Month: March 2014

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The SAT Redesign: What You Need To Know

The news broke recently that the College Board is once again changing the SAT. These new changes, scheduled to be implemented in spring 2016, represent a pretty large departure from the SAT of the past. The College Board states that this new SAT will “ask students to apply a deep understanding of the few things shown by current research to matter most for college readiness and success.” Here are the changes that will have the biggest effect on test preparation, as I see them:

An Increased Focus on Evidence-Based Analysis

The new SAT will place a higher priority on analysis based on evidence. In the critical reading and writing sections students will now be asked to support their answers with evidence, including citing portions of the passages. In effect, the new SAT will require students not only to know the correct answer, but to be able to explain why the answer is correct, and point to specific evidence in the passage that supports their choice. The essay will also now focus on supporting claims with evidence (which I will discuss more below in the Essay section).

An Increased Focus on the Topics that Students Will Need in College

Vocabulary, Math concepts, and Passage selections will all focus on skills and topics that students are most likely to need for college and beyond. No longer will the SAT throw obscure words at students simply because they are difficult; the new test will focus on vocabulary such as “empirical” and “synthesis,” words more likely to be found in the college and career environments. The Math section will focus more narrowly in on a smaller range of topics more likely to be encountered in life, and will now include a calculator-forbidden section where students must rely on their own abilities rather than the calculator to solve problems. The passages selected for critical reading questions, previously chosen specifically to be obscure and offer no unfair advantage to students, will now be selected to be reminiscent of classroom work assignments in social studies, science, and history.

Data Analysis and the “Great Global Conversation”

Passages will now be chosen which incorporate both text and graphs or charts, challenging students to be able to synthesize data presented in many formats and analyze with equal skill graphs as well as written paragraphs. In addition, every test will include at least one passage chosen using their new concept of “The Great Global Conversation,” meaning that they will incorporate passages from the founding fathers and/or other notable historical figures. This is meant to draw students more into the current or historical conversations that matter in our country and encourage them to participate in those conversations.

The Essay

The essay is now optional rather than mandatory. Instead of a randomly-assigned prompt, all students will know the basic prompt ahead of time. Rather than asking for a student’s opinion on a question, the student will be assigned a random source document and asked to write an essay explaining how the author of the source document builds his or her argument. This will require students to be able to cite specific parts of the text and support their claims with evidence. While the basic prompt (“write an essay explaining how the author builds his argument”) remains the same, the source material in question will be random.

The Nitty-Gritty Changes

The Writing section has been abandoned, combined into one with the verbal now called “Evidence-based Reading and Writing”. Together with the Math section, this will return the scoring system to its pre-2005 1600-point scale. The essay, if taken, will be scored separately and will now be given 50 minutes instead of the current 25.

In addition, the new SAT has abandoned the penalty for wrong answers. Entirely. You now receive zero points for a wrong answer instead of the previous quarter-point off. The College Board states this as “encourag[ing] students to give the best answer they have to every problem,” but what this effectively means is that you are no longer penalized for guessing.

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I’ll post another blog entry in a few days with my thoughts on these changes, once I’ve had time to process them a bit more. For now, I think the initiative is a good one, though there may be ramifications of these changes that the College Board is unaware of.

One other thing to mention – in addition to the changes to the actual test, the College Board is now partnering with Khan Academy in an initiative to make sure that every child gets access to the test prep help they need, even those from less-wealthy families. I definitely approve of this initiative; it’s always bothered me that the SAT really does require special prep help to score well, when many students are unable to afford such help. For my part, I am always willing to adjust my rate for a student with a special circumstance, so if you need help, don’t hesitate to contact me!

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Mathematical Journeys: Undoing the Unknown Exponent

This journey is heavily inspired by the youtube mathematician Vi Hart, whose videos describing mathematical concepts through doodling in a notebook were the inspiration for much of my mathematical journeys series. I’ll put a link to her video on this topic at the end of the journey, and I highly encourage everyone to go check her out.

Let’s talk exponents.

But to do that, first we should talk about multiplication. Multiplication is a shortcut for adding a bunch of the same number together. If I gave you:

5 + 5 + 5 + 5 + 5 + 5 = ?

You could just add them normally, treating each of those 5’s as a size-5 step along the number line. But since each of these addition steps is the same size, a faster way to figure out the result would be to determine two things: the size of the step, and how many steps we have. Then we can multiply the size of step (in this case, 5) by the number of steps. In this case, we have a total of 6 size-5 steps, so we’d say:

5 * 6 = 30 Size of step (5) times number of steps (6) = total number (30).

