The symbol problem is a recurring theme in the SAT mathematics session. What I’m referring to as the symbol problem is a particular problem type that appears on almost every SAT. The way it works that is a symbol such as @, which has no normal mathematical significance, is introduced as a function that takes in two variables and outputs a result. This function is very similar to the addition, subtraction, multiplication, or division symbols as it is placed between two numbers and does some sort of an operation to them. This problem type can be highly confusing, particularly for students who have not seen an example of the problem before. It is certainly worthwhile to spend some time mastering this problem type, so it can be completed in an effortless fashion.
Usually, these types of problems will be presented in an algebraic way, by showing what happens to two variables when the operation is applied to them.
Examples:
Say that x@y = 2x – y. Usually, the problem will require the student to use this definition to solve for the result when two numbers are substituted into this definition. For example, it might ask the test-taker to evaluate 2@3.
This would be done by simply letting x = 2, and y = 3.
Then 2@3 is equal to 2(2) – 3, which is equal to 1. Thus the correct answer is 1, as the answer can be found by simply evaluating the expression.
Now, let’s try a more difficult example. Let x#y = x * y – (x + y). The question will likely require you to evaluate something like 4#3. The result would be 4#3 = 4*3 – (4 + 3), by substituting the expression into the given definition. Thus, by evaluating the result, the answer is 5.
Here is a final example. Let x&y = (x^2 + y^2) – (x + y). Evaluate 3&2.
The answer is found by evaluating (3^2 + 2^2) – (3 + 2), which is equal to 13 – 5, which is 8.
For anyone with computer science experience, these will be quite simple problems, as this definition of a function is effectively the same as f(x, y) = (x^2 + y^2) – (x + y), except with different notation. This is a great way to reinterpret the problem if this makes it easier for the student.
In essence, remember to not panic upon seeing a question of this type. Despite seeming quite confusing, after doing a few examples, these questions will make a great deal of sense, and require minimal effort on the part of the student. They all follow the same form, with slight differences in the exact definition of the function and the symbol involved. With this practice, and a few more practice questions done independently, I expect it will be possible for any student to obtain full marks on this question type.
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This article was written for you by Tobias, one of the tutors with Test Prep Academy.