Friday, September 30, 2016

W. O. D. B.

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(C)Copyright 2016, C. Burke.

(*) Which One Doesn't Belong?

Each of the four panels in the odd one out, but for different reasons. Unlike when Bob from Sesame Street sang that three of these things are kind of the same, there isn't just one answer. These puzzles are quite popular now, and there are webpages and books filled with them.

Come back often for more funny math and geeky comics.

Monday, September 26, 2016

August 2016 Common Core Algebra II Regents Part 2, continued

I don't usually post the answers to the Algebra II exam, but I've received enough requests that I've decided to post some of the open-ended questions.
Some questions were posted earlier here.

August 2016, Algebra II Part 2 (continued)

28.Using the identity sin2 O + cos2O = 1, find the value of tan O, to the nearest hundredth, if cos O is –0.7 and O is in Quadrant II.

Copy the identity. Substitute the value for cosine and then square it. Solve for sine. Remember, because it is in Quadrant II where the y values are positive, we need a positive answer for sine. Once we know sine and cosine, we can divide to find tangent.
sin2 O + cos2O = 1
sin2 O + (-0.7)2 = 1
sin2 O + .49 = 1
sin2 O = 0.51
sinO = SQRT(0.51) = .7141...
tanO = sinO / cosO = 0.714 / -0.7 = -1.02

You could also have gotten this answer by converting the identity, by dividing all the terms by cos20, which would then give you an equation with tan 2O in it. Again, remember that because it is Quadrant II where x is negative and y is positive, that the tangent must be a negative value.

29. Elizabeth waited for 6 minutes at the drive thru at her favorite fast-food restaurant the last time she visited. She was upset about having to wait that long and notified the manager. The manager assured her that her experience was very unusual and that it would not happen again.

A study of customers commissioned by this restaurant found an approximately normal distribution of results. The mean wait time was 226 seconds and the standard deviation was 38 seconds. Given these data, and using a 95% level of confidence, was Elizabeth’s wait time unusual? Justify your answer.

A 95% level of confidence means that the value should be within 2 standard deviations from the mean.
The mean is 226. Plus 1 deviation = 226 + 38 = 264. Plus another deviation 264 + 38 = 302 seconds.
Six minutes is 360 seconds, which would be unusual because it is more than two standard deviations away from the mean.

30. The x-value of which function’s x-intercept is larger, f or h? Justify your answer.

f(x) = log(x - 4)

The x-intercept of h(x) = 2, as shown in the table.
To find the x-intercept of f(x), set f(x) = 0.
log(x - 4) = 0
100 = x - 4
1 = x - 4
5 = x

f(x) has the higher x-intercept.

to be continued.

Any questions?

Comments, corrections welcome.

(x, why?) Mini: Plank

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(C)Copyright 2016, C. Burke.

Keeping in shape is a Plus for a Constant

Yeah, that's a stylized letter C, "stylized" meaning that I didn't like any of the capital Cs in any of the fonts that I had available, so I drew my own.

Come back often for more funny math and geeky comics.

Saturday, September 24, 2016

August 2016 Common Core Algebra II Regents Part 2

I don't usually post the answers to the Algebra II exam, but I've received enough requests that I've decided to post some of the open-ended questions.

August 2016, Algebra II Part 2

25.The volume of air in a person’s lungs, as the person breathes in and out, can be modeled by a sine graph. A scientist is studying the differences in this volume for people at rest compared to people told to take a deep breath. When examining the graphs, should the scientist focus on the amplitude, period, or midline? Explain your choice.

The scientist should focus on the amplitude because the height of the graph corresponds to the Volume of air in the lungs.

26. Explain how (31/5)2 can be written as the equivalent radical expression "fifth root of 9".

Look at the illustration below:

The question says "explain" so you MUST include an explanation for full credit. You can't just write an equation.
The exponents can be be switched so you have 3 square to the one-fifth power. Three squared is 9. The one-fifth power is the same as taking the fifth root of the number. So (31/5)2 is the same as the fifth root of 9.

27. Simplify xi(i - 7i)2, where i is the imaginary unit.

Subtract i - 7i to get -6i. Square (-6i) to get 36i2. Replace i2 with -1, so you have xi(-36). Rewrite as -36xi.

Illustrated below.

to be continued.

Any questions?

Comments, corrections welcome.

Thursday, September 22, 2016

August 2016 Common Core Geometry Regents, Parts 3 and 4

What follows is a portion of the Common Core Geometry exam. Other parts will be posted on other days. Illustrations will be added at a later time when they become available.

Part 1 is was posted here.
Part 2 is was posted here.

August 2016 Geometry Regents, Part III

32. Using a compass and straightedge, construct and label triangle A'B'C', the image of triangle ABC after a dilation with a scale factor of 2 and centered at B. [Leave all construction marks.]
Describe the relationship between the lengths of AC and A'C'.

Constructions are not easy to show on this blog, but look at the illustration.
The scale factor is two, so you need to double the lengths of the sides. Start at B, open the compass to point C and make an arc. Now move to C without changing the compass and make another arc. Using the straightedge, draw the line from B through C to the new arc. Label that point C'.
Repeat the process for A to create A'. Then use the straightedge to draw A'C'. Finally, at point B, label it B' as well. That point does not move.
The relationship between AC and A'C' is that A'C' = 2AC or AC = 1/2 A'C'. The image is twice the length of the original line.

33. The grid below shows triangle ABC and triangle DEF.
Let trianlge A'B'C' be the image of triangle ABC after a rotation about point A. Determine and state location B' if the location of C' is (8, -3). Explain your answer.

Is triangle DEF congruent to triangle A'B'C'? Explain your answer.

If you plot C'(8, -3), you will see that the triangle has rotated 90 degrees counterclockwise. In a rotation, distance is preserved. C was six units away from A, C' is six units away from A. Point B is one unit above C and four units to the right. After a 90 degree rotation, B' will be one unit to the left of C' and four units above it. B' is at (7, 1)

Triangle DEF is congruent to triangle A'B'C'. This can be shown by finding the lengths of the sides or by doing a series of rigid motions. If A'B'C' is reflected over the y-axis, the image will have points A"(-2, -3), B"(-7, 1), C"(-8, -3). If A"B"C" is translated by (0, -2), then the image will coincide will DEF.