Simple enough, right?

Well, exponents are a similar type of shortcut – except this time, it’s actually a shortcut for the shortcut! Exponents are a shortcut for multiplying a bunch of the same number together, in the same way that multiplication is itself a shortcut for adding a bunch of the same number together. So, if we had:

2 * 2 * 2 * 2 * 2 = ?

Again, we could multiply them normally, but that’d take a while. Since all the steps are the same size, we can take a notational shortcut and write this as an exponent. The size of step is the base (2), the number of steps is the exponent (5). So this case could be rewritten as:

2^5

It’s important to note that I said “notational shortcut” up there. This exponent has exactly the same value as that series of * 2’s above it, which in turn has the same value as:

{[(2 + 2) + (2 + 2)] + [(2 + 2) + (2 + 2)] } + {[(2 + 2) + (2 + 2)] + [(2 + 2) + (2 + 2)] }

But that is nearly impossible to read, let alone to work with – it took me several minutes just to work out where all the brackets went! So we use notational shortcuts to express these concepts – which at their heart are just complicated forms of counting – in a way that’s easier to work with. But it’s important to note that even the most complex high-school algebra is just a fancy way of writing some complicated counting. You could take the long way around if you really wanted to, but it’d be much too confusing for everyday manipulations, so we come up with shortcuts.

But back to exponents. Knowing our shortcut makes it relatively simple to figure out the total number when faced with a problem like:

2^5 = x

You just take 2 * 2 * 2 * 2 * 2, and you arrive at 32 as your answer. Again, the base is the size of step, the exponent is the number of steps. The only difference is that these are what Vi Hart calls “times-ish” steps – steps where you’re increasing the value not by adding, but by multiplying.

One of the basic truths of mathematics is that for every operation, there is a way to undo that operation. Want to undo a multiplication? Just divide. Want to undo an addition? Subtraction’s got you covered. (This, incidentally, is sometimes why students find themselves going around and around in an infinite loop within a problem – at some point they’re undoing an operation they’ve done elsewhere in the problem, and getting back to where they started.) To undo an exponent, generally taking a root will have you covered. For example:

We’ve arrived at 32 after 5 “times-ish” steps, but I want to know what size step we took to get there. Or, in math terms:

x^5 = 32

Roots have got you covered here. Just find the 5th root of 32 and you’ll have your answer:

(5th)√32 = 2

But wait! There are actually three different positions in the basic exponent equation, right?

b^p = r b = base, p = power, r = result

We’ve seen how to solve if our variable is the result or the base, but what if the variable IS the exponent? In other words, what if I know the size of step and the number I want to get to, but not how many steps it would take to get there?

2^x = 32

How do we undo THAT one?

This is a case where, because there are three parts to the original operation, we need a second way to undo it. THAT is where logarithms come in. Logarithms are the way to undo exponentiation to solve for the exponent itself.

b^p = r solve for p?

Log(b) r = p

Traditionally, we read the equation above as “The log in base b of r is p.” When you hear the word “log,” think “number of steps to get there.” So you could really read the equation as “The number of steps of size b to get to r is p.”

Most math teachers will give you a funky-looking image with arrows pointing in a circle around the exponent to show you which variable to place where, however those images have always struck me as more confusing than they are helpful. The way I remember which number goes where actually clues off of a later part of the sentence: the phrase “in base b.” Just remember, the base is the original number, the number that says how big the steps are. The logarithm is your way of figuring out the number of steps to a total, so the total should be on the same side of the equation as your base. That leaves the exponent by itself on the other side of the equation, where we can solve for it easily.

So for our example problem from earlier:

2^x = 32

log(2) 32 = x

So now we know what we’re doing. Generally we complete logarithms either by entering them on a scientific calculator or referencing a table, since it is very difficult to calculate the value of a logarithm by hand without using trial and error. Unfortunately this can lead to math teachers focusing on teaching students to simply memorize which numbers go where in the calculator, rather than actually teaching the theory behind what a logarithm IS and why we set it up the way we do. But hopefully you understand a bit more now than you did five minutes ago. Just remember, logarithms are nothing more than your way of undoing an exponent when the exponent itself is unknown.

~

Check out Vi Hart’s video here.

Extra Exposure on WyzAnt!

