the rate of decay of the substance. It is the base in the exponential function.
• 300 is the initial amount of the substance. It is the y-intercept of the function and the co-efficient of the base.

27. Given 2x + ax – 7 > -12, determine the largest integer value of a when x = -1.

Confused? Don’t feel bad. Make sure you use x = -1 and not a = -1.
Plug is -1 for x and simplify the inequality before you do anything else.
2(-1) + a(-1) – 7 > -12
-2 – a – 7 > -12
-a – 9 > -12
-a > -3
a < 3
Remember to flip the inequality symbol when you divide by -1. If a < 3, then the largest integer value of a will be 2.

28. The vertex of the parabola represented by f(x) = x² - 4x + 3 has coordinates (2, -1). Find the coordinates of the vertex of the parabola defined by g(x) = f(x -2). Explain how you arrived at your answer.

The notation for this is confusing. And when my students see this, I know that they’ll want to solve something because of the equal sign, but it’s a definition, not an equation.

Others will look at this and think it’s a recursive function because we just reviewed those a few days ago. Sigh.

For every value of x, g(x) will have the same value that the f() function had when x was 2 less than it is now. So the entire parabola will shift two places to the right. That means that the coordinates of the vertex with be (4, -1).

There are more complicated ways of achieving the same result, which, for 2 miserable points, I hope that they aren’t looking for.

29. On the set of axes below, draw the graph of the equation y = (-3/4)x + 3. Is the point (3, 2) a solution to the equation? Explain your answer based on the graph drawn.

This seems to be the easiest, most straightforward question, so far. Okay, so it’s a graph. Do the graph. You have a calculator to help you, if you need it. The y-intercept is (0, 3). The slope is -3/4 – down 3, 4 to the right, make another point, down 3, 4 to the right, make another point, … when you’re at the end of the graph, go back up the other direction. LABEL THE LINE

(3, 2) is not a solution. How do you show this using the graph? Put the point on the graph at (3, 2). Label it (3, 2). Respond: (3, 2) is not on the line so it is not a solution to the equation.

Do NOT plug (3, 2) into the equation to check. That’s not what they asked for, so they won’t give you points for it.

30. The function f has a domain of {1, 3, 5, 7} and a range of {2, 4, 6}. Could f be represented by {(1, 2), (3, 4), (5, 6), (7, 2)}? Justify your answer.

This one led to a bit of a discussion in the Math Department. One side was quite sure of their superiority of knowledge, and the other side still wasn’t satisfied with the explanation. To put it plainly: I think I know what they are asking, but I’m not entirely sure. And I’ve learned in the past, you can’t always go for what you think they want – sometimes, you have to go with what they ask.

The argument boils down to semantics, really. Or maybe it’s syntax. I don’t know. I’m not an English teacher. However, I have a problem with the word “could”. Seriously.

Is this question asking if the relation they gave fits the domain and range of f? If so, the answer is YES. Or is this question asking if the relation is the ONLY POSSIBLE FUNCTION f? If that’s the case, it’s NO. We don’t know how f is defined. There is no mapping function. It could be that this relation represents f, but it might not be. Is that what it’s asking? Literally, yes, that is what it says, word for word. And yet I’m still not sure if that’s what they mean, and I’m not sure that my students will catch that meaning as well. Nuance? I don’t know. Maybe I’m overthinking it.

Another way for me to put it is like this: Could A represent B if A is only a subset of B?

Unfortunately, not all my students are native speakers, so I hope there isn’t a problem.

One thing I know: “Yes” or “No” without a good explanation will be worth nothing.

UPDATE: I spoke with a teacher who has been to training on how to grade these exams, and he had an answer key with sample responses and their point values. Basically, the answer is YES for reasons given above. When I explained my concerns about the wording, he thought I was splitting hairs. To be honest, I agree with that. That said, the Regents has been know to split hairs in the past.

31. Factor the expression x⁴ + 6x² - 7 completely.

They changed it up a bit. Usually, a “factor completely” question has a Greatest Common Factor (GCF) component to it.

x⁴ + 6x² - 7 factors into (x² + 7) (x² - 1). If you think that this seems a little simplistic for “factor completely” instead of “factor into two binomials”, you are not wrong.

That’s because using the Difference of Squares Rule (x² - 1) can be factored into (x – 1)(x + 1), making the final answer:

(x² + 7) (x – 1)(x + 1) Note that (x² + 7) has no real roots and cannot be factored further.

32. Robin collected data on the number of hours she watched television on Sunday through Thursday nights for a period of 3 weeks. The data are shown in the table below. … Using an appropriate scale on the number line below, construct a box plot for the 15 values.

