Wednesday, January 24, 2018

Diamond Girl

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(C)Copyright 2018, C. Burke.

The rectangle should tell teh square to go fly a kite!

Don't do it. It's a trap-ezoid, for very loose definitions.




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Tuesday, January 23, 2018

August 2017 Common Core Geometry Regents, Part 1

While I wait for the January 2018 Geometry Regents exams to become available, and for the go-ahead to publish the questions and answers, allow me to revisit the August 2017 exams, which I haven't gotten around to dealing with until now.

August 2017, Geometry, Part I

Each correct answer will receive 2 credits. No partial credit.


1. A two-dimensional cross section is taken of a three-dimensional object. If this cross section is a triangle, what can not be the three-dimensional object?

(2) cylinder You can't take a cross section of a cylinder and get a triangle as a result. Vertical cross sections of cones and pyramids are triangles. Slicing off the corner of a rectangular prism will give you a triangle.


2. The image of triangle DEF is triangle D'E'F'. Under which transformation will the triangles not be congruent?

(4) a dilation with a scale factor of 3/2 centered at the origin A dilation will yield a similar triangle, but not a congruent one. However, choice (3) has a scale factor of 1, which won't change the size. (It won't do anything.)


3. The vertices of square RSTV have coordinates R(-1,5), S(-3,1), T(-7,3), and V(-5,7). What is the perimeter of RSTV?

(3) 4 (sqrt (20)) Use the distance formula to find the length of one side and then multiply by four, because it's a square with four equal sides.
d = sqrt( (-3 - -1)2 + (1 - 5)2)
= sqrt ( (-2)2 + (-4)2)
= sqrt ( 4 + 16 ) = sqrt (20), the length of ONE side of the square.


4. In the diagram below of circle O, chord CD is parallel to diameter AOB and mCD = 130.


What is mAC?

(1) 25
The semicircle ACDB is 180 degrees. Arc CD is 130. 180 - 130 = 50
The sum of arcs AC and DB = 50, but because the chord is parallel to the diameter, the two arcs are equal.
Therefore arc AC = 50 / 2 = 25.


5. Iin the diagram below, AD intersects BE at C, and AB || DE.


If CD = 6.6 cm, D# = 3.4 cm, CE = 4.2 cm, and BC = 5.25 cm, what is the length of AC, to the nearest hundredth of a centimeter?

(4) 8.25
Because of the vertical angles and the alternate interior angles, we know that the triangles are similar. That means that the corresponding sides are proportional.
CD / CE = AC / BC
6.6 / 4.2 = AC / 5.25
4.2 AC = (6.6)(5.25)
AC = (6.6)(5.25)/4.2 = 8.25


6. As shown in the graph below, the quadrilateral is a rectangle.


Which transformation would not map the rectangle onto itself?

(3) a rotation of 180° about the origin
Rotating 180 degrees about the origin would put the rectangle in quadrants II and III.
Rotating 180 degrees about the point (4, 0) would map it onto itself because (4, 0) is the center of the rectangle.


7. In the diagram below, triangle ACD has points B and E on sides AC and AD, respectively, such that DE||CD, AB = 1, BC = 3.5, and AD 18.


What is the length of AE, to the nearest tenth?

(4) 4.0
Since the lines are parallel, triangles ABE and ACD are similar and their sides are proportional.
AB / AC = AE / AD
1 / 4.5 = AE / 18
4.5 AE = 18
AE = 4.0


8. In the diagram below of parallelogram ROCK, m∠C is 70° and m∠ROS is 65°.


What is m∠KSO?

(4) 135°
Angle R is congruent to Angle C, so angle R is 70.
Angle KSO is the exterior angle for triangle ROS. The remote angles are 65 and 70, so 65 + 70 = 135.


9. In the diagram below, ∠GRS ∠ART, GR 36, SR 45, AR 15, and RT 18.


Which triangle similarity statement is correct?

(4) Triangle GRS is not similar to triangle ART.
45 * 15 = 675
36 * 18 = 648
The sides are not proportional (which should've obvious because 5*5=25 and 6*8=48).


