An ACT Essay Prompt Simulacrum

Scot : July 6, 2017 8:09 pm : Blog Posts

The ACT changed its essay format last year.  It is difficult to get much practice for this essay, because so far the ACT has only published two samples:  Intelligent Machines and Public Health and Individual Freedom (p. 54).  Here I provide a simulacrum, an unofficial sample prompt.  I made every effort to write this simulacrum as closely as possible to the official ACT essay prompts, including word count and tone.  Please feel free to download, print, use, and distribute this essay prompt as long as you keep the attribution at the footer, so future students can follow up and give me a call!

Download the ACT essay prompt simulacrum on PDF

Smart Phone Etiquette

An ACT Essay Simulacrum from Nth Degree Tutoring

Within the lifetimes of most people living today, smart phones have transformed the very nature of social interaction.  Today, we conduct a large fraction of our correspondence by text message and social networking.  This silent mode of conversation enables multiple simultaneous threads of dialogue, with people near and far.  What is the appropriate use of cell phones during face-to-face conversation?  A generation gap causes misunderstandings and frustration.  Do young people text more because they grew up with the technology, or because they haven’t learned good manners? We are in need of standards of etiquette.

Read and carefully consider these perspectives.  Each suggests a particular way of thinking about the appropriate use of smart phones during in-person conversation.

Perspective One Perspective Two Perspective Three
It is rude and childish to use a smart phone while talking to someone in person.  It consumes time and distracts from the conversation.  It also makes present company feel ignored and slighted. We should not exclude smart phones from live conversation as a matter of principle.  Sometimes they can enhance conversation.  That being said, most text-messaging and internet surfing is frivolous and can easily be moderated. It is old-fashioned to expect people to put their smart phones away.  Today’s lifestyle requires constant contact with outside people and information.  Those who can’t multi-task will just fall behind.

 

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And you thought the LSAT was hard! Logic problems involving others’ minds

Scot : January 1, 2017 9:49 pm : Blog Posts

black_white_hats

Sometimes to solve a puzzle, you have to think about what other people are thinking.

This Christmas vacation, don’t ask me how, but at one point our dinner discussion turned to a logic problem.  It’s an interesting puzzle, because it requires thinking about what other people do or do not know, and what they can or can not figure out based on their knowledge.  I realize that this problem has two forms, easier and harder, but they both involve the same backstory, something like the plot to the opera Turandot:

Prince Peter travels to a nearby kingdom to ask the king for the princess’s hand in marriage.  Unfortunately, two other princes are also there to make the very same request.  The king takes advantage of the competition to marry his daughter off to the smartest prince; he pits them against each other in a battle of wits.  The king seats the princes at a round table and blindfolds them.  “I have five hats,” he tells them.  “Three of them are white, and two are black.  I am placing one hat on each of your heads, and I will hide the other two.”  As he does so, he tells the princes that he will shortly remove their blindfolds.  “The princess will go to the first prince to correctly identify the color of his own hat,” he explains.  “If you guess incorrectly, I will kill you.  If you cheat by looking at your hat directly or in a mirror, I will kill you.  Don’t answer until you have correctly surmised the color of your own hat!”  He then has his assistants remove the blindfolds simultaneously.  The princes look at each other’s hats.  None of them offers an answer for several minutes.  Finally, Prince Peter laughs with delight.  “Of course!” he cheers.  “My hat is _______________ !!”  He and the princess live happily ever after.

The hard version of the question leaves off here, and simply asks, “What color was Peter’s hat, and what colors were the other princes wearing?”  You can try your hand at this question first, and if you’re stumped, peek at the clue in the easier version.

To view the clue, highlight the blank area below this line:

The “easier” (but still hard) version of the question adds, “Prince Peter saw that the other two princes were both wearing white hats.  What color was Peter’s hat?”

In order to arrive at the answer to this question (which I’m not going to post today), we have to give some thought to what the other princes would know / think, and how they would react, if they saw certain colors.  We have to assume that the princes are acting rationally (because of the high price for random guessing) but that the others have either less information or less intelligence than Peter.

