We have studied bacteria and know that they reproduce by fission or simple cell division. Streptococcus bacteria cause the well-known illness, strep throat. The bacteria can reproduce every 20 minutes. According to a published source, if these bacteria continue unchecked, at the end of 24 hours this colony of bacteria would have a mass of 2 million pounds! Needless to say, this could not really happen in your throat. If you inhaled a single strep bacterium today at noon and it reproduced, unchecked, every 20 minutes, at approximately what time would the colony outweigh you?
We had been studying exponential growth in math and monerans in science. I wanted to connect these two concepts in the students' minds. In reading a reference book on bacteria, I came across the weight statistic in the task. I thought that students would be particularly engaged in this task if it related to them personally, which is why I wrote the question as I did. This task connects two curricular areas: it encourages students to recognize exponential growth and to use the powers of two to find a solution. Students will use their knowledge of the powers of two or binary numbers, to solve this task once they discover that they are working with numbers too large for their calculators. They will eventually discover that the colony of bacteria weighing 2,000,000 pounds at the end of 24 hours will weigh 1,000,000 pounds 20 minutes earlier. They can then work backwards until they approach their own weight. 45 minute However, some of the students worked beyond that time to word process their write-up and use computer programs to create graphs and tables of their results. If students have not been taught binary numbers, this task may well be too frustrating for them. I believe it is a good test of their comprehension of this concept once they have been taught it. Many seventh grade students study bacteria in life science classes, so you might want to coordinate with the Science teacher on this task. The NCTM Addenda Series on Patterns and Functions has a series of lessons on exponential growth that I used prior to this task. You might want to take a look at that resource. I also realized that some of my less-able students would have a very difficult time with this task, so I wrote an alternative task for those with fewer abilities. I gave this out to those students who expressed high levels of frustration after an initial investigation. All of the Exemplars work is in response to the original task. I have included the modified task to give you an idea of one way to adapt a task to ensure equity of access for all students. Baffling Bacteria - modified The bacteria that cause strep throat can reproduce by fission every 20 minutes. If you inhale a strep bacterium at noon today, how many will have been produced by the time you go home at 2:00 o'clock? How many will there be by 6:00 o'clock tonight? How many at midnight? Your calculator can not calculate the number that will have been produced by noon tomorrow so can you find some way to express this number without the aid of the calculator?
Scientific calculators
The exact solution depends on the weight of the student. All my students would be outweighed between 7:00 am and 7:20 am the next morning.
The students in this group missed the concept of doubling and exponential growth. They may have had the concept of the time slots, but that was generally it. Many of them did have some mathematical representation that they had used in an effort to find a solution. The exemplar student understood about the 20-minute time blocks required for the bacteria to double. The student confused 20,000 with 2,000,000 as the number of bacteria at the end of 24 hours. Beyond that, the student did not understand the concept of doubling as evidenced by going "down by 1,000s".
There were a number of reasons why a student fell into the Apprentice category. Many of them simply made a number of calculation errors due to the fact that their calculators were not scientific and could not handle the size of the numbers. They were not able to recognize the errors of their thinking or did not recognize the exponential growth factor, so did not question their inaccurate solutions. The exemplar in this group was interesting to me, as I believe that this student understood the task and was able to get an accurate answer, although the student only "thinks" the solution is correct. It seems evident that either this student was coached in making the growth chart or was simply trying to regale us with great wisdom in trying to explain the rule of exponential growth - which is inaccurate and not followed. The graph is also rather interesting as it shows us nothing of importance in understanding the solution and was not used in finding the solution. The hand done chart seems to be what the student used to find the solution (in class) and is accurate. In this case the computer work detracted from the student's solution.
These students all tended to understand the task, but made mistakes in their calculations or had errors in reasoning that prevented them from getting correct solutions. The exemplar piece represents some good logic in dealing with numbers too large for the calculator. By a careless error, the calculations were all off by one 20-minute period, so the solution of 6:50 is inaccurate as is evident in the accompanying table. This student has some interesting mathematically relevant comments as well.
These students immediately recognized the connection between cell division and exponential growth. They began working backward at once and frankly solved the problem in record time. The challenge for them was to use good mathematical language and representation in describing their solutions. The exemplar of this group verified the result through calculating a 61-pound colony's growth from 7:00 am until noon to see if the result would be the two million pounds stated in the task. The result was very close. While this student had a brief write up, there was strong mathematical language and a good table showing the resulting doubling of the colony of bacteria.
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