Four years ago, I came up with this math trick. Take a look at it, and at the end I'll show you why it works!
Let's play a game. I’m going to let you make up a math problem, and I will be able to tell you the answer from here. I can’t see what you’re doing, I’m not even in the same room as you, but I will still be able to tell you the correct answer.
Trust me. I’m a professional. Ready?
Okay. First, pick a number. It can be any number you wish, large or small. Now add 5 to that number. Got it? Okay, now double your new number (multiply by 2). Alright, now subtract 4 from the double.
Next, divide your new number by 2. Now, finally, subtract your original number from this new quotient. Got it? Okay. Here comes the cool part. Ready?
The answer is 3. Nifty, huh? What’s that? How’d I do it? Oh, magic.
Okay, okay, it’s not magic. The answer will always be 3, no matter what number you pick. Let’s illustrate this by...
Well, the new school year has started, and that means SAT test dates are fast approaching. In fact, the first one is this coming weekend. To anyone taking the SAT on Saturday, good luck! Remember to get a good night's sleep on Friday!
If you are thinking about applying to college in the next few years, it might be time to schedule an SAT date! Remember, you can retake the test as many times as you need to, so don't be afraid to schedule an early date.
Also, remember that the big SAT Redesign will be kicking into effect in the Spring of 2016, so if you are in the class of 2016 you may want to start your testing early, to make sure you have time to retake the current style of test and not have to relearn everything
for a completely new test the following year.
This semester's SAT test dates and registration deadlines are as follows:
October 11th – Registration ends September 12th
November 8th – Registration ends October 9th
Computerized spell-check can be a handy time-saver when writing papers, and many students swear by it. However amazing it may be, though, spell-check is still just a computer program, and as such should not be considered a substitute for proofreading with
human eyes. As evidence, here are three common mistakes that spell-check won't catch.
Spell-check uses a dictionary to compare the words you type to existing words. Proper nouns, like names of people or places, usually won't be in the computer's dictionary, and so the spell-check will flag them as misspelled. This means that when you proofread,
you'll have to ignore the wavy underline under those names. But this can also backfire – what if you happened to misspell that name? The computer will underline it same as before, but your brain is already prepared to ignore underlining on that name so you
run the risk of not catching it yourself. This is one reason I advocate actually printing out a hard...
We recently passed the 30th anniversary of "the 2 Sigma Problem," which is the problem of achieving the effectiveness of personalized, one-on-one instruction at a large scale. As a tutor, how do you help multiple students at the
same time while retaining the benefits of personalized tutoring?
To me, one of the major benefits of personalized tutoring is that the tutor has the space, time, and flexibility to respond to the student's needs. If I am tutoring a student in math, we can spend as many sessions as we need on a given topic to make sure the
student understands it thoroughly. I can also try a lot of different methods to explain a topic, since not everyone learns the same way. If a spatial or visual learner is having trouble with division, I might bring in a bag of M&Ms and show them physically
the process of dividing up a pile into smaller piles. If a student is having trouble understanding probability, I might bring in a set of polyhedral dice...
Suppose I place you at one end of a long, empty room. Your task is to get to the door at the other end of the room. Simple, right? But what if I told you that this simple task is actually mathematically impossible?
Think about it – in order to traverse the whole room, you first have to get to the halfway point, right? You'll have to travel one-half of the way there. And before you can get to that halfway point, you have to travel one-quarter of the way there (halfway
to the halfway point). And before you can get to the one-quarter point, you have to travel one-eighth of the way there (halfway to the quarter-way point). Since you have to go half of each distance before you can go the full distance, you'll never actually
get anywhere. The task requires an infinite number of steps, and you can never complete an infinite number of steps since there will always be another one. Furthermore, in order to even start your journey you would need to travel a specific distance, and...
WWTK: What advice would you give students going back to school so they start the year strong?
This is a great question, and one that I've answered before on this blog. In general, I'd say the most important thing for starting the new year strong is starting the new year ORGANIZED. Go back and look through your notebooks from the previous year, but not
for content – look at them like a detective. What does your note-taking style say about you? Do you have spiral notebooks stuffed full of handouts with rumpled edges? Are your note pages just solid blocks of hurried scribbles that are all but impossible to
read? Did you have to add extra notebooks halfway through the year? And most importantly, how easy is it to find a specific piece of information in one of your notebooks?
Take the opportunity while summer's still going strong to head to an office supplies store and wander around. Really look at all the organization solutions, and try to imagine yourself...
ALERT: This week's Literature Spotlight contains spoilers for The Hunger Games trilogy. Read at your own risk!
This week my Bring Your Own Book club met for tea, and our topic for the month was Dystopias. I had offered to host this month, because dystopia is one of my absolute favorite genres. As I sat listening to the others recount various dystopian tales,
I was struck by a thought that had been niggling at me for weeks – there's a significant difference between a dystopian setting and a true dystopian novel. With the increasing popularity of brilliant YA novels such as
The Hunger Games and Divergent, it's becoming more and more common to see stories set in corrupt dystopian societies – but are these stories true dystopias, in the classic sense of the word? There's more to a dystopian novel than a corrupt
society setting – classic dystopias also share certain plot and character elements. When viewed in this way works such as
The Hunger Games seem to fit more...