Likewise, you could reflect A'B'C' over the line y = -1 to have it coincide in a single step.

34. As modeled below, a movie is projected onto a large outdoor screen. The bottom of the 60-foot-tall screen is 12 feet off the ground. The projector sits on the ground at a horizontal distance of 75 feet from the screen.

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

You have two right angles. The first has legs 12 and 75, and the second has legs of 72 and 75. There legs are Opposite and Adjacent to angles with the projector. You can use the Tangent function to find the angles and then find the difference of the two to get theta.

Tan x = 12 / 75
x = tan-1(12/75) = 9.09
Tan x = 72 / 75
x = tan-1(12/75) = 43.83
43.83 - 9.09 = 34.74 = 34.7 degrees.

August 2016 Geometry Regents, Part IV

35. Given: Circle O, chords AB and CD intersect at E

Theorem: If two chords intersect in a circle, the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other chord.

Prove this theorem by proving (AE) • (EB) = (CE) • (ED)

The procedure to show that theis theorem is true requires you to create two triangles, by drawing extra chords, and then showing that the two triangles are similar in shape. This will make the sides proportional, and by cross-multiplying, you will get the equation in the theorem.

1. Circle O, chords AB and CDGiven
2. AD and BC are drawnAuxiliary lines can be drawn
3. <1 = <2Vertical angles
4. <A = <CInscribed angles that intercept the same arc are congruent
5. Triangle ADE ~ triangle CBEAA
6. AE/CE = ED/EBCorresponding sides of similar triangles are proportional
7. (AE)•(EB) = (CE)•(ED)if a proportion the products of the means equals the product of the extremes

36. A snow cone consists of a paper cone completely filled with shaved ice and topped with a hemisphere of shaved ice, as shown in the diagram below. The inside diameter of both the cone and the hemisphere is 8.3 centimeters. The height of the cone is 10.2 centimeters.

The desired density of the shaved ice is 0.697 g/cm3, and the cost, per kilogram, of ice is $3.83.
Determine and state the cost of the ice needed to make 50 snow cones.

Density is mass divided by the Volume. We know the density. We can calculate the volume. Then we can find the mass. Once we have the mass of 1 snow cones, we can multiply to find the mass of 50 snow cones and then calculate the cost of 50 snow cones. Note: Use the PI key on your calculator. Do NOT use 3.14 -- it doesn't have enough decimal places. Also, do NOT round in the middle of this problem.

Volume of a cone = (1/3) (pi)(r)2h
V = (1/3)(3.141592...)(4.15)2(10.2)=183.96067022

Volume of a hemisphere = (1/2)(4/3)(pi)(r)3
Volume = (1/2)(4/3)(3.141592...)(4.15)3 = 149.693486552
Total Volume = 183.96067022 + 149.693486552 = 333.654156772

d = m / V, m = Vd = (333.654156772)(0.697) = 232.55694727g in 1 snow cone.
Multiply by 50 to 11627.8473635g in 50 snow cones.
Convert to kilograms by dividing by 1000: 11,6278473635kg of ice needed.
Multiply by $3.83: 44.5346554022 = $44.53 is the cost

Yes, this is an insane amount of work for one question, but none is it involves anything really crazy. You just have to stick with it to the end. Remember, in questions like this one, one little error will throw off the rest of the numbers. HOWEVER, if you are consistent all the way to the end, and give an answer that matches the work you did, you may only lose 1 point for a "little" error early on.


How did you do? Any questions? (I appreciate pointing out any "typos" in my problems. Thank you.)

Monday, September 19, 2016

Talk-Like-A-Pirate Day 2016

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(C)Copyright 2016, C. Burke.

Yarr! Have a Happy Talk-Like-a-PI-rate Day!
When is Talk Like a Pair-rot Day?

Come back often for more funny math and geeky comics.

Sunday, September 18, 2016


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(C)Copyright 2016, C. Burke.

Want to crack a few puns, you have to crack a few eggs.

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Friday, September 16, 2016


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(C)Copyright 2016, C. Burke.

And parallel cray-cray is something serious!

Tomorrow's Teachers of Tomorrow are an ongoing feature.

Come back often for more funny math and geeky comics.

August 2016 Common Core Geometry Regents, Part 2

What follows is a portion of the Common Core Geometry exam. Other parts will be posted on other days. Illustrations will be added at a later time when they become available.

Part 1 is was posted here.

August 2016 Geometry Regents, Part II

25. 5 Lines AE and BD are tangent to circles O and P at A, E, B, and D, as shown in the diagram below. If AC:CE = 5:3, and BD = 56, determine and state the length of CD

Because they are tangent lines, AC = BC and CD = CE. So if AC:CE = 5:3, then BC:CD = 5:3 also.
So 5x + 3x = 56
8x = 56
x = 7
CD = 3x = 3(7) = 21.

26. 6 In the diagram below, triangle ABC has coordinates A(1,1), B(4,1), and C(4,5). Graph and label triangle A" B" C" , the image of ABC after the translation five units to the right and two units up followed by the reflection over the line y 0.

First, remember that the line y = 0 is the x-axis, not the y-axis.
Move each point 5 units to the right and 2 units up. Point A (1, 1) will move to A'(6, 3), etc.
Next, flip the points over the x-axis, so point A'(6, 3) becomes A"(6, -3).
You're final image will look like this.

27. A regular hexagon is rotated in a counterclockwise direction about its center. Determine and state the minimum number of degrees in the rotation such that the hexagon will coincide with itself.

A regular hexagon will coincide with itself every 1/6 of a revolution. (1/6)(360) = 60 degrees.

28. In the diagram of ABC shown below, use a compass and straightedge to construct the median to AB. [Leave all construction marks.]

Construction marks are difficult for me to show using the tools I'm currently using, so I will describe the process.
The median is a line drawn from point C to the midpoint of AB, so you need to find the midpoint of AB. The way to do this is to construct the perpendicular bisector of AB, label the midpoint and then draw a line from C to this new point.

The red lines are created by opening the compass so that it is wider than half the distance between A and B. If you make the arcs to small, then they won't intersect. You will have to make the arc bigger and try again. Remember that both arcs MUST be the same size when you center on A and when you center it on B.