10. The line represented by the equation 4y = 3x + 7 is transformed by a dilation centered at the origin. Which linear equation could represent its image?

(1) 3x - 4y = 9
A dilation would not change the slope. The slope of the original line is 3/4.
Choice (1) has a slope of 3/4.


11. Given triangle ABC with m∠B = 62° and side AC extended to D, as shown below.


Which value of x makes AB = CB?

(4) 121°
For AB to be congruent to CB, angle A must be congruent to angle ACB.
So y + y + 62 = 180
2y + 62 = 180
2y = 118
y = 59
Using the Remote Angle Theorem, x = 62 + 59 = 121.


13. A rectangle whose length and width are 10 and 6, respectively, is shown below. The rectangle is continuously rotated around a straight line to form an object whose volume is 150π.
Which line could the rectangle be rotated around?

(3) the vertical line of symmetry
Which line is used will determine what the radius is and what the height is.
Quick check: if the height is 10, then r2 = 150 / 10 = 15, which isn't a perfect square.
if the height is 6, then r2 = 150 / 6 = 25, which is a perfect square, and the radius is 5.
Since the long side has a length of 10, then a radius of 5 means you need to use the vertical line of symmetry and not the short side.


14. If ABCD is a parallelogram, which statement would prove that ABCD is a rhombus?

(3) AC is perpendicular to BD
What makes a parallel also a rhombus is that the consecutive sides are congruent. This isn't given. Choices (1) and (2) are also true for parallelograms that aren't rhombuses. (Rhombi?) However, another property of rhombuses is that the diagonals are perpendicular, which is not true for other parallelograms.


15. To build a handicapped-access ramp, the building code states that for every 1 inch of vertical rise in height, the ramp must extend out 12 inches horizontally, as shown in the diagram below.


What is the angle of inclination, x, of this ramp, to the nearest hundredth of a degree?

(1) 4.76
tan x = opp / adj
tan x = 1 / 12
x = tan-1(1/12) = 4.763...

More to come. Check back later ...

August 2017 Common Core Algebra 1 Regents, Part 4

Continuing with the August 2017 Algebra 1 Regents.
Here are the Part I questions, answers, and explanations.
Here are the Part II questions, answers, and explanations.
Here are the Part III questions, answers, and explanations.

August 2017, Algebra 1 (Common Core), Part IV

A correct answer will receive 6 credits. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit.


37. Zeke and six of his friends are going to a baseball game. Their combined money totals $28.50. At the game, hot dogs cost $1.25 each, hamburgers cost $2.50 each, and sodas cost $0.50 each. Each person buys one soda. They spend all $28.50 on food and soda.
Write an equation that can determine the number of hot dogs, x, and hamburgers, y, Zeke and his friends can buy.

Graph your equation on the grid below.
Determine how many different combinations, including those combinations containing zero, of hot dogs and hamburgers Zeke and his friends can buy, spending all $28.50. Explain your answer.

Answer: There are 7 people. Each buys a soda, which are $0.50 each. .50 * 7 = $3.50.
$28.50 - 3.50 = 25.00 to spend on hot dogs and hamburgers.

The equation is 1.25x + 2.50y + 3.50 = 28.50 or 1.25x + 2.50y = 25.00
The price of a hot dog times the number of hot dogs plus the price of a hamburger times the number of hamburgers plus the price of the sodas equals $28.50. Note that the question states that all the money was spent and that you are looking for an equation, not an inequality.

The number of $1.25 hot dogs that can be bought for $25.00 is 20. The number of $2.50 hamburgers that can be bought is 10.
There will be 11 possible combinations, which appear on the graph below. You don't need to list them, but they are:
(20, 0), (18, 1), (16, 2), (14, 3), (12, 4), (10, 5), (8, 6), (6, 7), (4, 8), (2, 9), (0, 10).

Here is the graph:

End of Part IV
End of Exam

How did you do?

Reminder: I've been asked not to post the questions to newer exams for at least a week after it.
So I probably won't be able to write up the answers to today's tests until after February 1.
Thanks for stopping by my blog. Feel free to leave a comment.

Monday, January 22, 2018

Hot Dogs, Buns and Laundry

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(C)Copyright 2018, C. Burke.

