Dear Danielle,

I have been tutoring a kid who is trying to up his ACT score (currently at 29 trying to get at least 30). We focus on the “last 20” questions and efficiency since he doesn’t have much time left by the time he gets to those. This was one of the questions at our last session:

The sum of two positive numbers is 151. The lesser number is 19 more than the square root of the greater number. What is the value of the greater number minus the lesser number?
Choices: 19, 66, 85, 91, 121

Being in full system mode, I told him to identify the variables, set up the equations, and began to solve by substitution until I realized that this would take too long. Graphing was not an efficient option either. By looking at the answer choices, it was clear that the numbers must be “nice”. We used a table approach and only considered perfect squares as the greater number until we found the pair that satisfied our equations. So, we used a combination of algebra and guess/check.

I know this particular problem is out of the scope of 8th/9th grade, but should we be doing more with tables and guess/check to prepare them for systems like this one? Or, could you solve it quicker doing something else? Thoughts?

Signed, Somebody Awesome

Dear Somebody Awesome,

If one of my tutoring students and I were looking at that question in the context of ACT prep, I would tell them to skip it and guess (no wrong answer penalty on ACT) as soon as they “realized that this would take too long”. Maybe pick either 85 or 91 because those numbers are close to each other. And then I would make them solve the system anyway because I’m mean.

There are three ways I can think of that would be “efficient” if you have a graphing calculator (which you should definitely have on the ACT!)

1. Use the graphing calculator to solve the system by graphing. FAST. (it took me 1:23, including reading the problem and writing down the system. I had to change the viewing window and use trace)
2. We hate the square root, so solve the system by letting $$\sqrt{y}=z$$, substitute, then solve the quadratic in z. With a calculator it would be quick to get those zeros, I used the table function rather than the quadratic formula. (2:39) If you didn’t want to use a calculator the quadratic isn’t too bad to factor.
3. Set up a new system that’s easier to deal with

$$\begin{cases} x+y=151 \\ y-x=c \end{cases}$$

where c is one of the 5 choices. then by elimination you get $$y=\frac{151-c}{2}$$, use values of c to get y, then back substitute for x, then check against the third equation. This would probably take the longest. (it did. 4:17)

Anyway, the answer is 91, so if a student takes my original advice they’ve got a 50-50 shot and saved themselves a few minutes.

And that problem is a GREAT example of why the standardized testing industry is an unethical mobster (they certainly have a great racket!), and why I decided to never offer test prep services through my tutoring business. You can increase scores and make money hand over fist teaching “tricks” but the whole system is extortion. I’ll do it for already established students, but man. Ew. Gross. My general perspective is, if you know the mathematics then the ACT is a breeze. Content is king.

“I don’t like it, turn it into something I do like” is a fundamental mathematical “habit of mind”. Though as far as I know I don’t think any education researchers include it in their habits of mind lists! (remind me to tell you the firefighter joke, it’s my favorite) “I don’t like it, turn it into something I do like” is why you’d think to let $$\sqrt{y}=z$$, “I don’t like it, turn it into something I do like” is why you’d look at this system

$$\begin{cases} x+y=151 \\ y-x=c \end{cases}$$

instead of the other one. Not more tables, not more guess/check, mathematics education should not be preparation for standardized tests. I think this problem could be taught in a rich way, but not as a timed multiple choice question.