Draw the perpendicular bisector, which is the line through the two points where the arcs meet. This line will cross AB at its midpoint.

Draw the median by using the straightedge and marking the line between C and the midpoint.

29. Triangle MNP is the image of triangle JKL after a 120° counterclockwise rotation about point Q. If the measure of angle L is 47° and the measure of angle N is 57°, determine the measure of angle M. Explain how you arrived at your answer.

When the triangles rotate, M is the image of J, N is the image of K and P is the image of L. Size and shape are preserved in a rotation so <M = <J, <N = <K annd <P = <L.
&LtM + &LtN + <P = 180
<M + 47 + 57 = 180
<M + 104 = 180
<M = 76

30. A circle has a center at (1,–2) and radius of 4. Does the point (3.4,1.2) lie on the circle? Justify your answer.

If (3.4, 1.2) is on the circle then the following equation must be true.
(3.4 - 1)2 + (1.2 + 2)2 = 42 ?
(2.4)2 + (3.2)2 = 16 ?
5.76 + 10.24 = 16 ?
16 = 16. Check
(3.4, 1.2) lies on the circle

31. In the diagram below, a window of a house is 15 feet above the ground. A ladder is placed against the house with its base at an angle of 75° with the ground. Determine and state the length of the ladder to the nearest tenth of a foot.

You have a right triangle. You have an angle. You have the opposite side. You want to know the hypotenuse. O-H means you need to use the sine ratio. (Make sure your calculator is in DEGREE mode.
sin 75 = 15 / x or as a proportion: sin 75 / 1 = 15 / x.
cross-multiply: (sin 75) x = 15
x = 15 / sin 75 = 15.5291427062
The ladder is 15.5 feet to the nearest tenth.


How did you do? Any questions? (I appreciate pointing out any "typos" in my problems. Thank you.)

Monday, September 12, 2016

What I Did on Summer Vacation

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(C)Copyright 2016, C. Burke.

And that's how I work in time-sensitive strips a month after I should have.

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Friday, September 09, 2016

Carat vs. Karat

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(C)Copyright 2016, C. Burke.

And this: ^^^^^^^^^^^^^^^^^^^^^^^^ is 24 carets!

For newer readers: Judy is a colleague in the English Dept and one of Michele's best friends. She's appears infrequently, in strips such as this one.

Come back often for more funny math and geeky comics.

August 2016 Common Core Geometry Regents, Part 1

What follows is a portion of the Common Core Geometry exam. Other parts will be posted on other days. Illustrations will be added at a later time when they become available.

August 2016 Geometry Regents, Part I

1. In the diagram below, lines l, m, n and p intersect line r.

Which statement is true?

(2) l || p. If two exterior angles on the same side of a transversal are supplementary, then the lines are parallel. The sum of 112 and 68 is 180 degrees.

2. Which transformation would not always proudce an image that would be congruent to the original figure?

(2) dilation. A dilation will change the size of an image, so it will not be congruent. The others only change the orientation.

3. If an equilateral triangle is continuously rotated around ones of its medians, which 3-dimensional object is generated?

(1) cone. One point will stay in play and the other two will form a circle.

4. In the diagram below, m<BDC = 100, m<A = 50, and m<DBC = 30.

Which statement is true?

(2) Triangle ABC is isosceles. BDC = 100, DBC = 30, so <C = 50, because 100 + 30 + 50 = 180. Angle A is also 50. Beacuse angle A = angle C, it is an isosceles triangle.

5. Which point shown in the graph below is the image of point P after a counterclockwise rotation of 90o about the angle?

(1) A. Counterclockwise from Quadrant IV brings you to Quadrant I. Point P is close to the x-axis, so the image will be close to the y-axis, which is where A is. Point B looks like a reflection across the x-axis.

6. In triangle ABC, where <C is a right angle, cos A = SQRT(21) / 5. What is sin B?

(1) SQRT(21) / 5. Because Sin B = Cos A

7. Quadrilateral ABCD with diagonals AC and BD is shown in the diagram below.

Which information is not enough to prove ABCD is a parallelogram?

(3) AB = CD and BC || AD. Either two pairs of parallel sides are needed, or two pair of congruent sides, or one pair that is both parallel and congruent would prove it. However, one pair of parallel sides and a different pair congruent is not sufficient. That information could be true is ABCD was, for example, an isosceles trapezoid.

8. An equilateral triangle has sides of length 20. To the nearest tenth, what is the height of the equilateral triangle?

(3) 17.3. You could guess this one. Draw a triangle and an altitude. Choice (1) 10 is the length of the base of the right triangle you just created. Choice (4) 23.1 is longer than the hypotenuse. It's either (2) or (3), but you know that it will be much bigger than the base. In fact, it will be the base TIMES SQRT (3), which is approximately 1.732. So 10(1.732) = 17.32.

Using the Pythagorean Theorem: x2 + 102 = 202
so x2 + 100 = 400
so x2 = 300
x = 17.3205..., which is 17.3 to the nearest tenth.

9. Given: Triangle AEC, triangle DEF, and FE perpendicular to CE

What is a correct sequence of similarity transformations that shows triangle AEC ~ triangle DEF?

(4) a counterclockwise rotation of 90 degrees about point E followed by a dilation with a scale factor 2 centered at point E. You can see that the figure rotated only 90 degrees, not 180 -- it would be upside down then -- so eliminate choices (1) and (3). Next, a translation would move DEF away from ACE, which didn't occur. However, the size did change, which indicates a dilation.

10. In the diagram of right triangle ABC, CD intersects hypotenuse AB at D.

If AD = 4 and DB = 6, which length of AC makes CD perpendicular to AB?

(1) 2*SQRT(6). If CD is perpendicular to AB, then CD is an altitude. The Right Triangle Altitude Theorem says that the product of AD times DB must equal the length of CD2. Since 4 * 6 = 24, CD must be SQRT(24), which reduces to 2*SQRT(6) in simplest form.

11. Segment CD is the perpendicular bisector of AB at E. Which pair of segments does not have to be congruent?

(4) DE, CE. AB is not said to bisect CD, so you cannot assume that DE and CE are congruent. AE and BE are congruent by definition. The other points would form congruent right triangles, with the corresponding hypotenuses being congruent.