Household Math 101: Division of Chores

Don't ask a math person a question if you don't want to know the answer.

For anyone wondering, Mike was referring to non-chlorine bleach but try fitter all that, without resorting to brand names!

I've also discovered that I dislike making close-up versions of my characters even more than when I simply "zoom in" on their pixelated images.




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Thursday, January 18, 2018

August 2017 Common Core Algebra 1 Regents, Part 3

Continuing with the August 2017 Algebra 1 Regents.
Here are the Part I questions, answers, and explanations.
Here are the Part II questions, answers, and explanations.

August 2017, Algebra 1 (Common Core), Part III

Each correct answer will receive 4 credits. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit.


33. Loretta and her family are going on vacation. Their destination is 610 miles from their home. Loretta is going to share some of the driving with her dad. Her average speed while driving is 55 mph and her dad’s average speed while driving is 65 mph.

The plan is for Loretta to drive for the first 4 hours of the trip and her dad to drive for the remainder of the trip. Determine the number of hours it will take her family to reach their destination.

After Loretta has been driving for 2 hours, she gets tired and asks her dad to take over. Determine, to the nearest tenth of an hour, how much time the family will save by having Loretta’s dad drive for the remainder of the trip.

Answer: Loretta drives 55 mph for 4 hours. Distance = rate X time, d = rt
Loretta drives d = (55)(4) = 220 miles of the 610 miles of the trip.
Her Dad will drive the remaining 610 - 220 = 390 miles
Using d = rt again, 390 = (65)t
t = 6 hours her Dad drives.

Note: you will neither gain nor lose a point whether or not you capitalize the "d" in "Dad", but I'm a "Dad", so there. But I digress.

The total number of hours it should take it 4 + 6 = 10 hours of total driving time.

Second part: If Loretta only drives 2 hours, then they have only traveled 55 * 2 = 110 miles.
Dad has to drive 610 - 110 = 500 miles
500 / 65 = 7.69... or 7.7 hours
To the nearest tenth, the family will drive for 2 + 7.7 = 9.7 hours.
The family will save 0.3 hours in this situation.


34. The heights, in feet, of former New York Knicks basketball players are listed below


Using the heights given, complete the frequency table below

Based on the frequency table created, draw and label a frequency histogram on the grid below.

Determine and state which interval contains the upper quartile. Justify your response

Answer: The frequency table will look like the following. Note that you could have just the tally marks or just the numerical value, and you would receive credit for this portion of the question.


Note that they wanted a frequency histogram, not a bar graph, line graph or anything cumulative. It's important to leave a blank column between 0 and the first bar to indicate the gap, and you should include the little squiggle (which they are inconsistent about demanding).
Remember to add title, labels, scale, etc. Also, do not shade or draw anything to represent 0 for 6.6-6.7.

You can find the upper quartile by either writing out the numbers in order, then finding the median and quartiles. (There are 29 numbers, the 15th is the median, and the upper quartile is between the 7th and 8th numbers of the top half of data. This will give you 6.5.) Otherwise, you can say that 29 divided by 4 is 7.25, and if you could back from the last tally mark on the frequency table, you will find that the 7th and 8th marks are both in the interval 6.4-6.5.


33. Solve the following system of inequalities graphically on the grid below and label the solution S.

3x + 4y > 20 x < 3y - 18

Is the point (3,7) in the solution set? Explain your answer.

Answer: Remember to label the axis. The inequalities used are greater than and less than, so neither will have a solid line.
If you are using the graphing calculator to help with the graph, you need to rewrite the equations is slope-intercept form.

3x + 4y > 20
4y > -3x + 20
y > -3/4 x + 5
x < 3y - 18
x + 18 < 3y
(1/3)x + 6 < y
y > (1/3)x + 6

(3, 7) is NOT a solution because it is on the broken line, and the line is not part of the solution.


33. An Air Force pilot is flying at a cruising altitude of 9000 feet and is forced to eject from her aircraft.
The function h(t) = -16t2 + 128t + 9000 models the height, in feet, of the pilot above the ground, where t is the time, in seconds, after she is ejected from the aircraft.

