(x, why?): June 2017

Wednesday, June 28, 2017

Pi vs. Tau

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

Would you rather have
Pi in the Sky or Tau in the Scow?

Come back often for more funny math and geeky comics.

Monday, June 26, 2017

June 2017: Common Core Algebra Regents, Part 1

The following are the questions and answers (and commentary) for part of the New York State Algebra Regents exam. If you have any questions or comments (or corrections), please add them in the Comments section.

My apologies for typos, particularly if they are in the questions, because then the answers are subject to change.

Answers to Part II can be found here.

Answers to Part III can be found here.

Answers to Part IV can be found here.

June 2017, Algebra I (Common Core), Part I

1.To keep track of his profits, the owner of a carnival booth decided to model his ticket sales on a graph. He found that his profits only declined when he sol between 10 and 40 tickets. Which graph could represent his profits?

(3) (image omitted). If his profits declined, then the y values will decrease in that interval -- in other words, the graph will go down, but only between x = 10 and x = 40. In choice (1), his profits are flat; that is, the remain the same. In choice (4), there is a decline, but it is in the wrong place.

2.The formula for the surface area of a right rectanguar prism is A = 2lw + 2hw + 2lh, where l, w, and h represent the length, width, and height, respectively. Which term of this formula is not dependent on the height?

(3) 2lw. There is no height, h, in the term.

3.Which graph represents y = SQRT(x - 2)?

(4) image omitted. Hopefully, this was an easy one. Besides the fact that you had a graphing calculator available to you, only one of the four graphs was a square root graph. The "- 2" under the radical shifts the graph two units to the right away from the origin.

You should make a note of this. Shifting functions is a common theme and may show up again

4.A student plotted the data from a sleep study as shown in the graph below. (image omitted)
The student used the equation of the line y = -0.09x + 9.24 to model the data. What does the rate of change represent in terms of these data?

(2) The average number of hours of sleep per day increases 0.09 hour per year of age.
Negative correlation. Sleep goes down as age goes up.

5. Lynn, Jude, and Anne were given the function f(x) = -2x² + 32, and they were asked to find f(3). Lynn's answer was 14, Jude's answer was 4, and Anne's answer was +4. Who is correct?

(1) Lynn, only. Calculate (-2)(3)² + 32 = (-2)(9) + 32 = -18 + 32 = 14.
The other two tried to solve for the zeroes of the function instead substitution 3 into the function.

6.Which expression is equivalent to 16x⁴ - 64?

(3) (4x² + 8)(4x² - 8). Difference of Squares rule.
If you multiply the choices, choice (1) will have a middle term of -64x². Choice (2) doesn't even produce the terms in the original expression, plus it will have an x² term. Choice (4) also yields incorrect coefficients, but no middle term.

7.Vinny collects population data, P(h), about a specific strain of bacteria over time in hours, h, as shown in the graph below. (image omitted)
Which equation represents the graph of P(h)?

(1) P(h) = 4(2)^h. If you look at the graph, you'll notice that the y-values are doubling. It's exponential with a base of 2.
If you look at the choices, there is only one exponential function. The others are linear, quadratic and cubic (3rd power).
If you weren't sure if the graph was a parabola, then you could have looked at the y-intercept. In Choice (3), if you substitute 0 for h, then P(h) = 3(0)² + 0.2(0) + 4.2 = 4.2. However, the graph shows a point at (0, 4), so choice (3) is incorrect.

8.What is the solution to the system of equations below?

y = 2x + 8
3(-2x + y) = 12

(1) no solution. Since "no solutions" and "infinite solutions" are options, it might make sense to rewrite the second equation in slope-intercept form first.

3(-2x + y) = 12
-2x + y = 4
y = 2x + 4
The first equation has a slope of 2 and y-intercept of 8. The second has the same slope but a different y-intercept.
That makes them parallel, and there will be no solutions.

9.A mapping is shown in the diagram below. (image omitted)
This mapping is

(3) not a function, because Feb has two outputs, 28 and 29. Each element in the domain (input) can map to only one element of the range (output). It is okay for Jan and Mar to both map to 31.

10. Which polynomial function has zeroes at -3, 0 and 4?

(3) f(x) = x(x + 3)(x - 4). Flip the signs. Which of the choices will give you a result of 0 when you enter each of those values?
Choices (1) and (2) don't work for 0 (you can stop checking). In choice (4), -3 - 3 =/= 0, nor is 4 + 4.

11.Jordan works for a landscape company during his summer vacation. He is paid $12 per hour for mowing lawns and $14 per hour for planting gardens. He can work a maximum of 40 hours per week, and would like to earn at least $250 this week. if m represents the number of hours mowing lawns and g represents the number of hours planting gardens, which system of inequalities could be used to represent the given conditions?

(1) m + g < 40, 12m + 14g > 250.
The hours must be less then or equal to 40. The amount he makes is the sum of the number of hours times the pay for those hours for each of these jobs, and that should be greater than or equal to 250.

12.Anne invested $1000 in an account with a 1.3% annual interest rate. She made no deposits or withdrawals on the account for 2 years. If interest was compounded annually, which equation represents the balance in the account after the 2 years?

