17 Apr SAT Math: Top 5 Mistakes and Misconceptions to Avoid
It’s normal to make mistakes every now and then. This is especially true in subjects like mathematics, which are riddled with hidden rules and misconceptions. However, mistakes are also essential in the learning process, and we sometimes learn best from making them.
But hey, nobody says that you can only learn from your mistakes alone. How about we try to profit from others’ experience? Here are five of the most common mistakes and misconceptions in SAT Math that you want to avoid at all costs.
Wrong: (x + y)² = x² + y²
Correct: (x + y)² = x² + 2xy + y²
This is by far the most common mistake we’ve seen students make in SAT Math. In fact, this particular misstep is so prevalent that it even has its own Wikipedia page (just look up freshman’s dream)!
Students tend to take the erroneous shortcut of simply distributing the power 2 to each term inside the parenthesis, which leaves out the product 2xy.
We believe the misconception stems (quite naturally) from the following two distributive properties, both of which are totally valid in mathematics:
- 2(x + y) = 2x + 2y
- (xy)² = x²y²
Pay close attention to the differences between each of the above:
- In the first case, the parenthesis contains the sum x + y, but the number 2 acts as a factor, and thus can be distributed to both x and y. This is not the case for (x + y)², where the number 2 is a power rather than a factor.
- In the second case, the number 2 does act as a power, but the parenthesis contains the product xy. Here, exponent rules would dictate that we can distribute the power to both x and y. This is not the case for (x + y)², as the same exponent rule does not apply to sums.
It is worth pointing out that the test makers of College Board are well aware of this mistake, and so they often include the wrong result as one of the answer choices (see choice B of Question 1 below).
Therefore, always stay vigilant when working out your algebraic steps. This type of mistake may mislead you into identifying the wrong answer, which you may then be tempted to pick, given that it is also bound to show up among your answer choices.
Wrong: √9 = ±3
Correct: √9 = 3 only
This is another tricky facet of math knowledge that the College Board loves to test. Essentially, if x is a positive real number, then the symbol x is DEFINED as the positive square root (a.k.a. principal square root) of x. There is really nothing mysterious or ambiguous about this – the symbol x is positive just by convention or definition. We fix a definition for a notation, and then we abide by it. Simple as that!
What if we want to address the negative square root of, say, 9, or maybe both its positive and negative square root? Well, according to our definition of the symbol, this is how we write:
- –√9 = –3
- ±√9 = ±3
As an illustration, consider Question 2 below.
A natural first step would be to square both sides of the equation to get rid of the square root, which yields a quadratic equation
x + 2 = (x – 4)²
Sparing you the details, we should get x = 2 and x = 7 as the two possible solutions upon solving the equation. However, are both of them valid answers?
To determine this, we ought to substitute the value of x into the original equation. Let’s try x = 2:
√(2 + 2) = 2 – 4
√4 = –2
At this stage, we see that x = 2 is not a solution because √4 = 2 by definition.
On the other hand, x = 7 is a solution because the following is true by definition:
√(7 + 2) = 7 – 4
√9 = 3
Hence, the only valid answer to Question 2 is x = 7 (by the way, x = 2 is called an extraneous solution).
The moral of the story is that, in the case of a square root equation, always remember to plug your solution candidate back into the original equation and check whether it satisfies the definition of the positive square root symbol.
Wrong: The coefficient of x of a linear equation will always indicate the slope.
Correct: The coefficient of x of a linear equation will always indicate the the slope IF the equation is in slope-intercept form, i.e. y = mx + c
Slope-intercept form means that the linear equation is expressed in such a way that the variable y is isolated on one side of the equation. Never jump the gun when the linear equation is presented otherwise, as shown in Question 3.
A (very) popular incorrect answer to the above question is 5. We really should rewrite the equation into slope-intercept form first before we read the coefficient of x.
5x – 3y = 7 ⟹ y = (5/3)x– (7/3)
The correct answer is therefore 5/3.
Wrong: y is 20% more than x, so x is 20% less than y.
Correct: y is 20% more than x, but x is NOT 20% less than y.
We can verify the above using the following simple example:
- Increasing 10 by 20% gives 12.
- Decreasing 12 by 20% gives 9.6 (not 10)!
Here’s the reason: When we increase 10 by 20%, the 20% refers to that of 10. However, when we decrease 12 by 20%, the 20% refers to that of 12.
So how should we reverse the process? Well, algebra is the way to go!
If y is 20% more than x, that means
y = 1.20x
To understand what x is in terms of y, we simply divide both sides of the equation by 1.20 to get
x = y/1.20
Let’s practice this idea using Question 4:
From the relationship described in the question, we can write down the equation
y = 1.20x
Since we are given y = 144, we can solve for x by
x = y/1.20 = 144/1.20 = 120
and so the correct answer is B.
Mind you, if we erroneously decrease 144 by 20%, we will get 115.2, which is why answer choice A is included by the devious test maker!
Wrong: The median of a data set is the data value in the middle position.
Correct: The median of a data set is the data value in the middle position IF the data set is arranged in order!
This last mistake is conceptually simple, so let’s look at a question directly.
Spoiler alert: The answer is not 50, which is ridiculous considering the fact that it is the smallest value in the data set.
Let’s do this properly by first rearranging the data values in ascending order:
50, 51, 52, 55, 58, 67, 68
Now we see that the data value in the middle position is 55, which is the correct answer.
Remember, you earn points in SAT Math only by putting down the correct answers. This means that every single step of your work must be flawless, not in terms of presentation, but in terms of accuracy. Therefore, any form of mistakes and misconceptions can be more detrimental than you imagined. Hopefully, this article helps you raise your awareness and become a more meticulous test taker, bringing you closer to your dream score.