12. In triangle CHR, O is on HR, and D is on CR so that <H = <RDO

If RD = 4, RO = 6 and OH = 4, what is the length of CD?

(3) 11. It may not seem obvious at first that the two triangles are similar because DO is not parallel to CH. However, you are given one pair of angles congruent, and both triangles contain angle R. Therefore, they are similar.

The proper proportion to set up is: RD / HR = RO / CR. And HR = 4 + 6 = 10.
So 4 / 6 = 10 / CR
4*CR = 60
CR = 15
CD = CR - DR = 15 - 4 = 11.
Don't forget to subtract the length of RD from the side of the triangle because they are looking for CD.

13. The cross section of a regular pyramid contains the altitude of the pyramid. The shape of this cross section is a (3) triangle. If the cross section contains the altitude, then imagine a vertical sheet of paper slicing through the top of a pyramid. No matter how you line it up, the sides of the pyramid will form a triangle on the paper.

14. The diagonals of rhombus TEAM intersect at P(2, 10). If the equation of the line that contains diagonal TA is y = -x + 3, what is the equation of a line that contains diagonal EM?

(1) y = x - 1. The lines must be perpendicular, and that means the slope of EM must be 1 because the slope of TA is -1. This eliminates choices (3) and (4). Given P(2, 1), substitute 2 for x in the first equation and you get y = 1, which is correct.

15. The coordinates of vertices A and B of triangle ABC are A(3, 4) and B(3, 12). If the area of triangle ABC is 24 square units, what could be the coordinates of point C?

(3) (-3, 8) Area of a triangle is 1/2 b h. The base has a length of 12 - 4 = 8.
(1/2)(8)h = 24, so 4h = 24 and h = 6.
The third coordinate has to have an x value that is six more or six less than 3. 3 - 6 = -3, so choice (3) works.

16. What are the coordinates of the center and the length of the radius of the circle represented by the equation x2 + y2 - 4x + 8y + 11 = 0?

(1) center (2, -4) and the radius 3. They are making these more difficult. You need to rewrite this in standard form (x - h)2 + (y - k)2 = r2.
To do this, we need to complete the square ... twice.

x2 + y2 - 4x + 8y + 11 = 0
regroup: x2 -4x + y2 + 8y + 11 = 0
complete the squares: x2 -4x + 4 + y2 + 8y + 16 + 11 = 0 + 4 + 16
simplify: (x - 2)2 + (y + 4)2 + 11 = 20
simplify: (x - 2)2 + (y + 4)2 = 9
So the center is (2, -4) and the radius is 3.

If you didn't remember how to complete the square, you could have started by writing an equation for choice (1), expanding it, and seeing if you got the same equation. If it was wrong, it would have been a matter of signs, so if you started with an incorrect choice, you could have deduced the correct one.

17. The density of the American white oak tree is 752 kilograms per cubic meter. If the trunk of an American white oak tree has a circumference of 4.5 meters and the height of the trunk is 8 meters, what is the approximate number of kilograms of the trunk?

(2) 9694. d = m / V or d = m / (pi * r2 * h). C = 2*pi*r so r = C / (2 * pi) = 0.7162
752 = m / (pi*0.716122*8)
m = 752 * (pi*0.716122*8) = 9692.355
which is approximately 9694, with errors for rounding in the middle of the problem.

Yes, this was an unnecessarily complicated problem for a multiple-choice question. They could have give a diameter instead of the circumference, or even just given the radius itself.

(1) 150(0.85)m. The hot chocolate is getting cooler, and the temperature is getting lower. Therefore, the base must be less than one (exponential decay), so choices (2) and (4) are out. If m = 0, then 150(0.85)0 = 150(1) = 150, which fits the table. The correct choice is (1). In choice (3), m - 1 would give an exponent of -1, which would actually increase the temperature at m = 0.

18. Point P is on the directed line segment from point X(-6, -2) to point Y(6, 7) and divides the segment in the ratio 1:5. What are the coordinates of point P?

(4) (-4, -1/2). The ratio 1:5 represents 1x + 5x which is 6x. If the calculate the distance between the two x-coordinates, the x-coordinate of P is 1/6th to the right of the x coordinate of point X. If the calculate the distance between the two y-coordinates, the y-coordinate of P is 1/6th to the above of the y coordinate of point X (not point Y).

From -6 to 6 is 12 spaces, and 1/6(12) = 2; -6 + 2 = -4. The answer is choice (4). You can verify with y.
From -2 to 7 is 9 spaces, and 1/6(9) = 1.5; -2 + 1.5 = -1/2, which is choice (4).

19. Due to a typographical error in the test booklet, there is no correct answer for question 19. I'm repeating it here just in case a teacher accidentally assigns it and a student puts it into a search engine.

In circle O, diameter AB, chord BC and radius OC are drawn, and the measure of arc BC is 108o. Some students wrote these formulas to find the area of sector COB (omitted)
Which students wrote correct formulas?

No answer.

20. Tennis balls are sold in cylindrical cans with the balls stacked one on top of the other. A tennis ball has a diameter of 6.7 cm. To the nearest cubic centimeter, what is the minimum volume of the can that holds a stack of 4 tennis balls?

(4) 945. Volume = (pi)r2h, r = 6.7/2 = 3.35, h = 4(6.7) = 26.8
V = (pi)(3.35)^2(26.8) = 944.87... = 945

21. Line segment A'B', whose endpoints are (4, -2) and (16, 14) is the image of AB after a dilation of 1/2 centered at the origin. What is the length of AB?

(4) 40. You could either find the length and double it (the image is 1/2 the original) or you can double the x- and y-values and then find the length. Your choice. The distance between the x values is 16 - 4 = 12. The distance between the y-values is 14 - (-2) = 16. Using either the distance formula, or Pythagorean theorem, you will find that the distance between the two is 20. (You know that 12-16-20 is just a 3-4-5 right triangle times 4, right?)
Since the image is half the length of the original, the length of AB is 40.

22. Given: Triangle ABE and triangle CBD shown in the diagram below with DB = BE

Which statement is needed to prove triangle ABE = ACBD using only SAS = SAS?