(2) A = 1000(1 + 0.013)². Interest increases your value, so choices (1) and (3) are right out. The percent 1.3% must be converted to a decimal, which is 0.013. The correct answer is (2).

13. Which value would be a solution for x in the inequality 47 - 4x < 7?

(4) 11. You could solve the inequality, or you can plug in the choices to see what works. Note that 47 - 4(10) = 7, it is not less than 7. Don't let them catch you on that.

47 - 4x < 7
-4x < -40
x > 10
11 is greater than 10.

14.Bella recorded data and used her graphing calculator to find the equation for the line of best fit. She then used the correlation coefficient to determine the strength of the linear fit.
Which correlation coefficient represents the strongest linear relationship?

(1) 0.9. The closer to 1 or -1, the stronger the correlation, the closer to a straight line you get.

15.The height, in inches, of 12 students are listed below
61, 67, 72, 62, 65, 59, 60, 79, 60, 61, 64, 63
Which statement best describes the spread of these data?

(4) 79 is an outlier, which would affect the standard deviation of these data.

16.The graph of a quadratic function is shown below. (image omitted)
An equation that represents the function could be

(4) q(x) = -1/2 (x - 15)² + 25. The four choices are written in vertex form. The vertex is (15, 25).
Vertex form is y = a(x - h)² + k. h = 15 and k = 25. (Note that the minus is part of the form.)

17.Which statement is true about the quadratic function g(x), shown in the table below, and f(x) = (x - 3)² + 2?

(3) They have the same axis of symmetry. The vertex of the equation is (3, 2). The vertex of the table is (3, -5). These points are different but they both lie on the axis of symmetry x = 3.

18.Given the function f(n) defined by the following:

f(1) = 2
f(n) = -5f(n - 1) + 2 Which set could represent the range of this function?

(2) {2, -8, 42, -208, ...}. Frankly, the other choices make no sense. The first number has to be 2, so choices (3) and (4) are right out. Choice (1) is all positive, but the negative multiplier. Substitute 2, 3, and 4, and you'll get the rest of the numbers in the list.

19.An equation is given below.

4(x - 7) = 0.3(x + 2) + 2.11 The solution to the equation is

(1) 8.3

4(x - 7) = 0.3(x + 2) + 2.11
4x - 28 = 0.3x + 0.6 + 2.11
3.7x = 2.71 + 28
3.7x = 30.71
x = 8.3

20.A construction worker needs to move 120 ft³ of dirt by using a wheelbarrow. One wheelbarrow load holds 8 ft³ of dirt and each load takes him 10 minutes to complete. One correct way to figure out the number of hours he would need to complete his job is

(4) (image omitted)
Choice (4) is the only one where all the units cancel out, except for hours.

21.One characteristic of all linear functions is that they change by

(3) equal differences over equal intervals. In other words, a constant rate of change.

22.What are the solutions to the equation x² - 8x = 10

(2) 4 + SQRT(26)

x² - 8x = 10
x² - 8x + 16 = 10 + 16
(x - 4)² = 26
x - 4 = SQRT(26)
x = 4 + SQRT(26)

23.The formula for blood flow rate is given by F = (P1 - P2) / r, where F is the flow rate, P1 the initial pressure, P2 the final pressure, and r the resistance created by blood vessel size. Which formula can not be derived from the the given formula?

(3) r = F(p₂ - p₁). To solve for r, you would have to multiply the original equation by r, but then you would have to divide by F, which would put it in the denominator of a fraction.

24.Morgan throws a ball up into the air. The height of the ball above the ground, in feet, is modeled by the function h(t) = -16t² + 24t, where t represent the time, in seconds, since the ball was thrown. What is the appropriate domain for this situation?

(1) 0 < t < 1.5. The domain is the t axis, so choices (3) and (4) are no good.
If you substitute t = 1.5, then h(1.5) = 0, which is the ground. Everything after that would yield a negative number meaning that the ball went below the ground.

End of Part I

How did you do?
Comments, questions, corrections and concerns are all welcome.
Typos happen.

Sunday, June 25, 2017

June 2017: Common Core Geometry Regents, Part 4

My apologies for typos, particularly if they are in the questions, because then the answers are subject to change.

The answers to Part II can be found here.

The answers to Part III can be found here.

June 2017, Geometry (Common Core), Part IV

35.Quadrilateral PQRS has vertices P(-2, 3), Q(3, 8), R(4, 1) and S(-1, -4).
Prove that PQRS is a rhombus.
[The use of the set of axes is optional.]

This is NOT a two-column proof. You could write one, but it will not be worth any points if you don’t use the coordinates and the formulas to show the work. You need to back up everything you say with numbers. (And those numbers have to be correct.)

There are a number of ways to show that a quadrilateral is a rhombus. The easiest is that the four sides have equal lengths. You must have a concluding statement that says this and show the work. (Radical 50 or 5 radical 2)

You can show that the diagonals are perpendicular bisectors of each other. So you need to find the slopes of the lines and the midpoint of each line. The slopes must be inverse reciprocals and therefore are perpendicular (that last part is important!) and the midpoints must be the same point.

