How to Start a Critical Reasoning Question

Reply

There is a debate in the LSAT and GMAT communities about where to start a critical reasoning question (the LSAT calls them logical reasoning, but everything I say holds true for both). One camp holds that you should start by reading the paragraph of information (usually an argument). The other says that you should start by reading the question before doubling back to read the argument. I’m amazed the argument even exists.

Imagine if they released the “Where’s Waldo?” books without a title. Here’s a massive jumble of people. That’s nice. After you’ve spent a couple minutes looking at the page, someone tells you that you are supposed to be looking for Waldo. Sure, there’s a chance you stumbled upon Waldo when you were looking at everything else, but there’s a much greater chance that you spent your time focusing on a whole lot of other irrelevant things.

Read the question first. It allows you to focus on the correct things and to figure out what’s important and what isn’t. And that’s even more relevant because not all critical reasoning questions ask about the same thing. Some tell you that an argument is coming and ask you to figure out what would strengthen it. Some tell you an argument is coming and ask you to find a flaw. Some tell you there’s no argument, just a set of facts and ask you to find what must be true. Knowing which of the question types you’re going to get allows to focus on the proper approach to the problem, filter out the irrelevant information and find Waldo much more effectively.

 So a Circle Walks into a Bar…

Reply

It sounds like the beginning of a math joke, but it isn’t.

“So a right triangle is inscribed into a circle…”

That’s the premise of a couple interesting GMAT questions that I came across lately, so I thought I’d share the issues that these problems bring. First it’s important to define that term inscribed. It’s the kind of term that you may have come across several times without ever knowing what it means because the visual diagram that accompanies the problem has you covered. In geometry, when we talk about something being inscribed we mean that it is drawn inside another shape such that all of its corners touch the edge of the larger shape without going outside of it. When a shape is inscribed within a circle it’s a little like that shape has a custom-built bubble surrounding it.

Now back to our problem. So there’s a right triangle in a bubble. So What? Well that particular situation actually gives us a very important piece of information. Whenever a right triangle is inscribed in a circle, the hypotenuse of the triangle is the diameter of the circle. That’s a fantastic rule, and one you ought to remember, but when we get to the difficult end of the quant section where a question like this is likely to occur, we’re probably going to need more than that.

So what other concepts fit in with this rule? Well, our rule gives us a fantastic way to find the hypotenuse of the triangle if we know something about the circle (or vice-versa), so a nice extra step is when the GMAT asks about the length of one of the other sides of the triangle. When would we be able to find the length of the other sides of the right triangle knowing only the length of the hypotenuse? When it’s a special right triangle! So, be on the lookout for 30:60:90 triangles or 45:45:90 triangles. Even if these aren’t immediately apparent, remember that every distance from the center of the circle to the edge of the circle is a radius, and drawing one or more of these radii in often gives you more opportunity to solve.

Keep this fantastic rule and these tips in mind the next time you come across a similar problem!

 Parabolas

Reply

I recently came across a problem that required knowledge of parabola formulas. That surprised me. Although parabola problems show up relatively frequently, they usually require little more than logic. However, I thought that provided a nice opportunity to refresh on all things parabola.

A parabola is defined mathematically by this formula: y= ax^2 + bx + c. We see parabolas in nature most often when we look at projectiles, like a cannonball shot out of a cannon or a jump shot out of the hand of Steph Curry. Parabolas are generally u-shaped and are symmetrical about the vertex, which is either the highest or lowest point of the parabola, depending on the orientation.

Whether our parabola is cupped upward or downward is determined by the sign of the “a” term in the formula we saw above. When a is positive, the parabola will have a vertex at the bottom and open upward. When a is negative, the parabola will have a vertex at the top and open downward.

However, there is more than one way to define a parabola mathematically. We can also solve a parabola if we have the vertex and another point on the parabola. We do that by using the similar formula y = a(x-h)^2 + k. The coordinates of the vertex are (h,k).

So using that information, find the equation of a parabola with vertex (-2,1) containing the point (1, 19). The first thing we need to do is solve for a by inserting our points into the formula. We get:

19 = a(1-(-2)^2 + 1

19= a(3^2) + 1

19= 9a + 1

Now, we put our a into the formula with our vertex (h,k), but instead of using the x and y from a specific

point we’re going to solve for the generic x and y.

y= 2(x-(-2)^2 + 1

y= 2(x+2)^2 +1

Now expand:

y=2(x+2)(x+2) +1

y=2x^2 + 8x + 9

Now we’ve solved for the equation of this parabola and we could mathematically figure out all of the

points on this curve.

I hope that’s been a good refresher on parabolas!

 Graduation Attitude

3

It’s getting to be the graduation time of year when my Facebook feed is filled with smiling happy pictures of my former students who have gone on to earn college or graduate degrees. It’s wonderful to know that I was able to have a part in these stories, but more than that it offers perspective.

Ask any of those graduates what the biggest obstacle on the way to her degree was, and she won’t tell you it was the SAT or the GRE or the GMAT. That was just a first step. A stepping stone obstacle on the way to bigger and tougher challenges. Those who will be successful with those later obstacles, exhibit five key attitudes that help them achieve.

1. Be Positive- Sure skepticism is “cool” but a positive attitude breeds success. People that believe they can achieve something special are most often the ones who do.

2. Set Goals- It’s a long way between here and graduation. If you don’t set markers along the way to gauge your progress, it’s tough to succeed.

3. Build a Support System- When the going gets tough– whether now or later– you need people who care about you to be there. Whether that’s someone who will lend a sympathetic ear, help you work through a tough problem, or just take you out for ice cream and a break, successful people don’t do it alone.

4. Take the Long View- Minor disappointments are going to pop up along the way. Will you dwell on them, or will you shift your focus toward making the best of what’s ahead?

5. Keep Going- No matter what keep going. Don’t stop, and you will achieve your goals.

Now get to it and get back to studying!