Note: A picture of the table will be added later.

Put the 15 data values in order. The appropriate scale would be start at 1 and increment by .5.

The data are: 1, 1.5, 1.5, 2, 2, 2.5, 2.5, 3, 3, 3, 3.5, 4, 4, 4.5, 5.
Note: if you don’t have 15 values, you left something out. Also, your calculator will do all this for you -- but copy it ALL down on your paper anyway!

Your five-number summary is as follows: Min: 1, Q1: 2 (4th value), Median: 3 (8th value), Q3: 4 (12th value), Max: 5. Number the scale from 1 to 5, counting by .5. Plot these five points. Draw a box using Q1 and Q3, with a vertical line through the median. Draw whiskers from Q1 to min and Q3 to max.

Done.

And that will do it for Part 2, which is much longer than the Integrated Algebra Part 2.

These Are Not the Volumes You're Looking For

A friend, William Ricker, pinged me on a G+ post, which contained the following image, credited to Nathan W. Pyle:

I immediately shared it and replied, "You know that I'll have to calculate the value of d which makes that true." But, of course, I messed up and commented on my own post and not his. No matter. I went ahead and figured it out.

A sphere has a Volume of (4/3) pi * r ^ 3. A hemisphere would therefore be (2/3) * pi * r ^ 3.
Factoring out r ^ 2 from this leaves d ^ 2 = (2/3) * pi * r, meaning that d = SQRT( (2/3) * pi * r), which wasn't as bad as I might have expected.
The problem is that d = 2r, or r = (1/2)d. Which means that I had to start over because this is actually a stinkin' quadratic equation. It was not the expression I was looking for.

Anyway, long story shorter, given the limitations of this text medium, here's the answer: r = 0 (discard) or r = pi / 6, which makes d = pi / 3.

The check is left as an exercise to the reader.

T-Test

(Click on the cartoon to see the full image.)
(C)Copyright 2014, C. Burke.

This comment is null. At least, that's the hypothesis.

The most interesting thing about the t-test was that it was created by a chemist for the Guinness brewery in Dublin in 1908.

Also interesting are the other appearances of Belle and Mr. Whiskers which happened here, here, here, and here.

That means the last one was four years ago. Should I have left them in the past?

Or call this Old Character Re-appearance Week! and ride a nostalgia train?

Me and Ann B. Davis, Part II

In my exuberance to get a post up quickly and start a trend in the hopes of going viral with my woeful story of a lost picture of me and Ann B. Davis ("Alice" from The Brady Bunch, as if you didn't know), I forgot to upload the photo from my journal. It's attached at the end of this entry.

Yesterday's entry was the highest one in quite a while, especially for one not linked to a spam site (like my New Year's 2014 post, which doesn't deserve all the hits is gets). Hits came in from twitter and Facebook, but the big leader was definitely reddit. However, while there were nearly 1,000 hits yesterday, I had only one comment on reddit, and a couple of Favorites on twitter, along with a retweet or two.

Wherever Dean and Marcy are, they aren't connected to this trend. By the way, I will be really, really embarrassed if I go back to that journal and find that I flubbed one of there names. I'm doing this from memory, but I remember them. Not only that, at the time, I was writing about 5 or 6 pages in my journal per day on the ride home. Well, I wrote about this trip for at least a week afterward so I wouldn't leave out the details. I put the journal back, but I can dig it out again.

Anyway, here's the promised picture:

Hey, Internet: Where's My Picture of Me and Ann B. Davis?

The news of the death of actress Ann B. Davis was a bit of a shock. For those of a certain age, she was "Schultzy" on the The Bob Cummings Show. I'm NOT that age, having grown up instead in the golden age of first-run episodes of The Brady Bunch. Totally tangential, and not to make light of her passing because it does sadden me, but a burning question has been brought once again to the forefront of my noggin from the deep recesses where it had been locked away for many years:

Where is my picture of me and Ann B. Davis

There's a short story here, but I'll be quick about it:

Sometime back in 1993 -- Who am I kidding: it was May 11! I have the ticket stub right next to me! -- I won tickets to a Brady Bunch Reunion Cruise, sponsored by WPLJ-FM radio (95.5 FM, NYC) and The Spirit of New York. The announced guests on the cruise were Barry Williams, Susan Olsen and Ann B. Davis, aka "Greg", "Cindy" and "Alice", respectively, if you grew up on a different planet.

It was a little of an odd evening for me. My wife couldn't make the cruise, and as I would be traveling home late by subway, I didn't want to ask anyone I'd feel obligated to take home at that hour of the night. Especially if there might be by alcohol involved.