We interrupt your regularly-scheduled weekly posting for a special update!

A few weeks ago I was asked by the WyzAnt team to write a pair of articles offering an overview of the SAT Writing section and the SAT Essay. Those articles have just gone live on WyzAnt’s “Lessons” page! You can check them out at the links below:

SAT Writing Overview
SAT Essay In-Depth

The articles have a byline linking to my profile page on the site, so it’s great exposure for drawing new students to me. If you’re here, you don’t really need to use WyzAnt – you can just contact me directly at the email in the sidebar. But it’s still a great chance for you to experience a taste of my teaching style, so enjoy!

~Ellen

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WWTK: Mastering Challenging Subjects

WyzAnt Wants To Know: How did you go about mastering a challenging subject in school?

When I was in high school, I was fairly skilled at most subjects. One that always seemed to be an effort, though, was French language classes. They just seemed to have a lot of little parts to memorize and drill, so many irregular verbs, so many special cases. I held on, but it always seemed like a lot of work.

That is, until my senior year. Due to my ballet training, the IB-level French class that I should have been in didn’t fit into my schedule. As a compromise, the IB 2 French professor agreed to give me an independent study in French, which would be a self-guided project. I decided I wanted to read a selection of plays by the French playwright Moliere, in their original French, and write a paper about each play, also in French. Over the course of that year, I learned more about the ins and outs of French as a language than any number of irregular verb drills would have taught me, and I chalk it up largely to the addition of context.

I find context to be a very powerful tool to enhance your understanding of a subject. In math class, if you know what you’re doing in a broader sense, and why, and how it fits into related math topics, you’ll be better able to understand the concepts. In history classes, if you explore the ramifications of a piece of legislation in the context of an economic or political situation as well as in the context of the nation’s place on the world stage, you’ll have a better understanding of the full scope of the topic itself. And with me and French, reading my way through a play written in the language in question gave me a broader sense of how the language came together and was used. By reading something with a plot, that native French speakers might read and laugh at (I mostly read his comedies), I began to see the context for all of those grammar specifics we’d worked on in previous years, and that random assortment of rules became a natural way of composing thoughts. I got used to thinking in French, which goes a long way towards being able to converse in another language.

Having context for the work you’re doing can connect that work to its proper place in the world and give you a more thorough understanding of it as a result. Next time you’re struggling with a subject, take a step back and ask yourself “Why am I learning this? How can it help me in the future, and how does this subject affect other subjects I’m learning or have learned?” You might find that subject is more fun than you thought.

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Mathematical Journeys: What’s a radian?

Buckle up readers, it’s Trig time!

Trigonometry can be scary to many students, and in my opinion, a lot
of that is because one of the most confusing concepts in trigonometry
occurs right at the very beginning, in the form of the Unit Circle
and Radians.

Let’s start at the beginning. Give yourself a circle with a radius of 1. Now center that circle on the origin of a coordinate plane, so that the line of the circle itself passes through the points (1,0) (0,1) (-1,0) and (0, -1). Got that?

Now, this circle is referred to as the Unit Circle, because the radius is one unit and it is therefore easier for us to do various manipulations and calculations with it.
Now choose any point on the circle (we’ll call the coordinates of
that point (x,y)), draw the radius to it (which will still be a
length of 1), and drop a line back perpendicular to one of the axes.
Do that and you’ll have a right triangle with the radius as the
hypotenuse, meaning it has a hypotenuse of 1. This becomes very
useful later on for side length calculations involving the
Pythagorean Theorem. But for right now, let’s talk angles.

So in basic geometry you should have learned that there are 360 degrees in a circle. If we want to find out the measure of the angle located at the origin in our right
triangle above (what is usually referred to as a central angle), it
makes sense that we could figure it out based on what portion of the
circle the angle is taking up. If the last slice of pie was taking
up 1/8 of the full pan, we’d know that that angle was 1/8 of 360
degrees, or 45 degrees. That’s why we bother with the circle in the
first place, rather than simply dropping the right triangle randomly
in the middle of a blank piece of paper. It gives us some context to
work with, so that we can think of our angles as parts of a whole.

So what is a Radian?

