Bloody Lips and Second Chances

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I had my biggest parental fail yet last week. I was in the process of unbuckling my son out of his high chair when I got distracted and turned my head. When I turned back, he’d tumbled forward and taken a spill. He and I were both pretty shaken up by the whole experience, but he’s on the mend and will be just fine.

In the days following, I’ve been a little paranoid and extra careful. But I wonder: had I left him in that dangerous position before? It could be that this was the first lapse I’d had and it turned out poorly, or it could be that I’d made this mistake before and gotten lucky because nothing bad had happened.

I don’t have a nanny cam running in my home 24-7 to monitor my parental performance, so I can’t really know the answer to that question. But if I did, this accident might not have happened. There’s a benefit to being able to review your performance, even if the end result turned out just fine.

Let’s carry that lesson over to the test prep world. Sure, it makes sense to spend most of your review time going over problems you missed so that you’re better able to understand what went wrong and how you can do better next time. However, it does make sense to spend some time going over the questions you get correct as well. At minimum it can reinforce good habits, and potentially you may spot mistakes that didn’t end up costing you a correct answer on this problem. However, you may not be so lucky next time.

For instance, if asked for the square root of 9, and given an answer choice of 3 you won’t be penalized for forgetting that -3 was also an option. However, if you review that correct answer and realize your mistake you’re less likely to make the same mistake where your error is the difference between a right answer and a wrong one.

As I’ve seen this week it’s helpful to learn from a wrong choice, but it’s even better if you can learn without suffering the consequences.

 A Word of Caution When Plugging in Numbers

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The positive numbers m and p are variables in the following equation:

4m + 5mp + 10 = 50m

What is p in terms of m?

A) 46/5 m

B) 9m + 5

C) 46/5 – 2/m

D) 5/m + 2

E) m/5 + 46

There’s always going to be an algebraic solution to this problem, but this one is… a little ugly. The solution doesn’t come out neatly, so there’s a fair amount of potential to make a mistake or lose confidence as you’re working through it. I hope you respond to that by saying, “Don’t worry! I see variables in the answer choices, so I can just plug in my own numbers!” That’s a wonderful thought, but here is how that might go wrong if you try it.

“Well, I know I’m supposed to pick numbers that are easy to calculate, so let me just use m = 2 and p = 3. If we plug those in to our original equation we get:

4(2) + 5(2)(3) + 10 = 50(2)

8 + 30 +10 = 100

48 = 100

Huh???

When plugging in your own numbers into a problem, it’s important to know that your numbers need to follow the rules set up in the problem. But don’t worry, that doesn’t mean you need to magically pick a pair of numbers that works out of the air. Notice, in this case we don’t see both variables in the answer choices, only m. So, let’s pick a value for m, plug that into the equation and see what value it spits out for p.

4(2) + 5(2)p + 10 = 50(2)

8 + 10p + 10 = 100

10p = 82

p = 41/5

So now we have a pair of numbers that works for this equation. We know that when m=2, p=41/5. Now we can just test that pair against our answer choices to see which equation gives us the value of p that we found.

A) 46/5 m; 92/5… Nope

B) 9m + 5; 23… Nope

C) 46/5 – 2/m; 41/5… YES

D) 5/m + 2; 9/2… Nope

E) m/5 + 46; 232/5… Nope

Plugging in numbers can be a valuable strategy, but make sure you’re using it correctly!

 Not Just Abstract Variables

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“In a sport where a percent of players have a body fat percentage below b, c times out of d a randomly selected player will be above the threshold for obesity.”

How many times did you have to read that until it made sense? Two? Three? More? Try this version:

“In a sport where 50 percent of players have a body fat percentage below 15, 1 time out of 3 a randomly selected player will be above the threshold for obesity.”

Much easier to understand, right? All we did is define those variables as you can see from a quick chart:

a = 50

b = 15

c = 1

d = 3

The difference in the readability of those two sentences should tell you a lot about the usefulness of picking numbers. When you have answer choices that are written in terms of variables, such as (ad/c) +b” rather than trying to parse exactly what that means it’s much easier to plug in your variables (50*3/1 +15) and see if that matches what you need it to. If it does, you’ve likely come across the correct equation.

Those who love pure equations often scoff at this approach and need to see the logic behind every equation. However, in a timed, multiple choice test you simply don’t have the luxury of going deep into every single problem. And the reality of the situation is that a deep understanding isn’t rewarded any more than a lucky guess. All that matters are correct answers.

So, the next time you notice variables in your answer choices, or are struggling to understand what’s going on through all the variables, try assigning values and see how much it helps!

 Improbable Isn’t Impossible

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We had a rough weekend. What should have been a relaxing weekend away was hampered by my wife becoming extremely ill on Saturday night. After a long, slow, traffic-filled car ride home on Sunday, my 11-month old son had a very difficult night of illness. I was recounting the story to my mom and she declared “It’s contagious then.”

My mom’s reasoning follows a version of Occam’s Razor, which states that with competing hypotheses in play, you should choose the one that requires the fewest assumptions. In this situation (one person gets sick then another person gets sick) a contagious bug is the most likely solution. But it’s likely, not certain.

My son’s sickness could have been the result of a long car ride, or a hot day, or some completely unrelated bug. Those aren’t likely solutions, but they are possible. When you’re working on data sufficiency, it’s very important to remember that distinction.

Imagine a problem where you’re asked to determine whether some value is even. Maybe the first three values you check yield 2, 4 and 8 respectively. At that point you can conclude that it’s likely the value will always be even (and given limited time you may just need to select that answer). However, you don’t KNOW the value will always be even. Here are some things to keep in mind:

1. Test fractional values- Positive fractional values get smaller when squared. That’s a unique property that can yield different results and help you find that a statement is insufficient.

2. Test negative values- Don’t forget about the negatives! They often produce different results.

3. Test 0- Zeroes are not only useful to check, they’re usually very easy to calculate!

4. Test 1- Just like 0, ones are easy to plug in and have several properties that can help give you different results.

Remember, just because something is probably true, doesn’t mean it must be true and testing different kids of numbers can help you strengthen your hypothesis.