(3) AD = CE. You are given DB = BE. Angle B is congruent to itself by the Reflexive Property. You need to know that AB = BC, but since you already know that DB = BE, then if AD = CE, you can conclude that AB = BC by the Addition Postulate. That will give you SAS = SAS.

23. In the diagram below, BC is the diameter of circle A.

Point D, which is unique from points D and C, is plotted on circle A. Which statement must always be true?

(1) Triangle BCD is a right triangle. Angle D will be an inscribed angle and the arc it intercepts is a semi-circle, which is 180 degrees. Therefore, angle D must be half of that, or 90 degrees. So BCD must be a right triangle, always.

24. In the diagram below, ABCD is a parallelogram, AB is extended through B to E, and CE is drawn.

If CE = BE and m<D = 112o, what is m<E?

(1) 44o. If angle D = 112, then angle A is supplementary, or 180 - 112 = 68. Angle CBE is congruent to A, so it is also 68 degrees. Because BE = CE, triangle BCE is an isosceles triangle and the base angles are congruent, so angle BCE is also 68 degrees. Angle E = 180 - 68 - 68 = 112 - 68 = 44 degrees.


How did you do? Any questions? (I appreciate pointing out any "typos" in my problems. Thank you.)

More Geometry problems.

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Tuesday, September 06, 2016

August 2016 Common Core Algebra 1 Regents, Part 1

What follows is a portion of the Common Core Integrate Algebra exam. Other parts will be posted on other days. Illustrations will be added at a later time when they become available.

Part II is available here.
Parts III & IV are available here.

August 2016 Algebra Regents, Part I

1. The graph below shows the distance in miles, m, hiked from a camp in h hours.
Which hourly interval had the greatest rate of change?

(1) hour 0 to hour 1. The line is the steepest between 0 and 1 hour, which means is has the greatest slope, or the greatest rate of change. In the first hour, 2 miles were hiked, then only 1.5 miles, then 1 mile, then 0.5 miles. In the last hour, the hikers did not move away from camp at all.

2. The solution of an equation with two variables, x and y, is

(3) the set of all ordered pairs, (x, y), that make the equation true. That's the definition of a solution to an equation. It is not about solving for the zeroes, as in choices (1) and (2).

3. Which statistic can not be determined from a box plot representing the scores on a math test in Mrs. DeRidder's algebra class?

(4) the score that occurs most frequently. You cannot tell the mode from a box plot (also known as a box and whisker plot). You can only see the quartiles, including the minimum, median and maximum.

4. Which chart could represent f(x) = -2x + 6?

(4). You don't need to calculate anything. You can figure this one out logically. When x = 0, f(x) must equal 6. Only choices (1) and (4) have (0, 6) for (x, f(x)). The rate of change is -2, but choice (1) has a positive rate of change, so it is incorrect. Choice (4) is the only one left.

If you put y = -2x + 6 into your graphing calculator and check the table, you will see that (4) has the correct ordered pairs.

5. If f(n) = (n - 1)2 + 3n, which statement is true?

(2) f(-2) = 3. You can skip (1) because in this function a positive number can't give a negative result. (-2 - 1)2 + 3(-2) = (-3)2 - 6 = 9 - 6 = 3.

6. The table below shows 6 students' overall averages and their averages in their math class.
If a linear model is applied to these data, which statement best describes the correlation coefficient?

(2) It is close to 1. You don't need to put this in your calculator. (You could if you truly weren't sure. When the top row is higher, the bottom is higher. When it is lower, the bottom is lower. There is a positive correlation, so choices (1) and (3) are out. Furthermore, the pairs are very close together, suggesting a strong correlation, which means that it would be closer to 1 than it would be to 0.5.

7. What is the solution to 2h + 8 > 3h - 6?

(1) h < 14.
Subtract 2h from both sides, you get 8 > h - 6
Add 6 to each side, you get 14 > h
Turn it around, h < 14

8. Which expression is equivalent to 36x2 - 100?

(2) 4(3x + 5)(3x - 5). You can tell by looking at it that it will be a Difference of Squares problem because it has no middle term. That means that the two factors will be conjugates; that is, the binomials will be the same except that one will have addition and one will have subtraction. That difference is what causes the middle terms to cancel out. So (1) and (3) are immediately eliminated.

3x times 3x is 9x2, which when multiplied by 4 has a product of 36x2, as given in the question. However, 9x times 9x is 81x2, which is already too big. Choice (4) is eliminated.

9. Patricia is trying to compare the average rainfall of New York to that of Arizona. A comparison between these two states for the months of July through September would best be measured in

(3) inches per month. This is a strange question. It assumes you know something about rainfall in Arizona. (Because it is a New York state exam, it's fair to assume that you know something about the rainfall in New York -- you live here.)

There isn't a lot of rainfall in Arizona, even when there may be in New York. Measuring the rainfall by the hour is just silly -- it doesn't rain for that many hours in an entire summer, so the ratio would be a tiny, tiny decimal. Rain, like snow, is usually measured in inches, not feet. It is not uncommon to hear forecasts of 10-14 inches of snow, as opposed to 1-2 feet of snow. Even when talking about the entire accumulation for the season, you are more likely to hear, say, 24 inches, rather than 2 feet.

So for a bunch of reasons, a few actually having to do with math, the best answer is inches per month.

10. Which function defines the sequence -6, -10, -14, -18, ..., where f(6) = -26?

(1) f(x) = -4x - 2. The sequence is going down by four. Only one choice, (1), has -4x in it.
Check: -4(6) - 2 = -24 - 2 = -26. Yes.

11. Which function has the greatest y-intercept?

(4) (the graph). The y-intercept of the graph is 5. In choice (1), 3(0) = 0, which is less than 5. In choice (2), if 2(0) + 3y = 12, then y = 4, which is less than 5. In choice (3), if there is a point on a line at (1, -4) and the slope is 2, then subtract 2 from the y value and you find that the line crosses the y-axis at (0, -6), which is less than 5.

12. What is the product of 2x + 3 and 4x2 - 5x + 6?

(3) 8x3 + 2x2 - 3x + 18. Notice that the first and last terms are the same for all four choices, so we don't have to worry about those.

(2x)(-5x) = -10x2 and (3)(4x2) = 12x2, -10 + 12 = -2. so we can eliminate choices (1) and (2).