If you only prove that the slopes are perpendicular, you MUST also show that the figure is a parallelogram. Otherwise, your proof is incomplete. A kite, for example, has perpendicular diagonals.

To show that it is NOT a square, you can show that two sides don’t have right angles: find the slopes and show that they are not perpendicular. You can show that the diagonals are not congruent: find the lengths of the two diagonals. You can even use Pythagorean Theorem to show that two sides and one diagonal do not form a right triangle.

Note that you did not have to simplify your radicals when you use the distance formula, but you did have to have accurate numbers.

Seem the image below.

Points: Basically, you got 4 points for proving and stating the figure was a rhombus and 2 points for proving that it was not a square, for a total of 6 points.

36. Freda, who is training to use a radar system, detects an airplane flying at a constant speed and heading in a straight line to pass directly over her location. She sees the airplane at an angle of elevation of 15^o and notes that it is maintaining a constant altitude of 6250 feet. One minute later, she sees the airplane at an angle of elevation of 52^o. how far has the airplane traveled, to the nearest foot?

Determine and state the speed of the airplane, to the nearest mile per hour.

You have to find the horizontal distance to the plane from the first sighting, and then the horizontal distance of the second sighting. And then subtract the two values to find how far the plane traveled.
Finally, you have to convert the number of feet it moved per second instead miles per hour.

This could also be done with the Law of Sines but I'll save that for another post, where I can go more in-depth into the problem.

Sketch a figure, or two, to show the plane's location. In each right triangle, you have the height of 6250, which is opposite the angle (either 15 or 52). And you need the ground distance, which will be adjacent to the angle. You don't need the hypotenuse in either triangle. That means using tangent.

So Tan 15^o = 6250 / x and Tan 52^o = 6250 / y. And the distance traveled will be x - y.

x = 6350 / tan 15 and y = 6250 / tan 52
x = 23325/32 and y = 4883.03
x - y = 18442 feet

No matter what number you got in the top half of the problem, you can get point for converting it into milers per hour correctly.
To change feet per second into miles per hour, multiply by 60 and divide by 5280.

18442.28... x 60 / 5280 = 1106536.94... / 5280 = 209.57... = 210 mph.

You might have run into problems if your calculator was in Radians mode. Hopefully, the negative numbers alerted you to a problem.

If you used sine or cosine (other than using the Law of Sines), you would have lost two points from the top portion of the problem.

End of Part IV

How did you do?
Comments, questions, corrections and concerns are all welcome.
Typos happen.

Saturday, June 24, 2017

June 2017: Common Core Geometry Regents, Part 3

My apologies for typos, particularly if they are in the questions, because then the answers are subject to change.

The answers to Part II can be found here.

June 2017, Geometry (Common Core), Part III

32. Triangle ABC has vertices A(-5, 2), B(-4, 7), and C(-2, 7), and triangle DEF has vertices at D(3, 2), E(2, 7), and F(0, 7). Graph and label triangle ABC and triangle DEF on the set of axes below.

Determine and state the single transformation where triangle DEF is the image of triangle ABC

Use your transformation to explain why triangle ABC = triangle DEF.

(image will be uploaded soon)
If you graph the two triangles, you will see that one is the reflection of the other. (Mirror images.) However, they are NOT reflected over the y-axis. To reflect over the y-axis, you would have to translate it first. But they want a single move.

You have to find the reflection line, which will be the vertical line halfway between points C and F, which occurs at x = -1.
So the transformation is r_{x = -1}.

Because a reflection is a rigid motion (which preserves distance and shape), DEF is congruent to ABC.

33. Given: RS and TV bisect each other at point X
TR and SV are drawn (image omitted)
Prove: TR || SV

Here’s the approach you need to take: If the lines are parallel, then the alternate interior angles along the transversals will be congruent. You can show that they are congruent by proving that the two triangles are congruent. SAS looks like the easiest approach.

Statement	Reason
1. RS and TV bisect each other at point X	Given
2. RX = XS and TX = XV	Definition of bisect
3. <TXR = <VXS	Vertical Angles are congruent
4. Triangle TXR = Triangle VXS	SAS
5. <T = <V	CPCTC (Corresponding Parts of Congruent Triangles are Congruent)
6. TR \|\| SV	If two lines are crossed by a transversal and the alternate interior angles are congruent, then the lines are parallel.

34. A gas stations has a cylindrical fueling tank that hold the gasoline for its pumps, as modeled below. The tank holds a maximum of 20,000 gallons of gasoline and has a height of 34.5 feet. (image omitted)
A metal pole is used to measure how much gas is in the tank. To the nearest tenth of a foot, how long does the pole need to be in order to reach the bottom of the tank and still extend one foot outside the tank? Justify your answer. [I ft³ = 7.48 gallons]

Before you start, what are you looking for? The height of the stick, with is the diameter plus 1 foot. When you use the Volume formula, you will get the radius. So you need to find the radius, then double it and add 1, and then round it to the nearest tenth of a foot. Do not round to the nearest tenth in the middle of the problem. You don’t need to carry as many decimal places as I’m showing. However, I left the numbers in the calculator, so I’m showing exactly what I did. There may be minor differences in your numbers if you round that won’t affect the final answer.

