Author: Scot

Nth Degree Tutoring is a premier service specializing in academic higher education.  I tutor math, science, English literacy, and standardized exams to students in high school, college, and beyond. I often get requests from families with younger children.  They ask a fair question: “Why don’t you work with younger students?” and they deserve a fair answer. 

In the tutoring industry, grades K – 8, high school, and college are distinct markets.  They have different demands, market rates, referral networks, etc.  To state it succinctly, I have made a business decision to serve some, but not all, of these markets.  This choice reflects on my strengths and limitations as a tutor.  Here are some specifics to consider:

• Grade K – 8 material is the easiest, and therefore this market is saturated with tutors.  It has the lowest price point.  You’ll be able to find good elementary math tutors for $20 – 30 / hr.  That rate is not practical for me, and there is no reason for you to pay me 2 – 3 times as much for the same service.
• I agree that some students are more mature than others. But everyone matures as they age.  In my experience, cohorts below 9th grade one-on-one, and below college in groups, pose non-academic challenges that other specialists will be able to handle better than I can.  I am not trained, certified, or skilled at working with any of the issues that are more highly associated with the K – 8 demographic, such as learning disabilities, attention deficit, dyslexia, emotional dysregulation, anxiety & depression, behavioral / talkative control, or boredom / negative attitudes about learning.  Younger parents are often looking for someone to “make learning fun”.  My lessons are always pleasant (maybe even fun by adult standards!) but I can’t claim to have the games, toys, or goofy antics to keep younger kids entertained.
• Parents of K – 8 students usually prefer house calls because their children don’t drive.  One of my greatest assets is a quiet, studious office.  Families with driving-age students will benefit most by coming to my location.

“But our child is so mature …” 

It is common for parents to request a special exception:  “Our child is so mature for his age!”  As you can see above, maturity is part, but not all, of the story.  I have made exceptions in the past, but I have learned from this experience that it is better to have a red line and stick to it.  Otherwise, “market creep” occurs very quickly.  Here’s a typical scenario: 

I tutor a 9th grade girl.  She has a 7th-grade brother, so I agree to tutor him too.  Then the parents start to refer me to his friends.  Another family will call and say, “We have a 6th-grade son.  We know that you work with a 7th-grader, so we figured you must work with 6th graders too?”  Before you know it, my name is all over the middle school.  You see where this is going. It just wouldn’t be fair for me to make exceptions for some families but not others.  

To keep it simple, I have established my red line at 9th grade.  I will consider 14-year-olds in the summer before 9th grade, although those summer lessons would be somewhat generic and not focused on any particular classes.    

Mark it on your calendar!

“Where shall we go for K – 8 tutoring in the meantime?”

I think you’ll find that K – 8 tutors come and go.  It is difficult for independent tutors to find clients, so the easiest way to connect with one is through a major agency.  Vacaville has a Kumon, Mathnasium, and Sylvan.  Kumon is a standardized program involving endless hours of repetitive drills, which is what many students need.  It will probably give you the best dollars-per-hour value.  

There is also something to be said for one-on-one lessons.  You can use Varsity Tutors to find one-on-one tutors online.  Agencies such as care.com or Tutor Doctor specialize in making house calls.  Their local availability is hit-and-miss.  You might also try local forums like Next Door, or simply ask around at your child’s school.  Working with a good tutor is much more important than being in the same room, so don’t shy away from Zoom lessons.

I hope to see you make the most of your child’s elementary and middle school years.  When you get to high school and those math classes start to challenge your childhood tutors, remember my number!  I am a professional (part-time) tutor with three decades of experience and another three decades of career path ahead of me.  I do not hire staff tutors; all lessons are with me personally.  All of my lessons for high-school students are individual, and I strongly recommend coming to my office in Downtown Vacaville for the best learning experience (I am also Zoom-ready).  Even if you find a good middle-school tutor, I am confident that you’ll benefit from a transition to Nth Degree when the time comes.