Determine and state the vertex of h(t). Explain what the second coordinate of the vertex represents in the context of the problem.

After the pilot is ejected, what is the maximum number of feet she was above the aircraft's cruising altitude? Justify your answer.

Answer: The vertex can be found from the axis of symmetry, which is x = (-b) / (2a), where a = -16 and b = 128.
x = -(128) / (2(-16)) = -128 / -32 = 4
y = h(4) = -16(4)^2 + 128(4) + 9000 = 9256
Oddly, in this particular problem, no actual work was needed, and if you found the answer in your calculator by graphing and finding the maximum, you would have been fine. I do not recommend that approach, except to check your math.

The coordinates of the vertex are (4, 9256) where the second coordinate is the height of the pilot above ground level in feet.

The maximum number of feet the pilot will rise above the plane is 9256 - 9000 = 256 feet.

Wednesday, January 17, 2018

(x, why?) Mini: Noah

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(C)Copyright 2018, C. Burke.

He built it in degrees.




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Tuesday, January 16, 2018

Center

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(C)Copyright 2018, C. Burke.

Maybe it should be point PU?




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Friday, January 12, 2018

Nonplussed

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(C)Copyright 2018, C. Burke.

At least there was nothing negative.




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August 2017 Common Core Algebra 1 Regents, Part 2

I recently realized that students are reviewing the August 2017 Algebra 1 Regents, and I never wrote up the open-ended problems.
Here are the Part I questions, answers, and explanations.

August 2017, Algebra 1 (Common Core), Part II




25. A teacher wrote the following set of numbers on the board:


Explain why a + b is irrational, but b + c is rational.

Answer: The sum a + b is irrational because the sum of an irrational number and a rational number is always irrational.
The sum of b + c is rational because the square root of 225 is 15, which is a rational number. The sum of two rational numbers is always rational.


26. Determine and state whether the sequence 1, 3, 9, 27,… displays exponential behavior. Explain. how you arrived at your decision.

Answer: The sequence displays exponential behavior because it has a common ratio.
3 / 1 = 3
9 / 3 = 3
27 / 3 = 3


27. Using the formula for the volume of a cone, express r in terms of V, h, and π. how you arrived at your decision.

Answer:

V = 1/3 π r 2 h
3 V = π r 2 h
3 V / (πh) = r 2
square root of (3 V / (πh)) = r



28. The graph below models the cost of renting video games with a membership in Plan A and Plan B.

Explain why Plan B is the better choice for Dylan if he only has $50 to spend on video games, including a membership fee.

Bobby wants to spend $65 on video games, including a membership fee. Which plan should he choose? Explain your answer.

Answer: According to the graph, Bobby can get 14 games under Plan B but he can only get 12 games under Plan A.
According to the graph, if Bobby plans to spend $65, then both plans will give him the same number of games because that is the point where the two plans are the same.


29. Samantha purchases a package of sugar cookies. The nutrition label states that each serving size of 3 cookies contains 160 Calories. Samantha creates the graph below showing the number of cookies eaten and the number of Calories consumed.


Explain why it is appropriate for Samantha to draw a line through the points on the graph.

Answer: Samantha should draw a line through the points on the graph because she could eat only 1 or 2 cookies, or even a part of a cookie.


30. A two-inch-long grasshopper can jump a horizontal distance of 40 inches. An athlete, who is five feet nine, wants to cover a distance of one mile by jumping. If this person could jump at the same ratio of body-length to jump-length as the grasshopper, determine, to the nearest jump, how many jumps it would take this athlete to jump one mile.

Commentary: I'm surprised that they didn't include the unit conversions necessary to complete this problem. However, they are in the back of the test book.

Answer: The grasshopper is 2 inches and can jump 40 inches, which is a ratio of 40:2, or 20 times his body length.
The person is 5'9", which converts to 5.75 feet (because 9/12 of a foot is .75 feet). Multiply this by 20: 5.75 * 20 = 115 feet per jump.
One mile is 5280 feet, so 5280 / 115 = 45.9, which rounds to 46 jumps would be needed.

