Coordinate Geometry: Sketch It Even When You Think You Don't Need To

Here's a scene I watch play out every week. Say you're strong at math. A GRE question gives you a line through two points, and you think, "I don't need a picture for this. It's just a formula." You run the slope formula in your head, get an answer in 40 seconds, click it and move on. Feels great. Then you review the section and find out you got it WRONG. Not because the math was hard. Because you flipped one sign.

That is NOT a math problem. That is a habit problem. And it's costing you some of the easiest points in the Quant section.

Why smart people skip the sketch

Quick terms first. The coordinate plane is the grid with an x-axis running left and right and a y-axis running up and down. Every point has an address, written (x, y). A line's slope is how steep it is: the change in y divided by the change in x, or "rise over run." If a line goes up as you move right, the slope is positive. If it goes down as you move right, the slope is negative. The x-intercept is where a line crosses the x-axis, and the y-intercept is where it crosses the y-axis.

You know all of that. THAT is the problem. Because you know it, you think drawing is for people who don't. So you skip it.

I'll bet that if you pull up the last 10 coordinate geometry questions you got wrong, MOST of them weren't hard questions. They were sign errors, swapped x and y values, or a point you pictured in one spot when it was really somewhere else. Go check. Don't read another word until you've looked.

Now think about time. Each of the two scored Quant sections on the GRE Quant section gives you 20 questions in 35 minutes. That's an average of 1 minute 45 seconds per question. A rough sketch takes about 10 seconds. You can afford it. What you can't afford is losing a question you KNEW how to do.

What a 10-second sketch actually does

A sketch is not art. It's two crossed lines for axes, a dot or two, and a rough line. No ruler. No grid. Its job is to give you a rough answer BEFORE you do the algebra, so that when the algebra spits out a number, you can tell if that number makes sense.

In about 10 seconds, the sketch tells you:

• Is the slope positive or negative?
• Roughly where does the line cross each axis?
• Which quadrant is each point in? (A quadrant is one of the four boxes the two axes cut the grid into.)

If your algebra and your sketch disagree, one of them is wrong. And it's almost never the sketch.

The sketch also catches distance mistakes

Distance questions are where the "I'll do it in my head" crowd really bleeds points. The distance between two points is the long side (the hypotenuse) of a right triangle you can draw under them. The trap is almost always a negative number.

The mistakes I see every week

1. Running the slope formula in your head. It's two subtractions and a division. That's exactly where a flipped sign hides. INSTEAD: write the two points stacked, one above the other, and subtract top minus bottom for BOTH the y values and the x values. Same order, every time.
2. Assuming every point lives in the top right box. Your brain loves positive numbers. The GRE loves putting points in the other three quadrants. INSTEAD: place every point on your sketch before you touch it. If negatives make you nervous, read up on the negative number trap, because it's the same trap wearing a different outfit.
3. Treating the on-screen calculator as your check. The calculator does exactly what you type. If you type 3 − 5 instead of 3 − (−5), it will happily give you the wrong answer with total confidence. INSTEAD: your sketch is your check. The calculator is only for arithmetic.
4. Sketching only when the question "looks hard." The questions you think are easy are the ones you rush, and rushing is where the silly mistakes come from. INSTEAD: sketch on EVERY coordinate question, including the ones you could do in your sleep. It's the same idea as breaking exponents down to primes: put the pieces on paper so a mistake has nowhere to hide.
5. Ignoring the answer choices. On multiple choice, the sketch often knocks out two or three answers before any math. INSTEAD: after your sketch, cross off every choice that can't be right. Then do the algebra only to choose between what's left.

The real choice you're making

You have to decide what you want MORE: to feel fast, or to score high. Skipping the sketch makes you feel fast. It doesn't make you accurate. And the GRE doesn't hand out points for confidence.

This is a standardized test. The same kinds of line and slope questions show up again and again. That means the same traps show up again and again, and you can train yourself to catch every one of them. Being good at math is a great start. But the strategist who draws the picture beats the mathematician who doesn't.

Here's your challenge for today. Pull 10 coordinate geometry questions you've already done. Redo every one, and before any algebra, write down your "sketch guess": is the slope positive or negative, and what's the rough range for the answer? Then count how many of your old mistakes the sketch would have caught. I'll bet the number is bigger than you want it to be.