A train leaves a station at 11:30 am heading west at a speed of 100 km per hour. A second, high-speed train leaves the same station 2 hours later traveling 180 km per hour on a parallel track. What time will it be when the second train passes the first train?
Do these questions make you panic?
I’ve been teaching math for 20 years, and a question about two trains leaving the station at different times can still give me a mini panic attack. When I read a question like the one above, my immediate reaction is “I have no idea how to solve that!” Along with that negative thought, I also feel a little sick in my stomach and a little lightheaded. Math can be intimidating, even for me! This problem has a high difficulty level with layers of complexity, and I honestly can’t solve it, or even know how to solve it, at first glance. Some of my students would give up before even starting on a question like this. However, students with growth mindsets are willing to at least start the question despite their uncertainty.
Focus on the first step only
My strategy for working through the initial anxiety sparked by a complicated problem is to focus on the first step only. James was a bright student in my college algebra class who struggled with severe anxiety. When I gave a test or quiz in that class, he would call me over to his desk multiple times, and his question would always be the same; in fact, it wouldn’t even be a question.
James: I don’t know how to do this! (pointing at the test with a look of panic)
Me: Which question don’t you know how to do?
James: All of them!
Me: Let’s take this one question at a time. Can you do number one?
James: No! I don’t understand any of these!
Me: Well just look at question number one. Can you just show me the first step for question one?
James does the first step of question one. He does the second step of question one. He keeps going and gets the right answer to question one because he knew how to do it all along.
Like many students with test anxiety, James tended to “go blank” during a test, even when he had paid attention in class, did his homework, studied for the test, and really understood the material. I was able to help him get past this problem by encouraging him to focus his attention on just the first step of just the first question. You probably won’t have a friendly teacher to guide you through this during the SAT, but you will be able to give yourself the same talk I gave to James (silently, of course!) when the test starts to feel overwhelming.
When in doubt, draw a picture
Going back to the problem about two trains, what I recommend for a first step is to draw a picture. This is my go-to strategy for many of the difficult questions in algebra, geometry, trigonometry, calculus, and physics. It works great on some of the trickiest word problems on the SAT. Drawing and labeling a picture guides your brain through the process of attending to each detail, one by one, and recognizing how each part interacts with the other parts. It is through the process of labeling your drawing with the known information that you can recognize which critical values are left unknown. Once the drawing is complete, you will know that the next step is to solve for one or more of those unknown values.
Please note that you don’t need to be a gifted artist to use this strategy. My students will be the first ones to tell you I have no artistic ability, but that doesn’t stop me from sketching out a few stick figures, blobs, and arrows to get myself started on the problem. If you do happen to be a gifted artist, go all out with your homework, but remember the SAT is a timed test!
Many tough problems on the SAT require you to have some sort of a leap of intuition and just “see” the key to solving the problem. If that leap of intuition doesn’t come, drawing a picture is the most likely activity to spark it. In the case of these two trains, you have to “see” that their distances will be equal at the moment that the one train passes the other train. If you follow the strategy of drawing the paths of these two trains as two parallel arrows and labeling each arrow with D = ______ R = ________ and T = ________, then you will be able to “see” the distances are equal because of something on your paper rather than something in your imagination. Drawing a picture of the situation allows even that intuitive leap to be broken down into steps.
Back to the question
A train leaves a station at 11:30 am heading west at a speed of 100 km per hour. A second, high speed train leaves the same station 2 hours later traveling 180 km per hour on a parallel track. What time will it be when the second train passes the first train?
A) 3:00 pm
B) 3:30 pm
C) 4:00 pm
D) 4:30 pm
No need for panic
Breathe. There is a strategy for this question. Remind yourself that you can start on the first step without knowing what the next steps will be. Start by drawing this one out. No artistic ability required.
Now, read through the paragraph again and fill in as much information as you can. In order to add the idea of “two hours later” to your drawing, use t and t-2. Make sure you only use one variable when filling in the drawing, or you will have trouble solving your equation later.
Next, it’s time to use the D=RT formula. Remind yourself that if you know any two out of the three variables in this formula, you can always figure out the last one with this formula. In this case, we can use it to find the distances for each of the trains in terms of t.
Finally, we can write and solve an equation. The equation is based on how you drew the two arrows. In this case, you should set the two distances equal to each other.
Once you work through the steps of solving this equation, you will know that t=4.5. Go back to your drawing and fill in this new information.
Finally, reread the question to check what you need as a final answer. What time will it be when the first train passes the second train? Your picture tells you that will be 4.5 hours after 11:30 am. That is 4:00 pm. (Count on your fingers if you have to – I did!) You can double-check by doing the same thing for Train 2. 1:30 pm plus 2.5 hours is also 4:00. The correct answer is C.
Celebrate your success
Often when I help one of my students fully understand a complex math problem, they happily exclaim “that was easy!” “No,” I will tell them, “Give yourself credit. That problem was crazy hard.” If you stuck with me through the whole problem, you had to combine concepts from physics, algebra, and our base-60 time system to arrive at a precise answer. In order to be successful, you needed to read and reorganize complex information, recall a formula, and create and solve a unique equation. If you followed the process, you have some serious math skills!
The truth is you can and will be able to solve extremely difficult problems, provided you have enough faith in yourself to take the first step. Don’t expect to know what all the steps are going to be at first. Start by reading the question, drawing a picture, and re-reading the question to unpack the data. With practice and experience, and by taking it one step at at time, you can master the most difficult questions the SAT or any other test throws at you.
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