First, convert gallons to cubic feet: 20,000 / 7.48 = 2673.79679
V = (pi) (r²)(h)
2673.79679 = (pi) (r²)(34.5)
r² = 2673.79679 / (34.5 * pi)
r² = 24.66944789
r = 4.96683
d = 9.93366
The stick is 10.9 feet long.

End of Part III

How did you do?
Comments, questions, corrections and concerns are all welcome.
Typos happen.

Friday, June 23, 2017

June 2017: Common Core Geometry Regents, Part 2

My apologies for typos, particularly if they are in the questions, because then the answers are subject to change.

June 2017, Geometry (Common Core), Part II

25. Given: Trapezoid JKLM with JK || ML Using a compass and straightedge, construct the altitude from vertex J to ML. {Leave all construction marks.]

There are several methods that work. One of them is to start at point J. Make an arc that cuts across LM in two places. You are allowed to use the straightedge to extend LM, if necessary. (If your arc goes outside the trapezoid, you MUST extend the line.)

From each of the points where the arc intersects LM, make an arc. At the two points of intersection, use the straightedge to draw the altitude, which will intersect point J.
Note: you only need the arcs to intersect once because you already have J.

26.Determine and state, in terms of pi, the area of a sector that intercepts a 40^o arc of a circle with a radius of 4.5.

The area of a sector of a circle is the area of the circle times the measure of the central angle divided by 360^o. (In other words, you multiply the area by the fraction of the circle represented.)

A = (40/360) * pi * (4.5)²
A = 2.25 pi or A = 9/4 pi.

You could also have found the area by formula A = ½ r² O, where O is the central angle, measured in radians.
40^o = 40 (pi ) / (180) = 2 pi / 9
A = ½ (4.5)² (2 pi / 9) = (1/2) (20.25) (2 / 9) pi = 2.25 pi, which is the same answer.

27. The diagram below shows two figures. Figure A is a right triangular prism and figure B is an oblique triangular prism. The base of figure A has a height of 5 and a length of 8 and the height of prism A is 14. The base of figure B has a height of 8 and a length of 5 and the height of prism B is 14.

Use Cavalieri’s Principle to explain why the volumes of these two triangular prisms are equal.
[Image omitted.]

First, yes, you needed to refer or appeal to the principle, even if you didn’t use the name.

Second, you had to be very specific about the language, particularly using words like base, length, and height.

Third, the triangle bases are NOT congruent, even if the area of the Bases is the same.

Fourth, the oblique prism has a height of 14. This is not the slant height, or the length of the sides. In other words, the surface areas are different, the prisms are not congruent, etc.

Those were some misunderstandings I came across speaking to students.

What was necessary to state was this: the areas of the base were equal and the heights of the prisms are the same, therefore the Volumes must be equal.

You could have mentioned that the area of the cross sections will be equal at every level, but it would not have been necessary.

You could have found the area of the triangles and the volumes of the prisms, but that was not necessary. And be careful if you calculate them incorrectly. (For instance, leaving out the ½, or using 1/3, which is for pyramids. A triangular prism is not a pyramid!)

28. When volleyballs are purchased, they are not fully inflated. A partially inflated volleyball can be modeled by a sphere whose volume is approximately 180 in³. After being fully inflated, its volume is approximately 294 in³. To the nearest tenth of an inch, how much does the radius increase when the volleyball is fully inflated?

You need to use the formula V = (4 / 3) (pi) (r³) for each volume, and solve for r.
Then subtract the two radii to find the increase. Do NOT find a ratio.
This was a lot of work for only 2 points, with plenty of places for an error to sneak in, but there was nothing “tricky” about it. It was just a lot of steps.
Look at the image below. The increase is 0.6 inches.

29. In right triangle ABC shown below (image omitted), altitude CD is drawn to hypotenuse AB.
Explain why triangle ABC ~ triangle ACD.

This is an explanation, not a proof. You need to have reasons, back up what you write, but you don’t need to be so formal.

Simplest answer: If you said it was true because of the Right Triangle Altitude Theorem (and stated what that says), that was sufficient. You didn’t have to prove it. You already know it’s a theorem. (You don’t have to prove Pythagorean Theorem every time you use it, right?)

Otherwise, you can prove it using AA, or AAA, but you didn’t need the third angle. If you do this, you need three things for two points: two pairs of congruent angles and a statement that the are similar because of AA.

Angle CDA is a right angle because of the altitude. Angle ACB is a right angle of the given triangle. Angle A is in both triangles. The two triangles have two pairs of angles that have the same measure so they are similar by AA.

30. Triangle ABC and triangle DEF are drawn below. (image omitted)
If AB = DE, AC = DF, and <A = <D, write a sequence of transformations that maps triangle ABC onto triangle DEF.