(2x)(6) = 12x and (3)(-5x) = -15x, 12 - 15 = -3. So the choice is (3).

13. The height of a rocket, at selected times, is shown in the table below.

Based on these data, which statement is not a valid conclusion?

(3) The rocket was in the air approximately 6 seconds before hitting the ground. It was still above the ground after 7 seconds. The other choices give information that can be found in the table. It was at 180 feet at 0 seconds (launch); it hit is maximum at 3 seconds; it was above the 300 feet for 2 seconds from about t = 2 to t = 4.

14. A parking garage charges a base rate of $3.50 for up to 2 hours, and an hourly rate for each additional hour. The sign below gives the prices for up to 5 hours of parking.
Which linear equation can be used to find x, the additional hourly parking rate?

(3) 2x + 3.50 = 14.50. Four hours cost $14.50, which is the first 2 hours plus 2 additional hours. The first 2 hours = $3.50 and the additional 2 hours are 2x.

Note that this is just one possible equation. A simpler equation would've been x + $3.50 = 9.00, but that was not an option.

15. Which function has a constant rate of change equal to -3?

(4) 2y = -6x + 10. Put the equation into slope-intercept form by dividing both sides by 2, and you get y = -3x + 5, which has a slope (rate of change) of -3.

In choice (1), the rate of change is increasing. In choice (2), the slope is -2. Choice three is not a linear function, so it does not have a constant rate of change, even though the change from x = 1 to x = 2 is, in fact, -3, but this does not continue.

16. Kendal bought x boxes of cookies to bring to a party. Each box contains 12 cookies. She decides to keep two boxes for herself. She brings 60 cookies to the party. Which equation can be used to find the number of boxes, x, Kendal bought?

(3) 12x - 24 = 60. If x is the number of boxes, then 12x is the number of cookies altogether. If she kept 2 boxes, then she took away 24 cookies leaving 60.

17. The table below shows the temperature, T(m), of a cup of hot chocolate that is allowed to chill over several minutes, m.
Which expression best fits the data for T(m)?

(1) 150(0.85)m. The hot chocolate is getting cooler, and the temperature is getting lower. Therefore, the base must be less than one (exponential decay), so choices (2) and (4) are out. If m = 0, then 150(0.85)0 = 150(1) = 150, which fits the table. The correct choice is (1). In choice (3), m - 1 would give an exponent of -1, which would actually increase the temperature at m = 0.

18. As x increases beyond 25, which function will have the largest value?

(1) f(x) = 1.5x. Exponential functions with bases greater than 1 will climb higher at a faster rate than quadratic or cubic functions. You can put all four of these functions in your calculator and see the results for 25 and greater.

19. What are the solutions to the equation 3x2 + 10x = 8?

(1) 2/3 and -4. You can enter y = 3x2 + 10x - 8 into your calculator and search for the zeroes of the function. Or you can check the numbers 2, -2, 4, and -4, in whatever order you like, to see which one makes the equation true. Because no number repeats, you don't need to worry about substituting the fractions into the equation.

20. An online company lets you download songs for $0.99 each after you have paid a $5 membership fee. Which domain would be most appropriate to calculate the cost to download songs?

(2) whole numbers greater than or equal to one. Rational numbers include fractions, and you can't download half a song (realistically, and financially, speaking). Integers less than zero are negative, which is also impossible. Whole numbers less than or equal to one means 1 and 0, only, which means you could only download at most 1 song ever.

21. The function f(x) = 3x2 + 12x + 11 can be written in vertex form as

(3) f(x) = 3(x + 2)2 - 1. Once again, you can put all the functions in your calculator and see which two are the same. You can save a little work if you realize that you need to divide 12 by 3 before you halve it -- so the parentheses have to say "(x + 2)" not "(x + 6)", and you can eliminate choices (1) and (2).

There is a quicker way for this particular problem if you see the relationship in the numbers -- but that is something you develop from doing a bunch of these kinds of problems, so I'll do the longer way first.

f(x) = 3x2 + 12x + 11, subtract 11 from both sides
f(x) - 11 = 3x2 + 12x, factor the 3 on the right side
f(x) - 11 = 3(x2 + 4x), half of 4 is 2, (2)2 = 4, (3)(4) = 12
.... so add 12 to both sides of the function
f(x) + 1 = 3(x2 + 4x) + 12, divide the 12 by 3 to get +4 inside the parentheses to complete the square
f(x) + 1 = 3(x2 + 4x + 4), factor the polynomial into a binomial squared
f(x) + 1 = 3(x + 2)2, subtract 1 from both sides to isolate f(x)
f(x) = 3(x + 2)2 - 1.

My "shorter" method comes from recognizes that 3 goes into 12 and if you add 1 more to 11 you get 12. If you factor 3 from 3x2 + 12x + 12, you get 3(x2 + 4x + 4), which you should recognize as the perfect square of (x + 2). And because 1 was added, 1 has to be subtracted from the final expression.

22. A system of equations is given below.

x + 2y = 5
2x + y = 4
Which system of equations does not have the same solution?

(4) x + 2y = 5, 4x + 2y = 12. If you double the second equation, you should have 4x + 2y = 8, not 12. In the other three choices, one of the equations is multiplied correctly.

23. based on the graph below, which expression is a possible factorization of p(x)?

(1) (x + 3)(x - 2)(x - 4). There are three zeroes, so there are three factors, eliminating choices (3) and (4). The zeroes are -3, 2, and 4, so the factors are the opposites (x - (-3)), (x - 2), (x - 4).

24. Milton has his money invested in a stock portfolio. The value, v(x) of his portfolio can be modeled with the function v(x) = 30,000((0.78)x, where x is the number of years since he made his investment. Which statement describes the rate of change of the value of his portfolio?

(2) It decreases 22% per year. The base is less than 1, so it is decay, decreasing. The rate it is decreasing is 1.00 - .78 = .22. Each year, he loses 22% of his money, and still has 78% of it remaining.

To be edited so I can add all of the images as time permits.


How did you do? Any questions?

Monday, September 05, 2016

The Interview

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

There was a shortage of teachers. There was an excess of teachers. There were positions to fill. There were no openings.

