Have you ever been taking a math test, got to a long word problem, and froze up because you had no idea where to begin? Even when you know how to do the math the problem is based on, tackling word problems can be difficult if you don’t know how to approach them.
By following these 4 easy tips, you can instantly improve your scores on practice tests, and even do better on longer multiple choice word problems in school!
We’ve all been there: you read a long paragraph and when you get to the end, you ask yourself, “What did I just read again?” This happens when you are practicing passive reading (in other words: just running your eyes over the words, without really putting it all together). Active reading is a technique you can use to make sure information sticks after you’ve read it just once.
To do this, as you read through a paragraph, after every long sentence (or every 2 short sentences) pause for a moment and summarize what was just said in a few short words to yourself, as if you were simplifying it and explaining it to a younger sibling. For example, if a word problem began:
“An alloy is a metal made by mixing and melting two or
more metals together. After the metals are mixed, the
alloy must be cooled slowly to avoid crystallization…”
You would say to yourself in your head something like, “So we’re mixing 2 metals together, then cooling them. Got it.” The problem may continue:
“Suppose a metallurgist heats a mixture of metals to a
temperature of 2,500°F and then removes the resulting
alloy from the furnace. The alloy will then cool at a
constant rate of 40°F every 15 minutes until it reaches
room temperature.”
After reading this part, you could think to yourself, “Ok, so these are the numbers we’re heating then cooling with. Got it.” We’ll cover what to do with the numbers later. For now, just understanding the gist of what’s being said is the key. The problem might finish with:
Which of the following functions represents the
temperature T of the alloy h hours after it was
removed from the furnace until it reaches room
temperature?
A. T(h) = − 15h + 2, 500
B. T(h) = − 40h + 2, 500
C. T(h) = − 160h + 2, 500
D. T(h) = − 600h + 2, 500
After reading this, you’d say to yourself, “Ok, so I’m picking out an equation that describes that cooling I was reading about before.” Now, notice that you know what the problem is talking about, and you have a clear goal for what you’re going to be doing to break it down and begin solving it! You’re already in a much better position than if you had just read every word back-to-back and got lost in what the problem was asking you to do.
This is something you should be doing while you’re practicing active reading like in the previous tip. This will allow you to quickly glance back at the text anytime you need to reference a number while you’re doing scratch work, so you don’t have to reread the entire paragraph again. It will also assist in active reading, since you’ll have to constantly be on the lookout for any number, variable, or unit as you’re reading through the first time.
When you finish reading the problem, your underlining might look something like this:
An alloy is a metal made by mixing and melting two or
more metals together. After the metals are mixed, the
alloy must be cooled slowly to avoid crystallization. Suppose a
metallurgist heats a mixture of metals to a
temperature of 2,500°F and then removes the resulting
alloy from the furnace. The alloy will then cool at a
constant rate of 40°F every 15 minutes until it reaches
room temperature. Which of the following functions represents the
temperature T of the alloy h hours after it was removed from
the furnace until it reaches room temperature?
A. T(h) = − 15h + 2, 500
B. T(h) = − 40h + 2, 500
C. T(h) = − 160h + 2, 500
D. T(h) = − 600h + 2, 500
Notice how if you just reread the underlined portions of the above problem, you can gather what’s going on, along with all the numbers and variables required to solve the problem: “Heats to 2,500°F, cools at a rate of 40°F every 15 mins. Function uses T for temperature and h for hours.”
Also, because we are careful to underline units, you’ll notice this problem describes one part in minutes, but then the answer uses the unit of hours. One of the ways test writers like to be tricky and see if you’re paying attention is to quietly change units halfway through the problem. These are some of the most missed problems as a result. If you want to get ahead of 90% of other test takers, simply paying attention to units in this way will prevent you from missing this kind of problem!
When a test writer is making a multiple choice question, they have to come up with 3-4 incorrect answers to go along with the correct answer. To keep the correct answer from being completely obvious, they will usually make the incorrect answers common mistakes that students will make when solving the problem. We can use this to our advantage!
If we know that messing up the units is a common mistake (that we won’t make because we’re following this guide!) then we can identify which answer would be chosen by students who weren’t reading actively and didn’t notice the unit change.
In our example problem, notice that answer choice (B) just uses the exact numbers from the problem without converting any unit changes. Before we even do any calculations, we can recognize that answer is incorrect and cross it out, since we know some type of unit conversion needs to be done. Even if we don’t know exactly what to do next, we’ve already improved our chances of getting this problem correct!
Additionally, we can sometimes use the answer choices to help us understand what’s going on if something is in common between all the answers. Notice how all four answers have “+ 2,500” in them. We should think back to what we underlined and think about what that number represents, because it’s obviously important to solving the problem. Since we underlined the important information, we know that was the temperature the alloy was originally heated to.
Also, notice how every answer choice includes some negative number times h. If we think back to the answer choice we eliminated, that negative number seems to indicate the rate the alloy is cooling at. So now we know we’re simply a unit conversion away from selecting the correct answer!
A good example of doing this is thinking of some easy numbers to plug in for h to begin. Usually, numbers like 0 and 1 are your friend here! If we plug in 0 for h, think about what that means: we’re thinking about the temperature of the alloy 0 hours after it began cooling. But “0 hours after” something is just the very beginning! And we know the temperature started at 2,500°F (remember, that is T!). So if we plug in h=0, we better get T=2,500. We can rule out any answer that doesn’t give us this!
On this particular problem, we can’t rule anything out by that, but you’d be surprised by how many questions this will help you on!
So next, we might try h=1, thinking about what that means: “1 hour after cooling began.” Well, we know from what we underlined that the temperature goes down by 40°F every 15 minutes. So we can think logically and realize the temperature is going to go down 40°F four times in one hour.
Subtracting 40 from 2,500 four times gives us 2,340. So if we plug in h=1, we better get T=2,340 in the correct answer, and we can rule out any answers which don’t give us this. Remember, since we already ruled out (B) before, we don’t even have to waste time calculating for that one. Just plug into the 3 possible remaining answer choices! You’ll notice only one of them works, and that would be your answer!
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Even though the writers of an SAT/ACT question like this one would expect you to understand linear functions, rates of change, and unit conversions in order to solve it, by following the 4 tips outlined in this post, we were able to simplify the problem, understand it better, and get to the answer with just some basic logic and arithmetic!
So next time you are faced with a long, tricky math problem, try applying the 4 tips we did here, and you’ll be surprised how much smoother it goes. By the way, did you figure out the correct answer to the question in this post? If so, comment below what it was!
Looking for more guidance in mapping your road to the SAT/ACT? Learn more about our test prep services here!
Daniel Brown
Test Prep Instructor
Vanguard College Prep