You can see that there needs to be a rotation and a translation. That answer isn’t good enough.
Consider this: if there was a coordinate plane, you would have been expected to give amounts, directions, etc. This is true here as well. You can do it using rigid motions.

Translate triangle ABC along vector CF, mapping C to point F. Then rotate ABC around point C until point A is mapped onto point D.

31. Line n is represented by the equation 3x + 4y = 20. Determine and state the equation of line p, the image of line n, after a dilation of scale factor 1/3, centered at the point (4, 2).
[The use of the set of axes below is optional.]

Explain your answer.

If you substitute (4, 2) into 3x + 4y = 20, you get 3(4) + 4(2) = 20, 12 + 8 = 20, 20 = 20.
Therefore, (4, 2) is a point on line n. (You also would have noticed this if you graphed the line.)

The dilation of a line centered at a point on the line will not affect the line at all. (One third of the infinite length is infinity. One third of its 0 width is zero. One third of its 0 distance from the center is still zero.)

So the equation for p is 3x + 4y = 20.
If you rewrote it in slope-intercept form for some reason, you would have y = -3/4 x + 5

End of Part II

How did you do?
Comments, questions, corrections and concerns are all welcome.
Typos happen.

End-of-Year Review 2017

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

If those were your final words, begging a pardon might be an effective strategy. But likely, not.

Come back often for more funny math and geeky comics.

Thursday, June 22, 2017

June 2017: Common Core Algebra Regents, Part 4

My apologies for typos, particularly if they are in the questions, because then the answers are subject to change.

The answers to Part II can be found here.
The answers to Part III can be found here.

June 2017, Algebra I (Common Core), Part IV

36.Michael has $10 in his savings account. Option 1 will add $100 to his account each week. Option 2 will double the amount in his account at the end of each week.
Write a function in terms of x to model each option of saving.

Michael wants to have at least $700 in his account at the end of 7 weeks to buy a mountain bike . Determine which option(s) will enable him to reach his goal. Justify your answer.

Option 1: f(x) = 100x + 10
Option 2: g(x) = 10(2)^x

f(7) = 100(7) + 10 = 710
g(7) = 10(2)⁷ = 10(128) = 1280
Both options will enable him to reach his goal.

Your answer to the second part is dependent upon the function you wrote in the first part. If you made a mistake in the beginning, you need to carry that through to the end.

37. Central High School had five members on their swim team in 2010. Over the next several years, the team increased by an average of 10 members per year. The same school had 35 members in their chorus in 2010. The chorus saw an increase of 5 members per year.

Write a system of equations to model this situation, where x represents the number of years since 2010.

Graph this system of equations on the set of axes below.

Explain in detail what each coordinate of the point of intersection of these equations means in the context of this problem.

Swim: y = 10x + 5
Chorus: y = 5x + 35

In the graph (below), the coordinates of the point of intersection are (6, 65). The six means six years after 2010, or 2016. The 65 means that there will be 65 members on the swim team and in chorus.

End of Part IV

How did you do?
Comments, questions, corrections and concerns are all welcome.
Typos happen.

Wednesday, June 21, 2017

June 2017: Common Core Algebra Regents, Part 3

My apologies for typos, particularly if they are in the questions, because then the answers are subject to change.

The answers to Part II can be found here.

June 2017, Algebra I (Common Core), Part III

33. The function r(x) is defined by the expression x² + 3x - 18. Use factoring to determine the zeroes of r(x).
Explain what the zeroes represent on the graph of r(x).

x² + 3x - 18 = 0
(x + 6)(x - 3) = 0
x + 6 = 0 or x - 3 = 0
x = -6 or x = 3 are the zeroes of the function

The zeroes of the function means that the graph will cross the x-axis at -6 and 3.

34. The graph below models Craig's trip to visit his friend in another state. In the course of his travels, he encountered both highway ad city driving.
Based on the graph, (image omitted) during which interval did Craig most likely drive in the city? Explain your reasoning.
Explain what might have happened in the interval between B and C.
Determine Craig's average speed, to the nearest tenth of a mile per hour, for his entire trip.

I would NOT want to grade this question. It assumes too much on the part of the students -- in particular, that you will travel more miles on the highway at a faster rate than in the city.

Second, you have to realize that the flat line between B and C means that the car is not moving at all (which is reasonable for the exam) but supposing why the car isn't moving. Did it stop on purpose? Is it stuck in traffic? Do cars get stuck in the city more than on the highway?

The answer that they are (probably) looking for is between points D and E, hours 5 and 7 when the rate of miles per hour has decreased, but the car is still moving. You wouldn't go as fast during city driving.

The 1.5 hours that the car was stopped was likely a stop in the trip and not driving at all.
Could be a rest stop. Could be a mall. Could be lunch. Could be a major traffic jam with a tree on the highway or a truck fire or a seven-car pile-up. I hope students get creative on this one!

Hint: to get the nearest tenth of a mile per hour, divide the total number of miles by the total number of hours: 230 miles / 7 hours = 32.8571428571, or 32.9 to the nearest tenth.