See you then!

Scot

The coronavirus epidemic affects us all in multiple ways. Some of us are more vulnerable or nervous or skeptical than others. I believe in respecting your choices while explaining mine. My personal choice was to limit all unnecessary interpersonal contact while any locality that I live in has more than a 1% active infection rate. That includes Rancho Park, Los Angeles City, Los Angeles County, California, the United States, and the world. As of July, 2020, the cities and counties of Los Angeles are infected above the 1% rate.

Therefore, my office will be closed to in-office lessons until we get back below that threshold. I am still available at all regular hours for remote services. That includes phone consultations, electronic document exchange, Skype “whiteboard webcam” lessons, and assistance with online courses.

Why 1%? Admittedly, it’s an arbitrary level. The way I look at it, when an epidemic is growing exponentially, 1% is within striking distance of large percentages. And when you go to a grocery store with more than 100 people in it, chances are the virus is in that store. I choose to play it safe, to protect myself, and to be part of the solution.

Last decade, when I taught math at Mt. St. Mary’s College, I made a video presentation about exponential growth. This lesson might surprise you. I hope you find it interesting and relevant to today’s pandemic.

When you first start preparing for a standardized exam, you usually take an entrance test.  It’s usually lower than what you had hoped for, and then you wonder, “How many points can I expect to gain?”  That is a very good question, and also a complicated one.  It doesn’t make sense to measure improvement in “points” or even in “percentiles”, because those units are not equal for all students (it’s much easier to gain 5 points or 10 percentile points at the lower end of the scale than the upper end).  The metric I use instead is the “Improvement Score”, IS, which is based on the percentage of students ahead of you that you can pull ahead of.  You can use your entrance diagnostic score and the average improvement score (IS = 50) to get a very good ballpark estimate of your exit score after an average program of study.

Use this spreadsheet:  How much will my score improve on the SAT / ACT / GRE / LSAT / GMAT?

View this video for an explanation:

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?  😛

 

 

I understand that large programs like Kaplan and Manhattan Prep. may offer you the chance to audit a class, i.e. to sit in on one session before committing to the full program. Unfortunately, I am not in a position to offer that option. These days, my classes are usually right at the minimum level of enrollment, so the number of students affects the duration of the whole course. The course is 3 weeks long if only 1 student enrolls, 4 weeks for 2 students, or 6 weeks for 3 or more students. Therefore, I’m afraid I have to know the enrollment numbers before we begin. This policy has made some students nervous about putting down a deposit.

First of all, let me explain why a deposit is necessary. Without it, I may get 5 or 6 students expressing interest in the course. Then maybe only 2 will show up. When that happens, I have to tell the students that the course will only be 4 weeks long, when they had been counting on 6. Collecting deposits before class ensures certainty.

Now, I know that several hundred dollars is a lot to ask, so there are compromise options.  I am available for free consultations throughout the week.  You also have the option of booking a one hour one-on-one lesson with me first. 

You should be advised that I do not refund or make up absences. However, bear in mind that the class is double-loaded. In addition to our class time, you also have a full math course component at home in the form of lecture videos. Those lectures alone would be almost an $800 value for a 1-on-1 student.

Your deposit is generally fully refundable before class begins. If you need to withdraw before Week 4, I even offer a pro-rated refund. The only exception is if exactly 2 or 3 students enroll. In that case, all students will be relying on each other to keep the course at its duration. If exactly 2 or 3 students enroll, then I will contact you immediately after the registration deadline and offer you a 24-hour period to withdraw. You might choose to wait until the next course to see if 3 or 4 people sign up.

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.

Finally, here’s the important issue: In all my years teaching GRE classes, I’ve only had one student withdraw because she was dissatisfied with the course — and that was for administrative reasons. She’s the student who complained about the deposit in the first place and inspired me to write this blog post! (Honestly, I never understood why it was such a big deal for her).

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.

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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