If you got as far as finding 115 feet (or 1380 inches), you would have gotten one point. You need to get to 46 jumps, rounded correctly, to get the other.


31. Write the expression 5x + 4x2(2x + 7) - 6x2 - 9x as a polynomial in standard form.

Answer: Use the Distributive Property, and then Combine Like Terms
Standard form means the term with the highest exponent goes first and then in decreasing order.
Remember that the sign goes with the term after it, and the first term here is positive (+).

5x + 4x2(2x + 7) - 6x2 - 9x
5x + 8x3 + 28x2 - 6x2 - 9x
8x3 + 28x2 - 6x2 + 5x - 9x
8x3 + 22x2 - 4x



32. Solve the equation x2 - 6x = 15 by completing the square

Answer: b = -6, so b / 2 = -3, which means that (x - 3) will be in the answer.
Also, (b / 2)2 = 9. Add 9 to both sides to begin.

x2 - 6x + 9 = 15 + 9
(x - 3)2 = 24
x - 3 = +(24)(1/2)
x = 3 +(24)(1/2)

It isn't necessary to reduce (radical (24)) into (2 X radical (6)).

End of Part II.
How did you do?
Corrections, comments, questions are welcome.

Thursday, January 11, 2018

The Roman One

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(C)Copyright 2018, C. Burke.

100 what 1 did there?




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Monday, January 08, 2018

New Mersenne Prime

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(C)Copyright 2018, C. Burke.

Joke's on Ken. You can buy a vowel!

First, here's a previous comic from 2008 about Mersenne primes. They've found a few more since then.

For those who don't know, Mersenne primes, they are a special subset of primes that could be written in the form of 2prime# - 1.

I'd say I'm waiting for the speculation about 2Mersenneprime# - 1, but I wouldn't be surprised if that's already been done.




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Thursday, January 04, 2018

Problem of the Day: Quadratic and Absolute Value Functions

During the break, an online friend, who knows I'm a math teacher (the user handle @mrburkemath is generally a tip-off), sent me the following math review problem:

"Given the functions h(x) = |x - 4| + 1 and k(x) = x2 + 3, which intervals contain a value of x for which h(x) = k(x)?
This was followed by a list of intervals in the form number < x < number. I've left these out as I have to assume the entire problem is copyrighted from a review book. I'm hoping my excerpt is covered by "fair use".

First thing I said what, in a purely multiple-choice format, just plug the two functions into a calculator (or use an online app, if you're at home without one) and look for the answers. Then select the intervals that include those values.

Otherwise, we can work it out. First thing to realize in that dealing with quadratics and absolute values, there can be, at most, two real number answers. (The problem didn't involve imaginary roots.)

The friend wasn't sure if she was supposed to set each equal to zero and solve, or set them equal to each other. It's in the question: h(x) = k(x).

So here is what we have:

h(x) = k(x)
|x - 4| + 1 = x2 + 3

First thing is to isolate the absolute value, by subtracting 1 from each side:


|x - 4| = x2 + 2

Many of you can skip the mini-review I'm going to do right now.
When solving an absolute value equation like |x - 4| = 7, you have to split the equation into two possibilities, one positive, one negative, and solve each.

x - 4 = 7x - 4 = -7

Applying that rule to this equation, you get
When solving an absolute value equation like |x - 4| = 7, you have to split the equation into two possibilities, one positive, one negative, and solve each.

x - 4 = x2 + 2x - 4 = -(x2 + 2)
x - 4 = x2 + 2x - 4 = -x2 - 2
0 = x2 - x + 6x2 + x - 2 = 0
No real roots Factor (x + 2)(x - 1) = 0
x = -2 or x = 1

If you check the discriminant (b2 - 4ac) for the first equation, you will get a number less than 0, meaning that there are no real roots.

Going back to the original problem, they wanted you to select any of the intervals that contained either -2 or 1 or both.

Had they asked for the points of intersection, the values of the functions for those x values, then you have to plug them in:
h(-2) = |-2 - 4| + 1 = |-6| + 1 = 6 + 1 = 7, (-2, 7)
h(1) = |1 - 4| + 1 = |-3| + 1 = 3 + 1 = 4, (1, 4)

Simple, right?