35. Given
g(x) = 2x² + 3x + 10
k(x) = 2x + 16
Solve the equation g(x) = 2k(x) algebraically for x, to the nearest tenth.
Explain why you chose the method you used to solve this quadratic equation.

g(x) = 2 k(x)
2x² + 3x + 10 = 2(2x + 16)
2x² + 3x + 10 = 4x + 32
2x² - x - 22 = 0

The Quadratic Formula is used in the image below:

Update: I cut the bottom from the image. Looks like I made a parenthesis error of some sort on the calculator and didn't catch it.

Calculate those fractions to the nearest tenth, and you get
x = 3.576... and x = -3.076, which round to x = 3.6 and x = -3.1

End of Part III

Answers to Part IV are here.

How did you do?
Comments, questions, corrections and concerns are all welcome.
Typos happen.

Tuesday, June 20, 2017

June 2017: Common Core Algebra Regents, Part 2

My apologies for typos, particularly if they are in the questions, because then the answers are subject to change.

Answers to Part III can be found here.

June 2017, Algebra I (Common Core), Part II

25. Express in simplest form: (3x² + 4x - 8) - (-2x² + 4x + 2)

5x² - 10.
Show something on the paper to indicate where you got this: line them up vertically; distribute the "-1" and combine like terms. Something, so you'll be sure to get both points. (Frankly, this is the kind of question that you should be able to do without showing any work. It can all be done in your head.)

26. Graph the function f(x) = -x² - 6x on the set of axes below.
State the coordinates of the vertex of the graph.

See the graph below. The vertex is at (-3, 9). You need to state the point and have a correct graph to get both points.

27. State whether 7 - SQRT(2) is rational or irrational. Explain your answer.

It is irrational because 7 is rational and SQRT(2) is irrational and the sum or difference of a rational and an irrational number is always irrational.

28. The value, v(t), of a car depreciates according to the function v(t) = P(.85)^t, where P is the purchase price of the car and t is the time, in years, since the car was purchased. State the percent that value of the car decreases by each year. Justify your answer.

The car's value decreases by 15% each year because 1.00 - .85 = .15, which is 15%.

29. A survey of 100 students was taken. It was found that 60 students watched sports, and 34 of these students did not like pop music. Of the students who did not watch sports, 70% liked pop music. Complete the two-way frequency table.

	Watch Sports	Don't Watch Sports	Total
Like Pop
Don't Like Pop
Total

Answer: see table below
Because 60 of 100 watched sports, then 40 did not, so the bottom row is 60, 40, 100.
Of 60, 34 did not like pop, so 26 did. First column is 26, 34, 60.
TWIST -- they used percentages in the next portion of the question.
Of the 40, 70% liked pop music. (.70)(40) = 28, and 40 - 28 = 12.
The second column is 28, 12, 40.
Add the totals for each row. Last column is 54, 46, 100.

	Watch Sports	Don't Watch Sports	Total
Like Pop	26	28	54
Don't Like Pop	34	12	46
Total	60	40	100

30. Graph the inequality y + 4 < -2(x - 4) on the set of axes below.

See graph below.

If you recognize point-slope form then you know that the slope of the boundary (broken) line is -2 and (4, -4) is a point on that broken line.

If you didn't recognize that, you could subtract 4 from each side and put

y = -2(x - 4) - 4
into your graphing calculator, and get the table of values.

Or use the Distributive property and created your own table from the following:

y < -2(x - 4) - 4
y < -2x + 8 - 4
y < -2x + 4

31. If f(x) = x² and g(x) = x, determine the value(s) of x that satisfy the equation f(x) = g(x).

Substitute x² = x
Subtract x² - x = 0
Factor x(x - 1) = 0
Find the zeroes: x = 0 or x - 1 = 0, so x = 0 or x = 1.

32. Describe the effect that each transformation below has on the function f(x) = |x|, where a > 0.
g(x) = |x - a|
h(x) = |x| - a

g(x) will shift f(x) a units to the right.
h(x) will shift f(x) a units down.
Both graphs will have the same shape.

End of Part II

How did you do?
Comments, questions, corrections and concerns are all welcome.
Typos happen.

a || b

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There are hand-writing issues, both with chalk and on electronic whiteboards.

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Thursday, June 15, 2017

Geometry Regents Questions? Post them here

As I type this, the New York State Geometry Regents is about 15 hours away

If you have a specific question about a topic that you would like to ask or have explained, post it as a comment to this thread. Then check in later for a response.

I'll check and update the comments periodically throughout the evening.

Update #1:
Anonymous said...
Could we go over completing the square for circles? Also, the equation of a circle?

Okay, "Anonymous". It's like this. The standard form of the equation of a circle is

(x - h)² + (y - k)² = r² Where (h, k) are the coordinates of the center of the circle and r is the radius.
Note: you can use the Distance formula or Pythagorean Theorem to see if another point they give is on the circle if the distance from the center to the point is the same as the radius.

As you might remember from Algebra, (x - h)² is the square of a binomial, and can be written as (x - h)(x - h). If you were to multiply that (think FOIL), you would get x² - 2hx + h². That expression would be a perfect square. If you have an incomplete expression, you need to complete it, so that you can factor it.