This problem reminds me of a moment when I was listening to the radio at about age 12.  The DJ announced that he would award a cash prize to the 10th caller.  My first thought was, “If I wait enough time for nine people to call, then I can call and be the tenth.”  But then I realized, “Wait a minute.  Everyone else will be playing by the same strategy!  They are all going to wait and try to be the 10th caller.  Since nobody will even start to call for ten minutes, I’ll wait for 20.”  This turned into an infinite regress: “But wait.  Everybody will think the same thing again, so they will all wait 10 minutes longer, so I should delay longer … on and on to eternity!”  I wondered how this game could possibly be won.  I was flabbergasted when the song ended four minutes later and there was already a winner!  People had rushed to call!  That wouldn’t make any sense unless they hadn’t thought it through — or unless they knew that at least nine other players wouldn’t think it through.  I learned that sometimes to win a game, you have to be irrational or to assume that you are playing against unintelligent or irrational competitors.

Finally, we come to the hardest logic problem I have ever heard.  In this problem, the rules are that each player is infinitely intelligent (but not clairvoyant).  The judge selects two different natural numbers, m and n.  (The natural numbers are the counting numbers:  1, 2, 3, 4, 5, …).  The judge reveals the numbers’ sum (m + n) to Player 1, and he gives their product (mn) to player 2.

“I do not know what the two numbers are,” says Player 1.

“Neither do I,” says Player 2.

“Oh, then I do know what the two numbers are!” says Player 1.

“Then so do I!” says Player 2.

What are the two numbers?

In an ideal world, I won’t have to reveal the answers to these puzzles because someone else will in the comments below.  Is that a rational assumption?  😛

 

 

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“I’m concerned about pre-paying for a course I’ve never seen”

Scot : February 25, 2016 2:03 pm : Blog Posts

Nth Degree Tutoring is a one-man shop, a proud local business owner serving the community of Westside Los Angeles.  Without national name recognition like Kaplan or Manhattan, one of my challenges is getting students to feel comfortable pre-paying for the GRE course.  I understand if you feel some hesitation about paying in advance for a course you’ve never heard of.

First, I’d like to explain the reasoning behind this.  I started out tutoring just one student at a time.  Due to the extremely lucky twist of fate that the name “Nth Degree” has the letters “GRE” in it, I suddenly found my online profiles attracting more searches for “GRE” than anything else.  When I started getting weekly calls from new GRE students, I thought it would make sense to put them together into a single class.  How do you encourage students to enroll together?  By lowering the price.  I slashed the hourly rate from $60 down to $25.  That price allowed me to be the industry’s price leader, offering an 18-week course for $500, but of course it only made sense for me if at least three students enrolled.

For the first few sessions, I would get calls from four or five students who all said, “That sounds great!  I’m definitely interested!”  Then only one or two students would actually show up.  That left me in an awkward position.  I had to either bill those students regular rates or accept very low payments for myself.  Either way, it wasn’t fair.  I soon realized that I needed a way to find out which students were “thinking about it” and which ones were committed.  I asked students to commit to the class with a deposit.

Now, I know that $550 is a lot to ask, so I give you many options.  Some students have called and complained about “having to prepay without even getting to talk to the instructor.”  Well, that’s just simply not accurate.  I am available for free consultations four days a week.  Of course, you also have the option of booking a one hour one-on-one lesson with me first.  Some students like to have a chance to work with me just a little bit before making the bigger decision about the whole class.  You may also break your tuition into two payments of $295 each (a little higher because I bear the risk of students’ not completing the course).

“What if I have to be absent?” you may ask.  That’s covered too.  I have video make-up lessons, so I’ll send you the links to what you need.

Furthermore, your deposit is even refundable — except in the rare circumstances when there are exactly three or ten students enrolled.

There are numerous ways to get a feel for the class before enrolling.  See my Yelp reviews (upper right corner of the website)most of which were written by GRE students.  You can even ask for contact information for a student who has taken my course.  My YouTube channel (also in the upper right) hosts several video lectures I have recorded, most of which are GRE-appropriate.

All in all, I think that asking for a prepayment is fair and justified.  It is also standard practice.  I don’t know of any test-prep courses that let you pay as you go.

If you don’t feel ready to commit to a deposit, then the answer is simple:  don’t sign up for the class yet!  Book a one-on-one lesson or a free 30-minute consultation so you can get to know me before making that decision.

 

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Why my 6-week classes cost less, and why that’s good

Scot : July 30, 2015 8:30 pm : Blog Posts

As a small, local test-prep company, one of my business strategies is to be a price leader.  To take a GRE class at a major test-prep company like Kaplan, Princeton Review, Manhattan, or Blueprint, you can expect to pay anywhere from $750 to $1,800.  You can take an equivalent class at Nth Degree for $550, including the book and handouts.