“Students often want to know how they'll use a subject "in the real world." Pick one of your subjects and tell us why it's important outside of the classroom.”
As it happens I wrote an article on this very topic as it relates to Algebra a few months back. You can check out that article
here. So since I've already answered this in relation to math, I'll discuss another of my topics today: writing.
It's true that once you finish college you'll probably never need to write another term paper. Unless your career path tends towards academics (or blog posting), regular paper-writing is probably not going to show up very much. But what will show up quite frequently
is the need to clearly and concisely articulate your thoughts and opinions in writing. In today's text-based world, first impressions are often written rather than spoken – whether that be a cover letter for a resume, a request for information about a position,
or a proposal for a new project...
There's a famous (and probably apocryphal) story about the mathematician Carl Friedrich Gauss that goes something like this:
Gauss was 9 years old, and sitting in his math class. He was a genius even at this young age, and as such was incredibly bored in his class and would always goof off and get into trouble. One day his teacher wanted to punish him for goofing off, and told him
that if he was so smart, why didn't he go sit in the corner and add up all the integers from 1 to 100? Gauss went and sat in the corner, but didn't pick up his pencil. The teacher confronted him, saying “Carl! Why aren't you working? I suppose you've figured
it out already, have you?” Gauss responded with “Yes – it's 5,050.” The teacher didn't believe him and spent the next ten minutes or so adding everything up by hand, only to find that Gauss was right!
So how did Gauss find the answer so fast? What did he see that his teacher didn't? The answer is simple, really – it's all about...
Bram Stoker's Dracula is a novel told in epistolary form – meaning the story is told entirely through documents, in this case journal entries and newspaper clippings. Epistolary is a very effective technique for writing certain types of stories,
and one that I feel is generally under-appreciated. In Dracula the epistolary form is used brilliantly to enhance the sense of mystery and suspense in the novel, and to add to the overall chilling effect of the story.
One of the ways in which epistolary form enhances the suspense is through the use of first person narration – from multiple sources. In a traditional first-person narration the reader follows a single protagonist, knowing only what they know and seeing only
what they see. This can be a welcome insight into a character's psyche, but can also be restrictive to the author since they cannot add outside information to the story. In epistolary form many characters can contribute first-person narratives to the novel...
This past weekend I went to see the long-awaited movie adaptation of John Green's bestselling novel “The Fault in Our Stars.” I'm a big fan of alternate-medium depictions of various art forms (movies based on books, theater, or games, books that expand
upon a movie or TV show, etc.) and I love to think about the ways in which a story is adapted for a new medium. Movies, TV, books, and live theater all have their own distinct methods of storytelling, and it's an enlightening exercise to think about how the
source material has to change to fit the new style. The Fault in Our Stars movie is one of the most faithful, and I think successful, adaptations I've seen in a long time. I'd like to take a moment to discuss a few of the ways in which I felt
they most successfully navigated the transition from book to movie. I'll refrain from spoilers in case any readers have not read the book or seen the movie yet.
The Fault in Our Stars, in book form, contains...
There's no such thing as the square root of a negative number. Right?
Since squaring a number is defined as multiplying it by itself, and multiplying a negative times a negative gives a positive, all squares should be positive. Right?
So any number you want to take the square root of should be positive to begin with. Right?
So what if it's not?
What do you do if you're chugging through a problem and suddenly find yourself confronted with
x = √(-9)
It seems like to finish this problem we'll need to take the square root of a negative number – but we can't, so what do we do? Drop the sign and hope nobody notices? Mark it as 'undefined' like dividing by zero? Give up? Cry?
Well, actually, we don't have to do any of that, because we've got an imaginary friend to help us.
i is a mathematical constant, whose sole definition is that i2 = -1. Or, in other words,
i = √(-1). i is an imaginary number...
Nobody likes doing homework in the summer. It's just a fact of life. My advice to students who want to stay sharp during the summer is to inject fun into your work and work into your fun. Find a way to connect your personal fun time back to the subjects
you're learning in school. The best way to accomplish this, in my opinion, is to look for school skills in unusual contexts. If you're interested in maintaining your English or writing, you can join a book club or arrange one with your friends. Take this summer
as an opportunity to read that book you've been dying to get to, and while you're reading think about it critically and talk about it with others. I'm part of a “Bring Your Own Book” club right now, where each month we are given a topic and have to find and
read a book that fits the topic. BYOBook clubs are a great chance to see a broad range of interpretations of a given theme and think about your reading in a larger context (what does the topic “animals as main characters”...
