SAT Math: What The...What Is That Crazy Symbol?!?

The SAT Math writers like to freak out test-takers by throwing some random symbol at them that they have never seen before. For example, can you tell me what 7▼5 is? Well, that’s obviously…wait, wha-!?

The good news is that no one knows what that crazy symbol means—but the question will always give you all of the information that you need to know. For example, an actual SAT question might look like this:

Let the operation ▼ be defined by x▼y = x² – 2y, for all integers a and b. Which of the following is equal to 7▼5 ?

(A)  29
(B)  42
(C)  35
(D)  39
(E)   51

This question still looks a little weird, but using the information they give us, it’s actually pretty simple! All we have to do is plug 7 and 5 into the equation given. If  x▼y = x² – 2y, then 7▼5 = 7² – 2*5 = 49 – 10 = 39. Our answer is (D).

Not too hard, right? Who cares that we’ve never seen the “▼” symbol used before. As long as the question tells us what the symbol means, we’re set. Let’s take a look at a more difficult “crazy symbol” SAT question:

For all numbers a and b, let the operation □ be defined by a □ b = a²- b. If k is a positive integer, which of the following can not be equal to zero?

I. k □ k

II. 2k □ k

III. k □ 2k

(A)  I only
(B)  II only
(C)  III only
(D)  I and III
(E)   II and III

We have a new crazy symbol “□”. Let’s make sure that we understand how this symbol works by trying out an example calculation. If a □ b = a²- b, then 3 □ 2 = 3² – 2 = 9 – 2 = 7.

Okay, that seems to make sense. Now what is the question asking us? We have options I, II, and III, and we need to determine which ones can not be equal to zero. Therefore, if we can find a value of k that makes one of our options equal to zero, it is not one of our answers. We need to go through all three options to figure this out. Let’s start with option I.

Option I

Here we have k □ k. According to the equation, k □ k = k² – k. Can we find an option for k that would make k² – k = 0? Remember that k has to be a positive integer.

k² – k = 0

By adding k to each side, we can rewrite this as:

k²= k

Now, divide both sides by k:

k = 1

Therefore, when k = 1, k² – k = 0. Option I will not be a part of our answer. Rule out answer choices (A) and (D).

Option II

Now for 2k □ k. Using our equation, we can say that 2k □ k = (2k)² – k. Can (2k)² – k = 0?

(2k)² – k = 0

Add k to both sides and square out the first term:

4k² = k

If we divide both sides by k:

4k = 1

This says k = 0.25, but that’s not a positive integer. The only other solution is k = 0, but that doesn’t work either (k has to be a positive integer, remember). Therefore, there is no possible way to make 2k □ k = 0. Option II will be a part of our answer. We can rule out answer choice (C), leaving us with either (B) or (E).

Option III

Here we have k □ 2k, which equals k² – 2k. Can this equal 0?

k² – 2k = 0

Or rewritten:

k² = 2k

Divide by k:

k = 2.

If k = 2, k □ 2k = 0. Option III cannot be a part of our answer choice.

Therefore, only option II cannot be equal to 0. (B) is our answer.

Let’s review how to deal with crazy symbols:

• Don’t freak out. You’re not the only one that doesn’t know what this symbol means. Read the question carefully and it will tell you everything you need to know.
• Understand the question. Sometimes these questions will just ask you to perform a simple calculation (like the first example in this post). Don’t make it harder than it needs to be.
• Rewrite the question without the crazy symbol. Using the information from the question, you can rewrite the equations that involve the symbol. Rewriting the question using only normal symbols will help you understand the question.

Scroll to the bottom of this page for another look at wacko SAT symbols.

When you're ready, get a bit of practice in with this crazy symbols quiz.

Got a math problem you can’t solve? Drop it in the comments or email it to testpreptutor@sparknotes.com.

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