I know what you’re thinking.  Test prep is important to you, and you don’t want to choose your service based strictly on the cheapest price tag!  Good point.  Let me explain and justify the difference.

The big difference between Nth Degree and a major test-prep company is size.  Nth Degree is a one-man operation.  I, Scot, am the business owner and the tutor.  I am hiring my time out directly to you.  Even if I get only three students, I earn $75 / hr — not bad.  With ten students, my income grows to $250 / hr.  That gives me every incentive to do a great job, because more students = more income for myself.  Because I can only afford so much rent, I can only accommodate ten students.  That’s good for you, because it guarantees small classes and individualized attention.

The other test-prep companies are agencies.  They have to cover the costs of management and operation expenses on top of the instructors.  Giants like Kaplan are mammoth national-scale corporations that spend millions upon millions of dollars in advertising and rents.  Their CEOs rake in huge salaries.  That explains why you would expect to pay them much more than me — it’s a measure of scale, not quality.  In fact, they like to pack you into large classes, the larger the better for their bottom line.  Would you rather study in a lecture hall or in a small seminar where you know everyone’s name and you can engage in deep analysis directly with the instructor?

At a test-prep agency, almost none of your tuition dollar goes toward hiring quality instruction.  As backward as it seems, agencies think of their instructors as costs that are to be minimized, rather than investments to help their business grow.  I have submitted applications to work for some agencies, but it is clear that they prefer entry-level tutors who will work for less.   Princeton Review offered me $15 / hr!  Even if I had gained a reputation for excellence and attracted dozens of students to their classes, I would not have earned one more dime.  Your instructors at the agencies are going to be a rapidly-revolving roster of graduate students who did well on their exams and now have this part-time job.  They are not going to be trained or experienced instructors.  By contrast, I am a professional educator with two decades of experience and a history of positive reviews.

Another issue is test questions and proprietary material.  Let’s say you are studying for the GRE.  The ETS, which writes the exam, publishes an official book of its test questions, and that’s what we use.  When you buy your book as part of my course, your money goes to ETS.  Major test-prep companies don’t like that.  They prefer to write their own proprietary test banks, because they sell them for profit.  That’s good for them, but it doesn’t give you the chance to practice with actual test questions!

So there you have it.  Nth Degree is your least expensive option because your tuition money is only going to one person, the one who is teaching you.  To compete with major agencies, I have no choice but to offer the best quality and the lowest price.

Hope to see you in class soon!

Scot

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Tests for Divisibility by 4 or 8

Scot : May 10, 2013 12:51 am : Blog Posts

There are many occasions in math problems when it is useful to know if one number is divisible by another.  For example, if you can tell at a glance that 3,521 is not divisible by 11, then you will not waste your time trying to divide 3,521 by 11!  Or if you need to reduce a fraction such as                         , it can be very useful to tell at a glance that 252 and 1,080 are both divisible by 9.  This can quickly facilitate the reduction of this fraction to , which can then quickly by reduced to  when 28 and 120 are both divided by 4.

There are certain slick tricks to tell at a glance if one number is divisible by another.  They’re called “Tests for Divisibility.”  Everybody knows the simplest test for divisibility.  If a number is even (i.e. if its ones digit is 2, 4, 6, 8, or 0) then that number is divisible by 2.  It doesn’t matter how big the number is or what the rest of its digits are.  This follows from the fact that 10 is divisible by 2, so after every cycle of 10, counting by 2’s starts over with 2, 4, 6, 8.  We can use this test to tell at a glance that 135,792 is divisible by 2 but 204,863 is not.

The tests for divisibility by 2, 5, and 10 all have to do with ones digits, so they are very simple, well-known, and intuitive.  Tests for divisibility by 3, 6, and 9 are slightly more complicated but also fairly well-known.  (They have to do with the sum of digits).  If you do some online research, you will probably even find the tests for divisibility by 7 and 11 – important because 7 and 11 are small prime numbers.

In my experience, I have never seen a particularly good test for divisibility by 4, and I’ve never seen a test for divisibility by 8 at all.  That bothered me, because 4 and 8 are such primary numbers (powers of 2, less than 10) that my gut told me there must be an easy algorithm for divisibility by these numbers.  The only test I’ve ever seen for divisibility by 4 is:  “If the last two digits form a multiple of 4, then the number is divisible by 4.”  For example, the number 5,128 is divisible by 4 because 28 is divisible by 4.  This follows because 100 is divisible by 4, so after every multiple of 100, counting by 4’s starts over with 04, 08, 12, … 88, 92, 96.  However, if you don’t know at a glance that 92 is divisible by 4 but 86 is not, then this shortcut won’t help you much with numbers like 592 or 37,086!  (For the record, if you want to be a seriously good math student, I do recommend that you memorize the “Extended Times Table” for all products up to 100.  That’s a blog for another day).