It takes practice to find your writing style, whether it be in fiction, research papers, or analytical essays. The best piece of writing is both grammatically correct and organized, but also contains the essence of the person who's writing it. When I correct
students' papers, I try to avoid suggesting alternate sentences in their entirety, since a paper written by you shouldn't sound like one written by me. Even if we are answering the exact same prompt in the exact same way, the tone and character of each paper
will be distinct, unique to each of us. Finding your style is a slow process, and generally comes about organically as a result of experience. Write more papers and you will begin to zero in on what makes a paper sing for you.
This is not to say that there aren't tips and techniques I can give to help you find your writing style. By far one of the most useful techniques in my own experience has been working with what I call “Finding your 'however'.” The name...
Crime and Punishment by Fyodor Dostoyevsky is a novel about guilt, morality and emotion. Throughout the novel many characters espouse the idea of reason and willpower over emotion – that if you have sufficient mental faculties you can prevent
emotion from getting in the way of your actions and behave truly rationally. The student Raskolnikov believes this with all his heart when he sets out to murder a pawnbroker for the good of the community. This concept is quickly proved to be fundamentally
flawed, however, as his inner guilt throws him into emotional turmoil and his brain attempts to protect him from the ugly truth of his actions. Raskolnikov displays several textbook examples of psychological defense mechanisms throughout the course of the
novel, proving that even the most thorough reasoning and intellect cannot prevent the emotional and psychological response to a crisis.
Psychological “defense mechanisms” are the brain's way of protecting itself from full awareness...
This week's Math Journey builds on the material in
The Function Machine. If you have not yet read that journey, I suggest you do so now.
In The Function Machine we discussed why graphing a function is possible at all on a conceptual level – essentially, since every x value of a function has a corresponding y value, we can plot those corresponding values as an ordered
pair on a coordinate plane. Plot enough pairs and a pattern begins to emerge; we join the points into a continuous line as an indication that there are actually an infinite number of pairs when you account for all real numbers as possible x values.
But plotting point after point is a tedious and time-consuming process. Wouldn't it be great if there was a quick way to tell what the graph was going to look like, and to be able to sketch it after plotting just a few carefully-chosen points?
Well, there is! Mathematicians look for an assortment of clues that help to determine the shape of a...
I hear a lot about math teachers from my students, and while every teacher is unique, some comments are repeated over and over. By far the most common one I hear is that their teacher didn't really explain something, or was incapable of elaborating when
questioned and simply repeated the same lecture again. As a tutor, my first priority is to make sure the student understands the material, and if they're still confused, to find another way to explain it so that it makes sense. In order to do that, I need
to have a thorough understanding of the concepts myself, so that I am not simply reading from a textbook but actually explaining a concept. In my years of tutoring math, I've developed a point of view and approach to math that I refer to as “teaching the concept,
not the algorithm.”
An algorithm is a step-by-step procedure for calculation. The term is used in math and computer science, but the concept of an algorithm is universal. I could tell you that I have an algorithm...
Prewriting often gets the short end of the stick with students rushing to get that paper written before its due date. Since many teachers don't require prewriting to be turned in with the paper, many students feel that it's a corner they can cut to save
time and launch straight into writing a first draft. In reality, prewriting is actually a great time-saver, particularly when you don't exactly know what you're going to talk about. It helps you to organize your thoughts, as well as make sure your points are
clear and your concept isn't too broad or too narrow. Prewriting is especially helpful in situations where you're given a very broad prompt – or even no prompt at all (as was the case with my IB World History term paper, whose prompt consisted of 'Write a
paper about something from 20th century world history'!)
Prewriting is usually defined broadly as anything you do before writing your paper, and can take many forms. This blog post will discuss a few of the most...
Since I've been tutoring English literature students, I've noticed a pattern: every time we read a book that I remember reading in my high school classes, I enjoy it far more as an adult than I ever did as a teenager. Time and time again I pick up a book
I remember hating in class, resigned to slog through it and discuss metaphor and symbolism with my student, only to find that I thoroughly enjoy it. Each time I come out of the unit with a fresh new appreciation for the work in question. As this happens more
and more I've come to the conclusion that there are whole worlds of theme and subtext in many novels that are only apparent to a reader who has reached adulthood, because they require the reader to have experiences beyond those of an average high-school student.
In today's Literature Spotlight I'd like to illustrate this point using a recently-transformed work for me, A Doll's House by Henrik Ibsen. One of the main themes in
A Doll's House is the idea of Nora's reluctance...
A few weeks ago I posted an article about the impending SAT redesign and the changes that have been announced. I mentioned at the end of that article that I'd be posting another one soon with my thoughts on the redesign, once I'd had time to think more
about them. Well, this is that article.
Overall, I think the motivation for the redesign is good – that the College Board's heart is in the right place and they're acknowledging some of the very real problems that the current SAT has. I'm very happy with their partnership with Khan Academy as well.
I'm happy to hear that they acknowledge that students really do need some kind of prep help for the SAT, and that if they're going to force every student who wants to apply to college to take it, they should be offering free prep help for everyone who wants
it. Not everyone can afford a private tutor, and money should not be a limiting factor in every student's ability to thoroughly prepare for the test. (That said, I am...