Well, a Radian is a unit of angle measurement, like degrees. Only this unit is written in terms of the distance around the circumference. So whereas degree measurements
are based on a portion of 360 (360 being the “whole” benchmark),
Radians are based on a portion of the circumference. They become
very handy for finding the measure of an angle when all you know is
the length of the arc that the angle cuts out of the circle. Say you
didn’t know what portion of the pie you had left, but you had a tape
measure and could measure the length of its crust. You’d be left with
what we call an Arc Length, or the distance around the circle that
the angle in question cuts off, and with the power of radians, you
could figure out how much of the pie you had left.

So let’s talk radians, shall we?

The circumference of a circle, you’ll remember from geometry, is 2πr, or 2 pi times the radius. Well, we know the radius of the Unit Circle, it’s 1, remember? That’s the definition of a Unit Circle. That means that the circumference of the Unit Circle (and of our tasty dessert) is simply 2π.

So if we wanted to know the measure of that mystery angle at the center of our last piece of pie, we could certainly find it in terms of what portion of 2π the length of that crust was. If it was half of 2π, we’d have half the circle, and so on. In fact, before we come back to our tasty treat, let’s look at some simple examples to give you an idea of what’s going on here.

I just mentioned half the circle, so let’s look at that first. Now, we
know that that angle would be 180 degrees, since it’s a straight line
and half of the full 360, but what portion of the circumference would
it be? What would the length of the curved portion of that
hemisphere be? Well, it’d be half of 2π,
wouldn’t it? So it would be 2π/2, or just a single π.

Let’s look at one-quarter of the circle. We know the angle would be 90
degrees, and the circumference would be one-quarter of 2π,
which is 2π/4,
or π/2. If we look at three-quarters of the circle, similarly we
find that the circumference is ¾ of 2π,
which is 6π/4, or 3π/2.

What we’ve just described are the measures of those angles in radians.
Don’t be confused into thinking that one is arc lengths and the other
is angle measurements; radians are a unit for describing angles in
terms of circumference, so it’s actually still the same thing. It’s
like describing the volume of a container in both cups and liters;
the actual volume hasn’t changed, but the metric by which you’re
measuring it has.

These are all good values to remember, by the way, as they make the
conversion from degrees to radians much easier. Commit these four to
memory – it will come in handy later, I promise!

90 degrees = π/2 radians
180 degrees = π radians
270 degrees = 3π/2 radians
360 degrees = 2π radians

Incidentally, you’ll now see why I harp so much on leaving values as fractions for
as long as possible. Many students like converting to decimals
because the decimal version often looks less intimidating, but a lot
of higher math disciplines (like this one) virtually require you to
be able to work with fractions. Take 3π/2, for example. Plug that
into your calculator, and you’ll get 4.712388980385, but if I handed
you that number on a test, you’d have absolutely no way of knowing
that it’s actually 3π/2 and therefore a 270-degree angle or
three-quarters of the circle, either of which are much more useful
things to know than that random string of numbers. This is
especially important wherever π or other irrational numbers come in,
since they are often non-repeating, non-terminating decimals which
would have to be rounded off and therefore render you an estimation
rather than an exact answer. 3π/2 is exact; 4.712388980385 is not.

But back to our tasty dessert conundrum. Say you measure the crust on
your remaining piece of pie, and it comes out to be π/4. (Don’t ask
me what kind of tape measure would give you that; maybe it was a
hand-me-down from your mathematician grandfather. Problems get
really funky if you try to take the π out of the picture here.) How
do you figure out what the angle is in degrees?

Well, by now you’ve probably figured out that π/4 is the measure of the
angle in radians. So to convert to degrees, what do you do?

The easiest solution is to set it up as a proportion. You remember
those, right?

(π/4)(part) / 2π(whole) = x (part) / 360 (whole)

And solve from there. Cross-multiply:

360π/4 = 2πx

Don’t worry, the π’s cancel each other out. Now do you see why I like
leaving things as fractions?

360/4 = 2x
90 = 2x
x = 45

So that π/4 radians converts to a simple 45 degree angle. And, if
you’re feeling industrious, that means that the piece of pie you have
left is 45/360 or 1/8th of the whole pie. Nifty, huh?

Although, to be fair, you could have figured out how much of the pie you had
without ever converting to degrees in the first place. You simply
know that you have:

(π/4)(part) out of 2π (whole), so

(π/4)/2π = π/4 * 1/2π = 1/(4*2) = 1/8

So 1/8th the total pie. The π’s cancel each other out again. Starting to
make sense to you? Always leave things as fractions!