I knew that the key to divisibility by 4 must lie in the simple fact that 4 goes into 20.  Thus, after every multiple of 20, the multiples of 4 cycle through the same pattern of ones digits 4, 8, 2, 6  (such as 24, 28, 32, 36, or 64, 68, 72, 76).  Notice that whenever the 1’s digit is divisible by 4, the 10’s digit is even.  For instance, 24, 48, and 84 are all divisible by 4.  Whenever the 1’s digit is not divisible by 4, the 10’s digit is odd (e.g. 32, 56, or 76).

This rule is still a little complicated, so I learned that I can state it differently, in a way that everyone can understand and memorize:

 

Test for Divisibility by 4

 

Double     the tens digit and add it to the ones digit.

 

If your answer is divisible by 4, then     so is the number you started with.

 

Here are some examples:

 

Question:  How can I tell if 156 is divisible by 4?

Answer:  Double the 5 and add it to the 6.  This gives 10 + 6, which is 16.  Since 16 is divisible by 4, it follows that 156 is divisible by 4.

 

Question:  How can I tell if 3,978 is divisible by 4?

Answer:  Double the 7 and add to the 8.  This gives 14 + 8, which is 22.  Since 22 is not divisible by 4, it follows that 3,978 is not divisible by 4.

 

Frankly, it isn’t hard to just memorize the multiples of 4 up to 100.  The real power of this test is that it provides us with the pattern to figure out the test for divisibility by 8.  The test is based on the similar fact that 8 goes into 200.  Thus, after every multiple of 200, counting by 8’s repeats a cycle such as 208, 216, 224, …, 376, 384, 392.  The last two digits always form a multiple of 4, but not necessarily a multiple of 8.  Whenever the last two digits form a multiple of 8, the hundreds digit is even.  When the last two digits do not form a multiple of 8, the hundreds digit is odd.  These facts are summarized in this nice test:

 

Test for Divisibility by 8

 

Quadruple     the hundreds digit, and then add to the number formed by the tens and ones     digits.

 

If your answer is divisible by 8, then     so is the number you started with.

 

 

Here are some examples:

 

Question:  How can I tell if 472 is divisible by 8?

Answer:  Quadruple the 4 and add to the 72.  This results in 16 + 72, which is 88.  Since 88 is divisible by 8, it follows that 472 is divisible by 8.

 

Question:  How can I tell if 3,860 is divisible by 8?

Answer:  Quadruple the 8 and add to to 60.  This results in 32 + 60, which is 92.  Since 92 is not divisible by 8, it follows that 3,860 is not divisible by 8.

 

Question:  How can I tell if 2,992 is divisible by 8?

Answer:  Quadruple the 9 and add to the 92.  This results in 36 + 92 = 128.  Uh-oh, a big number!  Repeat!

 

How can I tell if 128 is divisible by 8?

Answer:  Quadruple the 1 and add to the 28.  The result is 4 + 28, which is 32.  OK, this is divisible by 8.  Therefore, so is 128, and consequently so is 2,992!!

 

Technically, these tests can continue indefinitely for powers of 2, but beyond 8 they are no longer simple enough to be helpful.  For instance, we could express the test for divisibility by 16 using the same pattern:

Octuple the thousands digit, and then add to the number formed by the last three digits.  If your answer is divisible by 16, then so is the number you started with.

 

Just one last example will illustrate why this doesn’t help much.

 

Question:  How can I tell if 3,464 is divisible by 16?

Answer:  Octuple the 3 and add to the 464.  This results in 24 + 464, which is 488.

Repeat:  How can I tell if 488 is divisible by 16?  Octuple the thousands digit (0) and add to the 488.  This results in 0 + 488, which is still 488.  Unless you know the multiples of 16 up to 1,000 off the top of your head, this isn’t much of a shortcut!

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Welcome to Scot’s Blog!

Scot : April 12, 2013 4:19 pm : Blog Posts

This is the first post in the Blog.  Newer posts will appear above.  Thanks for visiting!

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