I briefly contemplated asking some young lady standing around waiting to get a glimpse of "Greg" by the gangway if she wanted to go dancing, but I found two problems with this. First, there was no guarantee I wouldn't be deserted the moment she got on board (or five moments after we would attempt polite dinner conversation). Second, they keep the gawkers far away from the ship out by South Street and well away from the pier. (Pier 11, if I remember correctly, down by Wall Street.)

Getting back to the story, I was on line alone, listening in to other people's conversations as we waiting to get our passes and board. I finally got mine, except it was someone else's -- they just checked my name of a list and gave me the next ticket. A photographer waited for each couple to start up the gangway to snap a souvenir picture. He asked the couple ahead of me, "Are you three together?" I nodded no, but the woman asked, "Would you like to be together?"

An interesting offer, but I declined, and I had a picture taken on my own. I don't remember if they'd waited for me, or we were just assigned seats for dinner, but the three of us shared a table for dinner. The "couple" turned out to be a mother and son. The guy's name was Dean, and he was around my age. She was Marcy; I don't know her age, but she looked like she'd been a young mother. Not that there's anything wrong with that. Dean and I each thought the other looked familiar. Having both grown up in Brooklyn with two million other people, it was possible. However, we only came up with one mutual acquaintance, and we couldn't think of a time we'd been together with him. (And it was only an acquaintance of mine, not a close friend or anything.)

Soooo ..... BRADY BUNCH! CRUISE!

The Spirit of New York is a nice dinner cruise ship, sailing from South Street along the East River, around the tip of Manhattan and into the Hudson River. The crew was friendly and professional. There were hors d'oeuvres to be eaten, drinks to be drunk, and views to be taken in. The ship sailed, and we went on deck to feel the sea breeze in our faces as we sang The Brady Bunch theme song with total strangers and a drag queen with a microphone as a second one -- a brunette wearing a green dress and carrying a videocamera the size of a small Buick on his (her?) shoulder -- filmed the merriment. They walked about the entire ship the entire evening, at one point being chased up from below decks. Honest mistake.

After dinner, the dancing started on one deck and the celebrities were signing autographs on the deck below. The line never got any shorter. Not until we got on it. A couple of girls got on line behind us and then that was in for the following hour of the cruise. I could've stayed on the dance floor and possibly Electric Slided (electically slid?) into newswoman Naomi DiClemente and had essentially the same place in line an hour later. But conversation with strangers is something New Yorkers do best. That is, when we're not totally ignoring total strangers, which we're pretty good at, too.

So I didn't have a camera on me (or if I did, it wasn't working), but I did have a journal on me. Back then, I had a mini-notebook on me all the time, and I tried to write in it every day. Usually, I was writing while riding home on the subway. I'm sure my poor penmanship suffered, but it's when I had the most solitude to write -- on crowded, evening rush hour subway trains. I worked far enough uptown that I usually had a seat, except when I had to give up a seat for a mother-to-be or the elderly. (Watch out for the Wednesday matinee crowd!)

When we finally got to meet them, Ann B. Davis, Alice, was at the first table. I asked her to sign my journal, opening it to a fresh page. She was impressed that I had a journal. At that point, Marcy asked if I wanted a picture with Ann. C'mon now: Who could say "no" to a picture with Ann B. Davis? Ann was obviously used to this, and had probably posed for dozens of pictures already that night. There's a table between us, so I leaned back and Ann leaned forward. Apparently, we weren't close enough because Ann pulled me back to narrow the gap as Marcy took the picture. (Or maybe she told Dean to take the picture? Could be.)

We met Barry and Susan. Marcy points out all the blank pages in the journal to "Cindy" and says that she needs to write her life story. Susan declines and adds a note in my journal that she's already written her life story. (Thanks, Marcy -- I had a thing for Cindy when I was, like 12 -- go and ruin it for me. Well, there was still Naomi ...)

The rest of the evening was fun ... and short because we really were on line a long time. When we pulled back into port, Marcy insisted that they give me a lift home, from lower Manhattan all the way out to Bensonhurst, my first apartment after I got married. I gave them my address, and they promised to send me a copy of the picture. Well, it never came. It's been twenty years, and still nothing. Granted, I moved out of that apartment within three years, so maybe it's sitting in a Dead letter pile at Bath Beach Station.

However, in the intervening years, the Internet has evolved, and who knows, maybe this will go viral and someone will see it. Maybe that someone will know a Dean who has a mother named Marcy. Maybe Marcy still has that picture of some oddball they met on a cruise, sitting someone in a shoebox with other pictures in the bottom of a closet behind from old mixtapes. Maybe the Internet can finally answer the question for me:

Where is my picture of me and Ann B. Davis?