Say they give you x² + y² + 6y = 16
x² is a perfect square. It's the same as (x - 0)².
However, y² + 6y is not a perfect square -- it doesn't have a constant term.
Do you see the 2hx in the expression above? The middle term of the complete square is double the number in the binomial, so you need to find half of it.
Half of 6 is 3. The final term of the completed square is h², h is 3, so h² is 9.
So you need to add 9 to both sides of the equation, which gives you

x² + y² + 6y + 9 = 16 + 9 This can be reduced to x² + (y + 3)² = 25
In this case, the center is (0, -3) -- because you flip the sign -- and the radius is the square root of 25, which is 5.
It is quite possible -- likely, even -- that they will give you a problem with a radius that has an irrational length.

Does this help/answer your question?

Update #2:
Anonymous said... Can you go over density problems

Density is mass divided by Volume. Imagine you have something that weighs 10 pounds. If it fits in the palm of your hand, it's pretty dense. If it's the size of your kitchen table, it's not very dense at all.
D = m / V, like those "dense" people you meet at the D.M.V. when you apply for a learner's permit.
They have to give you 2 of the three values, so that you can find the third one. However, they can make you figure out Volume.
Volume of a prism is the Area of its Base times its height.
A rectangle prism would be length X width X height
A cylinder would be pi * r² * h, etc.

After that, it's likely to just be an Algebra problem.

I don't have a specific example I can give you or that.

Update #3:
Anonymous said...
Can you go over proofs

Not really. I could spend a week on proofs. If you want something specific, I would check my old Regents exam posts.
Here are some general guidelines.
Look at the image. What do you see? What do things look like? You CANNOT go based on looks, but it might give you a direction to go in.
Don't "assume" anything. Either it's given, or you derived it from what was given.
Make a plan. How are you going to get there?

Does it involve proving triangles are congruent? Then you'll need SSS, SAS, ASA, AAS or HL. (Don't make a backward SSA of yourself!)
If you use any of those, you need to specify three pairs of things that are congruent. In the case of HL, make sure you state that all right angles are congruent. Seriously -- it needs to be stated.
If you are proving that two sides of a triangle or two angles are congruent, then the last reason will probably be CPCTC (Corresponding parts of congruent triangles are congruent).

They won't give you anything that you can't figure out. Two of the biggies are vertical angles are congruent, and the reflexive property (for sides or angles).

If it's a circle, remember that you can add extra radii, and that all radii of a given circle are congruent. Tangents are perpendicular to the radius, so you might see a right triangle in the circle. Similar triangles (use AA) inside the circle are also possible.
Make sure you state all the given, and if there's an illustration, mark off everything you know. It might give you ideas, or it might remind you what you haven't explicitly stated yet.

Obviously, you need to know your theorems. And there are a lot of them.

Does this help/answer your question?

Update #3:
Heaven said... Could we go over finding a section of a circle? And those circle problems dealing with an external point?

I assume you mean a "sector" of a circle, like a slice of pie? Think of slice of pi, if that helps.
The area of the sector of a circle is a fraction of the area of the entire circle.
The fraction that you need to multiply by is the central angle over 360 degrees. (Times pi r square)

They can also ask the reverse. They can tell you the area of the sector and the radius and have you come up with the central angle by working backward.
Think Algebra: inverse operations.

I'm not sure what you mean by "those circle problems dealing with an external point".
Do you mean finding the size of an angle from the arcs the lines intersect?
Do you mean the relationship between the lengths of the secants or tangents from an external point?
Do you have an example?

Final Update ... It's Friday morning
Anonymous Anonymous said... Can you go over finding a point on a circle, and also ratios of line segments?

Suppose you are given an equation like (x - 3)² + (y + 1)² = 20.
If you wanted to know if a point is on the circle, say (5, 3), substitute those values into the equation.
(5 - 3)² + (3 + 1)² =?= 20
2² + 4² = 20 ?
4 + 16 = 20
20 = 20, check
Therefore, (5, 3) is a point on the circle. Had it equaled anything other than 20, it would not have been on the circle.

If a point in on a line somewhere that isn't the midpoint, you need to use ratios to find its position.
For example, if given A(-1, 2) and B(7, 8) and you and P was a point such that the ratio of the lengths AP:PB was 3:1, where would P be?
First, 3 + 1 = 4, so P is 3/4 of the way from A to B.
Find the difference of the x values 7 - (-1) = 8, multiply it by 3/4, and you get +6.
Find the difference of the y values 8 - 2 = 6, multiply it by 3/4, and you get +4.5
Add those values to the coordinates of A to get P. P(-1+6, 2+4.5) gives you P(5, 6.5).
Yes, you can get a decimal.

Monday, June 12, 2017

RIP Adam "Batman" West

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Riddle me this: Was that a 'bat-joke' or a 'dad-joke' hidden in there?

I had afternoon reruns of Batman and Superman growing up, but Batman's shows were newer, in color, and had super villains in them. Superman had bad guys, but not Lex Luthor or ... whoever.

