There has been a lot of talk in education circles and amid social media recently about making algebra more "personal" in order to engage students and deepen their understanding and interest. I couldn't agree more! One of the primary goals I have for every lesson I teach is to avoid "naked math" which is math without context. For example, 1/4x + 25 is a "naked" expression when students do not know what the variable represents; as such, it is much harder for them to evaluate whether or not their answer makes sense. If, on the other hand, the mathematics is put into context, especially when that context is one they are interested in and can relate to, then they are able to "see" the real life application that the mathematics is modeling. So, if I tell you that x represents the number of text messages you send each month and that the expression represents the amount of your bill, then the previous expression suddenly becomes clear and meaningful. Now you understand that each text costs you 25 cents (1/4 of a dollar = 0.25) plus your regular phone plan which is $25. Now when I ask you a question in class, "What happens when x=10?", you immediately imagine sending 10 text messages in a month and ask yourself, "What would my cell phone bill be?" The output, $27.50, can be meaningfully associated with the input, 10.
But how can we make an entire lesson meaningful and present the mathematics in context? That's the problem I'm solving in today's blog.
This semester I am teaching an Algebra with Applications course and recently used the above "fact" from OMG Facts in a lesson on linear equations. We had done some initial work with one of the most common linear relationships: d = rt (Distance equals rate times time) when I put the following "fact" up on the board.
I then asked my students to play a little game of "Fact or Crap" and took a class vote. (Note: my students are adults so they are well acquainted with the notion of calling bullsh*!. If your students are younger, you may want to just play "Truth or Lie" instead.) In my class, the students were fairly equally divided on whether they thought this fact was true (and lively discussion started immediately). What can be done to resolve this stalemate? Why, MATH of course! One of the many beautiful things about mathematics is that, unlike many other fields, one does not have to take anyone else's word about something; instead, statements can be proven true (or not). So, that's exactly what we did next in class -- I asked, "How can we prove (or disprove) this statement?"
Students told me the important considerations in the problem:
- It depends on how fast you're traveling.
- It depends on what you're texting (how long the message is).
- It depends on how fast you can text.
Since we were very close to I-95, we decided on a rate of 70 mph. They all had cell phones (of course) so I asked them to pair up and collect data on text speed. In order for averaging to make sense, we came up with a specific "test text" that everyone would enter (exactly as typed) into their phone. They came up with this (which I loved because of the irony and humor):
I can't text now. I'm driving!
Their partner timed how long the process took (eyes up, eyes down, eyes up) and we recorded all the data on the board in order to determine time. We examined the data and discussed which measures of center might be best to use. In our case, we decided on mode as it had the added advantage of eliminating the really slow and really fast texters in the group. The most frequent time was 5 seconds.
As we reviewed Polya's process, we understood the problem (knew what we knew and what we needed to find), we had a plan (use the d=rt formula), and were ready to carry out the plan. We recognized that we had some problems in our units of measure: our rate was in miles per hour but our time was in seconds and our distance was in yards. My students decided we had to "convert stuff". They told me that a football field was 100 yards long or, equivalently, 300 feet. They also told me that there are 60 minutes in each hour, 60 seconds in each minute, and 5,280 feet in each mile.
Because of the context (and because they wanted to know if the "fact" was true or not), they were extremely engaged and actively working on making the appropriate conversions. We rounded to the nearest foot and found that 70 miles per hour is equivalent to traveling about 103 feet per second. Here are the steps:
- 70 miles per hr / 60 min per hr = 1 1/6 miles per minute
- 1 1/6 miles per minute * 5,280 feet per mile is approximately 6,160 feet per minute
- 6,160 feet per minute / 60 seconds per minute is approximately 103 feet per second
Since our class mode was 5 seconds, we then used d=rt to find that the OMG fact is TRUE as we would have traveled about 515 feet or over 171 yards (that's almost 2 football fields!).
Problem solved. We learned it's smart to use math and it's bad to text while driving. I will save the preaching for signs like this one:
Instead, I will advise you, know the facts and help others be educated about the math of texting as well. Here's one site that has a lot of good resources.
Back to my lesson, I will leave you with the same extension questions that I gave to my students; as always, I'd love to hear from you.
- We made reasonable assumptions about our highway speed and collected data to find a reasonable estimate of texting speed for a given message; what are possible assumptions that the OMG folks made to reach their "answer" of a distance of a football field?
- Is there a way we can use additional variables to graphically show how each piece of this problem (highway speed, text message length, texting speed) affect the 'answer' of distance traveled?
Ok, now go text someone the link to this blog! (but not while you're driving)
The Solver Blog
Author: Dr. Diana S. Perdue