It’s beneficial to get used to working with radians right off the bat, as
most higher math levels will work almost exclusively in radians
rather than degrees. Why? Well, radians are generally more useful
in wider contexts than degrees, partly because they are based on a
distance measurement rather than a somewhat arbitrary 360. You just
have to learn to take a deep breath and not be intimidated by working
with π…it’ll work itself out in the end, I promise. And if you
come up against a problem where the π’s don’t cancel themselves out,
it’s always permissible (and in fact preferable) to leave your answer
in terms of π, where it’s easier for the next mathematician to pick
it up and work with it further.

I hope you’ve enjoyed our mathematical journey, and that radians make a
bit more sense to you now. Stay tuned for more exciting journeys in
the future!

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Sloppy Math on Facebook, or “Who the heck would write a function like that anyway?”

Does this look familiar?

SOLVE IF YOU ARE A GENIUS! 99% OF PEOPLE WILL GET IT WRONG!

8 = 56
7 = 42
6 = 30
5 = 20
3 = ?

No doubt every time you’ve seen this on
Facebook, it’s followed by thousands upon thousands of responses,
each indignant that other people are getting the wrong answer.
Generally there are two or three different numbers that keep coming
up, with nobody able to see how anyone else could have gotten a
different answer from their own.

I hate these things.

These things are designed to be vague. There is no answer, or rather,
there are an infinite number of answers. The crux of the issue here is
that they don’t define the rule.

So these things are basically a weird
way of presenting a function. You remember functions from my
previous blog post, right? Well, essentially what this thing is
saying is “you take 8, do some mystery function to it, and you get
56. Et cetera, et cetera, what do you get when you use 3?” The
problem is that there are multiple rules that could apply here, so
you have no idea what function they actually gave you and therefore
cannot answer the question.

Here’s our example from above, showing
two different rules that could be used to create the list:

FIRST OPTION
y = x * (x-1)

56 = 8 * (8-1) = 8 * 7
42 = 7 * (7-1) = 7 * 6
30 = 6 * (6-1) = 6 * 5
20 = 5 * (5-1) = 5 * 4

So
y = 3 * (3-1) = 3 * 2 = 6

This is the answer that a lot of people
get because it’s a bit more obvious. It’s also the first answer that
jumps out at me. But there’s a second option:

SECOND OPTION
n1 = 7
nt = n1 – (t-1)
y = x * nt

This second option is more of a
programming-type sequence, but no less legitimate. In this case, nt
means “the value of n for any given term t.” The rule given for
finding nt works out to mean that after each term, the n
value counts down by 1. n1 would be set to 7 by the
puzzle, as is common with rules of this type. This means the rule is
no longer relative, but absolute. No matter what first x value you
choose, the first n is 7. The second n is 6, the third is 5, and so
on. By this logic the beginning is still correct:

56 = 8 * 7 First Term: t = 1, nt = n1 – (1-1) = n1 – 0
42 = 7 * 6 Second Term: t = 2, nt = n1 – (2-1) = n1 – 1
30 = 6 * 5 Third Term: t = 3, nt = n1 – (3-1) = n1 – 2
20 = 5 * 4 Fourth Term: t = 4, nt = n1 – (4-1) = n1 – 3

BUT THEN

y = 3 * 3 = 9 Fifth Term: t = 5, nt = n1 – (5-1) = n1 – 4, so 7 – 4, or 3

With this version you get 9 instead of
6. Why is that, when the other terms are all the same?

Notice they left out x = 4 in the
puzzle – it jumps straight from 5 to 3. If we were using the first
rule it wouldn’t matter, since the rule is only relative to the
current x value and will work no matter where in the sequence a given
number finds itself. You could shuffle the lines around to your
heart’s delight and the answers would still be the same. But this
version is what’s known as a “recursive” sequence, where the rule
depends on where the term is in the sequence, and moving the terms
around will change the results drastically. What looks like a
logical sixth step is actually the fifth term in the sequence.

If we put 4 in for x where it looks
like it should be, it changes how you find the result of each term
thereafter. Then we’d have:

20 = 5 * 4
12 = 4 * 3
6 = 3 * 2

So we’d get the same answer as the
first option. But that’s assuming they’ve left out a term, which you
can’t be sure of since they didn’t tell you that explicitly. To
assume that is simply to be a sloppy mathematician, and sloppy math
leads to incorrect math. Here is what they would have had to show
you for that to be the case:

n1 : 8 = 56
n2 : 7 = 42
n3 : 6 = 30
n4 : 5 = 20
n6 : 3 = ?