Bat-this, Bat-that. It was a part of growing up. I still tell my students when the bell rings, even though they don't get the reference, that I'll see them tomorrow.
"Same bat-time, same bat-channel."

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Friday, June 09, 2017

Failure

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Witty/Inspirational quote about how good Failure is goes here. Like "Yeah! Summer School! No sand in my toes!" Or something.

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Wednesday, June 07, 2017

The 0'Factor, Episode 17

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And things might get a little improper before that happens!

I knew I hadn't done one of these in a while. Didn't realize that it had been more than TWO YEARS since the last one.
Especially when you consider the "promo" version, without other characters, are easy to create. Coming up with a throwaway line, on the other hand, takes a little more time.

It also might be time for a new "set" because that desk was one of the earliest things I ever drew for this strip (nearly 10 years ago) that I still use, and for the life of me, I have no idea what fonts I use -- or even if I still have them available!

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Sunday, June 04, 2017

Geometry Problems of the Day

The following problems were taken from the GEOMETRY (COMMON CORE) Regents Exam given on Thursday, January 26, 2017.
Previous problems can be found here.

Part 1

23. A plane intersects a hexagonal prism. The plane is perpendicular to the base of the prism. Which two-dimensional figure is the cross section of the plane intersecting the prism?

(4) rectangle.
A hexagonal prism has a six-sided hexagon on its "bottom" and "top". Imagine a hexagonal building. The walls holding up the roof would be shaped like rectangles, going straight up, perpendicular to the ground.
Each of these walls part of planes that would be perpendicular to the base. So the answer is rectangle.

24. A water cup in the shape of a cone has a height of 4 inches and a maximum diameter of 3 inches. What is the volume of the water in the cup, to the nearest tenth of a cubic inch, when the cup is filled to half its height?

(1) 1.2
The equation for Volume of a cone is V = 1/3 π r²h, however, in this case, we only want 1/2 of the height. There are TWO problems with the radius. First, we're given the diameter of the top of the cone, not the radius. The radius of the top of the cone is 1.5, not 3. However, that's NOT the radius that we want. We need the radius of the circle that is halfway down the cone.
Luckily, the smaller cone and the larger cone are similar (have the same shape), so the radius is proportional. At half the height, the radius is also half, or 0.75.
Plug in these values and you have V = 1/3 (3.141592...) (0.75)² (2) = 1.178097..., which rounds to 1.2.

Did you get tripped up by that one?

That's the end of Part I. I hope you all did well.

Continue to the next problems.

Algebra 2 Problems of the Day

The following problems were taken from the ALGEBRA II (Common Core) Regents Exam given on Friday, January 27, 2017.
Previous problems can be found here

Part 1

21. Joelle has a credit card that has a 19.2% annual interest rate computations. compounded monthly. She owes a total balance of B dollars after m months. Assuming she makes no payments on her account, the table below illustrates the balance she owes after m months.

Over which interval of time is her average rate of change for the balance on her credit card account the greatest?

(4) month 60 to month 73
Look at the image below. Find the average rate of change by calculation (y₁ - y₂) / (x₁ - x₂).

22. Which graph represents a cosine function with no horizontal shift, an amplitude of 2, and a period of 2/3 π ?

(3) See below
Choices (2) and (4) are out because they start at 0 (sine graphs). Graph (1) shows a function with a period of 2/9 π, as it repeats three 3 in the space of 2/3 π. Choice (3) shows a function that repeats 3 times in the space of 2π, so it has a period of 2/3π.

Continue to the next problems

Thursday, June 01, 2017

Geometry Problems of the Day

The following problems were taken from the GEOMETRY (COMMON CORE) Regents Exam given on Thursday, January 26, 2017.
Previous problems can be found here.

Part 1

21. In the diagram below of circle O, GO 8 and m∠GOJ = 60°. What is the area, in terms of π, of the shaded region?

(4) 160π / 3.
Since 60° is 1/6th of the 360° degree in the complete circle, then the unshaded region of the circle is (1/6) πr² = (1/6) π8²
and the shaded portion would be (5/6) πr² = (5/6) π8² = (5 * 64π) / 6 = (5 * 32π) / 3 = 160π/ 3.

22. A circle whose center is the origin passes through the point (5,12).
Which point also lies on this circle?

(3) (11, 2 sqrt(12))
The equation of the circle is x² + y² = r². We can find r using the Distance Formula or Pythagorean Theorem: 5² + 12² = r².
25 + 144 = 169 = r²
r = 13 (which you really should have known. Look up Pythagorean triples.)

Which of the other points creates a right triangle with a hypotenuse of 13?
(10, 3) definitely do not -- they don't even create a triangle with a side of 13. (-12, 13) can't because the hypotenuse is longer than the legs (plus it would have to be -12 and either 5 or -5).
Check 11² + (2sqrt(12))² = 121 + 4(12) = 169 = 13². Looks good.
Check (-8)² + (5sqrt(21))² = 64 + 25(21) = way too much. No good.

Continue to the next problems.

Bubblegum

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These things have a way of blowing up in your face. Kind of like bubblegum.

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