Basically, there are lousy schools no one wants to be at, and great schools no one would want to leave. There are schools that do things by the book. There are schools that play games with the books.

I thought about doing another Big Back-To-School Production Number, but there are a couple of problems

  1. I need to think of a good one that's recognizable, or easily referenced
  2. I need time to write the words for the lyrics
  3. I need time to create (or find) artwork to support it
  4. I usually need to edit them down to a manageable size
  5. And I have to start thinking about all of that long before I actually do start thinking about all of that.

Good luck to all the teachers going back, or who have already gone back. Have a great year!

Come back often for more funny math and geeky comics.

Saturday, September 03, 2016

August 2016 Common Core Algebra 1 Regents, Parts 3 and 4

What follows is a portion of the Common Core Integrate Algebra exam. Other parts will be posted on other days. Illustrations will be added at a later time when they become available.

Part II is available here.

August 2016 Algebra Regents, Part III

33. The data table below shows the diameter of grains of sand and the slop of the beach for 9 naturally occurring ocean beaches.
Write the linear regression equation for this set of data, rounding all values to the nearest thousandth.

Using this equation, predict the slope of a beach, to the nearest tenth of a degree, on a beach with grains of sand having a median diameter of 0.65 mm.

First, a naturally occurring ocean beach is one that isn't artificial or man-made. The sand comes onto the shore because of nature. This is not important for solving the problem, but don't let it throw you if you don't know what it is.

Second, I cannot stress this enough, if you do not round everything correctly, you will lose 1 of the 3 points for a -- yes, I will go there -- STUPID reason. "Nearest thousandth" means three decimal places.

Put the data into LISTs on the graphing calculator. Put the x values in L1 and the y values in L2. Double check your values, making sure you have all of them entered correctly, and that the x and y values match those on the table. (If you switch two x values without also switching the y values, you will get the wrong answer. It's like you mixed up coordinates!)

Press STAT. Right arrow to CALC. Press 4 for LinReg(ax + b)4. Hit Enter. (Depending upon your calculator's operating system, you may have to hit Enter again.)

You should get the following:

33. Shawn incorrectly graphed the inequality -x - 2y > 8 as shown below.

Explain Shawn's mistake.
Graph the inequality correctly on the set of axes below.

Shawn shaded the graph incorrectly. If you pick (0, 0) as a test point,
-(0) - 2(0) is not greater than or equal to 8, 0 < 8. He should have shaded below the line.

One possible error, but we cannot be sure of this because we don't have his work, is that Shawn forgot to flip the sign when rewriting the equation in slope-intercept form when he divided by -2.

You can graph the equation either by finding the x-intercept and the y-intercept or by rewriting the equation in slope-intercept form, which can then be put in your calculator.

If -x - 2(0) = 8
the -x = 8
and x = -8, so there is a point at (-8, 0)

If -(0) - 2y = 8
the -2y = 8
and y = -4, so there is a point at (0, -4)

Draw a solid line. That includes these points. The slope is down 4, right 8, which is -4/8 or -1/2.

Checking a test point, (0, 0), which is not on the line, we can see (as mentioned above) that (0, 0) is NOT a solution to this inequality, so it cannot be in the shaded region. That means that the other side of the graph, below the line, needs to be shaded. You can check this by picking another point, say (0, -10).

-(0) - 2(-10) > 8?
0 + 20 > 8?
20 > 8. Check.

Your graph should like as follows:

35. A drama club is selling tickets to the spring musical. The auditorium holds 200 people. Tickets cost $12 at the door and $8.50 if purchased in advance. The drama club has a goal of selling at least $1000 worth of tickets to Saturday's show.

Write a system of inequalities that can be used to model this scenario.

If 50 tickets are sold in advance, what is the minimum number of tickets that must be sold at the door so that the club meets its goal? Justify your answer.

Let x be the number of advance-sale tickets, and let y be the number of at-the-door tickets. The total number of tickets must be less than or equal to 200 people.

So x + y < 200.

The amount of money raised by selling x advance tickets is $8.50x. The amount of money raised at the door is $12y. The total of these two amounts needs to be more than $1000.

So 8.50x + 12.00y > 1000

That is your system of inequalities.

If 50 tickets are sold in advance, then the second inequality becomes

8.50(50) + 12.00y > 1000
425 + 12.00y > 1000
12.00y > 575
y > 47.9166666...

So y has to be 48 or higher. Therefore, the minimum number of tickets sold at the door is 48.

Pointswise: if you had the correct system of inequalities, you scored 2 points, 48 was worth another point, and the work to justify 48 is the final point.

36. Janice is asked to solve 0 = 642 + 16x - 3. She begins the problem by writing the following steps:

Line 1 0 = 642 + 16x - 3
0 = B2 + 2B - 3
0 = (B + 3)(B - 1)

Use Janice's procedure to solve the equation for x.

Explain the method Janice used to solve the quadratic equation.

Janice simplified the problem by substituting the letter B in place of 8x, which is a factor of both 64x2 and 16x.
(8x)2 = 64x2 and 2(8x) = 16x. These substitutions will give you the original equation back.

Following Janice's logic:
B + 3 = 0 or B - 1 = 0
So B = -3 or B = 1
However, B = 8x
So 8x = -3 or 8x = 1
Therefore x = -3/8 or x = 1/8.

If you didn't do this kind of substitution, you likely would have been forced to use the Quadratic Formula to solve this equation, and there would have been BIG numbers to worry about.

If you solved it using another method, you lost a point.

August 2016 Algebra Regents, Part IV

37. For a class picnic, two teachers went to the same store to purchase drinks. One teacher purchased 18 juice boxes and 32 bottles of water, and spent $19.92. The other teacher purchased 14 juice boxes and 26 bottles of water, and spent $15. 76.

Write a system of equations to represent the costs of a juice box,j, and a bottle of water, w.

Kara said that the juice boxes might have cost 52 cents each and that the bottles of water might have cost 33 cents each. Use your system of equations to justify that Kara's prices are not possible.

Solve your system of equations to determine the actual cost, in dollars, of each juice box and each bottle of water.

The first part is a straightforward system of equations, similar to the inequalities we did in the earlier problem.

The equations are

18j + 32w = 19.92
14j + 26w = 15.76

To show that Kara is incorrect, substitute 52 for j and 33 for w in BOTH equations.

