A Fun Question Through Different Lenses

Today I had three of my five classes meet and they all met before lunch. So, now, I am writing a blog post after lunch!

 

Here is the problem –

Doris enters a 100-mile long bike race. The first 50 miles are along slow dirt roads, while the second half of the race is on smooth roads. Assume that Doris is able to travel at a constant rate of speed on each surface.

  1. If Doris’ speed on the first 50 miles of the race is 10 miles per hour, what must be her speed during the second half of the trip so that her average speed over the whole trip is 13 1/3 miles per hour?
  2. If Doris’ speed on the first 50 miles is12.5 miles per hour, what must be her speed on the second half of the trip so that her average speed over the whole 100 miles is 25 miles per hour?

 

A fairly standard question and I admit I stole it directly from a text I use ( a pretty cool Differential Calculus book for my non-AP class) and I originally intended to just use it with my Calculus Honors class this morning. After the discussion with them, I decided it was worthy of the attention of my Geometry section and my Discrete Math class as well. It is kind of fascinating to me to think about the different receptions that the problem had in each class. In all three classes I asked them to think about the first question and wait until we discussed it before moving on. I KNEW what mistake was going to be made. There was no doubt in my mind that the answer to question 1 would be 16 2/3 and the students did not disappoint me. They focused on the information given and processed it in a logical way given their use of the word average when considering two measurements. What they did not focus on was time and it is logical that they did not because it is not explicitly mentioned in the problem at all. My hope going in with my Calc Honors kiddos was that this would be an object lesson in weighted averages. What it turned into in my other two classes was a bit of a primer in how to think through a word problem instead of just automatically applying some mathematical operation on a set of numbers in front of you. In each class after the initial incorrect solution was offered I presented the following question. “I have a 90 on my first test and I score 100 on my second. What is my average? Now, I have a 90 average after four tests and I score 100 on my fifth. Is my average the same in each situation?” I had hoped, overoptimistically, that this would prompt thought about time but in all three classes I ended up explicitly urging them to think about time. In Calculus Honors and in Discrete this led most (maybe all) students to a correct conclusion. Not so with my younger Geometry students who were still a bit wary of the problem. In fact, one student asked me if all of my questions were going to be like this. I need to be a little more careful about these last minute decisions to follow my muse and approach a problem. I need to be more thoughtful about how appropriate the question is for the audience at hand. I still believe that this is a meaningful question for my Geometry kiddos to wrestle with, but perhaps I would have served them better by not making this the opening question on the second day of school…

Both groups of older kids arrived at the correct conclusion to the second question and they were amused by the result. Only two days in but of the eight classes I have had, I’d say 7 were successful with one uncomfortable miss.

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