One of the biggest differences between studying for a math class and preparing for the SAT is that, in a math class students are expected to understand (or even just memorize) certain processes or algorithms. Success on a math test in a math class often feels like looking at a problem, recognizing that it is a problem very similar to problems you’ve repeatedly practiced in your homework, and repeating the process you’ve learned. In fact, on a math test, if you solved a problem by simply plugging numbers in (rather than using the process taught you you by your teacher) you’ll likely get the problem marked as wrong (even if you got the correct answer!)
And that is understandable. A math teacher is trying to teach students how to use a particular tool. If the student solves a problem by using a tool other than what the teacher taught, the student has not demonstrated mastery of the tool. If a student is being taught how to use a hammer, and the student connects two pieces of wood with a screw, the student really hasn’t demonstrated master of the hammer.
There is, though, a BIG problem with telling students that the only way to solve a problem is the “right” way: that is not how the real world works. Success in our personal and professional lives depends on our ability to look at a problem we do not know how to solve, and find a way to solve it with what we know at the time. If we wait until we remember the “right” way to solve a problem, we might be waiting a very long time.
The result: students coming in for test prep and class support sessions with problems they could not solve, who haven’t tried anything to solve them, and who wait, with hands folded in lap, for instructors to teach them a process for them to memorize.
Waiting for someone to tell you what to do is not going to lead to success on the SAT, and it is not going to lead to success in life, personally or professionally. The SAT doesn’t care HOW you get the correct answer. They just care that you get the correct answer.
Here is an example from an SAT practice exam on the college board website:
Now, a student might look at this problem and say “Wow, this looks really complicated. I’ve got no idea how to do this...oh well.” That’s the thinking of a student who is looking for the “right” want to do the question.
Watch how asking questions and “trying something” can lead to the correct answer, even if the student doesn’t understand the “right way” to solve the problem.
First question: what concepts is this problem dealing with?
This problem is dealing with variables, right? Well, what is a variable? You might answer something like “It’s a letter that can be replaced with different numbers.” What does that suggest we can do with these variables?
Replace them with numbers of course! So let’s try plugging numbers in for m and p,
What numbers should we plug in?
"I don’t know."
And again, we have an opportunity for a student who doesn’t know the “right” numbers to plug in to throw up their hands in exasperation. How about trying...any numbers? Let’s try m=3 and p=7.
Now the problem turns into something simpler:
Armand sent 3 text messages per hour for 5 hours. What can we determine here? Oh, he sent 15 text messages!
Tyrone sent 7 text messages per hour for 4 hours. What can we determine here? Oh, he sent 28 text messages!
So it looks like he sent a total of 43 text messages.
Ok, so what? None of the answers say 43.
Well, what DO you know about the answers? They have variables. What can we do with variables? Plug in numbers. What numbers?
Oh, m=3 and p=7!
We plug those numbers in to each answer choice and we find that C gives us 43! C is the answer.
A discerning student might notice that answer C also used the same operations we did. But the more important point is this: we did not need to understand the algebra behind this problem to solve it by plugging in numbers. All we had to understand is that if you send 3 text messages per hour for 5 hours, that’s 15 text messages.
Too often, we see students leave points on the table on their SAT exams, practice tests, and homework assignments that could be solved very simply if only they asked questions about the problem and tried something!
This is not to say that learning the algebraic way to solve this problem is not valuable. It absolutely is! Ideally, when you see a problem like this on your SAT, you’ve mastered the algebraic method of solving it. If so, great! Solve it using algebra. But if you haven't, and you solve this problem “the long way”, yes, you should absolutely go back and review the necessary algebra. But, because you WILL see math problems on your test you don’t recognize, you MUST practice doing something, anything, with a problem to learn about it. Otherwise, you’ll be throwing away potentially easy points like these!
Here is another example:
This is one of the problems students miss the most on the diagnostic test, in my experience
Maybe you look at this problem and say “Ugh there are fractions within fractions. This looks so confusing. There is no way I’ll be able to simplify this”
Maybe you say, “I have never seen this before in my life.”
So you move on, because you “don’t know how to solve” the problem.
What if, instead, you said, “Hmm, these equations have variables. Maybe I can try plugging in numbers to the answer choices?”
What number should we try plugging into the answer choices? It says x > 3 How about 4?
And we see that if we plug in x = 4, the value of this expression is 42/13.
OK, but none of our answers are 42/13, so how does that help?
Look at the answers. What do you see? Variables. What can you do with variable? Plug in numbers. What number? x=4...the same number we plugged in a minute ago!
Is that going “to work”? No idea...let’s try it and see!
Even as you were plugging numbers in, you might have been saying to yourself, “I have no idea if this if going to work”. But when you plug 4 into the answer choices and you see 42/13 pop out of answer choice B, you’ll realize that, even if you didn’t feel like you knew what you were doing, plugging in numbers was enough to find the answer.
Now, if you looked closer are the answer choices, you might have also realized the following:
-C and D were not going to give a fractional value from plugging in x=4, s, so we probably don't need to try those ones.
-If you did A first and got 13/42, you might have noticed that B was the reciprocal of A, which means B must be correct (since the reciprobal of 13/42 is 42/13, which was our answers)
Even if you didn't notice those things, though, you can still get this question 100% correct by doing nothing more than plugging numbers in.
YES, after solving this problem by plugging in numbers, you need to go back and review complex fractions if that was a concept you were not comfortable with.
That said, in your test prep, you ALSO need to be practicing “what do I do when I don’t know the answer?” And waiting until the internet or a teacher/tutor gives you the knowledge you “need” is not practicing that.
In conclusion:
Big idea: when you are faced with an SAT problem that deals with variables, and you have no idea how to proceed, try plugging in numbers to see if you can learn something about the problem. You will surprise yourself with how often doing this alone will solve the problem!
Bigger idea: When you “don’t know how” to solve a problem in your practice, you MUST practice doing as much as you can. Try anything you can think of to learn something about the problem. This will be an invaluable skill if you want to squeeze more points out of the math section.
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