Standardized test math doesn't behave like normal math. On a normal math test, your knowledge of the concepts and material is being tested, using (hopefully) fair test questions. On a standardized test, though, they're looking for you to think outside the box, to apply math concepts and algorithms to unusual situations, and to really understand what they're looking for and find the quickest way to go about it. Let's take a question from a recent GRE student's lesson:
If 4x – 5y = 10 and 6y – 3x = 22, then what is x + y?
Now, this is a set of two equations with two variables each, so it looks to me like a perfect candidate for solving as a system. If I were solving this one on a regular math test, I'd start off trying the substitution method, since I'm more comfortable with that one. So let's explore that one first:
I'll start by solving the first equation for y:
4x – 5y = 10
- 5y = 10 – 4x
y = (-10/5) – (4/-5)x
y = -2 + (4/5)x
For this week's Ellen's Choice, it's time to run down another month of Reading Challenge books! Once again, it's a long one, so skim through for the titles in bold if you just want to see what I've read.
Book 11: “Red Seas Under Red Skies” by Scott Lynch
“a book with a color in the title”
Wow - Locke Lamora is at it again! Consider me officially hooked on this series now - yet another winner of a fantasy-action-con novel from Scott Lynch. In this one, Locke gets roped into being a real-life pirate - not by choice, unfortunately. Some great characterization and world-building ensues, with badass female pirates running a tight ship and standing toe-to-toe against much greater foes. Reading as Locke and Jean went full pirate-cliche for a con was incredibly entertaining, particularly since it's contrasted with the nowhere-near-cliche depictions of the actual pirates they're working with. I also really enjoyed getting to see more of the mysterious world where...
For this week's Ellen's Choice, I'm going to run down a month's worth of Reading Challenge books. It's going to be a long one, so feel free to skim through for the titles in bold if you just want to see which books I read and not my reviews of each one.
Book 5: "Catch-22," by Joseph Heller
"A book with a number in the title"
This one was to tackle “A book with a number in the title”. Of course, I could not pass up a chance to finally read a book that I’ve been wanting to read for years now – “Catch 22″!
The title has since become synonymous with the idea of a self-contradicting statement, a predicament you can’t get out of because of its self-referential nature. Catch 22 follows the experiences of a bomber pilot in WWII whose superior officer keeps raising the number of missions required to complete a tour so that nobody can go home. The protagonist’s efforts to worm his way out of flying missions repeatedly bump...
I recently read a new-ish novel by one of my favorite authors, the incomparable Terry Pratchett, that provided me with some much-needed food for thought.
The Long Earth, a collaboration between Pratchett and Stephen Baxter, centers around the invention and distribution of a simple contraption enabling its user to 'step' between an infinite number of parallel dimensions. Each of these dimensions is slightly different from every other, possibly depicting a series of 'what if?' alternate Earths, and the entirety together is referred to as 'The Long Earth.' One of the most curious things about the Long Earth, however, is that none of these alternate Earths have any humans on them – no cities, no civilizations, simply wild and beautiful vistas with plenty of local wildlife and a few enigmatic 'humanoid' races that are rarely seen. Forget space travel – mankind can simply step across the Long Earth and find millions of pristine new worlds to conquer!
The novel brings up quite...
Book 4: "a book of short stories"
I checked off "A book of short stories" and read this anthology edited by Neil Gaiman and Al Sarrantonio.
Gaiman begins the book with an introduction discussing how they collected the stories for this volume. He talks about the idea of four words that indicate a great story: "And then what happened?" This collection is stories that made him and Sarrantonio want to keep turning pages to see what happened next. I definitely see that common thread throughout the anthology. I also enjoyed the fact that each story was from a different author, and that most of the authors were ones I'd never read before. I'll almost certainly look up more of their work now that I've gotten a taste of what they can do.
The stories are all over the place in terms of setting, plot, characters, and writing style, though all have at least a little bit of that Gaiman-esque supernatural about them. I found that most...
Book 3: "A Play"
I threw this little beauty in my backpack a few months ago for a long day out teaching lessons. I've had Arthur Miller's "Death of a Salesman" on my shelf for a while, but I'd never actually read it cover to cover. I love reading plays, so if I can swing it there might be a few more of them in this year's challenge, but this one fulfills the "play" requirement.
I was fascinated by a couple of things while reading. The first was the subtle way that Willy Loman's mental state is depicted through intercutting of scenes. Willy is losing his mind due to stress, and struck me as perhaps having some kind of dementia. He lives in the past, drifting off into memories of happier times when things aren't going his way in the present. Several times in the play, there are two completely different conversations going on, one in the present where Willy is physically seated, and the other in the past where he is actually participating...
