How to Take a Test
Richard Reynolds
Department of Mathematics and Statistics; ArizonaStateUniversity
Question1: A person slams on the brakes of an automobile. The following table gives the speed at one-second intervals:
Time (seconds) / 0 / 1 / 2 / 3 / 4 / 5Speed (ft/sec) / 60 / 40 / 25 / 15 / 7 / 0
Give the upper and lower estimate for the distance the car has traveled during the five-second interval.
Answer 1: 147; 87
Answer 2: 60+40+25+15+7 = 147; 40+25+15+7+0 = 87
Answer 3: 60+40+25+15+7 = 147ft; 40+25+15+7+0 = 87ft
Answer 4: 1(60+40+25+15+7) = 147ft; 1(40+25+15+7+0) = 87ft
Answer 5:t= 1 second for each interval. Since D = VT
(distance is speed times the time spent at that speed), the upper estimate is given by using the upper bound for velocity in each interval:
(60 ft/s)(1s) + (40 ft/s)(1s) + (25 ft/s)(1s) + (15 ft/s)(1s) + (7 ft/s)(1s) = 147 ft
and the lower estimate is given by by using the lower bound for velocity in each interval:
(40 ft/s)(1s) + (25 ft/s)(1s) + (15 ft/s)(1s) + (7 ft/s)(1s) + (0 ft/s)(1s) = 87 ft
The above is a question from one of my tests. The answers are samples of the answers that I received. There used to be a game show “Name That Tune”, in which the object was to be able to name a tune in fewer notes than your opponent thought he could name the tune. Some students play that game with test questions. They can give the answer in fewer steps then most people need to show to get the answer. That is not the object of my tests. The work to get the answer is very important to me.
Do not leave it to me to fill in the details of your work! Assume I do not know how to find the answer, and you are not only showing me the answer, but how you got the answer. Alternatively, I want you to write your answers so that every other student in the same mathematics course (no matter what section and what level student) would understand how you got your answer.
Very often, when students go over their past tests while studying for the final, they cannot even remember how they got the answer.
Answer 1 above tells me nothing. Did you see those numbers on someone else’s paper?
Answer 2 would leave a student wondering if time had any part in finding the distance.
Answer 3 also says nothing about time, but at least units were included.
Answer 4 finally includes time. But I still wonder if the student just memorized one type of problem. What if the time interval wasn’t 1 second, or equal intervals? Would they be able to do the following variation? Also, although ‘obvious’, label which is the upper bound / lower bound.
Question2: A person slams on the brakes of an automobile. The following table gives the speed at different time intervals:
Time (seconds) / 0 / 1 / 1.5 / 2.25 / 3 / 4Speed (ft/sec) / 45 / 30 / 23 / 15 / 8 / 0
Give an upper and lower estimate for the distance the car has traveled during the four-second interval.
Answer 5 shows an understanding that the above question can also be answered. It can also be looked at months later, and be understood by anyone who learned how to do it in the first place.
Lastly, if the first question was studied in class, the second question is a fair question. I am not restricted to regurgitating questions identical to what you have seen. Honestly, both questions are very similar.