By showing you that it jumps straight
from the fourth term to the sixth, they’re letting you know the
counter should have ticked down one additional time, and at that
point you will be able to solve the problem. But not without that
additional piece of information.

So here I’ve shown two legitimate ways
to solve this badly-written problem, getting two completely different
answers. Since the issue is that the problem is not specific enough,
the only prudent answer to these questions is always “There is not
enough information given.”

I hate these things because they are
intentionally vague. The people who make these problems (or at the
very least the people who post them on Facebook) likely don’t realize
just how complicated the problem is. Most people will only see one
solution, so it will appear easy and they won’t be able to understand
how someone else got 9 when it’s “obviously” 6. This leads to
arguing futilely over something which, to any mathematician, is just
a problem that’s so badly-written it’s useless. No mathematician
worth her salt would ever write a sequence that way anyway, so why
argue about it?

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WWTK: Preparing to go back to School

Got this topic from WyzAnt this morning: How should students prepare to
go back to school if they only have a few minutes to spare each day?

Good question. I think it’s important to spend some time thinking
about the big picture of the coming year and getting organized, so
that you start out on the right foot. I believe it’s a very
personal question, since as a student you have to decide what you
want to get out of the coming school year as opposed to the previous
one. What areas did you feel you were lacking? What are
you most excited about? What are you least excited about or
most dreading? And why?

Here’s something that doesn’t occur to a lot of people: what format are you
learning in? I’m talking about two distinct things here – your
supplies and setup, and the way you approach classes. Let’s look at
supplies and setup first. Think about your usual note-taking
setup – is there anything you’d like to change? If you’ve been
using spiral notebooks but are finding you have to keep flipping back
and forth a lot while studying, for example, maybe you should try
switching to the flexibility of a three-ring binder. That way
you can reorganize your notes to your heart’s content, putting the
important notes together after the fact so they’re easier to refer
to. This is also handy if you have a teacher who gives a ton of
handouts; you can punch holes in the handouts and put them into
exactly the right place in the binder. What about the paper
you’re using? If you feel like you’re wading through reams of
notes to find the important bits, try switching to annotation-ruled
paper (paper with a little blank box down one side) and put the large
bullet points in the box and the detailed notes on the full sheet.
You can also put stars or asterisks in the box next to important
information so it’s easier to find. Then studying will be a
quick matter of scanning the box and referring to the full sheet when
necessary.

Don’t overlook the little things, either. One of the best decisions I ever
made in my academic experience was the year I switched from
traditional spiral notebooks to ones with the spiral on the top edge
rather than the left (sometimes called “steno” pads or
“stenographer’s” notebooks). It sounds like a small thing, but
it meant my hand no longer hurt from sitting on the wires when the
notebook was flipped over. That in turn made it easier for me to
concentrate, since I wasn’t thinking about how uncomfortable writing
was. Along the same lines, if you use a three-ring binder, consider
getting a clipboard and putting your looseleaf paper on it to write
instead of leaving it in your notebook. You’ll avoid having to write
around those rings when you’re on the back side of a sheet of paper
(or disrupting class by opening and closing the rings repeatedly to
remove each sheet), and you can simply transfer the completed pages
to the binder after class. I have found that both of these methods
have the added benefit of making me feel like I’m really taking
notes, rather than simply scribbling down everything the teacher
says. I can sit up and look around more easily, looking directly at
the teacher as they talk and then glancing down to write my short
reminder note about the topic.

This brings me to my second point: how are you approaching the class
itself? Making decisions about this sort of thing is much easier if
you know a bit about how you personally learn best. There are three
basic types of learners: visual, auditory, and tactile. Most of us
are combinations of these three, but usually one is dominant in some
way. Not everyone responds best to the “scribble down everything
he says without any pronouns” method of note-taking, and the
important thing is to make sure you remember the material, however
you need to do that. For example, I myself am a mostly visual
learner; if I can see someone solve a problem I can more easily solve
it myself, and I tend to think of things in a visual way. This made
me a stronger student in subjects where the material was visual, such
as mathematics and arts, and a bit weaker in things like history.
However, I have quite a bit of tactile learner in me as well, so I
found that in history classes, the act of taking notes was enough of
a reminder to me. I didn’t actually need to look back through my
notes that much, since writing them down locked them into my brain.
Even if we got a handout with all of the information on it and I was
tempted to just refer to it, I made myself write out anything
important since I knew I’d remember it better that way. Auditory
learners learn best when they hear something said to them, so if that
sounds like you, you might try bringing a tape recorder to class (or
use the record function on a smartphone) to record the teacher’s
lectures. Then you can play it back for yourself if you need to
refer to it later. Of course, some schools have rules about
electronic devices, so ask your teacher first if it’s okay to bring
it in and explain why you want to.