Toni Morrison's “Beloved” is a beautiful, poetic, and haunting work about love, motherhood and the lengths to which a mother will go to protect her children. The runaway slave woman Sethe kills one of her children (and attempts to kill the others) in order to save its life, and in doing so destroys the beautiful world she's tried so hard to create for the rest of her family. The baby's ghost haunts Sethe's world through the enigmatic character of Beloved, a character with at least three distinct possible interpretations. Is Beloved the ghost of the “already-crawling? baby,” or a simple runaway slave who just happens to call herself “Beloved,” or perhaps not even there at all? The novel is intentionally written to keep our interpretations vague, and each interpretation comes with its own commentary on the relationships in the novel.
On the one hand, Beloved could be a real, physical girl, not related to the family, who ran away from an abusive slave owner and found her way...
Book 2: "A Nonfiction book"
I came across this one while looking for an e-book to put on my iPad so I'd have something to read on those full-to-bursting-backpack days. It completes the item for "a nonfiction book," and wow...I tore through this book faster than I think I've read a nonfiction in a very long time. My favorite part about this one was Bill Nye's writing style: conversational, informal, strewn with jokes (such as the running gag of his old boss as a lesser-evolved life form), and unquestionably him. Fans of the Bill Nye the Science Guy show from the 90's (like myself) will be able to hear his voice quite clearly in their heads as they read - it's like an adult version of his show in book form!
I was also really impressed with the content of the book - I'd expected it to be much more about the debate with Creationist Ken Ham (the impetus for Bill writing this book in the first place). But he actually only touches on the debate...
I received this problem from a friend, who was having trouble while helping her nephew with it. It turned out to be quite a doozy, so I'm presenting it as today's Math Journey to show how the process we used last time works even with a gnarly, complicated problem.
Solve using the Addition Method:
3x – 3y + 4z = – 15
3x + y – 3z = – 8
23x – y – 4z = 0
As we discussed last month, the basic idea behind solving a system of equations is to use one equation to solve another for a specific variable, and to do that enough times that you can eventually rewrite one of those equations with only one variable in it, and solve from there. The way I learned to do this is the “substitution” method, where you solve one equation for one variable, plug the expression in for that variable in a second equation, et cetera until you're down to one variable. The addition method works a little differently, but it's the same basic goal: eliminate enough of the variables...
For the next few Ellen's Choices, unless I have a brilliant idea that must be discussed immediately, I'm going to start recapping the books I've read for my 2015 Reading Challenge. The Challenge is an ambitious attempt to read one book a week for the entire year, selecting each book based on prompt items in a checklist.
Book 1: "A Book You Started But Never Finished"
For my first Reading Challenge book, I tackled "A book you started but never finished" - and boy, am I glad that I did! I've had
Stormdancer by Jay Kristoff on my shelf for years, after I bought it and read the first few chapters, but got busy and lost interest. Turns out, this unassuming little book is an amazingly unique steampunk-dystopian-epic fantasy! The steampunk dystopia is thorough and riveting - I mean, c'mon, the sky is RED! And in amongst all the civil unrest, plots to overthrow society, and men clanking around in insectoid mechanical suits, there's...
Narrative Point of View, sometimes called Narrative Perspective, describes the position of the narrator in relation to the story. Commonly-used points of view include First Person, where the narrator is a main character in the story, describing the events using “I,” and Third Person, where the narrator is a separate entity describing the events of the story using “he” or “she”. Within Third Person there are two sub-categories dealing with how much information the narrator chooses to give. Third Person Omniscient places the narrator above the story, where they can provide narration of events that the main characters are unaware of. A good example of this is the Harry Potter series, where the books sometimes show scenes of the Malfoys, Snape, or Voldemort – things that Harry and his friends would have no way of knowing about. In contrast, Third Person Limited places the narrator inside the main character's head but not AS the main character – events are still described as “he did this”...
Settle in, folks, today's a long one.
The Function Machine, we learned that functions can be depicted as curves graphed on a coordinate plane. In
What Does the Function Look Like?, we learned how to tell the general shape of a function's graph based on characteristics of its equation, and vice versa. Today, we'll be focusing on linear equations (meaning any equation that graphs into a straight line).
The defining characteristic of a linear equation is that the highest power of x in the equation is x to the first. This denotes that for every y value, there is exactly one corresponding x value. Of course, there is always exactly one corresponding y value for every x, but this is one of those “square is a rectangle; rectangle is not necessarily a square” moments. We know there's exactly one y for every x because we choose our x's independently and the y's are dependent on them. There can't be more than one y for any given x; you've only got one output slot...
So-called “Young Adult” fiction seems to have gotten a bad rap lately among parents of teenage students. It seems as though adults tend to view YA as somehow “lesser” to other works, particularly as compared to the classics students are assigned in high school. I suspect this is because “Young Adult” as we conceptualize it today is a relatively recent invention – most bookstores and libraries didn't even have a YA shelf until the mid-1990s. When we were teenagers, there was no “Young Adult” section at the bookstore – there was “Children's” and there was “the rest of the store,” usually organized by genre. So as teens, too old for the Children's section, we chose books from the rest of the store based on genre or author. I enjoyed sci-fi and fantasy, for example, so I found each next reading experience in the Sci-Fi/Fantasy aisle of the store, reading greats like Larry Niven, Neil Gaiman, and Isaac Asimov.
