Using Eyeballing to Solve Equations

When I began teaching ACT Math four years ago, I primarily emphasized teaching students the “mathematical” ways of solving questions — that is, solving questions in ways that their math teachers would be proud of. As a math major, I felt — and still do — that it’s important for students to understand the concepts underlying the math that they’re doing. While I still encourage a conceptual understanding, I’ve also learned that -- on standardized tests like the ACT and the SAT -- students benefit most from having multiple ways to solve any given question. In fact, these tests routinely reward creative solving-problem. To that end, one surprisingly powerful technique that test-takers can use on the ACT Math — one that typical math teachers would probably not approve of — is eyeballing.

Take a look at this math question, taken from a real ACT, as featured in The Real ACT book, test 2:

This question can be solved in multiple ways:

The standard “math” way to solve it is to recognize that the two angles on the line are supplementary and so must add up to 180 degrees. We then set up the equation (4x + 6) + 2x = 180, and solve to get that x = 29. Since the question is asking for the measure of the smaller angle, which is 2x, we then double this to get (D) 58 degrees.

Another approach to the question is to approximate the measure of the smaller angle by simply eyeballing it: it looks to be slightly more than 45 degrees. We then go to the answer choices. The figures on the ACT Math are drawn roughly to scale, so what answer choices can we eliminate? Well, (A), (B), and (C) are all far too small. We can also eliminate (E) because we know that it’s possible to set up an equation to solve for the smaller angle.

Interestingly, for this particular question, eyeballing the figure to arrive at the answer is actually faster than solving the question algebraically. In addition, eyeballing avoids a common mistake students make when solving this question algebraically. That is, many students set up the equation and correctly solve for x, finding that x = 29. They forget, though, that the question is asking for the measure of the smaller angle (which is 2x), and they choose (C). (Note: Solving this question algebraically is still great as a primary strategy and can be done very quickly if you’re comfortable with the algebra.)

While eyeballing can be helpful, it should be thought of more as an extra tool rather than as a primary problem solving-strategy. The technique is only relevant for questions with figures, and, even on such questions, it often can’t be used by itself to narrow down to one answer. However, it 1) can be the most efficient way of solving certain questions, 2) will often allow you to eliminate at least two answers on many other questions if you need to make an educated guess, and 3) provides a way to double-check your work if you solve the question using a more standard math approach. For example, in the question above, if a student decides to solve the question algebraically, he or she can then quickly glance at the figure to see whether the answer makes sense given the scale.

As a final example, consider this question, again taken from a real ACT, as featured in The Real ACT book, test 4:

This question appears among the last 10 questions of an ACT Math section. At this stage of the Math section, many students are pressed for time and/or are unsure of how to solve certain questions using standard math approaches. In situations like this, when a student needs to make an educated guess, eyeballing can come in handy.

The question asks for the height of the building. Based on the given length of the shadow, which is 24 yards, and given the scale, we know that the height of the building is slightly more than 24 yards. We can use this to eliminate (F), (G), and (H). In a situation when we might need to guess, (the correct answer is (K)), we’ve very simply and quickly increased our chances.

Let us know if you ever use the eyeballing strategy on the ACT!