Taking the time to think about these small things can make all the
difference later on. I’ve tried many different note-taking systems
over the years, and some work better for me than others. What works
for me might not work for you. You just have to experiment until you
find one you like. Take a field trip to the office supply store, and
just wander around the school supplies looking at all the options.
Maybe something will jump out at you. Eventually, you’ll find the
method that works best for you.

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Ellen’s Choice: Occam’s Razor and SAT Math: Why the SAT Messes With Your Head

The SAT messes with your head. Don’t feel embarrassed, it messes with everyone’s head. It’s designed to. The SAT is a test of your critical reasoning skills, meaning it’s actually far more about logic and figuring out the correct course of action than it is about actually knowing the material. This is nowhere more evident than on the Math section.

The SAT Math trips up so many students because they expect it to behave like a math test. The truth is, the SAT Math is about figuring out how to answer each problem using as little actual math as possible. It’s all about working quickly, and the questions are structured such that they conceal the quick logic and context-based route behind the facade of a more complicated math question. They’re trying to psych you out; to make you think the problem is harder than it is. In math class you’re taught to be thorough, to show your work and not leave out any steps. On the SAT, it’s not only possible but downright preferable to solve as many questions as you can without ever picking up your pencil.

Let’s take an example that caught me on the first go-round.

If a + b = 3 and ab = 4, what is 1/a + 1/b ?

Since you have two equations with two variables, your first instinct might be to treat them as a system of equations. Namely, solve one for a, plug the result in for a in the other one, get it into quadratic form and solve from there, then use the newly-found roots in place of a and b in the final equation. If you take the time to go through the problem this way, you find that your answer is almost one of the options, but the sign is wrong (your answer is negative when the answer choice is positive). This can frustrate you to no end and cause you to waste precious minutes of your testing time checking and rechecking your math. The intended course of action is much simpler.

What the SAT wanted you to do with that problem is to just look at the last expression, the one you’re trying to find. How would you normally add two fractions with different denominators? You’d find the lowest common denominator and multiply, right? So just treat the variables as you would numbers:

1/a + 1/b = b/ab + a/ab = (a + b)/ab

By now you should have noticed that both the numerator and denominator of this final expression are identical to the first two equations they gave you. All you have to do is plug them in.

(a + b)/ab = 3/4

And that’s literally it. They knew you’d over-think it if they set it up to look like a system of equations rather than simply an algebraic manipulation where they happened to give you the answers in the problem. They also knew you’d likely taken advanced algebra and trig, and would probably forget that the test doesn’t cover those topics. The realization that it looks like a system of equations, and that you know how to solve those, would push you into starting the (somewhat long) procedure and it wouldn’t occur to you to find a simpler way. Since their goal is to trip you up, make you waste time, psych you out, and mess with your head, they’ll do everything they can to camouflage the quick shortcut behind big scary problems.

It’s helpful to remember Occam’s Razor; the simplest solution to a problem is usually the best one. Always look for the common-sense answer, and remember not to over-think it. This is also why I think it’s beneficial for a student to go into the SAT with an attitude of healthy indignation and skepticism; if you’re always looking for the traps, you’ll be less likely to fall into them. Rather than getting frustrated, if you can see the trick question coming and confront it with a “Seriously?!” you’ll be better prepared to deal with it.

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Mathematical Journeys: The Function Machine

y = f(x)

I can’t tell you how many times I’ve had students come to me profoundly confused about their entire math unit, all because their teachers never fully explained this concept. Teachers throw this equation up on the board without discussion as if it explains everything – which it does, but only if you know what it means. So let’s discuss!

First off, it’s important to remember that this is not just an equation; it’s an indication of a larger concept. We’ll get to that in a minute, but let’s start at the beginning.

Imagine that I have a little machine which I set on the table in front of you and turn on. You place a number in the slot in the top, and the machine begins to hum and churn. After a few moments, a drawer opens at the bottom and you pull out a different number. You can repeat this with any number you like, any number of times.