Young Adult, though, is a completely different animal. It's...
Back in the day, where you wanted to go to college dictated which standardized test you took. Colleges in the midwest generally required the ACT, while those on the coasts wanted the SAT. These days, the score conversions are commonplace enough that most colleges will accept either one. So how do you choose which one to take? Well, there are a few differences to keep in mind.
Most of the differences between the tests are matters of format. The SAT is comprised of ten sections ranging from 12 to 35 minutes each. The sections alternate between reading comprehension, math, and writing, and the whole test begins with a 25-minute timed essay. One of the ten sections is an “experimental” section, which is not scored as part of your test and is a chance for the test-makers to try out new ideas on a group of students. The ACT, in contrast, is four 75-minute sections, one for each subject. The ACT does not include an essay, but it does include a “science” section (which...
What are your 5 outside the box tips that help make your tutoring lessons fun?
1. Have a sense of humor about learning.
I like to use humor in my tutoring, to keep students engaged and interested in the material. I've found that it's easy to zone out during a lesson, and the classes I've retained the most information from myself have been ones where the teacher employed humor. In a writing class, a teacher explained the importance of context to spelling with the quip “You need to remember which witch is which, or you'll suddenly have a lady with a black hat appear in your paper.” My high-school calculus teacher helped us remember the SOH-CAH-TOA trig function sequence by telling us a long joke about a native american who stubbed his toe and was advised by the village elder to “Soak-a-toe-a.” And later on, in a materials science class in college, our professor explained the molecular physics properties of a certain material with a joke about coal trying to rob aluminum...
Since it's Thanksgiving week, let's think about pie for a second. No, not mathematical pi, just actual real edible pies. For Thanksgiving I'm in charge of making dessert, so I'll be bringing two pies, one pumpkin and one apple. Let's say that I sliced the apple pie into 12 pieces, and the pumpkin pie, since it held together better, into 18.
Fast forward to the end of the evening. My pies were a big hit, and I have almost none left. In fact, all I have is three pieces of apple and four pieces of pumpkin. I want to combine the remaining slices into a single pie pan, so that they take up less space in the fridge. How do I figure out if my remaining pie will fit in one pan?
Well, let's start by writing down the remaining amounts of pie in the form of fractions. Remember, one of the definitions of a fraction is parts of a whole, so let's apply that definition to figure out our starting fractions.
The apple pie was cut into 12 pieces, and we have three out...
War of the Worlds, by H.G. Wells, is classic science fiction. Written in 1924, it depicts the catastrophic and totally unexpected near-extinction of humanity by aliens from Mars. One of the main themes running through
War of the Worlds is the idea that mankind's assumptions about their world, the universe and the nature of life are constantly being challenged. The main reason the martians' landing is so catastrophic to humankind is because the humans, by and large, have been lulled into a false sense of security. They believe they are capable of overcoming anything, that they are the most powerful beings in the universe, and as such are completely unprepared for the martians' attack.
Humans at the beginning of H.G. Wells's novel are portrayed as very self-satisfied. Even when confronted with the landing of the first martian cylinder, humanity is quick to dismiss the event as a mere curiosity. The story on the eve of the first day was “dead men from Mars,” (P. 14) and...
Ragtime, by E.L. Doctorow, is a story about the American dream. Set in New York during the “period of Ragtime” between the turn of the 20th century and the beginning of World War I,
Ragtime tells the story of three different families struggling to find their place in this new America.
Doctorow makes use of an unusual writing style in Ragtime. He eschews the use of quotation marks and line breaks during dialogue, making the visual appearance of the novel one of long, blocky paragraphs. In addition, Doctorow writes the novel in third person from the perspective of not one but all of the main characters, allowing us to see the innermost thoughts and feelings of everyone in the story in turn. The characters have various degrees of name specificity, ranging from simply “Mother” and “Father” to “Sarah” (nobody knows her last name) to “Coalhouse Walker Jr.” All of these stylistic decisions come together to make a surprisingly fluid novel where actions speak much louder than words...
“What advice would you give students to prepare for their first session with a new tutor?”
This is a great question! Overall, I think the most important piece of advice I can give is to put some thought into exactly what you want to get out of your tutoring sessions. Many people come to tutoring simply because their grades (or test scores) are low, and they're hoping that private tutoring can “fix” the problem. Which it probably can, but if that's all you bring to the table then your tutor has to work that much harder to figure out exactly how to go about helping you.
Before you arrive at your first meeting, spend some time thinking about your classes. Which subjects in school do you feel most comfortable with, and which ones least comfortable? Think over your answers like a detective – what common themes do you see that could be the real root of the problem? Were you easily able to ace an open-ended, discussion-driven English class, but this year your teacher...
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...