18(.52) + 32(.33) = 19.92?
9.36 + 10.56 = 19.92 (check)

14(.52) + 26(.33) = 15.76?
7.28 + 8.58 = 15.76?
15.86 =/= 15.76 (does not check)

For the correct solution, you need to find the least common multiple of 14 and 18. (Yes! I hate the numbers in this problem!) The LCM is 126.

You need to multiply the first equation by 7, and the second equation by -9.

126j + 224w = 139.44
-126j - 234w = -141.84
add the equations
-10w = 2.40
w = .24.
A bottle of water = $.24.

14j + 26(.24) = 15.76
14j + 6.24 = 15.76
14j = 9.52
j = .68
A bottle of juice = $.68.

Partial credit would be given. A computational error would cost one point. Two or more computational errors would cost 2 points.


How did you do? Any questions?

Thursday, September 01, 2016

August 2016 Common Core Algebra 1 Regents, Part 2

What follows is a portion of the Common Core Integrate Algebra exam. Other parts will be posted on other days. Illustrations will be added at a later time when they become available.

August 2016 Algebra Regents, Part II

25. Graph the function y = -SQRT(x + 3) on the set of axes below.

Put the equation into a graphing calculator, or make a table of values. The negative sign tells you that the graph will be inverted (upside down). The +3 under the radical sign tells you that the graph will shift three spaces to the left. Any value of x that results in a negative number under the radical can be ignored -- those values are NOT in the domain of this function. The smallest possible values is -3 because -3 + 3 = 0, and the square root of 0 is 0.

Your graph should look like the one below. Be sure to graph the following points: (-3, 0), (-2, -1), (1, -2) and (6, -3).

26. Richard is asked to transform the graph of b(x) below.

The graph of b(x) is transformed using the equation h(x) = b(x - 2) - 3. Describe how the graph of b(x) changed to form the graph of h(x).

The -2 inside the parentheses will shift the graph two places to the right. The -3 outside the parentheses will shift the graph three places down.

So the point (0, -2) will shift to (2, -5).

27. Consider the pattern of squares below

Which type of model, linear or exponential, should be used to determine how many squares are in the nth pattern? Explain your answer.

The first pattern has 2 squares, the next has 4, the one after has 8. The rate of change is not constant, so it is not linear. However, there is a constant ratio, each pattern is twice as big as the one before it, so the pattern is exponential.

You didn't need to give the function but it would be something like f(n) = 2n.

28. When multiplying polynomials for a math assignment, Pat found the product to be -4x + 8x2 - 2x3 + 5. He then had to state the leading coefficient of this polynomial. Pat wrote down -4. Do you agree with Pat's answer? Explain your reasoning.

First, a reminder: You will NOT score a point for answering just "Yes" or "No" to any question like this. HOWEVER, you WILL LOSE a point if you don't include "Yes" or "No" in the answer. In this case, the answer is "No".

The leading coefficient is the coefficient of the term with the highest exponent because in standard form the term with the highest exponent comes first, and then the term with the second highest, etc.

In standard form, this expression is -2x3 + 8x2 - 4x + 5. Note that the sign goes with the term the follows it. Also, for clarity when reading, the negative in front of the 4x became a minus sign. It could have been written as + (-4x). Likewise, the minus in front of 2x3 became a negative sign for the same reason.

The leading coefficient for this expression is -2.

For full credit, answer "No, I don't agree" and explain that in standard form the term with the highest coefficient is the leading term, so the leading coefficient is "-2" because -2x3 is the term with the highest coefficient.

29. Is the sum of 3*SQRT(2) and 4*SQRT(2) rational or irrational? Explain your answer.

Sorry for the lack of radical signs on my keyboard.

Each of the two expressions is irrational because they cannot be written as fractions. The sum of two positive irrational numbers is always irrational.

I added positive because there is one exception to the rule that the sum of two irrational numbers is always irrational: if the numbers are opposites, then the sum is zero, which is rational. But it is obvious that they are not opposites.

Another way to state it is that the sum of the two numbers is 7*SQRT(2), which is irrational because the product of a rational number and an irrational number is irrational.

Sue wrote her answer in point-slope form and Kathy wrote her answer in slope-intercept form. However, you still have to prove that they both have the correct answer. You can calculate that the slope is (4 - 1) / (-3 - 6) = 3 / (-9) = -1/3, which is in both equations.

Sue used the slope -3 and the point (-3, 4) to write her equation.
If you distribute the (-1/3) on the right side, and then add 4 to both sides to isolate the y, you will get Kathy's equation.

If this didn't occur to you, but you checked both points in both equations and found out that they all worked, you should get full credit.

30. The graph below shows two functions, f(x) and g(x). State all the values of x for which f(x) = g(x).

-3 and 1.

Do not give the points of intersection. They are only asking for the x values.

I'll be honest with you: I don't like this question. It feels like they should be asking for more, but they aren't. They only want the two x-values, and each one is worth 1 point.

If you give the coordinates you lose a point. So if your entire answer is (1, 3), you scored zero.

31. Find the zeros of f(x) = (x - 3)2 - 49, algebraically.

You can put this in your graphing calculator to find the answers in advance, if you like. And those answers will be worth 1 of the 2 points.

Algebraically, set the function equal to 0:

(x - 3)2 - 49 = 0
(x - 3)2 = 49
take the square root of both sides -- don't forget to use +/-
x - 3 = +/- 7
split the equation into two parts to finish, one with +, one with -
x - 3 = 7 or x - 3 = -7
x = 10 or x = -4

32. Solve the equation below for x in terms of a.

4(ax + 3) - 3ax = 25 + 3a

Solve for x in terms of a means that you have to isolate x on the left side of the equation and have the numbers, operations and the letter a on the right side. You will not get a numeric answer.

4(ax + 3) - 3ax = 25 + 3a
4ax + 12 - 3ax = 25 + 3a
ax + 12 = 25 + 3a
ax = 13 + 3a
x = (13 + 3a)/a

Do NOT cancel out the "a" characters. You can't; it doesn't work like that.


How did you do? Any questions?

I hope you did well because, overall, they weren't difficult question (in my opinion). They could have been a little trickier, even for just two points.