Now this is a single-purpose machine, which means it has one rule that it uses to transform the starting number into the final number. When you insert a number, it applies the rule to it, and presents the result. For example, say my machine’s rule is that it takes the input number and adds 4 to it. If you give it a six, it will give you a 10. If you give it a 3, it will give you a 7. No matter what number you give it, it will add four and then give you the result. In math terminology, we call machines like this “functions.”

Definition: A “function” is a machine that transforms one value into another value via a set rule.

The notation for a function is

f(x)

where x is the input value. When you see an equation in the form

f(x) = x + 4

That is simply defining the rule of the function. In this case, this is the mathematical notation for our x + 4 machine from earlier. A common question type when learning about functions is as follows:

f(x) = 3x + 6. What is f(5)?

Don’t panic! This question is simply asking you to send a number of their choosing through your machine. In this case, the questioner has handed you a five and a machine with the rule 3x + 6, and wants you to give him his result. You just substitute 5 for the x in the rule and solve it. In our case,

3(5) + 6
15 + 6
21

So the answer is 21.

An important thing to note at this point is that the “f” in f(x) is NOT a variable like x or y. “f” is an OPERATOR, like a plus sign or a division bar. It’s an indication of an operation being performed. You could replace the “f” with literally anything, so long as you define what the rule of the machine is so you know it’s a function. The full notation for this operator is actually “f()”, and you put your starting value inside the parentheses. You’ll see this a lot on the SAT, where they include questions which replace the “f” with a series of nonsense symbols in order to test your understanding of the basic concept. They may even leave out the parentheses to confuse you further. On the SAT, you might see a question like this:

@ is a function such that @x = 2x – 3. What is @6?

Don’t be thrown by the use of an @ sign; they’re just trying to confuse you. Read the rest of the question, and you’ll see that they specifically define it as a new operation, a function with a rule they’ve given you. It’s the same kind of question we just did; they want you to plug the 6 in to the rule. Heck, the operator could be a little picture of an elephant for all you care; what’s important is that they define the rule of the function so that you know what effect it has on the input value.

Now, what about that pesky f(x) = y ?

Okay, bear with me, we’re going on a journey here. You have your machine, and you know that each input number you put into it will present you with a corresponding result number. We’ve been able to pick out individual pairs of corresponding values, but what if we wanted to represent the entirety of this transformation as a graph, something that would show all possible pairs at once, even the nitpicky ones in between integers?

NOTE: For the time being, we’re going to restrict our discussion to only rules of the first power (that means no exponents). Things get a bit more complicated if you’ve got powers in your rule, so we’ll save that for later.

If your rule is a first-power rule, then for every input value there’s going to be exactly one output value. Let’s take our original x + 4 machine again. Say I put in a number 2, and it gives me a 6. Now, if I take the 2 and the 6 out of the machine, I’ll have a pair of corresponding numbers that I can plot as a point on a two-dimensional graph (one axis for the starting values, and the other for the results). We’ve been calling our starting value x in the equation, so let’s make the starting value our x coordinate. That would make our resulting value the y coordinate. So our example above gives us the ordered pair (2, 6). Plug in some more x values, and you’ll get more y values and be able to plot more points, and the shape of the graph will begin to appear. In other words, the result of the transformation is the y value for any given value of x.

And THAT can also be written as f(x) = y.

So f(x) = y is not actually an equation as much as an indication of a concept. Basically, we’re saying that we can replace the phrase “f(x)” with the variable “y”, because we know that the number we’d get would be the corresponding y coordinate for the x we started with. So we can rewrite our initial function as:

y = x + 4,

knowing that it’s still a function, just written in a way that’s more conducive to graphing.

By the way, that last equation might look a bit familiar for those of you who’ve worked with slope-intercept format when graphing lines. It’s y = mx + b. In fact, all first-power functions form a line when graphed, and all can be reworked to fit into slope-intercept format (though some get kind of unwieldy when you do). So now we see that there are two simple ways to graph a function: plotting several points and then connecting the dots, or working it into slope-intercept form and graphing it from there.

Functions are a lot of fun when you know what you’re doing, and you can get into all sorts of complex graphs using higher power functions with the same concepts in mind. Just remember: a function is nothing more than a machine that turns x’s into y’s by following a set rule. Hopefully this journey helped you understand functions a bit better, and gave you the desire to get back in there and finish your homework!